The Study of Epidemic and Endemic Diseases usingMathematical ModelsbyJummy Funke DavidB.Tech., Ladoke Akintola University of Technology, 2011M.Sc., The University of British Columbia, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Interdisciplinary Studies)The University of British Columbia(Vancouver)February 2020c© Jummy Funke David, 2020The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:The Study of Epidemic and Endemic Diseases using Mathematical Mod-elssubmitted by Jummy Funke David in partial fulfillment of the requirements forthe degree of Doctor of Philosophy in Interdisciplinary Studies.Examining Committee:Fred Brauer, MathematicsSupervisorViviane Dias Lima, Faculty of MedicineCo-supervisorPriscilla (Cindy) E. Greenwood, MathematicsSupervisory Committee MemberDaniel Coombs, MathematicsUniversity ExaminerPaul Gustafson, StatisticsUniversity ExaminerJulien Arino, Mathematics at University of ManitobaExternal ExamineriiAbstractMathematical models used in epidemiology provides a comprehensive understand-ing of disease transmission channels and they provide recommendations for meth-ods of control. This thesis uses different mathematical models (direct and indirecttransmission models) to understand and analyze different infectious diseases dy-namics and possible prevention and/or elimination strategies.As a first step in this research, an age of infection model with heterogeneousmixing and indirect transmission was considered. The simplest form of SIRP epi-demic model was introduced and served as a basis for other models. Most math-ematical results in this chapter were based on the basic reproduction number andthe final size relation.The epidemic model was further extended to incorporate the effect of diffusionusing a coupled PDE-ODE system. We proposed a novel approach to modellingair-transmitted diseases using a reduced ODE system, and showed how the reducedODE system approximates the coupled PDE-ODE system.A deterministic compartmental model of the co-interaction of HIV and infec-tious syphilis transmission among gay, bisexual and other men who have sex withmen (gbMSM) was developed and used to examine the impact of syphilis infectionon the HIV epidemic, and vice versa. Analytical expressions for the reproductionnumber and necessary conditions under which disease-free and endemic equilibriaare asymptotically stable were established. Numerical simulations were performedand used to support the analytical results.Finally, the co-interaction model was modified to assess the impact of combin-ing different HIV and syphilis interventions on HIV incidence, HIV prevalence,syphilis incidence and all-cause mortality among gbMSM in British Columbiaiiifrom 2019 to 2028. Plausible strategies for the elimination of both diseases wereevaluated. According to our model predictions and based on the World Health Or-ganization (WHO) threshold for disease elimination as a public health concern, wesuggested the most effective strategies to eliminate the HIV and syphilis epidemicsover a 10-year intervention period.The results of the research suggest diverse ways in which infectious diseasescan be modelled, and possible ways to improve the health of individuals and reducethe overall disease burden, ultimately resulting in improved epidemic control.ivLay SummaryThe work highlights different ways of modelling infectious diseases transmittedindirectly through virus transferred by air, contaminated hands or object (host-source-host models in Chapters 2 and 3) and directly through sexual contacts (person-to-person models in Chapters 4 and 5). In particular, the main contribution of thiswork are the developed epidemic models with heterogeneous mixing and indirecttransmission, the epidemic model designed using a coupled PDE-ODE system,and the co-interaction model of HIV and syphilis infections. The models exhibitmany of the features we expect to see in more complex models, and respectivelyhighlights the core differences between sudden occurrence (epidemic models) andconstant presence (HIV and syphilis co-interaction models) of diseases in the en-vironment, and possible ways in which these diseases could be eliminated in thecommunity.vPrefaceChapter 1 gives the basic background of infectious disease models, motivationsand their impact on public health policies, and there are no original results in thisChapter. I was primarily responsible for all the works in Chapers 2, 4 and 5, andModels in 2.3, 2.31, 3.31, 3.33, 4.4, B.1 are the main contributions.Chapter 2. A version of this material has been published as Jummy FunkeDavid (2018) Epidemic models with heterogeneous mixing and indirect transmis-sion, Journal of Biological Dynamics, 12:1, 375-399. I was primarily responsiblefor all areas of model and concept formation and analysis, as well as manuscriptcomposition under the supervision of Dr. Fred Brauer.Chapter 3. A version of this joint work between the author and Sarafa Iyani-wura is under review, and we were both responsible for the study and model design,and drafting of the manuscript. I was responsible for most part of the analysis ofthe reproduction number and the final size relation, while Sarafa Iyaniwura was pri-marily responsible for the asymptotic analysis. Drs Fred Brauer and Michael Wardassisted with study concept and critical revision of the manuscript for importantintellectual content. This chapter extends the work done in Chapter 2.Chapter 4. A version of this chapter is under review. I was primarily respon-sible for all major areas of the model formulation, concept formation and analysis,as well as a draft of the manuscript under the supervision of Drs Fred Brauer andViviane Dias Lima. Drs Fred Brauer, Viviane Dias Lima and Jielin Zhu assistedwith model design, data acquisition, and critical revision of the manuscript.Chapter 5. A version of this work is in preparation for publication and I wasprimarily responsible for the study concept, analysis, and drafting of the manuscriptunder the supervision of Drs Fred Brauer and Viviane Dias Lima. Drs Fred Brauerviand Viviane Lima assisted with the study concept. Drs Fred Brauer, Viviane DiasLima and Jielin Zhu assisted with model design and data acquisition.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Natural history of disease in humans . . . . . . . . . . . . . . . . 11.2 Brief introduction to mathematical epidemiology . . . . . . . . . 31.3 Formulation and examples of some disease models . . . . . . . . 51.3.1 Simple epidemic model . . . . . . . . . . . . . . . . . . 71.3.2 Simple endemic model . . . . . . . . . . . . . . . . . . . 91.4 Qualitative analysis . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.1 Epidemic model . . . . . . . . . . . . . . . . . . . . . . 101.4.2 Endemic model . . . . . . . . . . . . . . . . . . . . . . . 121.5 Quantitative analysis: . . . . . . . . . . . . . . . . . . . . . . . . 13viii1.6 Human epidemiological data, model fitting and parameter estimation 162 Epidemic models with heterogeneous mixing and indirect transmission 192.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 A two-group age of infection model with heterogeneous mixing . 222.3.1 A special case: heterogeneous mixing and indirect trans-mission for simple SIRP epidemic model . . . . . . . . . 242.3.2 Reproduction numberR0 . . . . . . . . . . . . . . . . . 302.3.3 The initial exponential growth rate . . . . . . . . . . . . . 322.3.4 The final size relation . . . . . . . . . . . . . . . . . . . . 342.4 Variable pathogen shedding rates . . . . . . . . . . . . . . . . . . 342.4.1 Reproduction numberR0 . . . . . . . . . . . . . . . . . 362.4.2 The initial exponential growth rate . . . . . . . . . . . . . 382.4.3 The final size relation . . . . . . . . . . . . . . . . . . . . 382.5 Heterogeneous mixing and indirect transmission with residence time 402.5.1 Reproduction numberR0 . . . . . . . . . . . . . . . . . 432.5.2 The initial exponential growth rate . . . . . . . . . . . . . 452.5.3 The final size relation . . . . . . . . . . . . . . . . . . . . 482.5.4 Numerical simulations . . . . . . . . . . . . . . . . . . . 512.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 A novel approach to modelling the spatial spread of airborne dis-eases: an epidemic model with indirect transmission . . . . . . . . . 543.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . 563.3.1 Non-dimensionalization of the coupled PDE-ODE model . 583.3.2 Asymptotic analysis of the dimensionless coupled PDE-ODE model . . . . . . . . . . . . . . . . . . . . . . . . . 603.4 One-patch model . . . . . . . . . . . . . . . . . . . . . . . . . . 663.4.1 The basic reproduction numberR0 . . . . . . . . . . . . 683.4.2 The final size relation . . . . . . . . . . . . . . . . . . . . 70ix3.4.3 Numerical simulation for one-patch model . . . . . . . . 713.5 Two-patch model . . . . . . . . . . . . . . . . . . . . . . . . . . 763.5.1 Reproduction numberR0 . . . . . . . . . . . . . . . . . 783.5.2 The final size relation . . . . . . . . . . . . . . . . . . . . 803.5.3 Numerical simulation for two-patch model . . . . . . . . 833.6 Effect of patch location on the spread of infection . . . . . . . . . 883.6.1 Effect of patch location on the basic reproduction number 883.6.2 Effect of patch location on the final size relation . . . . . . 923.6.3 Numerical simulation for two patch model with effect ofpatch location . . . . . . . . . . . . . . . . . . . . . . . . 953.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994 A co-interaction model of HIV and syphilis infection among gay, bi-sexual and other men who have sex with men . . . . . . . . . . . . . 1024.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.3 Model formulation and description . . . . . . . . . . . . . . . . . 1054.4 Syphilis sub-model . . . . . . . . . . . . . . . . . . . . . . . . . 1094.4.1 Endemic equilibrium points . . . . . . . . . . . . . . . . 1114.4.2 Global stability of the endemic equilibrium for syphilis-only model . . . . . . . . . . . . . . . . . . . . . . . . . 1124.4.3 Sensitivity analysis ofReS . . . . . . . . . . . . . . . . . 1124.5 HIV sub-model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.5.1 Disease free equilibrium point . . . . . . . . . . . . . . . 1144.5.2 Effective reproduction numberReH . . . . . . . . . . . . 1144.5.3 Global stability of the disease-free for HIV-only model . . 1174.5.4 Endemic equilibrium points . . . . . . . . . . . . . . . . 1184.5.5 Global stability of the endemic equilibrium for HIV-onlymodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.5.6 Sensitivity analysis ofReH . . . . . . . . . . . . . . . . . 1214.6 Analysis of the HIV-syphilis model . . . . . . . . . . . . . . . . . 1244.6.1 Disease free equilibrium point (DFE) of the full HIV-syphilismodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124x4.6.2 Effective reproduction numberRe . . . . . . . . . . . . . 1254.6.3 Global stability of the disease-free of the full HIV-syphilismodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.6.4 Endemic equilibrium point of the full HIV-syphilis model 1254.7 Numerical simulations of the full model . . . . . . . . . . . . . . 1274.8 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . 1345 Assessing the combined impact of interventions on HIV and syphilisepidemics among gay, bisexual and other men who have sex withmen in British Columbia: a co-interaction model . . . . . . . . . . . 1365.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.3.1 HIV-syphilis transmission model . . . . . . . . . . . . . . 1405.3.2 Modeling scenarios . . . . . . . . . . . . . . . . . . . . . 1425.3.3 Main outcomes . . . . . . . . . . . . . . . . . . . . . . . 1435.3.4 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . 1435.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1455.4.1 Status Quo . . . . . . . . . . . . . . . . . . . . . . . . . 1455.4.2 TasP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1455.4.3 PrEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1465.4.4 Condom use . . . . . . . . . . . . . . . . . . . . . . . . 1465.4.5 Test & Treat syphilis . . . . . . . . . . . . . . . . . . . . 1475.4.6 Combining two interventions . . . . . . . . . . . . . . . . 1475.4.7 Combining three interventions . . . . . . . . . . . . . . . 1485.4.8 Conditions for the elimination of the HIV and syphilis epi-demics . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.4.9 Sensitivity analyses . . . . . . . . . . . . . . . . . . . . . 1535.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1585.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1606 Conclusions and future directions . . . . . . . . . . . . . . . . . . . 161Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164xiA Supporting information for the co-interactional model used in Chap-ter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184A.1 The proof of Lemma 4.4.2 . . . . . . . . . . . . . . . . . . . . . 184A.2 The proof of Lemma 4.4.4 . . . . . . . . . . . . . . . . . . . . . 185A.3 The proof of Lemma (4.5.4) . . . . . . . . . . . . . . . . . . . . 186A.4 The proof of Lemma (4.6.2) . . . . . . . . . . . . . . . . . . . . 187B Supporting information for the co-interactional model used in Chap-ter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190B.1 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . 190B.1.1 Model equations . . . . . . . . . . . . . . . . . . . . . . 190B.1.2 Model parameters and variables . . . . . . . . . . . . . . 192B.1.3 Model assumptions about PrEP uptake in BC . . . . . . . 195B.1.4 Model calibration . . . . . . . . . . . . . . . . . . . . . . 195B.2 Model outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . 198xiiList of TablesTable 2.1 Model variables, parameters and their descriptions. . . . . . . . 25Table 2.2 Model variables, parameters and their descriptions. . . . . . . . 42Table 2.3 Parameter values and their sources. . . . . . . . . . . . . . . . 50Table 3.1 Model variables and their descriptions . . . . . . . . . . . . . 72Table 3.2 Parameter descriptions and values for the Two-patch model. . 84Table 4.1 Model variables and their descriptions . . . . . . . . . . . . . 105Table 4.2 Model parameters and their interpretations. . . . . . . . . . . . 128Table 5.1 Scenarios for the interventions examined in the study . . . . . 144Table B.1 Model parameters and variables. Abbreviations: PrEP: Pre-Exposure Prophylasis, gbMSM: Gay, bisexual and other menwho have sex with men, STIs: Sexually Transmitted Infections,ART: Antiretroviral Therapy . . . . . . . . . . . . . . . . . . 192Table B.2 Estimates of the number of PLWH and the number of annualnew HIV infections from PHAC. Abbreviation: PLWH: Peopleliving with HIV . . . . . . . . . . . . . . . . . . . . . . . . . 196Table B.3 Published data on cases of HIV and syphilis infections fromBC-CFE and BCCDC respectively. Abbreviation: BCCfE: BritishColumbia Centre for Excellence for HIV/AIDS; BCCDC: BritishColumbia Centre for Disease Control . . . . . . . . . . . . . . 196Table B.4 Model outcomes under TasP interventions . . . . . . . . . . . 199Table B.5 Model outcomes under Test & Treat syphilis interventions . . . 201xiiiTable B.6 Model outcomes under PrEP and condom use interventions . . 202Table B.7 Model outcomes under the combination of different interventions 203Table B.8 HIV prevalence and incidence rates, syphilis incidence rates,mortality rate among PLWH under different interventions . . . 205xivList of FiguresFigure 1.1 Figure (1.1a) explains the onset of a disease from the infectionstage to outcome (Removed) stage, while figure (1.1b) givesthe pathways through which diseases are tranmsitted . . . . . 2Figure 1.2 SIR model flow chart [25] . . . . . . . . . . . . . . . . . . . 7Figure 1.3 Results of numerical solutions of the SIR (figure 1.3a) andSEIR (figure 1.3b) epidemic model which predict the rate ofchange of susceptible, exposed, infected and removed over time,and compare quantitative behaviours of the two models. Thesimulations show basically the effect of exposed period on thebehaviour of the model . . . . . . . . . . . . . . . . . . . . . 15Figure 2.1 Dynamics of I1 and I2 when we vary p11, p12, p21, p22 and haveno movement (p11 = p22 = 1, p12 = p21 = 0), half populationsmoving (p11 = p22 = p12 = p21 = 0.5), and all populationsmoving (p11 = p22 = 0, p12 = p21 = 1). The figure on the leftpanel shows that the prevalence in patch 1 reaches its highestwhen in extreme mobility case (blue line) and is lowest whenthere is no mobility between patches (red line). The figure onthe right panel show the opposite of this senario in patch 2(high risk). . . . . . . . . . . . . . . . . . . . . . . . . . . . 51xvFigure 3.1 The dynamics of infected I(t) for different diffusion rates ofpathogen D and D0, and other parameters as in Table 3.1. (a)shows the result obtained from the reduced ODE (3.35) withinitial conditions (S(0), I(0),R(0), p(0))= (249/250,1/250,0,0),while (b) is the result of the dimensionless coupled PDE-ODEmodel (3.34) with initial conditions (S(0), I(0),R(0),P(0)) =(249/250,1/250,0,0) . . . . . . . . . . . . . . . . . . . . . 72Figure 3.2 The dynamics of proportion of infected individuals I(t) usingdifferent diffusion rates of pathogen, and all other parametersas in Table (3.1). (a) shows the result obtained from the systemof ODEs (3.35) with initial conditions (S(0), I(0),R(0), p(0))=(249/250,1/250,0,1), while (b) is the result of the dimension-less coupled PDE-ODE model (3.34) with initial conditions(S(0), I(0),R(0),P(0)) = (249/250,1/250,0,1) . . . . . . . 74Figure 3.3 Surface plots of the basic reproduction number R0 (3.38) forthe one-patch model (3.35) plotted with respect to the diffusionrate of pathoegns D0 and some dimensionless parameters ofthe SIR model. (a) is for D0 and the transimission rate β , while(b) is for D0 and the shedding rate σ . The parameters used aregiven in Table 3.1 . . . . . . . . . . . . . . . . . . . . . . . . 75Figure 3.4 The dynamics of proportion of infected individuals I(t) usingdifferent diffusion rates, and all other parameters as in Table3.1. (a) shows the results for patches 1 and 2 obtained from thereduced ODE (3.46) with initial conditions (S1(0), I1(0),R1(0))=(249/250,1/250,0), (S2(0), I2(0),R2(0))= (250/250,0,0) andp(0) = 0, and (b) shows similar results obtained with the cou-pled PDE-ODE model (3.45) for the same initial conditionsin the patches as the ODEs and P(0) = 0 for the diffusingpathogens. In both plots, the solid lines represents patch 1,while the dashed lines are for patch 2 . . . . . . . . . . . . . 85xviFigure 3.5 The dynamics of infected I(t) using different diffusion rates ofpathogen, and all other parameters as in Table 3.1. (a) showsthe results obtained for patches 1 and 2 from the reduced sys-tem of ODEs (3.46) with initial conditions (S1(0), I1(0),R1(0))=(249/250,1/300,0), (S2(0), I2(0),R2(0))= (250/250,0,0), andp(0) = 1, while (b) shows similar results obtained from thecoupled PDE-ODE model (3.45) with the same initial condi-tions for the ODEs in the patches and P(0)= 1 for the diffusingpatheogens. In both plots, the solid lines represent of patch 1,while the dashed lines are for patch2 . . . . . . . . . . . . . . 86Figure 3.6 The dynamics of infected I(t) using different diffusion rates,and all other parameters as in Table 3.2. (a) and (b) show theresults obtained from the reduced system of ODEs (3.46) forpatches 1 and 2, with initial conditions (S1(0), I1(0),R1(0)) =(299/300,1/300,0), (S2(0), I2(0),R2(0)) = (250/250,0,0),and p(0) = 1, while (c) and (d) show similar results obtainedby solving the coupled PDE-ODE model (3.45) with the sameinitial conditions for the ODEs in the patches and P(0) = 1 forthe diffusing patheogens . . . . . . . . . . . . . . . . . . . . 87Figure 3.7 Surface plots of the basic reproduction number R (3.67) (sec-ond row) and its O(ν) term R1 (3.68) (first row) with respectto the distance of the patches from the centre of a unit diskr, for different values of the transmission rates β1 and β2 forpatches 1 and 2, respectively. The parameters used are givenin Table (3.2) except for pc = 450, with diffusion rate D0 = 5.For each of the graphs, β1 (vertical axis) is plotted against r(horizontal axis). The value of β2 changes for each columnfrom left to right in increasing order. The term R1 show howthe leading-order basic reproduction number R0 is perturbedby the location of the patches . . . . . . . . . . . . . . . . . . 96xviiFigure 3.8 The dynamics of infected I(t) for different ring radius r. (a)and (b) show the results obtained from the reduced ODE (3.75)for patches 1 and 2, with initial conditions (S1(0), I1(0),R1(0))=(299/300,1/300,0), (S2(0), I2(0),R2(0))= (250/250,0,0), andp(0) = 1, while (c) and (d) show similar results obtained fromthe coupled PDE-ODE model (3.45) with the same initial con-ditions for the ODEs in the patches and P(0) = 1 for the dif-fusing patheogens. The diffusion rate of pathogens is fixed atD0 = D = 5, while all other parameters are as given Table (3.2) 98Figure 4.1 Diagram of the HIV/Syphilis co-interaction model . . . . . . 107Figure 4.2 Syphilis reproduction number ReS as a function of testing andtreatment rate σ1, with all parameters as in Table B.1 exceptβS = 5.0. The red dashed line indicates the reproduction num-berReS = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Figure 4.3 Impact of increasing testing rate α1, treatment rate ρ2 and rateof treatment failure ν1 on HIV reproduction numberReH , withall parameters as in Table B.1 except for βH = 0.4. The redline shows whenReH = 1 . . . . . . . . . . . . . . . . . . . 123Figure 4.4 Number of HIV infected individuals (green) and syphilis in-fected individuals (red) based on initial condition (4.42) andparameters in Table B.1, with different transmission rates andreproduction number: βH = 0.02,βS = 0.1,Re = 0.139 (left);βH = 0.4,βS = 5.0,Re = 2.780 (right) . . . . . . . . . . . . . 132Figure 4.5 Using the initial condition in (4.42) with βH = 0.02 and βS =5.0, the figure shows dynamics of HIV mono-infected individu-als (UH +AH +TH) (A), co-infected individuals (USH +ASH +TSH) (B), and syphilis mono-infected individuals (IS) (C). . . . 133Figure 4.6 Using the initial condition in (4.42) with βH = 0.4 and βS =0.1, the figure shows dynamics of HIV mono-infected individu-als (UH +AH +TH) (A), co-infected individuals (USH +ASH +TSH) (B), and syphilis mono-infected individuals (IS) (C). . . . 133xviiiFigure 4.7 Prevalence of HIV and syphilis with βH = 0.4 and βS = 5.0(ReH = 2.780> 1,ReS = 1.245> 1,Re = 2.780> 1). (a) Fig-ure 4.7a shows the prevalence of HIV with syphilis at the ini-tial stage of the epidemic (initial condition (4.42), blue dashedline) and without syphilis (initial condition (4.43), red solidline). (b) Figure 4.7b shows the prevalence of syphilis infectionwith HIV at the initial stage of the epidemic (initial condition(4.42), blue dashed line) and without HIV (initial condition(4.44), red solid line). . . . . . . . . . . . . . . . . . . . . . . 134Figure 5.1 Diagram of the HIV/Syphilis co-interaction model . . . . . . 141Figure 5.2 HIV incidence rate under different intervention scenarios incomparison to the WHO threshold for disease elimination asa public health concern at the end of 2028. WHO: WorldHealth Organization; GBMSM: gay, bisexual and other menwho have sex with men; TasP: treatment as prevention; PrEP:pre-exposure prophylaxis; Test & Treat: test and treat syphilis. 150Figure 5.3 Syphilis incidence rate under different intervention scenariosin comparison to the WHO threshold for disease eliminationas a public health concern at the end of 2028. WHO: WorldHealth Organization; GBMSM: gay, bisexual and other menwho have sex with men; TasP: treatment as prevention; PrEP:pre-exposure prophylaxis; Test & Treat: test and treat syphilis. 151Figure 5.4 Results for the reduction in HIV point prevalence, the cumu-lative number of HIV incident cases, and all-cause mortalitycases among PLWH (first row), and the cumulative numberof syphilis incident cases (second row) among gbMSM livingwith HIV after 10 years of TasP, PrEP, condom use, and Test& Treat (syphilis) interventions . . . . . . . . . . . . . . . . 152xixFigure 5.5 Results of the sensitivity analyses for the top ten parameterswith the highest sensitivity coefficients based on the scenar-ios for PrEP use. Row 1: cumulative number of HIV incidentcases at the end of 2028; Row 2: cumulative number of syphilisincident cases at the end of 2028; PrEP: pre-exposure prophy-laxis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154Figure 5.6 Results of the sensitivity analyses for the top ten parameterswith the highest sensitivity coefficients based on the scenariosfor TasP. Row 1: cumulative number of HIV incident cases atthe end of 2028; Row 2: cumulative number of syphilis inci-dent cases at the end of 2028; TasP: HIV treatment as prevention155Figure 5.7 Results of the sensitivity analyses for the top ten parameterswith the highest sensitivity coefficients based on the scenariosfor Test & Treat. Row 1: cumulative number of HIV incidentcases at the end of 2028; Row 2: cumulative number of syphilisincident cases at the end of 2028; Test & Treat: test and treatsyphilis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156Figure 5.8 Results of the sensitivity analysis for the parameters with themost uncertainty based on the available literature. Row 1: Per-cent change in the cumulative number of HIV incident casesin comparison to the Status Quo at the end of 2028; Row 2:Percent change in the cumulative number of syphilis incidentcases in comparison to the Status Quo at the end of 2028 . . . 157Figure B.1 PHAC estimates of PLWH and annual new HIV infections (blueerror bars) and model simulations (solid red line) during theperiod 2011−2018 . . . . . . . . . . . . . . . . . . . . . . . 197Figure B.2 Annual HIV and syphilis diagnoses (blue points) and modelsimulations (solid red line) during the period 2011−2018 . . 197xxAcknowledgementsFirst, I would like to express my sincere gratitude to my supervisor Professor FredBrauer, for his tremendous encouragement, patience, and motivation towards thecompletion of my PhD studies. This has indeed been a long journey and I wouldn’thave made it this far without your insightful ideas, countless useful discussions andespecially your believe in my ability. I also like to thank my co-supervisor Profes-sor Viviane Dias Lima, for her cordial help, technical assistance, guidance andsupport. Her valuable discussion and data interpretation helped me to better under-stand applications of the models to public health. Special thanks to my committeemember Professor Priscilla Greenwood for her insightful comments, encourage-ment and enlightenment towards widening my research from an interdisciplinaryperspective, and for always listening to my complaints.My profound gratitude to Professor Leah Edelstein-Keshet for constantly check-ing up on me and for being instrumental to my coming to Canada and working withFred. Many thanks to Professor Daniel Coombs for his expertise, valuable advice,and for always encouraging me to think critically when he was my course instruc-tor and graduate advisor at UBC. I am grateful to Professor Michael Ward for hiscountless suggestions during graduate courses and for collaborating on one of ourprojects. Also, my appreciation goes to Professor Carlos Castillo-Chavez, who firstintroduced me to co-interaction of infectious disease models through Fred. Manythanks to Professor Linda Allen for many useful forwarded posts and ads. I sin-cerely appreciate Professor Julien Arino for recommending my dissertation to theGraduate Studies.I will like to appreciate Dr. Jielin Zhu, Dr. Ignacio Rozada, Sarafa Iyani-wura who have immensely contributed to this success. I thank my fellow col-xxileagues at UBC Mathematics department, Interdisciplinary Studies, Institutes ofApplied Mathematics and British Columbia Centre for Excellence in HIV/AIDS(BCCfE). Over the years friendships were made and intercontinental relationshipsbuilt. Ditha, you were the best distraction I always wished for at BCCfE.To my husband and my daughter (my Hero Seyi Oyajumo, and my PrincessValerie Oyajumo): I know we have been through a lot as a family, but smiles onyour faces encouraged and gave me strength to carry through. You both have beenmy support at the toughest moment of this study. Special thanks to my family, mybrothers and sister for supporting me and for believing in my dreams. Words cannot express how grateful I am to my mother, Mosunmola David and mother-in-lawOlolade Oyerinde, for all the sacrifices you made during my PhD program.Whenever I felt the PhD journey would never end, my families at RCCG GraceChapel have always ignited the passion to hope for a better future. Thank you toPastor Bayo and Pastor Ola Adediran for their support towards destiny fulfilment.I will end by saying, all my life God has been faithful. I am a living testimony anda product of grace. I thank my God and my good good Father. Thank you JESUS.xxiiDedicationTo my late father David Osevwe who died during my PhD program, and to theLord Almighty, the author and the finisher of my faith.xxiiiChapter 1Introduction1.1 Natural history of disease in humansThe epidemiologic triad of human diseases as in figure (1.1b) results from the inter-action of a host (human), an infectious or other agent (e.g., virus, bacterium), andthe environment in which the exposure is being promoted (e.g., contaminated watersource) [25, 83]. In general, diseases such as influenza, measles, rubella (Germanmeasles), and chicken pox that are transmitted via viral agents generate durableimmunity against reinfection, while diseases such as tuberculosis, meningitis, andgonorrhoea that are transmitted via bacteria produce no immunity against reinfec-tion. Many other human diseases (e.g., malaria, West Nile virus, HIV/AIDS) aretransmitted through infection of vectors or agents (usually insects) by a second host(human) and from infected vectors or agents to another host (indirect transmissionpathway) [25]. Infectious diseases are generally transmitted via direct (person-person) and/or indirect (person-host-person) pathways [23, 25, 83].1(a) Natural history of disease [83] (b) Epidemiologic triad of a disease [83]Figure 1.1: Figure (1.1a) explains the onset of a disease from the infectionstage to outcome (Removed) stage, while figure (1.1b) gives the path-ways through which diseases are tranmsittedIn the web of disease transmission, both clinical and subclinical cases of dis-ease are important, although most subclinical cases are invisible (asymptomatic).Cases of polio in prevaccine days were one of the subclinical cases, where manywho contacted polio infection were not clinically ill, but were still capable ofspreading the virus to others [83]. From figure (1.1a), we denote this U (unex-plained) as the interval from the exposed period to the time of clinical onset ofthe disease). The rate of spread of disease is related to the virulence of the organ-ism (the rate of production of disease by the organism), site in the host body, andcharacteristics of host body (in terms of immune response) [83].The Latent/Exposed period is the time when an individual is infected, shows nosigns or symptoms and cannot transmit the disease. The incubation period is theinterval from infection to the time of the clinical illness. During this time, an in-fected individual shows no symptoms or signs of the disease and this time dependson the organism, site in the body and the dose of the infectious agent received atthe infection time (large dose shortens the incubation period [83]). The length ofthe incubation period for a given disease is characterized by the infective organ-ism. The epidemiological problem here is that, during the latter part of incubationperiod (denoted as U in figure (1.1a)), a person can transmit the disease to oth-ers (e.g., common childhood disease) [83], and many mathematical models and aswell as epidemiological data may not take this period into account, and this posesa problem of when to quarantine, isolate or even treat an infected person. This is2common to influenza infections, where it is known as pre-symptomatic infection,in which infected individuals become infective/infectious before the appearance ofdisease symptoms [7, 83].The infectious period, as in figure (1.1a), is the time during which an infectedindividual is clinically ill, shows signs or symptoms and can transmit the disease[25, 43, 110], and the earlier part of this period overlaps with the incubation period.Mathematical modelers commonly refer to this period as the infective period, whilepublic health professionals refer to it as infectious period [7, 25, 43]. The termsinfective or infectious period will be used interchangeably in this essay. The epi-demiological outcome of exposure is recovery, death (mostly used in epidemiologybecause it is easy to measure), critical/severe illness and so on [83]. We will focusmainly on epidemics and partially on endemic scenarios. Epidemic is the suddenoccurrence of disease in a region above the normally expected level [25, 43, 83].Some epidemic outbreaks and events of concern to people include but are not lim-ited to the 2002 SARS outbreak, Ebola virus and avian flu [25]. Endemic is theconstant presence of disease within a particular region [25, 43, 83]. Prevalence ofdiseases such as HIV, malaria, cholera, and typhus are endemic in most less de-veloped countries and in many parts of the world [25]. World wide epidemics arereferred to as Pandemics, and surveillance in public health helps in detecting anunusual outbreak above the normal level [83].1.2 Brief introduction to mathematical epidemiologyMathematical models have been extensively used to study disease transmissiondynamics in human populations and to extrapolate from epidemiology data in pre-dicting risk. We can similarly say that mathematical epidemiology is the use ofmathematical techniques to understand the spread of infectious diseases in humanpopulations [6]. Mathematical modeling in epidemiology gives us comprehensionof the disease transmission channels and then recommends methods of control asin [25] and as well broadly discussed in [26]. Models also help to identify mea-surement errors (information bias which may overestimate or underestimate thetrue association of exposure and outcome [83]) in the the experimental data [25].Models are used to evaluate the number needed to treat (NNT) and how extensive3a vaccination plan must be to prevent epidemics [3, 25, 83, 86, 112].Epidemiological experiments are often difficult or may be unethical to carryout when diseases are involved, i.e., placing some groups on drug (treated group)and others as control group (placebo or untreated group) may be impossible whendiseases are involved. Blinding is sometimes unethical in this situation and if donemay produce some irregularities in the data which may distort the true result [7,25, 83]. Models are often used to identify unclear behaviour in experimental data[7, 25]. Some of the models include, but are not limited to, deterministic models,and stochastic models [86, 150]. We can also have heterogeneous models to includedifferent behaviour of people and the possibility of having superspreaders (peoplethat transmit infection to many people in the population) [22]. Many modellingpatterns that have been used include, but are not limited to, ordinary differentialequations, partial differential equations, integral equations, branching processes,and chain stochastic models [3, 6, 7].One of the basic results in mathematical epidemiology is the exhibition ofthreshold behaviour by most mathematical epidemic models. This is symbolicallywritten as R0 (the basic reproduction number) and in epidemiological terms, itmeans the average number of secondary infections caused by an average infective.If the basic reproduction number R0 is less than one, this means that the diseasedies out and greater than one means that there is a possibility that an epidemic willoccur [7, 25, 169]. This idea, consistent with observations and quantified throughepidemiological models has been constantly used to estimate the efficacy (how avaccine works under ideal conditions [83]), effectiveness (how a vaccine works inreal life [83]), and efficiency (the cost benefit ratio of an effective vaccine [83]) ofvaccination policies and the possibility that a disease will be eradicated or elimi-nated [25].When we incorporate factors aimed at controlling the spread of disease intoa model, we use instead the control reproduction number, denoted as Rc, sincecontrol measures decrease reproduction number and therefore decrease the numberof secondary infections caused by a single infective. Models give a methodical wayto estimate Rc, which is very important to determine the public health measuresnecessary for disease control and impact on infection transmission [7, 25].In modeling of disease transmission, there is always an issue of simple mod-4els omitting most details, and designed only for analyzing general qualitative be-haviours, while detailed and complex models are designed for specific situationsand prediction making. The use of detailed and complex models for theoreticalpurposes is limited since they are difficult to handle and can not be solved analyt-ically; Complex models with high strategic value and numerical simulations areneeded for detailed planning by public health professionals and policy decisionmakers. Simpler models such as systems of small number of differential equationsare the building blocks of most complex models, and may give some useful con-clusions [6, 7, 22, 25, 43, 99]. There is need for collaboration to build models thataddress the right questions for complex and timely decision making [7]. We there-fore focus this introductory chapter on simple compartmental models to establishbasic concepts.1.3 Formulation and examples of some disease modelsDifferent models have been used in different forms to answer some public healthquestions and in public policy making. To gain a broader knowledge, we willlimit the scope to simple compartmental models and begin the model formula-tion with epidemic models (models with no demographic effects), and later extendto incorporate demographic effects to explore endemic scenario. The Kermack-McKendrick model, which comes with simple assumptions on rates of flow be-tween different classes of individuals in the populaion is the form of the compart-mental model we will mostly consider. The simple models we will formulate willhelp answer some questions of interest to public health professionals. For exam-ple, how detrimental do we expect an epidemic to be? [25, 43, 83]. The Kermack-McKendrick model assumes complete homogeneous mixing between susceptiblesand infectives, but at the beginning of a disease outbreak, this assumption is notvalid for a stochastic process. Examining the network of person to person con-tacts is more realistic for the description of the disease outbreak [7, 86]. The useof network models led to greater improvements in the understanding of epidemicdevelopment [7]. Network models are able to show that even if the basic reproduc-tion number is greater than 1, there is a possibility that only a minor outbreak withno full-blown epidemic may occur [7].5An outbreak or epidemic is investigated when three critical variables are known,i.e., the time the exposure took place, the time the disease began and the incuba-tion period of the disease. One variable can be calculated when the other two areknown [83]. The study population is divided into different compartments, with as-sumptions about the nature and time transfer rate from one compartment to another.Susceptibles are individuals who have no full immunity against the infectious agentand therefore can become infected when exposed. Infectives are individuals cur-rently infected and can transmit the infection to susceptible individuals they are incontact with. Removed individuals are individuals who have immunity against in-fection, and have no effect on transmission dynamics whenever they are in contactwith other individuals [25].The term SIR will be used to describe diseases that confer immunity againstre-infection (e.g., recovery from measles), which denote that the transition of indi-viduals is from the susceptible class S to the infectious class I and to the removed(outcome) class R. On the other hand, the term SIS will be used to describe dis-eases with no immunity against re-infection (e.g., common cold, syphilis), whichdenote the transition of individuals from the susceptible class S to the infectiousclass I and then back to the susceptible class S again. Similarly, we have otherpossibilities as SEIR (diseases such as tuberculosis, SARS, flu) and SEIS, whichinclude an exposed period between being infected and becoming infectious/infec-tive, and SIRS models (diseases such as syphilis), with temporary immunity onrecovery from infection, and SI models (diseases such as HIV, and herpes), withno recovery from infection [25, 43].To begin with, our models are formulated as differential equations with timet (the indepedent variable) and transfer rates between compartments expressed inmathematical terms as derivatives with respect to time of the sizes of the modelcompartments. It is possible to generalize to models in which transfer rates de-pend on the compartments sizes over the past and at the instant of transfer aswell. These will lead to more general types of functional equations (differential-difference equations, integral equations, or integro-differential equations [25]) andwill not be considered in this essay.6Figure 1.2: SIR model flow chart [25]1.3.1 Simple epidemic modelDuring the course of an epidemic, there is an initial increase in the number of newinfections, which leads to decrease in the number of susceptibles, and thereforedecreases the number of new infections. This decrease in the number of new in-fection, as a result of decrease in the number of susceptible individuals, slows thespread of disease and may eventually end the epidemic [7]. We assume a determin-istic epidemic process here and for SIR epidemic model, the population under studyis divided into three classes S, I and R. Three papers written by W.O. Kermack andA.G. McKendrick in 1927, 1932, and 1933 proposed simple compartmental mod-els to describe the transmission of communicable diseases, and the first paper gavethe description of epidemic models (often called the Kermack-McKendrick epi-demic model). The simple epidemic model that will be considered in this studywill be the special case of the proposed model by Kermack and McKendrick in1927 [6, 25, 43], which is given asdSdt= −β IS,dIdt= β IS−σ I, (1.1)dRdt= σ I.The flow chart in figure (1.2) shows the transmission dynamics between com-partments. Model (1.1) assumes mass action incidence, i.e., individual makes con-tact enough to transmit infection with βN others per unit time and the total sizeof the population is assumed to be N. The model focuses on the dynamics of asingle epidemic outbreak and therefore assumes no entry into the population andthat departure only exist through death from the disease (no demographic effects).7The model assumes that infectious individuals leave the infective class at rate σ Iper unit time with recovery rate σ , which give the mean infective period 1/σ .The total study population N = S(0) initially and in the absence of an infection.The probability that an infectious individual made arbitrary contact with a sus-ceptible individual, who then transmit infection is given by S/N and we thereforehave the number of new infections in unit time per an infectious individual to be(βN)(S/N) = βS, which gives the rate of new infections (βN)(S/N)I = βSI, withthe transmission rate (per capita) β . Note that for an SIR disease model, the to-tal population is N = S+ I +R. We may neglect the differential equation for thenumber of removed individuals asdSdt= −β IS,dIdt= β IS−σ I, (1.2)since R does not appear in (1.1), the equation for R˙ or dR/dt has no effect on thetransmission dynamics of S and I [25]. This SIR model is standard and now beingdiscussed in many introductory calculus textbook [25]. The numberR0 = βN/σ isknown as the basic reproduction number, and it is the important to consider in theanalysis of any infectious disease epidemic model. The first infectious individualis expected to infect R0 = βN/σ individuals and this determines the occurrenceof an epidemic at all [25, 169]. The SI epidemic model (e.g., syphilis) is given asdSdt= −β IS,dIdt= β IS−αI, (1.3)where α is the disease induced mortality. Diseases such as herpes and all chronicinfections [6] (e.g. HIV) are some of examples of SI cases. Similarly, the SISepidemic model is given asdSdt= δ I−β IS,dIdt= β IS− (α+δ )I, (1.4)8where α is the disease induced mortality and δ is the disease recovery rate with noimmunity and we also have the SEIR epidemic model asdSdt= −β IS,dEdt= β IS−νE,dIdt= νE−σ I, (1.5)dRdt= σ I,where the exposed individuals leave the exposed class at rate νE per unit time withexposed rate ν , which give the mean exposed period 1/ν .1.3.2 Simple endemic modelWe assume a deterministic endemic process here and for SIR endemic model (modelwith demography), the population under study is also divided into three classes S,I and R. Since an epidemic generally has a much shorter time scale than the de-mographic time scale, births and deaths which were omitted in the description ofepidemic will be discussed here with the use of longer time scale. Many endemicdiseases have caused millions of deaths each year in many parts of the world. Forendemic diseases, public health professionals are mostly interested in the numberof infectives at a given time, the rate of rise of new infections, possible control mea-sures, and methods to eradicate the disease in a population. The simple endemicSIR model that will be considered is given asdSdt= µN−β IS−µS,dIdt= β IS− (α+σ +µ)I, (1.6)dRdt= σ I−µR.For the sake of simplicity, model (1.6) assumes mass action contact rate, similar tothe case of epidemic models previously considered. We have the disease recovery9rate to be σ and disease induced mortality to be α . For simplicity, we may assumeequal birth and death as µ and no disease induced mortality (α), so that N is con-stant. Since S+ I+R = N, we can determine R if S and I are known, and thereforehave model (1.6) to be written asdSdt= µN−β IS−µS,dIdt= β IS− (α+σ +µ)I. (1.7)An endemic model that describes diseases with no immunity against re-infection(SIS model for bacteria diseases such as common cold [7]) is given asdSdt= µN+δ I−β IS−µS,dIdt= β IS− (α+δ +µ)I, (1.8)with disease recovery with no immunity to be δ .Using model (1.2), we can write the initial exponential growth rate ϒ asϒ= σ(R0−1). (1.9)Measuring ϒ makes it easier to estimate the basic reproduction number (R0) inequation (1.9).1.4 Qualitative analysis1.4.1 Epidemic modelModel (1.2) with initial conditions S(0) = S0, I(0) = I0, S0 + I0 = N onlymakes sense when S(t) and I(t) are nonnegative, and then the system ends wheneither of S(t) or I(t) reaches zero. We notice that S˙ < 0 for all t and I˙ > 0 on thecondition that S > σ/β , which then increases I and decreases S for all t. This de-crease in S eventually decreases I, and I tends to zero. Infective I decreases to zero(no epidemic) whenever S0 < σ/β , and on the other hand if S0 > σ/β , I increasesinitially to a maximum reached when S = σ/β and then decreases to zero, which10denotes an epidemic. The basic reproduction number for model (1.2) is denotedas R0 = βS0/σ , which determines whether an epidemic will occur. The infectiondies out wheneverR0 < 1, and an epidemic occur wheneverR0 > 1.The basic reproduction number is defined as the number of secondary infec-tions caused by the introduction of a single infective into a totally susceptible pop-ulation of size N ≈ S0 during the period of infection of the single infective intro-duced. In this scenario, βN contacts are made by an infective in unit time, withall being with susceptibles and producing new infections and with mean infectiveperiod 1/σ , which gives the basic reproduction number to be R0 = βN/σ ratherthan βS0/σ . We can also explain this evident difference by looking at two differ-ent ways in which epidemic begins. Epidemic may begin by either a member of apopulation under study with I0 > 0 and S0+ I0 = N or by a visitor from outside ofthe study population with S0 = N.The naive way to solve a two-dimensional autonomous system of differentialequations like model (1.2) is to find equilibria and determine stability by linearizingabout each equilibrium. Nevertheless, model (1.2) has a line of equilibria (i.e.every point with I = 0 is an equilibrium) and it is impossible to use this methodsince the linearization matrix produces a zero eigenvalue at each equilibrium. Wetherefore use a different method to analyze the system (1.2). The sum of equationsS and I in (1.2) givesd(S+ I)dt=−σ I.We can see that (S+ I) decreases to a limit, and since (S+ I) is a nonnegativesmooth function, we could show that its derivative approaches zero, from whichcan be concluded thatI∞ = limt→∞ I(t) = 0.Integrate the sum of the two equations of (1.2) from 0 to ∞ to haveσ∫ ∞0I(t)dt = S0+ I0−S∞ = N−S∞,∫ ∞0I(t)dt =N−S∞σ, (1.10)which implies that∫ ∞0 I(t)dt < ∞.11Divide the first equation of (1.2) by S and integrate from 0 to ∞ to havelogS0S∞= β∫ ∞0I(t)dt,and by substituting equation (1.10), we havelogS0S∞= βN−S∞σ=βNσ[1− S∞N]=R0[1− S∞N]. (1.11)Equation (1.11) is known as the final size relation. It gives an estimate of thetotal number of infections over the course of the epidemic from the parameter inthe model [23], and as well shows the relationship between the basic reproductionnumber and the size of the epidemic. The final size of the epidemic (N− S∞) isalways described in terms of the attack rate/ratio (1−S∞/N). We can generalize thefinal size relation (1.11) to epidemic model with more complex compartment thanthe simple SIR model (1.2), including model (1.5) with exposed periods, modelswith treatment, models involving quarantine of suspected individuals and isolationof diagnosed cases. For example, an epidemic with proportion of susceptiblesS0 = 0.999, and S∞ = 0.35 as in figure 1.3a and substituting into equation 1.11,gives the the estimate β/α = 1.61 andR0 = 1.61.1.4.2 Endemic modelWe can determine a disease free equilibrium (DFE) of the endemic model (1.7) bysetting S˙ = I˙ = 0:µN−β IS−µS = 0β IS− (α+σ +µ)I = 0. (1.12)We therefore have the disease free equilibrium (DFE) as (S, I) = (N,0), and the en-demic equilibrium point (EEP) as (S, I)=((α+σ +µ)β,µ(βN− (α+σ +µ))β (α+σ +µ)),which exists only when (α+σ +µ)< βN.We can analyze the stability of the above equlibria by the theorem below as;Theorem 1.4.1. Let the basic reproduction number be R0 =βNα+σ +µ, then12R0 < 1 shows that EEP does not exit, and for all positive initial conditions, wehave limt→∞(S(t), I(t)) = (N,0) and the disease dies out. Also, ifR0 > 1, then for allpositive initial conditions,limt→∞(S(t), I(t)) =((α+σ +µ)β,µ(βN− (α+σ +µ))β (α+σ +µ))=( 1R0N,µβ(R0−1)),and the disease persists in the population.We can interpret the basic reproduction number R0 =βNα+σ +µ(the aver-age number of cases produced when a case is introduced into a totally susceptiblepopulation) as the product of• β , the probability of contracting the disease when a potentially infectingcontact occurs,• 1α+σ +µ, the mean time spent in the infectious class when subject to thecompeting risks of natural death, recovery and disease induced death.1.5 Quantitative analysis:The SIR model, which is one of the easiest and basic of all epidemiological models,depends on calculating the percentage of the population in each classes (suscepti-ble, infected and removed/recovered) and determining the transmission rates be-tween these classes. Considering the simplest form of a single epidemic (ignoringbirths and deaths) as in equation (1.1), there are only two transitions: infection (in-dividuals progress from susceptible to the infected class) and recovery (individualsprogress from infected to the recovered class). For simplicity, it is often assumedthat individuals infected with a disease do recover at a constant rate [100], whereasgenerally assumed from epidemic data that the per capita rate of a given suscepti-ble individual being infected is proportional to the prevalence of the infection in thepopulation [83]. To make headway with the simple model in (1.1) needs modellersto estimate two parameters (the infection transmission rate β and recovery rate σ )which demonstrates the basic relationship between models and statistics. The in-terchange between models and statistics is that only models with good statisticalestimated parameters from epidemiological data can be used for prediction.13Once the two parameters have been estimated, the SIR model then predictsan epidemic which follows the pattern in figure (1.3a): the number of cases (asin colour red) initially increases until the percentage of Susceptible individuals(as in colour blue) have been adequately consumed. This process continues untilthe epidemic can no longer be maintained and eventually decreases the number ofcases, and increases the number of individuals being removed (as in green colour),which then leads to extinction of infection (as seen in figure (1.3a) which showshow the red curve goes to zero). The numerical simulations of the SIR model (1.1)shown in figure 1.3a produce three general predictions that are of importance topublic health and have policy implications. Predictions from this simple model aresupported by many more complicated models with numerous parameters [4, 99].For example, if figure (1.3a) assumes the numerical simulation for total proportionof population N = 1, with S0 = 0.999, I0 = 0.001 and with β = 0.3, σ = 0.187.We can therefore predict:1. The value of R0 = β/σ < 1 denotes an epidemic that is destined to quickfailure due to inability of the epidemic to sustain the transmission dynamics,whereas R0 > 1 denotes possibility of an epidemic. For the example shownin figure (1.3a),R0 is estimated to be approximately 1.6 which is dependenton both the population and infection.2. In general, the proportion of susceptible population at the end of epidemicbecomes very small for large values ofR0, but for the scenario ofR0 = 1.6,approximately 62% of the population is expected to be infected during anepidemic. More complicated model with many parameters may change theprecise value of the proportion infected, but the general idea continues tohold.3. Susceptibility could be reduced through vaccination and therefore decreasethe spread of infection in the population. Epidemic could as well be pre-vented by vaccinating only some part of the population, and endemic infec-tion could also be eradicated or pandemic prevented if a proportion 1−1/R0of the population is successfully immunized (number needed to treat) [4].For our example, we would need to immunize approximately 38% (NNT) of14the population to eradicate endemic and prevent pandemic. This value canbe reduced if vaccination is sensibly targeted with more complicated model[4, 99].(a) SIR epidemic model (b) SEIR epidemic modelFigure 1.3: Results of numerical solutions of the SIR (figure 1.3a) and SEIR(figure 1.3b) epidemic model which predict the rate of change of sus-ceptible, exposed, infected and removed over time, and compare quan-titative behaviours of the two models. The simulations show basicallythe effect of exposed period on the behaviour of the modelWe can similarly consider an epidemic scenario (including an exposed class)as in equation (1.5), and there are only three transitions: exposure (individualsprogress from susceptible to the exposed class), infection (individuals progressfrom exposed to the infected class) and recovery (individuals progress from in-fected to the recovered class). To make headway with the simple model in (1.5),modellers need to estimate three parameters: the infection transmission rate β ,exposed rate ν , and recovery rate σ .Once the three parameters have been estimated, the SEIR model then predictsan epidemic which follows the pattern in figure (1.3b): the number of cases (asin colour red) initially increases but lower than in SIR model from figure (1.3a).The epidemic was sustained for approximately 100 and 130 days in SIR and SEIRmodel respectively (as in figure (1.3a) and (1.3b), which shows how the red curvegoes to zero). The numerical simulations of SEIR model (1.5) shown in figure 1.3balso produces three general predictions like the SIR model (1.1).15From the previous example, if figure (1.3b) assumes the numerical simulationfor total proportion of population N = 1, with S0 = 0.999, E0 = 0, I0 = 0.001 andwith β = 0.3, σ = 0.187, ν = 0.5. We therefore have a similar basic reproductionnumberR0 = β/σ estimated to be 1.6 in both models. The proportion of suscepti-ble population at the end of epidemic in both models is not significantly different asthe same number of population would need to be treated (NNT= 38%) to preventone less case.In our example, the SIR and SEIR gave a very similar result and therefore SIRshould be preferred since only two parameters are needed, and of course this is justan example to show the impact of including an exposed class. We can improvethe authenticity and predictive accuracy of a model but also increase the number ofparameters needed to estimate, by considering more complicated models which in-corporate heterogeneous mixing and possibility of superspreaders [22], metapopu-lation studies [8], age of infection [19], residence time [13, 22] and mixing patternsthrough network models [150].1.6 Human epidemiological data, model fitting andparameter estimationHuman data are the most desirable and are highly prioritized, but unfortunately,completely reliable epidemiological data are rarely available. Even when epidemi-ological studies have been conducted, they usually have incomplete and unreliableexposure histories. Data are considered to be inadequate evidence in humans ifno satisfactory epidemiological studies exist. For better predictions, more data isneeded to refine the models being used. For example, it may be possible to decideoptimal allocation of resources for treatment from a model when there are enoughdata to know susceptibility to infection for several different age groups [7].Building a model that describes the transmission dynamics of an infectious dis-ease will strongly depend on parameters and available data to make proper estima-tion of unknown parameters and possibly predictions. Nevertheless, this procedurecomes with some fundamental challenges since models are based on unobservableoccurrences at the time of modelling, such as the transmission of infection betweeninfectives and susceptible individuals, the start and end of an infectious period (the16unexplained scenario, U in figure (1.1a) [83]), and the serial interval (the time in-terval between sequential infectious individual in a series of transmission). Butdata are based on observable occurrences that are usually collected by means ofepidemiological and clinical evidences [7]. The clinical serial interval may differfrom the serial interval from the model. There is also a problem of difference interminology, as public health professionals use the word incubation period of aninfection (the time from the period of infection/exposure to the clinical onset ofthe disease as in figure (1.1a) [7, 83]), while modellers use the word latent/exposedperiod (the time from the period of infection/exposure to the period of being infec-tious as in figure (1.1a) [7, 83]). Inconsistent or inappropriate use of these wordsmay lead to confusion and may not be appropriately accounted for in the model.For example, the case of influenza where we have an infectious pre-symptomaticperiod, which implies a shorter latent period than the incubation period. In this sit-uation, individuals become infectious before showing symptoms or signs and thispose a problem denoted as U (unexplained) in figure (1.1a).Another problem is the bias that may arise from data collection [7]. Admin-istrative factors such as delay in report (report bias), and misclassification bias(inconsistencies in classifying clinical cases) may distort and complicate the anal-ysis of clinical data. A disease such as influenza which has an infectious pre-symptomatic period and is therefore undiagnosed or not reported, or has differencesin reportability from one location to another, may present a complicated or dis-torted clinical data analysis [7]. Modellers may describe the course of a disease andestimate some key transmission parameters (e.g., the basic reproduction number)by fitting models to data. Nevertheless, it is unreasonable to fit curves to data if themodel does not produce a curve that has the same qualitative features as the data,and many times a model curve may not give the correct image of observations [7].In addition, there is also a problem of differences in reported cases (symptomaticcases) and actual cases (includes both symptomatic and asymptomatic cases) ofinfection. The curve produced from epidemic data represents the reported cases,while simple modelling will produce a curve that represents the actual cases of in-fection. Proper distinction between these two is very important and necessary toobtain appropriate results. Data collected from an epidemic is commonly used toestimate the basic reproduction number based on the observed initial exponential17growth rate of infectious cases. Measuring the initial exponential growth rate (ϒ)in equation (1.9) makes it easier to estimate the basic reproduction number (R0).However, a different model (SEIR) is needed if there is an exposed/latent pe-riod between being infected and being infectious, and this will change the rela-tion between the initial exponential growth rate and the basic reproduction number[6, 7, 25]. Therefore, the use of a simplified model can lead to incorrect estimatesof important parameters [7]. Some limitations such as a balance between predic-tive power of the model, its level of complication and the type of questions to beaddressed are inherent to the model structure itself. We therefore need to decideon which parameters are needed to be included or excluded from the model basedon their relevance and effect on the correctness of predictions [110]. The accuracyof the data used for estimating parameters of the model determines how useful amodel will be [7, 99]. In the case of limited data, sensitivity and uncertainty analy-ses may be done to determine the most important information for reliable estimateof outcomes [7, 99, 155]. Uncertainty analysis is done to investigate the effect ofunknown parameters or missing data on model outputs, while sensitivity analysisis done to investigate how model outputs vary with changes in input parameter val-ues [7, 155]. These two methods are now commonly used in decision analysis andare now being used in infectious disease modeling. These methods help to identifyparameter values that most influence model estimates [7, 155].While data early in the disease outbreak are usually misleading, we need au-thentic data to develop models that compare management policies for disease out-break. We can as well do the uncertainty and sensitivity analyses to know whichparameters mostly impact the model projections. To easily design public healthplanning, control policy and decision making based on each country, quantitativemodelling techniques would need multi-disciplinary collaborations among expertsfrom different disciplines such as clinicians, public health professionals, labora-tory technologists, epidemiologists, statistical and mathematical modelers. Simi-larly, knowledge translation activities are crucial part of modelling and therefore,modelers need to involve knowledge translation activities to demonstrate and com-municate the relevance of their results in plain language and in the context of publichealth. For detailed explanation of the main epidemic models contributed, pleasesee Chapters 2 & 3, and for endemic models, see Chapters 4 & 5 of this thesis.18Chapter 2Epidemic models withheterogeneous mixing andindirect transmission2.1 SynopsisWe developed an age of infection model with heterogeneous mixing in which indi-rect pathogen transmission is considered as a good way to describe contact that isusually considered as direct and we also incorporate virus shedding as a functionof age of infection. The simplest form of SIRP epidemic model is introduced andit serves as a basis for the age of infection model and a 2-patch SIRP model wherethe risk of infection is solely dependent on the residence times and other environ-mental factors. The computation of the basic reproduction number R0, the initialexponential growth rate and the final size relation is done and by mathematicalanalysis, we study the impact of patches connection and use the final size relationto analyze the ability of disease to invade over a short period of time.2.2 IntroductionEpidemic model of infectious diseases had been extensively investigated by propos-ing and investigating mathematical models ([13, 23, 25, 29, 169, 178] and refer-19ences therein). Diseases such as cholera and some airborne infections are pathogenicmicroorganism diseases that are usually transmitted directly via host-to-host [178],or indirectly by virus transferred through objects such as contaminated hands, en-vironments or objects such as shelves and handles [18, 23, 127, 173]. Pathogensheds by infected individuals may stay outside of human hosts for a long periodof time. However, alternative transmission pathways as a result of the behavior ofhost may constitute to the spread of infection, such as drinking contaminated wa-ter, touching handles that have been exposed to a virus, eating contaminated foodand so on [178]. Brauer [23] proposed a SIVR epidemic model with homogeneousmixing, which is an extension of the SIR model by the addition of a pathogen com-partment V to describe the indirect transmission pathway (host-source-host). Thebasic reproduction number and the final size relation was derived and investigatedto determine the impact of indirect transmission pathway on disease spread. Sim-ilarly, Bichara et al. [13] proposed an SIR epidemic model in two patches withresidence times, which describes patches with residents who spent a proportion oftheir time in different patches to analyze the direct transmission pathway (host-host). They derived the basic reproduction number, final size relation and investi-gated how residence times influence them. Tien and Earn [165] developed a SIWRdisease model which extended the SIR model by the addition of a compartment Wthat describes direct and indirect transmission pathways.We have based most mathematical results in this chapter on the final size rela-tion of epidemic models in an heterogeneous environment. This relation had beenextensively discussed in [13, 19, 20, 23, 28] using different models to predict howbad an epidemic could be during a disease outbreak. For example, consider a sim-ple compartmental model, which comes with simple assumptions on rates of flowbetween different classes of individuals in the population (the special case of theproposed model by Kermack and McKendrick in [100–102]) given asdSdt= −β IS,dIdt= β IS−ρI, (2.1)dRdt= ρI.20The final size relation to the simple model in (2.1) islogS0S∞= β∫ ∞0I(t)dt,=βNρ[1− S∞N], (2.2)= R0[1− S∞N],where S0 denotes the initial size of the susceptible class, N the size of the en-tire population, β effective contact rate, ρ removed rate, and R0 =(βNρ)thebasic reproduction number. The first infectious individual is expected to infectR0 =(βNρ)individuals and this determines if an epidemic will occur at all. Theinfection dies out whenever R0 < 1, and an epidemic occur whenever R0 > 1.Equation (2.2) which is known as the final size relation and gives an estimate ofthe total number of infections over the course of the epidemic from the parameterin the model [19, 23], and can similarly show the relationship between the basicreproduction number and the size of the epidemic. The final size (N−S∞) is usu-ally described in terms of the attack rate/ratio (1−S∞/N). Note that the final sizerelation in (2.2) can be generalized to epidemics model with more complex com-partments than the simple model in (2.1). Papers [13, 19, 20, 23, 28] extensivelydiscussed details of age of infection models and their final size relations, and wewill use these techniques to derive the final size relations throughout the paper.We intend in this work to incorporate an epidemic model with age of infectionand indirect transmission pathway in which pathogen is shed by infected individu-als into the environment, acquired by susceptible individuals from the environment,and transmitted in an heterogeneous mixing environment. We will further inves-tigate the nature of the epidemic when variable virus shedding rate and residencetime are taken into consideration. A Lagrangian method is used to monitor theplace of residence of each population at all times [13, 29, 36, 57]. We propose thatthis may be an alternative way to study disease epidemic in an heterogeneous mix-ing environment. The rest of this chapter is structured as follows. In section 2.3,we introduce the age of infection model in an heterogeneous mixing settings andanalyse the model succinctly. The analysis of the age of infection model follows21similar steps from the simpler version analyzed in 2.3.1. We describe in section 2.4how variable pathogen shedding rates are incorporated. In section 2.5, we formu-late a 2-patch model with residence time and determine the nature of the epidemicwhen populations in one patch spend some of their time in another patch. We anal-yse the patchy model for different scenarios numerically in the last part of section2.5 and devote section 2.6 to a summarized conclusion. Note that the same analyticapproach, a standard way to analyze disease transmission models will be used ineach section.2.3 A two-group age of infection model withheterogeneous mixingWe consider two subpopulations of sizes N1, N2, each divided into susceptibles S1and S2 and infectives I1 and I2 with a pathogen class P. We assume that Susceptibleindividuals become infected only through contact with the pathogen sheded byinfectives. Pathogen P is shed by infected individuals I1 and I2 at a rate r1 and r2respectively as in [95, 178]. The model assumes that the epidemic occurs within ashort period of time.Considering the age of infection, we define ϕ1(t) and ϕ2(t) as total infectivityin classes I1 and I2 at time t respectively, ϕ10(t) and ϕ20(t) represent the totalinfectivity at time t of all individuals already infected at time t = 0, A1(τ) andA2(τ) are the mean infectivity of individuals in classes I1 and I2 at age of infectionτ and Γ(τ) the fraction of pathogen remaining τ time units after having been shedby an infectious individual. This is an extension of [23] from homogeneous mixing22to heterogeneous mixing, and we therefore have the equation as in [28] asdS1(t)dt= −β1S1(t)P(t),ϕ1(t) = ϕ10(t)+∫ ∞0[−dS1(t− τ)dt]A1(τ)dτ,dS2(t)dt= −β2S2(t)P(t), (2.3)ϕ2(t) = ϕ20(t)+∫ ∞0[−dS2(t− τ)dt]A2(τ)dτ,P(t) =∫ ∞0(r1ϕ1(t− τ)+ r2ϕ2(t− τ))Γ(τ)dτ.We can replace (2.3) by the limit equationdS1(t)dt= −β1S1(t)P(t),ϕ1(t) =∫ ∞0[−dS1(t− τ)dt]A1(τ)dτ,dS2(t)dt= −β2S2(t)P(t), (2.4)ϕ2(t) =∫ ∞0[−dS2(t− τ)dt]A2(τ)dτ,P(t) =∫ ∞0(r1ϕ1(t− τ)+ r2ϕ2(t− τ))Γ(τ)dτ,with a choice of initial function ϕ10(t) and ϕ20(t). Asymptotic theory of integralequations in [111] assures that the asymptotic behaviour of (2.3) is synonymousto that of the limit equation (2.4) for every initial function with limt→∞ϕ10(t) =limt→∞ϕ20(t) = 0 [28, 111]. We assume that∫ ∞0 Γ(τ)dτ < ∞, where the functionΓ is monotone non-increasing with Γ(0) = 1, and that∫ ∞0 A(τ)dτ < ∞, where A isnot necessarily non-increasing.In order to evaluate the basic reproduction number, the initial exponential growthrate, and the final size relation in terms of the model parameters, it makes sense tostart with the simplest form of the limit equation (2.4) as was done in [20, 21, 28]by considering a special case in Section (2.3.1). For this special case, we assumethat there is no age of infection, so that we approximate the model (2.4) by a com-23partmental model in (2.5).2.3.1 A special case: heterogeneous mixing and indirect transmissionfor simple SIRP epidemic modelThe age-of-infection model includes models with multiple infective. For exam-ple, consider the standard SIRP epidemic model with pathogen P being shed byinfected individuals I1 and I2 at a rate r1 and r2, respectively, and these pathogendecay at rate δ . Pathogen shed outside of the host organism can persist and repro-duce but the decay rate δ is bigger than the reproduction rate [95, 178]. Infectedpopulations are removed at rate α . The indirect transmission model is thereforewritten asdS1dt= −β1S1P,dI1dt= β1S1P−αI1,dR1dt= αI1,dS2dt= −β2S2P, (2.5)dI2dt= β2S2P−αI2,dR2dt= αI2,dPdt= r1I1+ r2I2−δP,with initial conditionsS1(0)= S10, S2(0)= S20, I1(0)= I10, I2(0)= I20, P(0)=P0, R1(0)=R2(0)= 0,in a population of constant total size N = N1+N2 whereN1 = S1+ I1+R1 = S10+ I10 and N2 = S2+ I2+R2 = S20+ I20.Again, model (2.5) is an extension of [23] from homogeneous mixing to heteroge-neous mixing in the population.24Model (2.5) will be analyzed using the method of Kermack-McKendrick epi-demic model [23, 25].Table 2.1: Model variables, parameters and their descriptions.Variables DescriptionSi Population of susceptible individualsIi Population of infected individualsRi Population of recovered individualsP Pathogen shed by infected individualsParameters Descriptionβi Effective contact rateα Removed rate for infected individualsri Pathogen shedding rate for infected individualsδ Infectivity loss rate for pathogenNote: For all i = 1,2.Lemma 2.3.1. Let f (t) be a nonnegative monotone nonincreasing continuosly dif-ferentiable function such that as t→ ∞, f (t)→ f∞ ≥ 0, then d fdt → 0.Summation of equations S1 and I1 in (2.5) givesd(S1+ I1)dt=−αI1 ≤ 0.We can see that (S1 + I1) decreases to a limit, and by Lemma 2.3.1 we couldshow that its derivative approaches zero, from which we can infer that I1∞ =limt→∞ I1(t) = 0.Integrate this equation to have α∫ ∞0 I1(t)dt = S1(0)+ I1(0)−S1(∞) = N1(0)−S1(∞), ∫ ∞0I1(t)dt =N1(0)−S1(∞)α, (2.6)which implies that∫ ∞0 I1(t)dt < ∞.Similarly, sum S2 and I2 in (2.5) asd(S2+ I2)dt=−αI2 ≤ 0,25and by Lemma 2.3.1 and integrating, we have∫ ∞0I2(t)dt =N2(0)−S2(∞)α, (2.7)which implies that∫ ∞0 I2(t)dt < ∞.Reproduction numberR0Here, we use the next generation matrix approach [169] to find the basic reproduc-tion number. Note that we have three infectious classes I1, I2,P, and the Jacobianmatrix ofFi =(F1,F2,F3), evaluated at the disease free equilibrium point (DFE)DFE=(S10,0,0,S20,0,0,0)=(N1(0),0,0,N2(0),0,0,0) is given byF =(∂Fi∂x j)i, j=0 0 β1N1(0)0 0 β2N2(0)0 0 0 ,where x j = I1, I2,P for j = 1,2,3 and i = 1,2,3.The Jacobian matrix of Vi = (V1,V2,V3), evaluated at the disease free equilib-rium point DFE, isV =(∂Vi∂x j)i, j= α 0 00 α 0−r1 −r2 δ ,FV−1 =β1N1(0)r1αδβ1N1(0)r2αδβ1N1(0)δβ2N2(0)r1αδβ2N2(0)r2αδβ2N2(0)δ0 0 0 .Remark 1. Since we can not calculate the basic reproduction number for our two-group model (2.5) by knowing secondary infections, we therefore used the methodof next generation matrix in [169] to find the basic reproduction number as thedominant eigenvalues of FV−1 (the spectral radius of the matrix FV−1). And it isgiven as26R0 =r1β1N1αδ+r2β2N2αδ.R0 can be written asR0 = β1R1+β2R2, whereR1 =r1N1α1δandR2 =r2N2α2δ.The first term in this expression represents secondary infections caused indi-rectly through the pathogen since a single infective I1 sheds a quantity r1 of thepathogen per unit time for a time period 1/α , and this pathogen infects β1N1 sus-ceptible individuals per unit time for a time period 1/δ , while the second termrepresents secondary infections caused indirectly through the pathogen since a sin-gle infective I2 sheds a quantity r2 of the pathogen per unit time for a time period1/α and this pathogen infects β2N2 susceptible individuals per unit time for a timeperiod 1/δ . The following easily proved Theorem will be used to summarize thebenefit of the basic reproduction numberR0.Theorem 2.3.2. For system (2.5), the infection dies out whenever R0 < 1 andepidemic occur wheneverR0 > 1.The initial exponential growth rateThe initial exponential growth rate is a quantity that can be compared with ex-perimental data [21, 27]. We can linearize the model (2.5) about the disease-freeequilibrium S1 =N1, I1 = R1 = 0,S2 =N2, I2 = R2 = P= 0 by letting u1 =N1−S1,u2 = N2−S2 to obtain the linearizationdu1dt= β1N1P,dI1dt= β1N1P−αI1,dR1dt= αI1,du2dt= β2N2P, (2.8)dI2dt= β2N2P−αI2,dR2dt= αI2,dPdt= r1I1+ r2I2−δP.27The equivalent characteristic equation is given bydet−λ 0 0 0 0 0 β1N1(0)0 −α−λ 0 0 0 0 β1N1(0)0 α −λ 0 0 0 00 0 0 −λ 0 0 β2N2(0)0 0 0 0 −α−λ 0 β2N2(0)0 0 0 0 α −λ 00 r1 0 0 r2 0 −δ −λ= 0.This equation can be reduced to a product of four factors and a third degree poly-nomial equation(λ 4)det−α−λ 0 β1N1(0)0 −α−λ β2N2(0)r1 r2 −δ −λ= 0.The initial exponential growth rate is the largest root of this third degree equationand it reduces toG(λ ) = (α+λ )2(δ +λ )− (α+λ )(β1r1N1+β2r2N2), (2.9)G(λ ) = (α+λ )2(δ +λ )− (α+λ )αδR0 = 0. (2.10)We can measure the initial exponential growth rate, and if the measured value is ξ ,then from (2.10) we obtain(α+ξ )2(δ +ξ )− (α+ξ )αδR0 = 0, (2.11)and we haveR0 =(α+ξ )(δ +ξ )αδ. (2.12)Equation (2.12) gives a way to estimate the basic reproduction number from knownquantities, and ξ = 0 in (2.12) corresponds toR0 = 1, which confirms the thresholdbehaviour for the calculated R0. We can obviously see that λ > 0 in (2.10) isequivalent to R0 > 1. Estimating the final epidemic size after an epidemic has28passed is possible, and this also makes it feasible to choose values of α and β1β2that satisfy (2.11) such that the simulations of the model (2.5) give the observedfinal size. In summary, we have the following Theorem;Theorem 2.3.3. For eigenvalue λ > 0 in (2.10), we haveR0 > 1 denoting epidemicoccurrence, and ξ = 0 in (2.12) which corresponds to R0 = 1 also confirms thethreshold behaviour forR0.The final size relationThe final epidemic size is achieved from the solutions of the final size relationshipwhich gives an estimate of the total number of infections and the epidemic size forthe period of the epidemic from the parameters in the model [13, 20]. The approachin [19, 20, 23] is used to find the final size relation in order to evaluate the numberof disease cases and disease deaths in terms of the model parameters. It is assumedthat the total population sizes N1,N2 of both groups are constant.Integrate the equation for S1 and S2 in (2.5);logSi0Si∞= βi∫ ∞0P(t)dt ∀i = 1,2. (2.13)Integrate the linear equation for P in (2.5) to haveP(t) = P0e−δ t + r1∫ t0e−δ (t−s)I1(s)ds+ r2∫ t0e−δ (t−s)I2(s)ds. (2.14)Next, we need to show thatlimt→∞∫ t0e−δ (t−s)Ii(s)ds = limt→∞∫ t0 eδ sIi(s)dseδ t= 0 ∀i = 1,2. (2.15)If the integral in the numerator of (2.15) is bounded, this is obvious; and if un-bounded, l’Hospital’s rule shows that limt→∞ Ii(t)/δ = 0 [23], and (2.14) impliesthatP∞ = limt→∞P(t) = 0.Integrate (2.14), and interchange the order of integration, then use (2.6) and (2.7)29to have ∫ ∞0P(t)dt =r1δ∫ ∞0I1(t)dt+r2δ∫ ∞0I2(t)dt, (2.16)which implies that∫ ∞0 P(t)dt < ∞.Substitute (2.16) into (2.13) to havelogSi0Si∞= βi(r1δ∫ ∞0I1(t)dt+r2δ∫ ∞0I2(t)dt+2P0δ), ∀ i = 1,2,and now the final size relationlogSi0Si∞= βi(r1N1α1δ{1− S1(∞)N1}+r2N2α2δ{1− S2(∞)N2}+2P0δ),= βi(R1{1− S1(∞)N1}+R2{1− S2(∞)N2}+2P0δ), ∀ i = 1,2,is from the substitution of (2.6) and (2.7) which implies Si∞ > 0. If the outbreakbegins with no contact with pathogen, P0 = 0, and then the final size relation iswritten aslogSi0Si∞= βi(R1{1− S1(∞)N1}+R2{1− S2(∞)N2})∀ i = 1,2.Note that the total number of infected populations over the period of the epidemicin patch 1 and 2 are respectively N1−S1∞ and N2−S2∞ which are always describedin terms of the attack rate(1− S1∞N1)and(1− S2∞N2)as in [19].Following the steps used in section (2.3.1), we can compute the reproductionnumber, the exponential growth rate and the final size relation from equation (2.4)as;2.3.2 Reproduction numberR0Having analyzed the special case in Equation 2.5, We will use a similar approachfor the model in Equation 2.4. We have 3 infected classes ϕ1, ϕ2, P in Equations302.4 and following the approach of [169], the next generation matrix is 0 0 β1N1∫ ∞0 A1(τ)dτ0 0 β2N2∫ ∞0 A2(τ)dτr1∫ ∞0 Γ(τ)dτ r2∫ ∞0 Γ(τ)dτ 0 ,andR0 is the largest root ofdet −λ 0 β1N1∫ ∞0 A1(τ)dτ0 −λ β2N2∫ ∞0 A2(τ)dτr1∫ ∞0 Γ(τ)dτ r2∫ ∞0 Γ(τ)dτ −λ= 0. (2.17)The basic reproduction number for the model (2.4), which is the number of sec-ondary infections caused by a single infective in a totally susceptible population isgiven byR0 = r1β1N1∫ ∞0A1(τ)dτ∫ ∞0Γ(τ)dτ+ r2β2N2∫ ∞0A2(τ)dτ∫ ∞0Γ(τ)dτ, (2.18)which can be written as β1R1+β2R2, whereR1 = r1N1∫ ∞0A1(τ)dτ∫ ∞0Γ(τ)dτ,represent secondary infections caused by an infectious individual in I1 indirectlyby the pathogen shed andR2 = r2N2∫ ∞0A2(τ)dτ∫ ∞0Γ(τ)dτ,represent secondary infections caused by an infectious individual in I2 indirectlyby the pathogen shed. We summarize the analysis and impacts of R1 and R2 inthe following Theorem.Theorem 2.3.4. Disease dies out wheneverR0 < 1 (i.e. R1 < 1 andR2 < 1) andepidemic occur wheneverR0 > 1 (i.e. R1 > 1 andR2 > 1).312.3.3 The initial exponential growth rateIn order to avoid the difficulties caused by the fact that there is a three-dimensionalsubspace of equilibria ϕ1 = ϕ2 = P = 0 and following the approach of [21], weinclude small birth rates in the equations for S1 and S2, and equivalent proportionalnatural death rates in each of the compartment to give the systemdS1(t)dt= µN1−µS1−β1S1(t)P(t),ϕ1(t) =∫ ∞0[−dS1(t− τ)dt]e−µτA1(τ)dτ,dS2(t)dt= µN2−µS2−β2S2(t)P(t), (2.19)ϕ2(t) =∫ ∞0[−dS2(t− τ)dt]e−µτA2(τ)dτ,P(t) =∫ ∞0[r1ϕ1(t− τ)+ r2ϕ2(t− τ)]e−µτΓ(τ)dτ.We then linearize (2.19) about the disease-free equilibrium S1 = N1, ϕ1 = 0, S2 =N2, ϕ2 = 0, P = 0 by letting u1 = N1−S1, u2 = N2−S2 to obtain the linearizationdu1(t)dt= −β1N1P−µu1,v1(t) =∫ ∞0β1N1P(t− τ)e−µτA1(τ)dτ,du2(t)dt= −β2N2P−µu2, (2.20)v2(t) =∫ ∞0β2N2P(t− τ)e−µτA2(τ)dτ,P(t) =∫ ∞0[r1v1(t− τ)+ r2v2(t− τ)]e−µτΓ(τ)dτ,32and form the characteristic equation, which is the condition on λ that the lineariza-tion have a solution u1 = u10eλ t , v1 = v10eλ t , u2 = u20eλ t , v2 = v20eλ t , P = u0eλ t ,det−(λ +µ) 0 0 0 −β1N10 −1 0 0 β1N1♥0 0 −(λ +µ) 0 −β2N20 0 0 −1 β2N2♠0 r1∫ ∞0 e−(λ+µ)τΓ(τ)dτ 0 r2∫ ∞0 e−(λ+µ)τΓ(τ)dτ −1= 0,where ♥= ∫ ∞0 e−(λ+µ)τA1(τ)dτ and ♠= ∫ ∞0 e−(λ+µ)τA2(τ)dτ .We have a double root λ =−µ < 0, and the remaining roots of the character-istic equation are the roots ofdet −1 0 β1N1∫ ∞0 e−(λ+µ)τA1(τ)dτ0 −1 β2N2∫ ∞0 e−(λ+µ)τA2(τ)dτr1∫ ∞0 e−(λ+µ)τΓ(τ)dτ r2∫ ∞0 e−(λ+µ)τΓ(τ)dτ −1= 0.Since this is true for all sufficiently small µ > 0, we may let µ −→ 0 and concludethat in a scenario where there is an epidemic, equivalent to an unstable equilibriumof the model, then the positive root of the characteristic equationdet −1 0 β1N1∫ ∞0 e−λτA1(τ)dτ0 −1 β2N2∫ ∞0 e−λτA2(τ)dτr1∫ ∞0 e−λτΓ(τ)dτ r2∫ ∞0 e−λτΓ(τ)dτ −1= 0, (2.21)is the initial exponential growth rate and this isr1β1N1∫ ∞0e−λτA1(τ)dτ∫ ∞0e−λτΓ(τ)dτ+r2β2N2∫ ∞0e−λτA2(τ)dτ∫ ∞0e−λτΓ(τ)dτ = 1.(2.22)We can obviously see from equations (2.18) and (2.22) that epidemic occurs onlyif λ > 0 which is equivalent to R0 > 1. In summary, we have a simple Theoremas;Theorem 2.3.5. An epidemic occurs if and only if λ > 0, which is equivalent toR0 > 1.332.3.4 The final size relationIntegrate the equations for S1 and S2 in (2.4) to havelogSi0Si∞= βi∫ ∞0P(t)dt ∀ i = 1,2. (2.23)Interchanging the order of integration, using S1(u) and S2(u) for u < 0, and byLemma (2.3.1) to have∫ ∞0ϕi(t)dt = [Ni−Si∞]∫ ∞0Ai(τ)dτ ∀ i = 1,2,∫ ∞0P(t)dt = r1∫ ∞0ϕ1(τ)∫ ∞0Γ(τ)dτ+ r2∫ ∞0ϕ2(τ)∫ ∞0Γ(τ)dτ= r1[N1−S1∞]∫ ∞0A1(τ)dτ∫ ∞0Γ(τ)dτ+r2[N2−S2∞]∫ ∞0A2(τ)dτ∫ ∞0Γ(τ)dτ.Substitute into (2.23) to havelogSi0Si∞= βi(r1[N1−S1∞]∫ ∞0A1(τ)dτ∫ ∞0Γ(τ)dτ+r2[N2−S2∞]∫ ∞0A2(τ)dτ∫ ∞0Γ(τ)dτ),logSi0Si∞= βi(R1[1− S1∞N1]+R2[1− S2∞N2])∀ i = 1,2. (2.24)Note that the final size of the epidemic, the total number of members of the pop-ulation infected over the course of the epidemic in patch 1 and 2 are respectivelyN1−S1∞ and N2−S2∞ and are often described in terms of the attack rates(1− S1∞N1)and(1− S2∞N2)respectively.2.4 Variable pathogen shedding ratesWe describe a more realistic model that allows the pathogen shedding rates r1 andr2 depend on age of infection of the shedding individual. We need a more complex34model that allows the shedding rates decrease to zero. We therefore let Q1(w) andQ2(w) be rates at which virus is being shed for infectives with age of infection w,and Γ(c) be the proportion of infectivity remaining for virus already shed c timeunits earlier.We can reasonably assume that infectivities (Q1(τ) and Q2(τ)) which are func-tions of infection age, are effective viruses at time t shed by infectives I1 and I2 withage of infection τ at time t.Then, it makes sense to make changes of A1(τ) = Q1(τ) and A2(τ) = Q2(τ) inthe equation for ϕ1 and ϕ2 in (2.4).A more general equation for P need to be developed while equations for S1 andS2 from (2.4) remain unchanged and the idea follows from [23].Let the number of individuals with age of infection w at time t be i(t,w), whichmay include individuals with zero infectivity who do not infect any more.Therefore i(t,w) = i(t−w,0) =−S′i(t−w).Consider infectives that are infected at time t−c, 0≤ c≤∞ with infection agev, 0≤ v≤ c and contribution of their virus at time t.At time t− c+ v, we havei(t− c+ v,v) = i(t− c,0) =−S′i(t− c).Their shedding rates are Q1(v) and Q2(v), and the viruses remaining at time t areQ1(v)Γ(c− v) and Q2(v)Γ(c− v). We therefore haveP(t) =∫ ∞0∫ c0[−S′1(t− c)]Q1(v)Γ(c− v)dvdc+∫ ∞0∫ c0[−S′2(t− c)]Q2(v)Γ(c− v)dvdc=∫ ∞0∫ ∞v[−S′1(t− c)]Γ(c− v)dcQ1(v)dv+∫ ∞0∫ ∞v[−S′2(t− c)]Γ(c− v)dcQ2(v)dv=∫ ∞0∫ ∞0[−S′1(t− z− v)]Γ(z)dzQ1(v)dv+∫ ∞0∫ ∞0[−S′2(t− z− v)]Γ(z)dzQ2(v)dv.35The general model becomesdS1(t)dt= −β1S1(t)P(t),ϕ1(t) =∫ ∞0[−dS1(t− τ)dt]Q1(τ)dτ,dS2(t)dt= −β2S2(t)P(t), (2.25)ϕ2(t) =∫ ∞0[−dS2(t− τ)dt]Q2(τ)dτ,P(t) =∫ ∞0[∫ ∞0[−dS1(t− z− v)dt]Γ(z)dz]Q1(v)dv+∫ ∞0[∫ ∞0[−dS2(t− z− v)dt]Γ(z)dz]Q2(v)dv.The equation for P can be substituted into equations for S1 and S2 in the model(2.25) to have two single equations for S1 and S2 asdS1(t)dt= −β1S1(t)(∫ ∞0[∫ ∞0[−dS1(t− z− v)dt]Γ(z)dz]Q1(v)dv+∫ ∞0[∫ ∞0[−dS2(t− z− v)dt]Γ(z)dz]Q2(v)dv),anddS2(t)dt= −β2S2(t)(∫ ∞0[∫ ∞0[−dS1(t− z− v)dt]Γ(z)dz]Q1(v)dv+∫ ∞0[∫ ∞0[−dS2(t− z− v)dt]Γ(z)dz]Q2(v)dv).2.4.1 Reproduction numberR0We will find the basic reproduction number for (2.25) by beginning with new infec-tives and calculating the virus shed over the period of the infection. The effective36viruses at time t are given as∫ t0Qi(w)Γ(t−w)ds =∫ t0Qi(t− c)Γ(c)dc ∀ i = 1,2,and total infectivities over the period of the infection are∫ ∞0∫ t0Qi(t− c)Γ(c)dcdt =∫ ∞0[∫ ∞cQi(t− c)dt]Γ(c)dc=∫ ∞0[∫ ∞0Qi(v)dv]Γ(c)dc=∫ ∞0Qi(v)dv∫ ∞0Γ(c)dc ∀ i = 1,2.The basic reproduction number can therefore be written asR0 = β1N1∫ ∞0Q1(v)dv∫ ∞0Γ(c)dc+β2N2∫ ∞0Q2(v)dv∫ ∞0Γ(c)dc, (2.26)and we haveR0 = β1R1+β2R2,whereR1 = N1∫ ∞0Q1(v)dv∫ ∞0Γ(c)dc and R2 = N2∫ ∞0Q2(v)dv∫ ∞0Γ(c)dc,and follows from Theorem 2.3.4.372.4.2 The initial exponential growth rateThe linearization of (2.25) at the equilibrium S1 =N1, S2 =N2, ϕ1 = ϕ2 = 0, P= 0,aredS1(t)dt= −β1N1(∫ ∞0[∫ ∞0[−dS1(t− z− v)dt]Γ(z)dz]Q1(v)dv+∫ ∞0[∫ ∞0[−dS2(t− z− v)dt]Γ(z)dz]Q2(v)dv),anddS2(t)dt= −β2N2(∫ ∞0[∫ ∞0[−dS1(t− z− v)dt]Γ(z)dz]Q1(v)dv+∫ ∞0[∫ ∞0[−dS2(t− z− v)dt]Γ(z)dz]Q2(v)dv).The characteristic equation shows a situation when the linearization have solutionsS1(t) = S10eλ t and S2(t) = S20eλ t , which areβ1N1(∫ ∞0e−λvQ1(v)dv∫ ∞0e−λcΓ(c)dc+∫ ∞0e−λvQ2(v)dv∫ ∞0e−λcΓ(c)dc)= 1,(2.27a)β2N2(∫ ∞0e−λvQ1(v)dv∫ ∞0e−λcΓ(c)dc+∫ ∞0e−λvQ2(v)dv∫ ∞0e−λcΓ(c)dc)= 1.(2.27b)Theorem 2.4.1. The disease dies out and there is no epidemic when λ < 0 (i.e.when R0 < 1) in equation (2.27), but disease persists when λ > 0 (i.e. whenR0 > 1) which corresponds to an epidemic.Combining (2.26) and (2.27) we haveR0 =∫ ∞0 Q1(v)dv∫ ∞0 Γ(c)dc+∫ ∞0 Q2(v)dv∫ ∞0 Γ(c)dc∫ ∞0 e−λvQ1(v)dv∫ ∞0 e−λcΓ(c)dc+∫ ∞0 e−λvQ2(v)dv∫ ∞0 e−λcΓ(c)dc.2.4.3 The final size relationIntegrate the equations for S1 and S2 in (2.25) to obtain the final size relation,logSi0Si∞= βi∫ ∞0P(t)dt. (2.28)38But we know that∫ ∞0P(t)dt =∫ ∞0∫ ∞0[∫ ∞0[−dS1(t− z− v)dt]Γ(z)dz]Q1(v)dvdt+∫ ∞0∫ ∞0[∫ ∞0[−dS2(t− z− v)dt]Γ(z)dz]Q2(v)dvdt.Interchange the order of integration, integrate with respect to t to obtain∫ ∞0P(t)dt =∫ ∞0∫ ∞0[∫ ∞0[−dS1(t− z− v)dtdt]Γ(z)dz]Q1(v)dv+∫ ∞0∫ ∞0[∫ ∞0[−dS2(t− z− v)dtdt]Γ(z)dz]Q2(v)dv=∫ ∞0∫ ∞0[S1(−z− v)−S1∞]Γ(z)dzQ1(v)dv+∫ ∞0∫ ∞0[S2(−z− v)−S2∞]Γ(z)dzQ2(v)dv=∫ ∞0∫ ∞0[N1−S1∞]Γ(z)dzQ1(v)dv (2.29)+∫ ∞0∫ ∞0[N2−S2∞]Γ(z)dzQ2(v)dv= [N1−S1∞]∫ ∞0Γ(z)dz∫ ∞0Q1(v)dv+[N2−S2∞]∫ ∞0Γ(z)dz∫ ∞0Q2(v)dv= R1[1− S1∞N1]+R2[1− S2∞N2].Using (2.29) in (2.28) and by Lemma (2.3.1), we obtain,logS10S1∞= β1(R1[1− S1∞N1]+R2[1− S2∞N2]),logS20S2∞= β2(R1[1− S1∞N1]+R2[1− S2∞N2]). (2.30)392.5 Heterogeneous mixing and indirect transmission withresidence timeHere we examined SIRP two patch model which included an explicit travel ratebetween patch. We divide the environment into two patches, and the populationin each patch is divided into Susceptible, Infective and Removed with differentpathogens in each patch. This model considers patches with residents who spendsome of their time in another patch or in a different environment.The model is considered for a short period of time and therefore assumes norecruitment, birth or natural death. We assume that the rate of travel of individualsbetween the two patches depends on the status of the disease, and individuals donot change disease status during travel. The disease is assumed to be transmitted byhorizontal incidence βiSiPi(i= 1,2) with the same removal rate and infectivity lossrate for infected individuals in both patches. We assume that one of the patches hasa larger contact rate β2 > β1, with short term travel between the two patches andthat each patch has a constant total population with p11 + p12 = 1, p21 + p22 = 1,where pi j(i, j = 1,2) is the fraction of contact made by patch i residents in patch j[13, 20].A Lagrangian method is followed to keep track of individual’s place of resi-dence at all times. This model with direct transmission of infection is the startingpoint of [13, 29].40Two-patch SIRP model with residence timedS1dt= −β1 p11S1(p11P1+ p21P2)−β2 p12S1(p12P1+ p22P2),dI1dt= β1 p11S1(p11P1+ p21P2)+β2 p12S1(p12P1+ p22P2)−αI1,dR1dt= αI1,dP1dt= r1I1−δP1,dS2dt= −β1 p21S2(p11P1+ p21P2)−β2 p22S2(p12P1+ p22P2), (2.31)dI2dt= β1 p21S2(p11P1+ p21P2)+β2 p22S2(p12P1+ p22P2)−αI2,dR2dt= αI2,dP2dt= r2I2−δP2,with initial conditionsS1(0)= S10, S2(0)= S20, I1(0)= I10, I2(0)= I20, P1(0)=P10, P2(0)=P20, R1(0)=R2(0)= 0,in a population of constant total size N = N1+N2 whereN1 = S1+ I1+R1 = S10+ I10 and N2 = S2+ I2+R2 = S20+ I20.Since this is an indirect transmission model, each of the p11S1 susceptibles fromGroup 1 present in patch 1 can be infected by pathogens shed by members of Group1 and Group 2 present in patch 1. Similarly, each of the p12S1 susceptibles fromGroup 1 present in patch 2 can be infected by pathogens shed by members of Group1 and Group 2 present in patch 2. The infective proportion in patch 1 is given byp11P1(t)+ p21P2(t) and in patch 2 is p12P1(t)+ p22P2(t).Therefore, the rate of new infections of members of patch 1 in patch 1 isβ1 p11S1(p11P1+ p21P2).41Table 2.2: Model variables, parameters and their descriptions.Variables DescriptionSi Population of susceptibles in patch iIi Population of infectives in patch iRi Population of removed in patch iPi Pathogens shed by infectives in patch iParameters Descriptionβi Effective contact rate in patch i.α Removed rate for infected individuals.ri Pathogen shedding rate for infected individuals.δ Infectivity loss rate for pathogen.p11 The fraction of contact made by patch 1 residents in patch 1p12 The fraction of contact made by patch 1 residents in patch 2p21 The fraction of contact made by patch 2 residents in patch 1p22 The fraction of contact made by patch 2 residents in patch 2.The rate of new infections of members of patch 1 in patch 2 isβ2 p12S1(p12P1+ p22P2).Similarly, the rate of new infections of members of patch 2 in patch 1 isβ1 p21S2(p11P1+ p21P2).The rate of new infections of members of patch 2 in patch 2 isβ2 p22S2(p12P1+ p22P2).From the sum of the equations for S1, S2, I1 and I2 in (2.31), we haved(S1+ I1)dt=−αI1 ≤ 0.We can see that (S1+ I1) decreases to a limit, and by Lemma 2.3.1 we could show42that its derivative approaches zero, from which can be deduced thatI1∞ = limt→∞ I1(t) = 0.Integrate this equation to giveα∫ ∞0I1(t)dt = S1(0)+ I1(0)−S1(∞) = N1(0)−S1(∞),∫ ∞0I1(t)dt =N1(0)−S1(∞)α, (2.32)implying that∫ ∞0 I1(t)dt < ∞. Similarly,d(S2+ I2)dt=−αI2 and we have∫ ∞0I2(t)dt =N2(0)−S2(∞)α, (2.33)implying that∫ ∞0 I2(t)dt < ∞.2.5.1 Reproduction numberR0Note that we have four infectious classes I1,P1, I2,P2, and the Jacobian matrix ofFi = (F1,F2,F3), evaluated at the disease free equilibrium point,DFE=(S10,0,0,0,S20,0,0,0)=(N1(0),0,0,0,N2(0),0,0,0) is given byF =(∂Fi∂x j)i, j=0 (β1 p211+β2 p212)N1(0) 0 (β1 p11 p21+β2 p12 p22)N1(0)0 0 0 00 (β1 p11 p21+β2 p12 p22)N2(0) 0 (β1 p221+β2 p222)N2(0)0 0 0 0 ,where x j = I1,P1, I2,P2 for j = 1, . . . ,4 and i = 1, . . . ,4.The jacobian matrix of Vi = (V1,V2,V3), evaluated at the disease free equilib-43rium point DFE isV =(∂Vi∂x j)i, j=α 0 0 0−r1 δ 0 00 0 α 00 0 −r2 δ .The dominant eigenvalues of FV−1 which is the spectral radius of the matrixFV−1, gives the basic reproduction number for Epidemic from the model (2.31)as;R0 =N+H±√(N+H)2−4β1β2(p11 p22− p12 p21)2N1(0)N2(0)r1r22αδ, (2.34)whereN= (β1 p211+β2 p212)N1(0)r1,andH= (β1 p221+β2 p222)N2(0)r2.Note that in the special case of proportionate mixing where we have p11 = p21 andp12 = p22, so that p12 p21 = p11 p22, the simplified basic reproduction number from(2.34) is given asR0 =(β1 p211+β2 p222)N1(0)r1+(β1 p211+β2 p222)N2(0)r2αδ. (2.35)Similarly for the case of no movement between patches, we have:p11 = p22 = 1, p12 = p21 = 0,so that the simplified basic reproduction number from (2.34) is given asR0 = ρ(FV−1) = max(r1β1N1αδ,r2β2N2αδ). (2.36)44R0 in (2.36) can be written asR0 = max(R1,R2),where R1 =r1β1N1αδ(the reproduction number for patch 1) and R2 =r2β2N2αδ(thereproduction number for patch 2). Theorem (2.3.4) gives the summary of thisanalysis.2.5.2 The initial exponential growth rateThe initial exponential growth rate is a quantity that can be compared with experi-mental data [21, 27]. We can linearize the model (2.31) about the disease-free equi-librium S1 =N1, I1 =R1 =P1 = 0,S2 =N2, I2 =R2 =P2 = 0 by letting u1 =N1−S1,u2 = N2−S2 to obtain the linearizationdu1dt= β1 p11N1(p11P1+ p21P2)+β2 p12N1(p12P1+ p22P2),dI1dt= β1 p11N1(p11P1+ p21P2)+β2 p12N1(p12P1+ p22P2)−αI1,dR1dt= αI1,dP1dt= r1I1−δP1,du2dt= β1 p21N2(p11P1+ p21P2)+β2 p22N2(p12P1+ p22P2), (2.37)dI2dt= β1 p21N2(p11P1+ p21P2)+β2 p22N2(p12P1+ p22P2)−αI2,dR2dt= αI2,dP2dt= r2I2−δP2.45The equivalent characteristic equation be reduced to a product of four factors anda fourth degree polynomial equationλ 4det−α−λ (β1 p211+β2 p212)N1 0 (β1 p11 p21+β2 p12 p22)N1r1 −δ −λ 0 00 (β1 p11 p21+β2 p12 p22)N2 −α−λ (β1 p221+β2 p222)N20 0 r2 −δ −λ= 0.The initial exponential growth rate corresponds to the largest root of this fourthdegree equation and it reduces toG(λ ) = (α+λ )2(δ +λ )2− (α+λ )(δ +λ )((β1 p211+β2 p212)r1N1+(β1 p221+β2 p222)r2N2)+β1β2r1r2N1N2(p11 p22− p12 p21)2.We can write the initial exponential growth rate in a simplified form using(2.35) asG(λ ) = (α+λ )2(δ +λ )2− (α+λ )(δ +λ )αδR0 = 0. (2.38)Estimating the initial exponential growth rate from data is possible, and if the esti-mated value is ξ , then from (2.38) we obtain(α+ξ )2(δ +ξ )2− (α+ξ )(δ +ξ )αδR0 = 0, (2.39)and we haveR0 =(α+ξ )(δ +ξ )αδ. (2.40)Equation (2.40) gives a way to estimate the basic reproduction number from knownquantities, and ξ = 0 in (2.40) corresponds to R0 = 1, which confirms the properthreshold behaviour for the calculatedR0. Estimating the final epidemic size afteran epidemic has passed is possible, and this makes it feasible to choose values ofα and β1β2 that satisfy (2.39) such that the simulations of the model (2.31) givethe observed final size.46In the case of no movement, the initial exponential growth rate is given asG(λ ) = (α+λ )2(δ +λ )2− (α+λ )(δ +λ )(β1r1N1+β2r2N2)+β1β2r1r2N1N2,and simplified using (2.36) asG(λ ) = (α+λ )2(δ +λ )2− (αδ )(α+λ )(δ +λ )(R1+R2)= 0. (2.41)Estimating the initial exponential growth rate from data is also possible, and if theestimated value is ξ , then from (2.41) we obtain(α+ξ )2(δ +ξ )2− (αδ )(α+ξ )(δ +ξ )(R1+R2)= 0, (2.42)and we haveR1+R2 =(α+ξ )(δ +ξ )αδ. (2.43)On the one hand, if R1 >R2, it means disease is more effectively spread in patch1 and infection in patch 2 is therefore driven to extinction. Then the basic repro-duction number from (2.43) becomesR0 =R1 =(α+ξ )(δ +ξ )αδ. (2.44)On the other hand, if R2 > R1, it means disease is more effectively spread inpatch 2 and infection in patch 1 is therefore driven to extinction. Then the basicreproduction number from (2.43) becomesR0 =R2 =(α+ξ )(δ +ξ )αδ. (2.45)Equations (2.5.2) & (2.45) give a way to estimate the basic reproduction numberfrom known quantities, and by Theorem (2.3.3) and ξ = 0 in either of these equa-tions corresponds toR0 = 1, which confirms the proper threshold behaviour for thecalculated R0. Estimating the final epidemic size after an epidemic has passed isalso possible, and this makes it feasible to choose values of α and β1β2 that satisfy(2.42) such that the simulations of the model (2.31) give the observed final sizewhen there is no movement between patches.472.5.3 The final size relationIntegrate the equation for S1 and S2 in (2.31);logS10S1∞= β1 p211∫ ∞0P1(t)dt+β1 p11 p21∫ ∞0P2(t)dt+β2 p212∫ ∞0P1(t)dt+β2 p12 p22∫ ∞0P2(t)dt,logS20S2∞= β1 p11 p21∫ ∞0P1(t)dt+β1 p221∫ ∞0P2(t)dt (2.46)+β2 p12 p22∫ ∞0P1(t)dt+β2 p222∫ ∞0P2(t)dt.Integrate the linear equation for P1 and P2 in (2.31) to haveP1(t) = P10e−δ t + r1∫ t0e−δ (t−s)I1(s)ds, (2.47)P2(t) = P20e−δ t + r2∫ t0e−δ (t−s)I2(s)ds.Next, we need to show thatlimt→∞∫ t0e−δ (t−s)Ii(s)ds = limt→∞∫ t0 eδ sIi(s)dseδ t= 0 ∀ i = 1,2. (2.48)This is clear if the integral in the numerator of (2.48) is bounded, and if unbounded,l’Hospital’s rule shows that the limit is limt→∞ Ii(t)/δ = 0 [23]. And (2.47) impliesthatPi∞ = limt→∞Pi(t) = 0.But integrate (2.47), interchange the order of integration, and use (2.32) and (2.33)to have ∫ ∞0P1(t)dt =r1δ∫ ∞0I1(t)dt, (2.49)∫ ∞0P2(t)dt =r2δ∫ ∞0I2(t)dt.implying that∫ ∞0 Vi(t)dt < ∞.Substitute (2.49) into (2.46) to have48logS10S1∞= β1 p211r1δ∫ ∞0I1(t)dt+β1 p11 p21r2δ∫ ∞0I2(t)dt+β2 p212r1δ∫ ∞0I1(t)dt+β2 p12 p22r2δ∫ ∞0I2(t)dt,logS20S2∞= β1 p11 p21r1δ∫ ∞0I1(t)dt+β1 p221r2δ∫ ∞0I2(t)dt (2.50)+β2 p12 p22r1δ∫ ∞0I1(t)dt+β2 p222r2δ∫ ∞0I2(t)dt.Now substitute (2.32) and (2.33) into (2.50) and using Lemma (2.3.1), gives thefinal size relationlogS10S1∞= (β1 p211+β2 p212)(r1N1αδ){1− S1(∞)N1}+(β1 p11 p21+β2 p12 p22)(r2N2αδ){1− S2(∞)N2},logS20S2∞= (β1 p11 p21+β2 p12 p22)(r1N1αδ){1− S1(∞)N1}(2.51)+(β1 p221+β2 p222)(r2N2αδ){1− S2(∞)N2}.which implies Si∞ > 0.Equation (2.51) can as well be written as logS10S1∞log S20S2∞= M11 M12M21 M22 1−S1(∞)N11− S2(∞)N2 , (2.52)whereM= (β1 p211+β2 p212)r1N1αδ (β1 p11 p21+β2 p12 p22)r2N2αδ(β1 p11 p21+β2 p12 p22) r1N1αδ (β1 p221+β2 p222)r2N2αδ .In a situation where we have no movement between patches, the final size relation49can be written aslogS10S1∞=(β1r1N1αδ){1− S1(∞)N1},logS20S2∞=(β2r2N2αδ){1− S2(∞)N2}. (2.53)which implies Si∞ > 0.Equation (2.53) can as well be written as logS10S1∞log S20S2∞= M11 M12M21 M22 1−S1(∞)N11− S2(∞)N2 , (2.54)whereM =β1r1N1αδ 00 β2r2N2αδ .Table 2.3: Parameter values and their sources.Symbol Value ReferencesN1(0) 200N2(0) 300β1 0.3 [13]β2 1.2 [13]α 1.87 [178]r1 0.1 [178]r2 1 [178]δ 0.2550Figure 2.1: Dynamics of I1 and I2 when we vary p11, p12, p21, p22 and haveno movement (p11 = p22 = 1, p12 = p21 = 0), half populations moving(p11 = p22 = p12 = p21 = 0.5), and all populations moving (p11 = p22 =0, p12 = p21 = 1). The figure on the left panel shows that the prevalencein patch 1 reaches its highest when in extreme mobility case (blue line)and is lowest when there is no mobility between patches (red line). Thefigure on the right panel show the opposite of this senario in patch 2(high risk).Note that the eigenvalues of FV−1 (the next generation matrix) are the same as theeigenvalues of the matrices M (the final epidemic size) andM (the final epidemicsize for no movement between patches). In a special case where the epidemiologi-cal system cannot be controlled, we have the dominant eigenvalue to beR0.2.5.4 Numerical simulationsWe run simulations to gain deeper understanding of the role of residence time ondisease dynamics.We simulate for Susceptible populations S1(0) = 199 in patch 1 with one in-fective and similarly for S2(0) = 298 in patch 2 with two infective. We assume thatpatch 2 has higher risk with β2 = 1.2 and patch 1 has lower risk with β1 = 0.3. Wehave the parameter values and their sources in table 2.3.From our simulations in figure 2.1,we observe that:1. For the case of no movement between patches (no mobility), that is, p11 =51p22 = 1 and p12 = p21 = 0, the system behaves as two separated patcheswhere we have the disease prevalence to be at its highest in patch 2.2. For the symmetric case in which p11 = p12 = p21 = p22 = 0.5, the systemhas the same level of disease prevalence in both patches.3. The case where everyone move from their patch to the other patch (highmobility), that is p11 = p22 = 1 and p12 = p21 = 0, the system has the highestdisease prevalence in patch 1.Our numerical results is similar to [13] where direct transmission pathway is con-sidered as a form of disease spread. Our results show that considering indirecttransmission pathway is of great importance and disease spread may be difficult tocontrol (the case of cholera) if otherwise, as in figure 2.1.2.6 ConclusionIn this chapter, we proposed and studied an epidemic model in which infection istransmitted when viruses are shed and acquired through host (population)-source(environment)-host (population) in heterogeneous environments. For the threemodels developed, we calculated the reproduction number, estimated the initialexponential growth rate and obtained the reproduction number in terms of param-eters that can be estimated. The final size relation was also analyzed to find thenumber of disease cases and disease deaths in terms of the model parameters.We examined an SIVR model with residence times and developed a 2-patchmodel where infection risk is as a result of the residence time and other environ-mental factors. With this approach, we studied the disease prevalence in hetero-geneous environment through indirect transmission pathways without needing tomeasure contact rates, and our analysis was also buttressed by numerical results.Our primary result shows that the number of populations being infected throughindirect transmission, which had been omitted in some other previous works isworth taking into account. The result of our numerical simulation is similar to oneof the results in [13] in which only a direct transmission pathway was considered.We were able to show how much worse the prevalence of a disease could be whenthe disease transmission is indirect.52We considered indirect transmission of viruses in heterogeneous mixing popu-lations, but considering direct and indirect pathways (the case of ebola), may give adifferent/better insight into the disease prevalence and how accurate treatment willbe apportioned.Despite these limitations, our models can be used to compare disease spread be-tween two populations with different contact rates, such as cities against villages,rich against poor populations and so on. The derivation of the age of infectionmodel could be extended to include direct transmission pathways. It is also possi-ble to extend the model with the residence times to incorporate treatment strategieswhich may reduce the contact rates and then lower the reproduction number. Inaddition, it may be more realistic to extend the model to incorporate multiple classof hosts and sources in order to compare the disease spread among different popu-lations and with different viruses.53Chapter 3A novel approach to modellingthe spatial spread of airbornediseases: an epidemic model withindirect transmission3.1 SynopsisWe formulated and analyzed a class of coupled partial and ordinary differentialequation (PDE-ODE) model to study the spread of airborne diseases. Our modeldescribes human populations with patches and the movement of pathogens in theair with linear diffusion. The diffusing pathogens are coupled to the SIR dynamicsof each population patch using an integro-differential equation. Susceptible indi-viduals become infected at some rate whenever they are in contact with pathogens(indirect transmission), and the spread of infection in each patch depends on thedensity of pathogens around the patch. In the limit where the pathogens are dif-fusing fast, matched asymptotic analysis is used to reduce the coupled PDE-ODEmodel into a nonlinear system of ODEs, which is then used to compute the ba-sic reproduction number and final size relation for different scenarios. Numericalsimulations of the reduced system of ODEs and the full PDE-ODE model are con-54sistent, and they predict decrease in the spread of infection as the diffusion rate ofpathogens increases. Furthermore, we studied the effect of patch location on thespread of infections for the case of two patches, our models predict higher infec-tions when the patches are closer to each other.3.2 IntroductionAirborne diseases are well studied in epidemiology and public health, and stillremain a serious public health concern today. Many airborne diseases are transmit-ted directly (host-host) and/or indirectly (host-source-host) through actions suchas coughing, sneezing and sometimes vomiting [131]. For example, viral diseases(measles, influenza) and bacterial infections (tuberculosis) are transmitted via air-borne route. In addition, there has been evidence that airborne transmission playsa significant role in the spread of many opportunistic pathogens causing severalacquired nosocomial (hospital) infections [11]. Some mathematical models havebeen used to study the transmission of airborne diseases using direct and indirecttransmission pathways. Noakes et al. in [131] studied the transmission of airborneinfections in enclosed spaces using an SEIR model to show how changes to bothphysical environment and infection control could be a potential limitation in thespread of airborne infections. Issarow et al. in [94] developed a model to predictthe risk of airborne infectious diseases such as tuberculosis in confined spaces us-ing exhaled air. Several approaches including but not limited to the framework in[176] have been used to study the dynamics of an SIS model with diffusion, [81] toassess the impact of heterogeneity of environment and advection on the persistenceand infectious diseases eradication, and [114] to evaluate population migration us-ing SIS epidemic models with diffusion. In addition, several PDE models such as[176, 180] were also used to study the effect of diffusion. However, despite allthese models and previous studies, it has largely been an open problem to evaluatethe effect of diffusion on the spread of infections between one or two populations.To our knowledge none of these works has assessed the impact of diffusion usinga coupled PDE-ODE SIR model with an indirect transmission pathway.In this chaper, we consider an airborne disease as any disease caused by pathogenand transmitted through the air. Such diseases include but are not limited to chick-55enpox, influenza, measles, smallpox, tuberculosis, among others. We focus on anindirect transmission pathways and derive fundamental quantities such as the basicreproduction number (R0) and the final size relation. To incorporate the limitationof the impact of diffusion among homogeneous and heterogeneous mixing popu-lation, we propose a coupled PDE-ODE model similar to the one used in [84] tomodel communication between dynamically active signaling compartments. Ourmodel extends the models presented in [23] and [50] by incorporating diffusion ofpathogens. This allows us to theoretically and numerically analyze how diffusionaffects the spread of air-transmitted diseases, in which the human populations areconfined to a distinct spatially segregated regions. The novelty of our approach isthat through a PDE-ODE system we model the spread of airborne diseases allow-ing allowing for person-air-person transmission. Overall, our modelling frameworkprovides an alternative way to describe the epidemics of airborne diseases.The outline of this chapter is as follows. In Section 3.3, a new coupled PDE-ODE model of epidemics is formulated, and this model is non-dimensionalized inSection 3.3.1. In Section 3.3.2, matched asymptotic expansion methods is usedto reduce the dimensionless coupled PDE-ODE model into a nonlinear system ofODEs in the limit where the pathogens are diffusing very fast. In Section 3.4, westudy the dimensionless coupled PDE-ODE model for a single population patchnumerically and compare the result to that of the reduced system of ODEs. Wealso use the reduced system of ODEs to compute the basic reproduction numberand final size relation. A similar study is performed for the case of two populationpatches in Section 3.5. In Section 3.6, we study the effect of patch location on thespread of infection for two population patches. The chapter concludes with a briefdiscussion in Section 3.7.3.3 Model formulationIn this section, we formulate and analyze a coupled PDE-ODE model for studyingthe spread of airborne diseases. This model is non-dimensionalized and later re-duced into a nonlinear ODE system in the limit where the diffusivity of pathogensis large.We begin by representing human populations by localized patches with par-56tially transmitting boundaries through which pathogens are shed into the atmo-sphere by infected individuals. These pathogens are assumed to diffuse and de-cay at constant rate in the air (bulk region), while the spread of infection in eachpatch depends on the density of pathogens around the patch. Pathogens are notexplicitly modelled in the patches; likewise the movement of individuals betweenpatches is not accounted for. A susceptible individual becomes infected by com-ing in contact with pathogens (indirect transmission pathway). Let Ω⊂ R2 be our2-D bounded domain of interest containing m population patches represented byΩ j for j = 1, . . . ,m, and separated by an O(1) distance from each other and fromthe boundary of the domain ∂Ω. In the region Ω \∪mj=1Ω j (bulk region) betweenthe patches, the spatio-temporal density of pathogensP(X ,T ) satisfies the partialdifferential equation (PDE) given by∂P∂T=DB∆P−δP, T > 0, X ∈Ω\∪mj=1Ω j; (3.1a)∂nX P = 0, X ∈ ∂Ω; DB ∂nX P =−r jI j, X ∈ ∂Ω j, j = 1, . . . ,m,(3.1b)where DB > 0 denotes the diffusion rate of pathogens in the bulk region, δ is thedimensional decay rate of pathogens, r j > 0 is the dimensional shedding rate ofpathogen by an infected individual in the jth patch, and ∂nX is the outward nor-mal derivative on the boundary of the domain Ω. The dynamics of the diffusingpathogens is coupled to the population dynamics of the jth patch using the integro-differential system of equations given bydS jdT=−µ jS j∫∂Ω j(P/pc) dSX ; (3.1c)dI jdT= µ jS j∫∂Ω j(P/pc) dSX −α jI j; (3.1d)dR jdT= α jI j, j = 1, . . . ,m, (3.1e)whereS j,I j, andR j denote the population of susceptible, infected, and removedindividuals in the jth patch, respectively, withN j(T ) =S j(T )+I j(T )+R j(T ).The parameters µ j and α j are the dimensional transmission and recovery rates,57respectively, for individuals in the jth patch, and pc is a typical value for the den-sity of pathogens. The integrals in (3.1c) and (3.1d) are over the boundary of thejth patch, and are used to account for all the pathogens around the patch. Theseterms show that the spread of infection within a patch depends on the density ofpathogens around it. It is important to emphasize that our model does not accountfor pathogens in the patches. The Robin boundary condition DB ∂nX P = −r jI jon the boundary of the jth patch accounts for the amount of pathogen shed intothe atmoshere by infected individuals in the patch. This condition shows that theamount of pathogens shed into the atmosphere from the jth patch depends on thepopulation of infected individuals within the patch.3.3.1 Non-dimensionalization of the coupled PDE-ODE modelIn this subsection, we non-dimensionalize the coupled PDE-ODE model (3.1). Thedimensions of the variables and parameters of the model are given as follow:[P] =pathogens(length)2, [DB] =(length)2time, [pc] = pathogens, [T ] = time,[X ] = length, [δ ] = [α j] =1time, [N j] = [S j] = [I j] = [R j] = individuals,[µ j] =lengthtime, [r j] =pathogensindividual × time× length , j = 1, . . . ,m.(3.2)where [γ] represents the dimension of γ . Assuming that the patches are circularwith common radius R, which is small relative to the length-scale L of the 2-Ddomain Ω, we introduce a small scaling parameter ε = R/L 1 and the followingdimensionless variablesP =L2pcP, S j =S jN j, I j =I jN j, R j =R jN j, x =XL, t = δ T.(3.3)In this way, S j, I j, and R j are the proportion of susceptible, infected, and removedindividuals in the jth patch, respectively, and P≡ P(x, t) is the dimensionless den-sity of the pathogens at position x at time t. Upon substituting (3.3) into (3.1), we58derive that the dimensionless spatio-temporal density of pathogens P(x, t) satisfies∂P∂ t=D∆P− P, t > 0, x ∈Ω\∪mj=1Ω j; (3.4a)∂nx P = 0, x ∈ ∂Ω; D∂nx P =−r j(N jLδ pc)I j, x ∈ ∂Ω j, j = 1, . . . ,m,(3.4b)where D ≡ DB/(δ L2) is the effective diffusion rate of the pathogens. From thesystem of ODEs ((3.1c) - (3.1e)) for the population dynamics of the jth patch, wederive the dimensionless systemdS jdt=−( µ jδ L)S j∫∂Ωε jP dsx;dI jdt=( µ jδ L)S j∫∂Ωε jP dsx−φ jI j;dR jdt= φ jI j, j = 1, . . . ,m,(3.5)where Ωε j = {x : |x j−x|< ε} represents the jth patch of radius ε 1 with centerat x j and boundary ∂Ωε j. It is important to remark that we have used the scalingdSX = Ldsx in the integrals on the boundary of the patches. Since the patchesare relatively small compared to the length-scale of the domain, we assume that(µ j/δ L) and r j (N jL/δ pc) are O(1/ε) in order to effectively capture the densityof the pathogen shed into the atmosphere. Hence, we setβ j2piε=µ jδLandσ j2piε= r jN jLδ pc, (3.6)such that β j and σ j are O(1). This rescaling enables us to write the dimension-less transmission and shedding rates, β j and σ j, respectively, as functions of thecircumference of the jth patch. Substituting (3.6) into (3.4) and (3.5), we have that59the dimensionless density of the pathogens P(x, t) satisfies∂P∂ t=D∆P− P, t > 0, x ∈Ω\∪mj=1Ωε j; (3.7a)∂nx P = 0, x ∈ ∂Ω; 2piεD∂nx P =−σ j I j, x ∈ ∂Ωε j, j = 1, . . . ,m,(3.7b)which is coupled to the dimensionless SIR dynamics of the jth patch through theintegro-differential equations given bydS jdt=−β jS j2piε∫∂Ωε jP dsx;dI jdt=β jS j2piε∫∂Ωε jP dsx−φ jI j; (3.7c)dR jdt= φ jI j, j = 1, . . . ,m,where β j, σ j and φ j are the dimensionless transmission, shedding and recoveryrates for the jth patch, respectively, and are given byβ j =2piεδLµ j, σ j =2piεδ pcr jN jL and φ j =α jδ. (3.8)In the next subsection, we study the dimensionless coupled PDE-ODE model (3.7)in the limit D = O(ν−1), where ν = −1/ loge(ε) and ε 1 using the method ofmatched asymptotic expansions.3.3.2 Asymptotic analysis of the dimensionless coupled PDE-ODEmodelHere, the dimensionless coupled PDE-ODE model (3.7) is analyzed in the limitD = O(ν−1), where ν ≡ −1/ loge(ε) for ε 1, using the method of matchedasymptotic expansions. This analysis is used to reduce the coupled model into anonlinear system of ODEs, which is then used to determine the basic reproductionnumber and final size relation of epidemics.60We begin our analysis by rescaling the diffusion rate of pathogens asD =D0ν, where D0 = O(1) and ν =− 1loge(ε) 1. (3.9)Substituting D = D0/ν into (3.7a) and (3.7b), we obtain∂P∂ t=D0ν∆P− P, t > 0, x ∈Ω\∪mj=1Ωε j; (3.10a)∂nx P = 0, x ∈ ∂Ω; 2piεD0ν∂nx P =−σ j I j, x ∈ ∂Ωε j, j = 1, . . . ,m,(3.10b)Since the pathogens shed by infected individuals go into the air through the bound-ary of the patches, one would expect the density of pathogens around each patchto be high relative to the regions far away from the patches. As a result of this,we construct an inner region at an O(ε) neighborhood of each patch, and intro-duce the local variables y = ε−1(x−x j) and P(x) = Q j(εy+x j), with |y| = ρ forj = 1, . . . ,m. Upon writing (3.10a) and (3.10b) in terms of the inner variables, weobtain for ε 1 the limiting inner problem∆ρ Q j = 0, t > 0, ρ > 1;2piD0ν∂ρ Q j = −σ j I j, ρ = 1, j = 1, . . . ,m,(3.11)where ∆ρ ≡ ∂ρρ +ρ−1∂ρ is the radially symmetric part of the Laplacian in 2-D. Inthis inner region, we expand Q j(ρ, t) asQ j = Q0 j +νD0Q1 j + . . . (3.12)Upon substituting this expansion into (3.11) and collecting terms in powers of ν ,we obtain the leading-order inner problem∆ρ Q0 j = 0, t > 0, ρ > 1; ∂ρ Q0 j = 0 on ρ = 1, j = 1, . . . ,m,(3.13)Observe that any constant or function of time is a solution to this problem, so that61Q0 j ≡ Q0 j(t). The next-order inner problem is given by∆ρ Q1 j = 0, t > 0, ρ > 1; 2pi ∂ρ Q1 j = −σ j I j on ρ = 1, j = 1, . . . ,m,(3.14)and its solution is readily calculated asQ1 j =(−σ j I j2pi)loge(ρ)+ c j, j = 1, . . . ,m, (3.15)where c j, for j = 1, . . . ,m, are constants to be determined. Substituting the solu-tions Q0 j and Q1 j into the inner expansion (3.12), and writing the resulting expres-sion in terms of the outer variables, we obtain a two term asymptotic expansion ofthe inner solutionQ j =(Q0 j(t)− σ jI j2piD0)+νD0[−σ jI j2piloge |x−x j |+ c j]+ . . . (3.16)Next, from (3.4a) and (3.4b), we construct the outer problem for the density ofpathogens, which is valid far away from the patches, as∂P∂ t=D∆P− P, t > 0, x ∈Ω\{x1, . . . ,xm}; ∂n P = 0, x ∈ ∂Ω, (3.17)where x1, . . . ,xm are the centres of the patches. In this region, we expand the outersolution asP = P0+νD0P1+ . . . (3.18)Substituting (3.18) into (3.17) and collecting terms in powers of ν , we obtain theleading-order outer problem given by∆P0 = 0, t > 0, x ∈Ω\{x1, . . . ,xm}; ∂nP0 = 0, x ∈ ∂Ω. (3.19)Observe that this problem is similar to the leading-order inner problem (3.13) andany constant or function of time satisfies it. As a result of this, we chose theleading-order outer solution to be P0 ≡ P0(t). The next order outer problem for P162is given by∆P1 = P0+P0t , x ∈Ω\{x1,x2, . . . ,xm}; ∂nP1 = 0, x ∈ ∂Ω. (3.20)Upon matching the inner solution (3.16) and the outer expansion (3.18), we obtainthe following required singularity behavior for the outer solution as x −→ x j:P0(t)+νD0P1+ · · · ∼(Q0 j(t)− σ jI j2piD0)+νD0[−σ jI j2piloge |x−x j |+ c j]+ . . . ,x −→ x j.(3.21)In this way, we obtain the matching conditionsP0(t)∼(Q0 j(t)− σ jI j2piD0)and P1 ∼−(σ j I j2pi)log |x−x j| as x −→ x j.(3.22)The first condition yields that Q0 j(t) = P0(t)+σ jI j/2piD0 for each j = 1, . . . ,m.The ODE for P0(t) is derived from a solvability condition on the problem for P1.To do so, it is convenient to write the singularity behaviour of P1 given in (3.22) asa delta function forcing for the PDE in (3.20). In this way, the outer problem forP1 is equivalent to∆P1 = P0+P′0+m∑i=1(−σi Ii)δ (x−xi), x ∈Ω; ∂nP1 = 0, x ∈ ∂Ω. (3.23)Integrating (3.23) over the domain Ω and using the divergence theorem, we obtainan ODE for the leading-order density of pathogens P0(t) in the bulk region givenbyP′0 =−P0+1|Ω|m∑i=1σi Ii. (3.24)This ODE is the solvability condition for the O(ν) outer problem (3.23).To solve the outer problem (3.23), we introduce the Neumann Green’s function63G(x;x j), which satisfies∆G =1|Ω| −δ (x−x j), x ∈Ω; ∂nG = 0, x ∈ ∂Ω; (3.25a)G(x;x j)∼− 12pi log |x−x j|+R j, as x −→ x j, and∫ΩGdx = 0, (3.25b)whereR j ≡R(x j) is the regular part of G(x;x j) at x = x j for j = 1, . . . ,m. Withoutloss of generality, we impose∫ΩP1 dx = 0, so that the spatial average of P in thebulk region is P0. Therefore, the solution to the outer problem (3.23) is written interms of the Neumann Green’s function G(x;x j) asP1 =m∑i=1σiIi G(x;xi). (3.26)Upon substituting (3.26) into the outer expansion (3.18), we obtain a two-termasymptotic expansion of the outer solution in the bulk region asP = P0+νD0m∑i=1σiIi G(x;xi)+ . . . . (3.27)Now, we expand (3.26) as x −→ x j, and substitute the singularity behaviour ofthe Neumann Green’s function G(x,x j) given in (3.25b) into the correspondingexpansion to getP1 ∼ σ jI j(− 12piloge |x−x j|+R j)+m∑i6= jσiIi G(x j;xi) as x −→ x j, j = 1, . . . ,m.(3.28)Matching the inner and outer solutions, we derive the constants c j in terms of theNeumann Green’s function asc j = σ jI jR j +m∑i 6= jσiIi G(x j;xi), j = 1, . . . ,m. (3.29)Thus, substituting (3.29) into (3.16), we derive a two-term asymptotic expansion ofthe inner solution Q j(ρ, t), valid in an O(ε) neighbourhood of the jth patch, given64byQ j =(P0(t)+σ j I j2piD0)+νD0[−(σ j I j2pi)logeρ+σ jI jR j +m∑i6= jσiIi G(x j;xi)]+ . . . ,j = 1, . . . ,m.(3.30)Lastly, by substituting the inner solution (3.30) into the SIR system in (3.7c) andevaluating the integrals over the boundary of the jth patch, we obtain a nonlin-ear system of ODEs for the leading-order density of pathogens in the bulk regioncoupled to the population dynamics of the jth patch. This system is given bydP0dt=−P0+ 1|Ω|m∑j=1σ j I j, (3.31a)which is coupled todS jdt=−β jS j(P0(t)+σ j I j2piD0)− νD0β jS jΨ j,dI jdt= β jS j(P0(t)+σ j I j2piD0)+νD0β jS jΨ j−φ jI j, (3.31b)dR jdt= φ jI j, j = 1, . . . ,m,Here, Ψ j = (GΦ) j is the jth entry of the vector GΦ, with Φ = (σ1I1, . . . ,σmIm)Tand G is the Neumann Green’s matrix whose entries are defined by(G ) j j =R j≡R(x j) for i= j and (G )i j =G(xi;x j) for i 6= j with (G )i j =(G ) ji.(3.32)The function G(x j;xi) is the Neumann Green’s function satisfying (3.25) andR j ≡R(x j) is its regular part at the point x = x j. In our analysis, D0 = O(1) and ν =−1/ loge(ε) 1. Therefore, as ε tends to zero, ν also approaches zero, and so tothe leading-order we can neglect theO(ν) terms in (3.31b). Replacing the leading-order density of pathogens P0 with p in (3.31), we derive the leading-order system65of ODEs given byd pdt=−p+ 1|Ω|m∑j=1σ j I j, (3.33a)dS jdt=−β jS j(p(t)+σ j I j2piD0), (3.33b)dI jdt= β jS j(p(t)+σ j I j2piD0)−φ jI j, (3.33c)dR jdt= φ jI j, j = 1, . . . ,m, (3.33d)Observe that we started with the dimensionless coupled PDE-ODE model (3.7) forstudying the spread of airborne disease with indirect transmission, and arrived atthe leading-order reduced system of ODEs (3.33) in the limit D = O(ν−1). Thissystem of ODEs also models indirect transmission of infections, even though, theterms with σ j I j/(2piD0) in (3.33b) and (3.33c) make it look like infection is trans-mitted from person-to-person through direct interaction. This term does not modeldirect transmission, rather, it accounts for the pathogens shed by infected individ-uals in a patch. The density of these pathogens depend on the scaled-diffusion rateD0 > 0. When the pathogens diffuse slowly (smaller values of D0), there is sig-nificant contribution from this term. This contribution decreases as the pathogensdiffuse faster (increasing D0). Moreover, in the limit D0 −→ ∞, this terms tendsto zero and (3.33) reduces to the model for well-mixed regime given in equation5 of [50]. In Sections 3.4 and 3.5, the reduced system of ODEs (3.33) is used tocompute the basic reproduction number and final size relation for one and two pop-ulation patches, respectively. The effect of the spatial locations of the patches andtheir interaction, as characterized by O(ν) terms in (3.31), is studied in Section3.6.3.4 One-patch modelIn the previous section, the method of matched asymptotic expansions was used toreduce the dimensionless coupled PDE-ODE model (3.7) to the nonlinear systemof ODEs (3.31), for the leading-order density of pathogens p and m populationpatches. In this section, we consider a single population patch located at the center66of a unit disk, and use the dimensionless coupled model (3.7) together with thereduced system of ODEs (3.33) to study the effect of diffusion on the spread ofinfection in the population.From (3.7), we derive that the density of pathogens P(x, t) for this one-patchscenario satisfies∂P∂ t=D∆P− P, t > 0, x ∈Ω\Ω0; (3.34a)∂n P = 0, x ∈ ∂Ω; 2piεD∂n P =−σ I, x ∈ ∂Ω0, (3.34b)Here Ω is a unit disk and Ω0 ⊂Ω is a disk of radius ε 1 representing the singlepopulation patch, which is located at the centre of the unit disk. The density ofpathogens P is coupled to theSIR dynamics of the population given bydSdt=− βS2piε∫∂Ω0P ds;dIdt=βS2piε∫∂Ω0P ds−φ I; (3.34c)dRdt= φ I,From the reduced system of ODEs (3.33), we obtain an ODE model for a singlepopulation patch given byd pdt=−p+ 1|Ω|(σ I),dSdt=−βSp−βS(σ I2piD0),dIdt= βSp+βS(σ I2piD0)−φ I,dRdt= φ I.(3.35)To study the spread of infection in the population and the effect of diffusionon the epidemic caused by the pathogens, we solve the coupled PDE-ODE model(3.34) numerically using the commercial finite element package, FlexPDE [161]67with several diffusion rate of pathogens. The full PDE results are then com-pared with results from the reduced system of ODEs (3.35), which is valid forD = D0/ν 1. In addition, the limiting ODE system (3.35) is analyzed using themethod of Kermack-McKendrick similar to that done in [23], [25]. To do so, thefollowing simple lemma is needed:Lemma 3.4.1. Let f (t) be a nonnegative monotone nonincreasing continuouslydifferentiable function such that as t→ ∞, f (t)→ f∞ ≥ 0, then d fdt → 0 as t→ ∞.Upon summing the equations for S and I in (3.35), we obtaind(S+ I)dt=−φ I ≤ 0. (3.36)This implies that (S+ I) decreases to a limit. It can be shown from Lemma 3.4.1that its derivative approaches zero, so that we can conclude that I∞ = limt→∞ I(t) = 0.In addition, by integrating (3.36), we obtain∫ ∞0I(t)dt =N(0)−S(∞)φ, (3.37)which implies that∫ ∞0 I(t)dt <∞, where S∞= limt→∞S(t) denotes the total susceptiblepopulations remaining after the outbreak. This simple property is employed whencomputing the final size relation below.3.4.1 The basic reproduction numberR0The calculation of the basic reproduction number is done using the next generationmatrix method similar to that done in [53, 169]. Note that there are two infectiousclasses I and p for this scenario. The Jacobian matrix F for new infections evalu-ated at the disease free equilibrium point, DFE=(S0,0,0)=(N(0),0,0) is given byF =(∂Fi∂x j)i, j=βσN(0)2piD0 βN(0)0 0 ,where the functions F1 ≡ dI/dt, F2 ≡ d p/dt in (3.35) and x j = I, p for j = 1,2.The Jacobian matrix V for transfer of infections between compartments, evaluated68at the disease free equilibrium point DFE isV =(∂Vi∂x j)i, j=φ 0− σ|Ω| 1 , and FV−1 =βN(0)σφ |Ω| +βN(0)σ2φpiD0βN(0)0 0 .Remark 2. In order to calculate the basic reproduction number for the model in(3.35), we use the next generation matrix method as in [53], [169] known to be thedominant eigenvalues of FV−1 (the spectral radius of the matrix FV−1), and givenasR0 =βN(0)σφ |Ω| +βN(0)σ2φpiD0. (3.38)Conveniently, we can decomposeR0 asR0 =R?+RD, whereR? =βN(0)σφ |Ω| andRD =βN(0)σ2φpiD0.The expression for R0 in (3.38) denotes the secondary infections contributedby indirect transmission (R?) and diffusion (RD). The term R? represents thesecondary infections caused indirectly through the pathogen since a single infectiveI sheds a quantity σ of the pathogen per unit time for a time period 1/φ and thispathogen infects βN susceptibles. The second term RD denotes the secondaryinfections caused by the pathogen diffusing in the bulk at the rate D0 since a singleinfective I sheds a quantity σ of the pathogen per unit time for a time period 1/φand this pathogen infects βN susceptible individuals in the patch. As the diffusionrate of pathogens become asymptotically large, that is, D0 → ∞, we observe thatRD→ 0. Therefore, in this limit, the basic reproduction number R0 in (3.38) canbe written asR∞0 = limD0→∞R0 =βN(0)σφ |Ω| =R?. (3.39)A more detailed discussion of equation (3.39) will be given below while explainingour numerical simulations. The implication of the basic reprodction numberR0 issummarized as follows in the readily proved result.Theorem 3.4.2. For system (3.35), the infection dies out whenever R0 < 1. In69contrast, an epidemic occurs wheneverR0 > 1.3.4.2 The final size relationThe final size relation gives an estimate of the total number of infections and theepidemic size for the period of the epidemic in terms of the parameters in the modelas similar to that done in [13, 20]. In other words, the final size relation is used toestimate the total number of disease deaths and cases from the parameters of themodel. Following the approach in [19, 20, 23], we integrate the equation for S in(3.35) to getlogS0S∞= β∫ ∞0p(t)dt+βσ2piD0∫ ∞0I(t)dt. (3.40)Similarly, integrating the equation for p(t) in (3.35) givesp(t) = p0 e−t +σ|Ω|∫ t0e−(t−s)I(s)ds. (3.41)Next, we need to show thatlimt→∞∫ t0e−(t−s)I(s) ds = limt→∞∫ t0 esI(s) dset= 0. (3.42)If the integral in the numerator of (3.42) is bounded, this relation is immediate.If the numerator is unbounded, L’Hopital’s rule yields that limt→∞∫ t0e−(t−s)I(s) ds =limt→∞ I(t) = I(∞), which vanishes as established above following equation (3.36).Therefore, (3.41) yields thatp∞ = limt→∞ p(t) = 0.By integrating (3.41), interchanging the order of integration, and then using (3.37)we get ∫ ∞0p(t)dt =σ|Ω|∫ ∞0I(t)dt+ p0, (3.43)which implies that∫ ∞0 p(t)dt < ∞. Upon substituting (3.43) into (3.40), we obtainlogS0S∞=βσ|Ω|∫ ∞0I(t)dt+βσ2piD0∫ ∞0I(t)dt+β p0,70so that, by using (3.37), we obtain the final size relationlogS0S∞=βσNφ |Ω|{1− S(∞)N}+βσN2piφD0{1− S(∞)N}+β p0,= R?{1− S(∞)N}+RD{1− S(∞)N}+β p0.This implies that S∞ > 0. In a situation where the outbreak begins with no contactwith pathogen, so that p0 = 0, the final size relation becomeslogS0S∞= R?{1− S(∞)N}+RD{1− S(∞)N},=(R?+RD){1− S(∞)N}, (3.44)= R0{1− S(∞)N}.Equation (3.44) is referred to as the final size relation, and this gives the relation-ship between the basic reproduction number and the epidemic size. Note that thetotal number of infected population over the period of the epidemic is N−S∞ andcan also be described in terms of the attack rate(1−S∞/N)as in [20].3.4.3 Numerical simulation for one-patch modelNext, we present some numerical simulations of the coupled PDE-ODE model(3.34) and the reduced system of ODEs (3.35) for the case of a single populationpatch located at the centre of a unit disk. The coupled model (3.34) is solvednumerically using the commercial finite element package FlexPDE [161], whilethe reduced ODE system (3.35) is solved using the numerical ODE solver ODE45in MATLAB [158]. For these models, simulations are done with different diffusionrates of pathogens in order to understand the effect of diffusion on the dynamicsof the infected population. The parameters used for these simulations are shown inTable 3.1.71Table 3.1: Model variables and their descriptionsParameter Description Dimensional(less) valuesµ (β ) dimensional (dimensionless)effective contact rate0.3 [13](computed using (3.8))r (σ) dimensional (dimensionless)pathogen shedding rate0.1 [178](computed using (3.8))α (φ) dimensional (dimensionless)removed rate for infected in-dividuals1.87 [178](computed using (3.8))δ dimensional decay rate ofpathogens0.25pc typical value for density ofpathogens0.01ε radius of the populationpatches0.02|Ω| area of the domain (unit disk) pi0 0.5 1 1.5 2 2.5 300.20.40.60.81D0 = 0.1D0 = 2D0 = 10D0 = 300(a) Simulation of the reduced system ofODE (3.35)0 0.5 1 1.5 2 2.5 300.20.40.60.81D = 0.1D = 2D = 10D = 300D0 = 300 (ODE)(b) Simulation of the coupled PDE-ODE(3.34)Figure 3.1: The dynamics of infected I(t) for different diffusion rates ofpathogen D and D0, and other parameters as in Table 3.1. (a) showsthe result obtained from the reduced ODE (3.35) with initial conditions(S(0), I(0),R(0), p(0)) = (249/250,1/250,0,0), while (b) is the resultof the dimensionless coupled PDE-ODE model (3.34) with initial con-ditions (S(0), I(0),R(0),P(0)) = (249/250,1/250,0,0)72Figure 3.1 shows the proportion of infected over time when an epidemic be-gins with one infective in a total population of 250 individuals, with susceptible,infected and recovered population given as S(0) = 249/250, I(0) = 1/250 andR(0) = 0, respectively. The initial density of pathogen used for both the coupledmodel and the reduced system of ODEs is p(0) = P(0) = 0, denoting that the out-break begins with no pathogen in the air, and that the only source of pathogen intothe system is the one shed by infected individuals. We observe from this figure thatthe proportion of infected individuals decreases with increases in the diffusion rateof pathogen. This shows that when pathogens diffuse slowly, they cluster aroundthe population as they are being shed. As a result, since human populations areconfined in a region, this in turn leads to more infections.However, when pathogens diffuse faster (diffusion rate increases), they diffuseaway from the population as they are being shed, which in turn, reduce the den-sity of pathogens around the patch. This effect lowers the population of infectedindividuals. Comparing Figures 3.1a and 3.1b, we notice that the proportion ofinfectives estimated by the two models are similar when D and D0 are small andwhen they are asymptotically large. Since D = D0/ν , a small value of D0 impliesthat D is also small. As a result of this, the two models would behave similar inthis limit even though the reduced system of ODEs (3.35) is only valid in the limitD = O(ν−1), where ν = − log(ε), with ε 1. The difference between the twomodels becomes more apparent with an increase in diffusion rate, as the number ofinfectives estimated by the system of ODEs is less as compared to that of the PDE-ODE model. This is because the system of ODEs is valid in the limit D=O(ν−1),and the spread of infection decreases as D increases. Lastly, as the diffusion ratesbecome asymptotically large, the solutions of the two models essentially coincide.This is because when D→∞, the problem becomes well-mixed, where the densityof pathogen is homogeneous in space, and the coupled PDE-ODE model can bereduced to a system of ODEs. Similarly, if we take the limit of the reduced systemof ODEs (3.35) as D0 −→ ∞, we have the model studied in [23]. This suggeststhat the model in [23] can be interpreted as the well-mixed limit of the coupledPDE-ODE model (3.34).730 0.5 1 1.5 2 2.5 300.20.40.60.81D0 = 0.1D0 = 2D0 = 10D0 = 300(a) Simulation of the reduced system ofODE (3.35)0 0.5 1 1.5 2 2.5 300.20.40.60.81D = 0.1D = 2D = 10D = 300D0 = 300 (ODE)(b) Simulation of the coupled PDE-ODE(3.34)Figure 3.2: The dynamics of proportion of infected individuals I(t) us-ing different diffusion rates of pathogen, and all other parametersas in Table (3.1). (a) shows the result obtained from the sys-tem of ODEs (3.35) with initial conditions (S(0), I(0),R(0), p(0)) =(249/250,1/250,0,1), while (b) is the result of the dimen-sionless coupled PDE-ODE model (3.34) with initial conditions(S(0), I(0),R(0),P(0)) = (249/250,1/250,0,1)The results in Figure 3.2 are similar to those in Figure 3.1 except that the initialconditions of the pathogens is taken as P(0) = p(0) = 1. This models the casewhere there is pathogen in the air at the beginning of the outbreak. The otherinitial conditions are the same as those used in Figure 3.1. These results showhow the presence of pathogens in the atmosphere at the beginning would affect thetransmission of infection. The epidemic takes off earlier when there are pathogensin the air at the beginning of the outbreak (Figure 3.2) compared to when there areno pathogens at the beginning (Figure 3.1). When the diffusion rate is small, themodel with (Figure 3.2) and without (Figure 3.1) pathogen at the beginning of theoutbreak have similar estimates. This is because the pathogens are barely movingwhen the diffusion rate is small, and as a result, it does not make much differencewhether they are present or not. As the diffusion rate of pathogens increases, thereseems to be significant differences in the two solutions, since, the epidemic takesoff earlier in Figure 3.2 as compared to Figure 3.1. Therefore, the presence ofpathogens in the air around the population patch increases the spread of infection74in the population, as expected intuitively.(a) (b)Figure 3.3: Surface plots of the basic reproduction number R0 (3.38) forthe one-patch model (3.35) plotted with respect to the diffusion rate ofpathoegns D0 and some dimensionless parameters of the SIR model.(a) is for D0 and the transimission rate β , while (b) is for D0 and theshedding rate σ . The parameters used are given in Table 3.1The surface plots in Figure 3.3 show the basic reproduction R0 (3.38) for theone-patch model (3.35) in terms of the diffusion rate of pathogens D0 and the di-mensionless transmission and shedding rates, β and σ , respectively. These resultsshow the effect of D0 on the basic reproduction number,R0. We observe from bothresults in this figure that R0 increases as the transmission and shedding rates in-crease, and decreases as D0 increases for a fixed value of the transmission and shed-ding rates. These results agree with the simulations in Figures 3.1 and 3.2, wherethe spread of infections decreases as the diffusion rate of pathogen increases. InFigure 3.3a, the largest R0 is obtained when D0 is small and the transmission rateβ is large. This is reasonable because when pathogens diffuse slowly, it would takelonger for them to diffuse away from the population, and as a result, they continueto cause infections in the population, and consequently this leads to a large basicreproduction number. Similarly, in Figure 3.3b, the largest R0 is obtained whenthe shedding rate of pathogen is large and D0 is small, because when infected indi-viduals shed pathogens at a high rate and the pathogens do not diffuse away fromthe population, they lead to more infections. When the transmission and shed-75ding rates are low, irrespective of the diffusion rate of pathogens, the reproductionnumber will be less than one and the epidemic will die out.In the next section, we perform a similar analysis for a scenario with two spa-tially segregated population patches.3.5 Two-patch modelIn the previous section, we studied the effect of the diffusion rate of pathogens onthe spread of infection in a single population. In this section, we consider a scenariowith two population patches, and use the dimensionless coupled PDE-ODE model(3.7) and the reduced system of ODEs (3.33) with m = 2 to study the dynamicsof infection in these populations. The patches are centered at x1 = (0.5,0) andx2 = (−0.5,0) in a unit disk.For this two-patch scenario, the density of pathogen P for the coupled PDE-ODE model satisfies∂P∂ t=D∆P− P, t > 0, x ∈Ω\{Ω1∪Ω2}; (3.45a)∂n P = 0, x ∈ ∂Ω; 2piεD∂n P =−σ1 I1, x ∈ ∂Ω0; 2piεD∂n P =−σ2 I2, x ∈ ∂Ω2,(3.45b)where Ω1 and Ω2 are the two population patches centered at x1 = (0.5,0) andx2 = (−0.5,0). This density of pathogen is coupled to the population dynamics ofthe two patches through the following ODE system:Patch 1 Patch 2dS1dt=−β1S12piε∫∂Ω1P ds;dS2dt=−β2S22piε∫∂Ω2P ds;dI1dt=β1S12piε∫∂Ω1P ds−φ1I1; dI2dt =β2S22piε∫∂Ω2P ds−φ2I2; (3.45c)dR1dt= φ1I1;dR2dt= φ2I2.The coupled PDE-ODE model (3.45) is solved numerically using FlexPDE [161]with different diffusion rate for the pathogens. The solutions are used to study theeffect of diffusion on the spread of the infection caused by the pathogens within the76population. From (3.33), we construct the reduced system of ODEs for the case oftwo patches asd pdt=−p+ 1|Ω|(σ1 I1+σ2 I2), (3.46a)Patch 1 Patch 2dS1dt=−β1S1 p−β1S1(σ1 I12piD0),dS2dt=−β2S2 p−β2S2(σ2 I22piD0),dI1dt= β1S1 p+β1S1(σ1 I12piD0)−φ1I1, dI2dt = β2S2 p+β2S2(σ2 I22piD0)−φ2I2,(3.46b)dR1dt= φ1I1,dR2dt= φ2I2,with initial conditionsS1(0)= S10, S2(0)= S20, I1(0)= I10, I2(0)= I20, R1(0)= 0, R2(0)= 0, p(0)= p0,in a population of constant total size N = N1 +N2, where N1 = S1 + I1 + R1 =S10 + I10 and N2 = S2 + I2 +R2 = S20 + I20. This system of ODEs is similar to themodel studied in [50], in which indirect transmission of diseases with no diffusionwas studied. We use this model to compute the basic reproduction number and thefinal size relation, and the method of Kermack-McKendrick epidemic model as in[23, 25] will be used to analyze the model.Summing the equations for S1 and I1, and then those for S2 and I2 in (3.46b),we obtaind(S1+ I1)dt=−φ1I1 ≤ 0. (3.47)d(S2+ I2)dt=−φ2I2 ≤ 0. (3.48)Here, we see that (S1+ I1) and (S2+ I2) decreases to a limit, and we can showfrom Lemma (3.4.1) that their derivatives approach zero. Therefore, we concludethat I1∞ = limt→∞ I1(t) = 0 and I2∞ = limt→∞ I2(t) = 0.Next, by integrating (3.47) we get φ1∫ ∞0 I1(t)dt = S1(0) + I1(0)− S1(∞) =77N1(0)−S1(∞), so that ∫ ∞0I1(t)dt =N1(0)−S1(∞)φ1, (3.49)which implies that∫ ∞0 I1(t)dt < ∞. Similarly, we integrate (3.48), to obtain∫ ∞0I2(t)dt =N2(0)−S2(∞)φ2. (3.50)Here, S1∞ = limt→∞ S1(t) and S2∞ = limt→∞ S2(t) denote the total susceptiblepopulation remaining in patch 1 and patch 2, respectively, after the outbreak.3.5.1 Reproduction numberR0Following a similar approach to that used in Section 3.4.1 for the one-patch model,we construct our system of infected classes asdI1dt= β1S1 p+β1S1(σ1 I12piD0)−φ1I1, (3.51a)dI2dt= β2S2 p+β2S2(σ2 I22piD0)−φ2I2 (3.51b)d pdt=−p+ 1|Ω|(σ1 I1+σ2 I2). (3.51c)Using the next generation matrix approach in [53, 169], the basic reproductionnumber is calculated as follows. We first introduce the three infectious classesI1, I2, p, and the Jacobian matrix of Fi = (F1,F2,F3), evaluated at the diseasefree equilibrium pointDFE=(S10,0,0,S20,0,0)=(N1(0),0,0,N2(0),0,0,0) given byF =(∂Fi∂x j)i, j=β1σ1N1(0)2piD00 β1N1(0)0β2σ2N2(0)2piD0β2N2(0)0 0 0 ,where x j = I1, I2, p for j = 1,2,3 and i = 1,2,3.The Jacobian matrix of Vi = (V1,V2,V3), evaluated at the disease free equilib-78rium point DFE isV =(∂Vi∂x j)i, j=φ1 0 00 φ2 0− σ1|Ω| −σ2|Ω| 1, andFV−1 =β1N1(0)σ1φ1|Ω| +β1N1(0)σ12φ1piD0β1N1(0)σ2φ2|Ω| β1N1(0)β2N2(0)σ1φ1|Ω|β2N2(0)σ2φ2|Ω| +β2N2(0)σ22φ2piD0β2N2(0)0 0 0.Remark 3. To calculate the basic reproduction number for the two-patch modelin (3.35), we use the method of next generation matrix in [53, 169] given as thedominant eigenvalues of FV−1 (the spectral radius of the matrix FV−1). A simplecalculation yields thatR0 =(|Ω|+2piD0)♣4piφ1φ2|Ω|D0 +√(|Ω|2+4pi|Ω|D0)♠2+4pi2D20♣24piφ1φ2|Ω|D0 , (3.52)where we have defined ♣ and ♠ by♣= β1N1(0)φ2σ1+β2N2(0)φ1σ2 and ♠= β1N1(0)φ2σ1−β2N2(0)φ1σ2.In the well-mixed limit D0 1, similar to that studied for one patch model inSection 3.4, the reproduction numberR0 in (3.52) reduces toR∞0 = limD0→∞=♣φ1φ2|Ω| ,79which implies thatR∞0 =β1N1(0)σ1φ1|Ω| +β2N2(0)σ2φ2|Ω| . (3.53)We observe that R∞0 can be decomposed as R∞0 = β1R1 + β2R2, where R1 =N1(0)σ1φ1|Ω| andR2 =N2(0)σ2φ2|Ω| .We can interpret the large D0 limiting value R∞0 of the reproduction numberas follows. The formula for R0 in Equation (3.53) denotes the contribution ofthe first and second patch. The term β1R1 represents the secondary infectionscaused indirectly through the pathogen since a single infective I1 sheds a quantityσ1 of the pathogen per unit time for a time period 1/φ1 and this pathogen infectsβ1N1 susceptibles. The second term β2R2 denotes the secondary infections causedindirectly through the pathogen since a single infective I2 sheds a quantity σ2 ofthe pathogen per unit time for a time period 1/φ2 and this pathogen infects β2N2susceptibles.A detailed qualitative explanation of R0 (for the case where D0 = O(1)) andR∞0 (for the well-mixed limit D0→ ∞), will be given below when discussing ourresults from numerical simulations. The following easily-proved Theorem summa-rizes the implications of the reproduction numberR∞0 .Theorem 3.5.1. For the well-mixed limit D0→ ∞ for the system (3.46), the infec-tion dies out wheneverR∞0 < 1, while epidemic occurs wheneverR∞0 > 1.3.5.2 The final size relationFollowing the same approach as in subsection 3.4.2, we integrate the equations forS1 and S2 in (3.46b) to getlogS10S1∞= β1∫ ∞0p(t)dt+β1σ12piD0∫ ∞0I1(t)dt. (3.54)andlogS20S2∞= β2∫ ∞0p(t)dt+β2σ22piD0∫ ∞0I2(t)dt. (3.55)80Similarly, integrating the equation for p in (3.46a), we obtainp(t) = p0e−t +σ1|Ω|∫ t0e−(t−s)I1(s)ds+σ2|Ω|∫ t0e−(t−s)I2(s)ds. (3.56)Next, we need to show thatlimt→∞∫ t0e−(t−s)I1(s)ds = limt→∞∫ t0 esI1(s)dset= 0. (3.57)and thatlimt→∞∫ t0e−(t−s)I2(s)ds = limt→∞∫ t0 esI2(s)dset= 0. (3.58)If the integral in the numerator of (3.57) and (3.58) are bounded, the result isimmediate. Alternatively, if these integrals are unbounded, then by l’Hospital’srule these two limits reduce to limt→∞ I1(t) and limt→∞ I2(t) = 0, which vanish sinceI1(∞) = I2(∞) = 0 as was shown above following (3.48) (see also [23]). As a result,(3.56) yields thatp∞ = limt→∞ p(t) = 0.Next, upon integrating (3.56), interchanging the order of integration, and then using(3.49) and (3.50), we get∫ ∞0p(t) dt = p0+σ1|Ω|∫ ∞0I1(t) dt+σ2|Ω|∫ ∞0I2(t) dt, (3.59)which implies that∫ ∞0 p(t)dt < ∞.We then substitute (3.59) into (3.54) and (3.55) to getlogS10S1∞=β1σ1|Ω|∫ ∞0I1(t)dt+β1σ2|Ω|∫ ∞0I2(t)dt+β1σ12piD0∫ ∞0I1(t)dt+β1 p0,andlogS20S2∞=β2σ1|Ω|∫ ∞0I1(t)dt+β2σ2|Ω|∫ ∞0I2(t)dt+β2σ22piD0∫ ∞0I2(t)dt+β2 p0,81which yields the final size relationlogS10S1∞=β1σ1N1(0)φ1|Ω|{1− S1(∞)N1(0)}+β1σ2N2(0)φ2|Ω|{1− S2(∞)N2(0)}+β1σ1N1(0)2piφ1D0{1− S1(∞)N1(0)}+β1 p0,andlogS20S2∞=β2σ1N1(0)φ1|Ω|{1− S1(∞)N1(0)}+β2σ2N2(0)φ2|Ω|{1− S2(∞)N2(0)}+β2σ2N2(0)2piφ2D0{1− S2(∞)N2(0)}+β2 p0,This expression can be written by usingR1 =N1(0)σ1φ1|Ω| andR2 =N2(0)σ2φ2|Ω| , aslogS10S1∞= β1(R1{1− S1(∞)N1(0)}+R2{1− S2(∞)N2(0)}+σ1N1(0)2piφ1D0{1− S1(∞)N1(0)})+β1 p0,andlogS20S2∞= β2(R1{1− S1(∞)N1(0)}+R2{1− S2(∞)N2(0)}+σ2N2(0)2piφ2D0{1− S2(∞)N2(0)})+β2 p0,where we used (3.47), which implies S1∞ > 0 and S2∞ > 0. In the case wherethe outbreak begins with no contact with pathogen, so that p0 = 0, the final sizerelation for patch 1 and 2 can be written aslogS10S1∞= β1(R1{1− S1(∞)N1(0)}+R2{1− S2(∞)N2(0)}+σ1N1(0)2piφ1D0{1− S1(∞)N1(0)}),(3.60)logS20S2∞= β2(R1{1− S1(∞)N1(0)}+R2{1− S2(∞)N2(0)}+σ2N2(0)2piφ2D0{1− S2(∞)N2(0)}).In th well-mixed limit of asymptotically large diffusion in which limD0→ ∞,the final size relation (3.60) becomeslogS10S1∞= β1(R1{1− S1(∞)N1(0)}+R2{1− S2(∞)N2(0)}),logS20S2∞= β2(R1{1− S1(∞)N1(0)}+R2{1− S2(∞)N2(0)}).(3.61)82This result can be written in matrix form aslogS10S1∞logS20S2∞= T11 T12T21 T221− S1(∞)N1(0)1− S2(∞)N2(0) , (3.62)whereT= β1R1 β1R2β2R1 β2R2 .Note that the total number of infected populations in patches 1 and 2 over the periodof the epidemic are respectively N1−S1∞ and N2(0)−S2∞, and can be described interms of the attack rate(1−S1∞/N1(0))and(1−S2∞/N2(0))as in [20].3.5.3 Numerical simulation for two-patch modelHere, we present numerical simulations of the dimensional coupled PDE-ODEmodel (3.45) and the reduced system of ODEs (3.46) for the case of two populationpatches. Our goal is to study the spread of infection between and within thesepopulations. For the coupled PDE-ODE model, our patches are centered at x1 =(0.5,0) and x2 = (−0.5,0) for patches 1 and 2, respectively.The results in Figure (3.4) show the proportion of infected individuals for thetwo patches in the case where the outbreak begins with no pathogen in the air(P(0) = p(0) = 0), only one infected individual in patch 1 (I1(0) = 1/250), andno infections in patch 2 (I2(0) = 0). The population patches are assumed to beidentical with parameters as given in Table (3.2).Figures 3.4a and 3.4b show the result obtained from the reduced system ofODEs (3.46) and the coupled PDE-ODE model (3.45) respectively, for differentdiffusion rates.Similar to the results for a single population patch, epidemic take-off is de-layed, and epidemic size decreases as the diffusion rate increases. When the dif-fusion rate is small, there is a delay in the epidemic take-off time in patch 2, andthis delay decreases as the diffusion rate increases. The observed delays are due83to the time it takes the pathogens shed in patch 1 to diffuse to patch 2, since thereare no pathogens in the air, and no infections in patch 2 at the beginning of theepidemic. As the diffusion rate increases, the time it takes the pathogens to dif-fuse from patch 1 to patch 2 decreases, thereby decreasing the delay in epidemictake-off in the second population. In the limit, where the diffusion rate becomesasymptotically large, the epidemics starts at approximately the same time in thetwo population patches for both models. Observe that the delay in epidemic take-off time in the second population is more obvious in the results from the coupledPDE-ODE model in Figure 3.4b than those of the reduced system of ODEs inFigure 3.4a. This is because the system of ODEs is valid in the limit where thediffusion rate of the pathogens D = O(ν−1), where ν = −1/ log(ε) with ε 1.In this limit, the pathogens are already diffusing fast enough to reduce the time ittakes them to travel from patch 1 to patch 2. This reduces the delay in take-off timeof the epidemic in patch 2. As D,D0 → ∞, the system becomes well-mixed, andthe predictions for the two model agree (D = D0 = 300).Table 3.2: Parameter descriptions and values for the Two-patch model.Parameter Description Patch 1, 2 valuesµ (β ) dimensional (dimensionless) ef-fective contact rate0.3, 1.2 [13](computed using (3.8))r (σ) dimensional (dimensionless)pathogen shedding rate0.1, 1 [178](computed using (3.8))α (φ) dimensional (dimensionless) re-moved rate for infected individ-uals1.87 [178](computed using (3.8))N1,N2 total population 300, 250δ dimensionless decay rate ofpathogens0.25pc typical value for density ofpathogens0.01ε radius of the population patch 0.02|Ω| area of the domain (unit disk) piWhen there is a pathogen at the beginning of the outbreak (P(0) = p(0) = 1),with only one infected individual in patch 1 (I1(0) = 1/250) and no infectives inpatch 2 (I2(0) = 0), epidemics starts in the two patches at approximately the same84time for both models, irrespective of the diffusion rate of pathogens. This occurssimply because there are pathogens in the air close to the second patch at timet = 0, which cause infections to spread into the population immediately. Theseresults are shown in Figure 3.5. The parameters and initial conditions used are thesame as those used for the results in Figure 3.4 except for the initial density ofpathogen p(0) = P(0) = 1.0 0.5 1 1.5 200.10.20.30.40.50.60.70.80.91D0 = 0.1D0 = 2D0 = 10D0 = 300(a) Simulation of the reduced system ofODE (3.46)0 0.5 1 1.5 200.10.20.30.40.50.60.70.80.91D = 0.1D = 2D = 10D = 300(b) Simulation of the coupled PDE-ODE(3.45)Figure 3.4: The dynamics of proportion of infected individuals I(t) using dif-ferent diffusion rates, and all other parameters as in Table 3.1. (a)shows the results for patches 1 and 2 obtained from the reduced ODE(3.46) with initial conditions(S1(0), I1(0),R1(0)) = (249/250,1/250,0), (S2(0), I2(0),R2(0)) =(250/250,0,0) and p(0) = 0, and (b) shows similar results obtainedwith the coupled PDE-ODE model (3.45) for the same initial conditionsin the patches as the ODEs and P(0) = 0 for the diffusing pathogens. Inboth plots, the solid lines represents patch 1, while the dashed lines arefor patch 2For a scenario with two distinct patches where the transmission of infectionsand shedding of pathogens are done at different rates (a more realistic scenario), thereduced system of ODEs (3.46) predicts slightly different results from the coupledPDE-ODE models (3.45). These results are shown in Figure 3.6. The dimensionaltransmission and shedding rates in patch 1 are µ1 = 0.3 and r1 = 0.1, respectively,while in patch 2 are µ2 = 1.2 and r2 = 1 respectively (their dimensionless equiv-85alents can be computed with (3.8)). Figures 3.6a and 3.6b show the proportion ofinfected individuals in patches 1 and 2, respectively, obtained using the reducedsystem of ODEs (3.46) with different values of D0, while Figures 3.6c and 3.6dshow similar results for the coupled PDE-ODE model (3.45). We observe fromthese figures that even though there are no infections in patch 2 when the outbreakbegins, there are still more infections occurring in the patch 2 than in patch 1. Thisoccurs simply because patch 2 has a higher shedding and transmission rate. Highershedding and transmission rates imply more pathogens are shed and transmittedfaster in patch 2 than in patch 1.0 0.5 1 1.5 200.10.20.30.40.50.60.70.80.91D0 = 0.1D0 = 2D0 = 10D0 = 300(a) Simulation of the reduced system ofODE (3.46)0 0.5 1 1.5 200.10.20.30.40.50.60.70.80.91D = 0.1D = 2D = 10D = 300(b) Simulation of the coupled PDE-ODE(3.45)Figure 3.5: The dynamics of infected I(t) using different diffusion rates ofpathogen, and all other parameters as in Table 3.1. (a) shows the resultsobtained for patches 1 and 2 from the reduced system of ODEs (3.46)with initial conditions (S1(0), I1(0),R1(0)) = (249/250,1/300,0),(S2(0), I2(0),R2(0)) = (250/250,0,0), and p(0) = 1, while (b) showssimilar results obtained from the coupled PDE-ODE model (3.45) withthe same initial conditions for the ODEs in the patches and P(0) = 1for the diffusing patheogens. In both plots, the solid lines represent ofpatch 1, while the dashed lines are for patch2When the diffusion rate is small or asymptotically large, the estimates fromthe two models agree qualitatively. However, this is not the case for intermediatediffusion rates. For these rates, the coupled PDE-ODE model shows no signifi-cant difference in the epidemic take-off times in patch 1 for different values of D,86although, the maximum number of proportion of infectives at a given time is differ-ent (see the results for D = 0.5 and D = 10 in Figure 3.6c). This is due to the factthat the transmission and shedding of pathogens are done at higher rates in patch 2relative to patch 1.0 0.5 1 1.500.10.20.30.40.50.60.70.80.91D0 = 0.1D0 = 0.5D0 = 10D0 = 300(a) Patch 1 for ODE model0 0.5 1 1.500.10.20.30.40.50.60.70.80.91D0 = 0.1D0 = 0.5D0 = 10D0 = 300(b) Patch 2 for ODE model0 0.5 1 1.500.10.20.30.40.50.60.70.80.91D = 0.1D = 0.5D = 10D = 300(c) Patch 1 for coupled PDE-ODE model,p =0.10 0.5 1 1.500.10.20.30.40.50.60.70.80.91D = 0.1D = 0.5D = 10D = 300(d) Patch 2 for coupled PDE-ODE model,p=0.1Figure 3.6: The dynamics of infected I(t) using different diffusion rates, andall other parameters as in Table 3.2. (a) and (b) show the results ob-tained from the reduced system of ODEs (3.46) for patches 1 and 2, withinitial conditions (S1(0), I1(0),R1(0)) = (299/300,1/300,0),(S2(0), I2(0),R2(0)) = (250/250,0,0), and p(0) = 1, while (c) and (d)show similar results obtained by solving the coupled PDE-ODE model(3.45) with the same initial conditions for the ODEs in the patches andP(0) = 1 for the diffusing patheogens87The pathogens shed in patch 2 diffuse to patch 1, thereby causing the epidemicin patch 1 to occur earlier than one would have expected if the populations wereidentical or the patches are farther away from each other. As the distance betweenthe two patches increases, the effect of the pathogens shed in patch 2 on the popula-tion in patch 1 decreases. This qualitative effect is discussed in detail in Section 3.6,where we study the effect of patch locations on the spread of infections.3.6 Effect of patch location on the spread of infectionSo far, we have studied the effect of the diffusion rate of pathogens on the spread ofinfections within and between populations, and we have not considered the effectof the location of the patches. In this section, we study the effect of patch locationon the dynamics of infections by analyzing the two-term (extended model) reducedsystem of ODEs, as given in (3.31), which involves the Neumann Green’s matrixcharacterizing the spatial interaction between patches. This extended ODE systemis then used to compute the basic reproduction number and the final size relation,which now depends on the location of the patches. In addition, we present somenumerical simulations for two patches equally-placed on a ring of radius of radiusr, with 0 < r < 1, concentric within a unit disk, and we study how the proportion ofinfected individuals changes as the distance between the patch locations is varied.3.6.1 Effect of patch location on the basic reproduction numberIn our analysis below we assume that our domain is a unit disk and that the patchesare equally-placed on ring of radius r, with 0 < r < 1, which is concentric withinthe disk. Here, we derive an expression for the basic reproduction number using(3.31), while following the analytical framework used in Section (3.5). From thereduced system of ODEs (3.31), we construct our infectious classes for m patchesasdI jdt= β jS j(P0(t)+σ j I j2piD0)+νD0β jS jΨ j−φ jI j, j = 1, . . . ,m,d pdt=−p+ 1|Ω|m∑j=1σ j I j.(3.63)88Here, Ψ j = (GΦ) j is the jth entry of the vector GΦ, with Φ = (σ1I1, . . .σmIm)T ,and G is the Neumann Green’s matrix define in (3.32). The resulting ODE systemis an (m+ 1) dimensional system of equations for the infected classes I1, . . . , Im,and the pathogens p. At the disease free equilibrium, we construct the Jacobianmatrix F for the new infections asF =(∂Fi∂x j)i, j=β1N1(0)D0(σ12pi+ν∂Ψ1∂ I1)νβ2N2(0)D0∂Ψ1∂ I2. . .νβmNm(0)D0∂Ψ1∂ Imβ1N1(0)νβ1N1(0)D0∂Ψ2∂ I1β2N2(0)D0(σ22pi+ν∂Ψ2∂ I2). . .νβmNm(0)D0∂Ψ2∂ Imβ2N2(0)....... . .......νβ1N1(0)D0∂Ψm∂ I1νβ2N2(0)D0∂Ψm∂ I2. . .βmNm(0)D0(σm2pi+ν∂Ψm∂ Im)βmN0m0 0 0 0 0,(3.64)where the functionsF j ≡ I′j for j = 1, . . . ,m, andFm+1 ≡ p′ are as given in (3.63),x j ≡ I j for j = 1, . . . ,m, and xm+1 ≡ p. Similarly, from (3.63), we construct theJacobian matrix V for the transfer of infections between compartments, evaluatedat the disease free equilibrium point asV =(∂Vi∂x j)i, j=φ1 0 0 . . . 00 φ2 0 . . . 0....... . .... 00 0 0 . . . φm− σ1|Ω| −σ2|Ω| . . . −σm|Ω| 1. (3.65)89For the case of two population patches, the Jacobian matrices (3.64) and (3.65)reduce toF =β1N1(0)D0(σ12pi+ν∂Ψ1∂ I1)νβ2N2(0)D0∂Ψ1∂ I2β1N1(0)νβ1N1(0)D0∂Ψ2∂ I1β2N2(0)D0(σ22pi+ν∂Ψ2∂ I2)β2N2(0)0 0 0 and V =φ1 0 00 φ2 0− σ1|Ω| −σ2|Ω| 1 .Upon calculating the inverse of the matrix V , and then multiplying by the matrix Ffrom the left, we construct our next generational matrix asFV−1 =β1N1(0)φ1D0(σ12pi+σ1D0|Ω| +ν∂Ψ1∂ I1)νβ2N2(0)φ2D0∂Ψ1∂ I2+σ2β1N1(0)φ2|Ω| β1N1(0)νβ1N1(0)φ1D0∂Ψ2∂ I1+σ1β2N2(0)φ1|Ω|β2N2(0)φ2D0(σ22pi+σ2D0|Ω| +ν∂Ψ2∂ I2)β2N2(0)0 0 0 .(3.66)The dominant eigenvalue of the next generation matrix is our desired basic repro-duction number. From a computation of the eigenvalues of (3.66), we derive a twoterm asymptotic expansion for the basic reproduction number given byR =R0+νR1+O(ν2), (3.67)where R0 ≡ R0 is the leading-order basic reproduction number given in (3.52),and the O(ν) termR1 is given byR1 =1φ1φ2D0N2+4piφ1φ2D0+(|Ω|+2piD0)H♠√(|Ω|2+4pi|Ω|D0)♠2+4pi2D20♣2 , (3.68)where the quantities N, H and , are defined byN≡ β1N1(0)φ2 ∂Ψ1∂ I1 +β2N2(0)φ1∂Ψ2∂ I2, H≡ β1N1(0)φ2 ∂Ψ1∂ I1 −β2N2(0)φ1∂Ψ2∂ I2,≡(β1N1(0))2σ2 ∂Ψ2∂ I1 +(β2N2(0))2σ1∂Ψ1∂ I2.(3.69)90Here, the variables ♠ and ♣ are as defined in (3.52), and Ψ j for j = 1,2 are asdefined in (3.31). Since there are only two patches, we can use (3.31) to constructΨ1 and Ψ2 explicitly asΨ1 = σ1I1R1+σ2I2 G(x1;x2), and Ψ2 = σ1I1 G(x2;x1)+σ2I2R2,whereR j ≡R(x j) is the regular part of the Neumann Green’s function G(xi;x j) atx = x j. Upon differentiating Ψ1 and Ψ2, with respect to I1 and I2, we obtain∂Ψ1∂ I1= σ1R1,∂Ψ1∂ I2= σ2 G(x1;x2),∂Ψ2∂ I1= σ1 G(x2;x1), and∂Ψ2∂ I2= σ2R2.(3.70)To evaluate these derivatives explicitly, as are needed in (3.69), we must determinethe Neumann Green’s function G(xi;x j) and its regular part R j as obtained bysolving (3.25) in the unit disk. These results are well-known (see equation (4.3) of[105]), and we haveG(x;ξ ) =− 12pilog |x−ξ |− 14pilog(|x|2|ξ |2+1−2x ·ξ )+ (|x|2+ |ξ |2)4pi− 38pi,R(ξ ) =− 12pilog(1−|ξ |2)+ |ξ |22pi− 38pi.(3.71)Since the patches are symmetrically located on a ring of radius r, with 0 <r < 1, we can take their centres as x1 = (r,0) and x2 = (−r,0) for patch 1 and 2,respectively, so that, |x1|= |x2|= r. Substituting x1 and x2 into (3.71), we concludethatG(x1;x2) = G(x2;x1) =12pi(− log(2r)− log(1+ r2)+ r2− 34),R(x1) =R(x2) =12pi(− log(1− r2)+ r2− 34).(3.72)91Upon substituting (3.72) into (3.70), and then using (3.70) in (3.69), we obtainN≡ ♣2pi(− log(1− r2)+ r2− 34), H≡ ♠2pi(− log(1− r2)+ r2− 34),≡ σ1σ22pi(− log(2r)− log(1+ r2)+ r2− 34)[(β1N1(0))2+(β2N2(0))2],(3.73)where ♣ and ♠ are given by♣= β1N1(0)φ2σ1+β2N2(0)φ1σ2 and ♠= β1N1(0)φ2σ1−β2N2(0)φ1σ2.(3.74)Therefore, theO(ν) term in the basic reproduction number (3.67) can be computedby substituting (3.74) and (3.73) into (3.68). By using the leading-order basicreproduction number R0 given in (3.52) together with R1 in (3.67), we arrive atan explicit two term asymptotic expansion for the basic reproduction R whichdepends on the location of the patches. Note that the dependence on the locationcomes intoR through only the O(ν) term, which involves the Green’s function.3.6.2 Effect of patch location on the final size relationIn the previous subsection, for a special two-patch configuration where the patchesare equally spaced on a ring concentric within the disk, we derived a two-termasymptotic formula for the basic reproduction number. In this subsection, we studythe effect of patch location on the final size of the epidemic.From (3.31), a two term asymptotic expansion of the reduced system of ODEsfor the case of two population patches is given byd pdt=−p+ 1|Ω|(σ1 I1+σ2 I2), (3.75a)92Patch 1 Patch 2dS1dt=−β1S1 p−β1S1(σ1 I12piD0)− νD0β1S1Ψ1dS2dt=−β2S2 p−β2S2(σ2 I22piD0)− νD0β2S2Ψ2,dI1dt= β1S1 p+β1S1(σ1 I12piD0)+νD0β1S1Ψ1−φ1I1, dI2dt = β2S2 p+β2S2(σ2 I22piD0)+νD0β2S2Ψ2−φ2I2,(3.75b)dR1dt= φ1I1,dR2dt= φ2I2where p(t) is the leading-order density of pathogens in the air, S j, I j,R j are thesusceptible, infected, and removed individuals, respectively, in the jth patch for j =1,2. Here, Ψ1 = σ1I1R1 +σ2I2 G(x1;x2) and Ψ2 = σ1I1 G(x2;x1)+σ2I2R2, andG(x1;x2) = G(x1;x1) is the Neumann Green’s function and R1 =R2 is it regularpart at the points x1 and x2. This system of ODEs depends on the locations of thepatches, and the dependence arises from the O(ν) terms.Following the same approach as in subsection (3.5.2), we integrate the S1 andS2 equations in (3.75b) to obtainlogS10S1∞= β1∫ ∞0p(t) dt+β1σ12piD0∫ ∞0I1(t) dt+β1νD0∫ ∞0Ψ1 dt. (3.76)andlogS20S2∞= β2∫ ∞0p(t) dt+β2σ22piD0∫ ∞0I2(t) dt+β2νD0∫ ∞0Ψ2 dt. (3.77)The integral of Ψ1 and Ψ2 are given by∫ ∞0Ψ1 dt = σ1R1∫ ∞0I1(t) dt+σ2 G(x1;x2)∫ ∞0I2(t) dtand ∫ ∞0Ψ2 dt = σ1 G(x2;x1)∫ ∞0I1(t) dt+σ2R2∫ ∞0I2(t) dtSimilarly, the integral of (3.75a) is given by (3.59). Upon substituting (3.59) into(3.76) and (3.77), and assuming that the outbreak begins with no epidemic (p0 = 0),93the final size relation is given bylogS10S1∞=β1σ1N1(0)φ1|Ω|{1− S1(∞)N1(0)}+β1σ2N2(0)φ2|Ω|{1− S2(∞)N2(0)}+β1σ1N1(0)2piφ1D0{1− S1(∞)N1(0)}+β1 νD0[σ1R1N1(0)φ1{1− S1(∞)N1(0)}+σ2G(x1;x2)N2(0)φ2{1− S2(∞)N2(0)}],andlogS20S2∞=β2σ1N1(0)φ1|Ω|{1− S1(∞)N1(0)}+β2σ2N2(0)φ2|Ω|{1− S2(∞)N2(0)}+β2σ2N2(0)2piφ2D0{1− S2(∞)N2(0)}+β2 νD0[σ2R2N2(0)φ2{1− S2(∞)N2(0)}+σ1G(x2;x1)N1(0)φ1{1− S1(∞)N1(0)}],which can be written aslogS10S1∞= β1(R1{1− S1(∞)N1(0)}+R2{1− S2(∞)N2(0)}+σ1N1(0)2piφ1D0{1− S1(∞)N1(0)}+νD0[σ1R1N1(0)φ1{1− S1(∞)N1(0)}+σ2G(x1;x2)N2(0)φ2{1− S2(∞)N2(0)}]),(3.78)logS20S2∞= β2(R1{1− S1(∞)N1(0)}+R2{1− S2(∞)N2(0)}+σ2N2(0)2piφ2D0{1− S2(∞)N2(0)}+νD0[σ2R2N2(0)φ2{1− S2(∞)N2(0)}+σ1G(x2;x1)N1(0)φ1{1− S1(∞)N1(0)}]),whereR1 = σ1N1(0)/(φ1|Ω|) andR2 = σ2N2(0)/(φ2|Ω|). Upon writing (3.78) inmatrix form, we havelogS10S1∞logS20S2∞=(A+B+νD0C)1− S1(∞)N1(0)1− S2(∞)N2(0) , (3.79)94where the matrices A, B, and C are defined as followsA=β1R1 β1R2β2R1 β2R2 , B=σ1β1N1(0)2piφ1D0σ2β2N2(0)2piφ2D0andC=σ1β1R1N1(0)φ1σ2β1G(x1;x2)N2(0)φ2σ1β2G(x2;x1)N1(0)φ1σ2β2R2N2(0)φ2.In this way, we have derived a two term expansion of the final size relation thatdepends on the location of the patches. Similar to the basic reproduction number(3.67), the dependence on the location comes into this result at the O(ν) termthrough the Green’s function and its regular part. The explanation of the final sizefollows from subsection (3.5.2).3.6.3 Numerical simulation for two patch model with effect of patchlocationNext, we present some surface plots of the basic reproduction number R in (3.67)and its O(ν) correction term R1, defined in (3.68) with respect to the locationof the patches. Our plots are for different values of the dimensionless transmissionrates β1 and β2 for patches 1 and 2, respectively. In addition, we show some numer-ical simulations of the reduced system of ODEs (3.75) and the coupled PDE-ODEmodel (3.45) for two patches. The system of ODEs is solved using the MATLABnumerical ODE solver ODE45 ode45 [158], while the PDE is solved using Flex-PDE [161]. Our goal is to numerically study the effect of patch location on thespread of infections and the final epidemic size.Figure (3.7) shows the surface plot of the O(ν) term of the basic reproductionnumber,R1 (3.68) (first row), and the basic reproduction numberR (3.67) (second95row) with respect to the location of the patches and the transmission rates of thetwo patches. For each of the results in this figure, the transmission rate β1 (verticalaxis) is plotted against the distance of the patches from the centre of the unit disk(horizontal axis). A fixed value of β2 is used for each column, with the valueincreasing from left to right (β2 = 0.1, 0.4, 0.8, 1.2). Since r is the distance fromthe centre of each patch to the centre of the unit disk, for each value of r, thedistance between the centre of the two patches is 2r. The O(ν) term R1 showshow the leading-order basic reproduction number R0 ≡R0 (3.53) is perturbed bythe locations of the patches. Note that we have omitted the surface plots of R0from this figure because it is independent on the location of the patches.(a) β2 = 0.1 (b) β2 = 0.4 (c) β2 = 0.8 (d) β2 = 1.2(e) β2 = 0.1 (f) β2 = 0.4 (g) β2 = 0.8 (h) β2 = 1.2Figure 3.7: Surface plots of the basic reproduction number R (3.67) (sec-ond row) and its O(ν) term R1 (3.68) (first row) with respect to thedistance of the patches from the centre of a unit disk r, for differentvalues of the transmission rates β1 and β2 for patches 1 and 2, respec-tively. The parameters used are given in Table (3.2) except for pc = 450,with diffusion rate D0 = 5. For each of the graphs, β1 (vertical axis) isplotted against r (horizontal axis). The value of β2 changes for eachcolumn from left to right in increasing order. The term R1 show howthe leading-order basic reproduction number R0 is perturbed by thelocation of the patches96The first row of Figure (3.7) show thatR1 may increase or decreaseR depend-ing on the location of the patches and the transmission rates. When the transmis-sion rate in patch 2, β2 = 0.1 (Figures 3.7e), we observe that R1 has high valueswhen r is small and β1 is high. In this case, where the two patches are close toeach other, the infection is transmitted at a high rate in patch 1. As the distancebetween the two patches increases (r increases), the value of R1 decreases for allvalues of β1 and it attains negative values for some values of β1. For this range,R1decreases the leading-order basic reproduction number R0. These figures showthat for all values of β2, there are more infections (R, &R1 high) when the twopatches are closer to each other (r small) as compared to when they are fartherapart (r large). In addition, as the transmission rate β2 increases (from left to rightin Figure 3.7), the surface plot of R1 in the rβ1 parameter space changes, and theeffect of the reflecting boundary of the unit disk becomes apparent. For each valueof β2, the corresponding effect of R1 on the overall basic reproduction number Ris shown in the second row of Figure 3.7. When β2 = 1.2, we observe that highervalues ofR1 occur for smaller values of r (when the two patches are close to eachother) and for values of r close to 1 (when the patches are close to the boundaryof the disk). When the patches become closer to the reflecting boundary of theunit disk, they see a reflection of themselves through the boundary. This leads toa feedback-effect whereby the reflection of pathogen from the boundary returnsback to the patches. This boundary effect, as evident in the surface plot of the basicreproduction numberR in Figure 3.7h, leads to a higher level of infections.The numerical simulations of the coupled PDE-ODE model (3.45) and the re-duced system of ODEs (3.75) for two patches in the unit disk are shown in Fig-ure (3.8). For a fixed diffusion rate of pathogens, these two models are solvednumerically for different locations of the patches. The location of each patch isgiven in terms of the parameter r, which measures distance between the centre ofthe patch and the centre of the unit disk. Both patches have the same parametersexcept for the transmission rates of infection, and the shedding rate of pathogens(see Table 3.2). In addition, both patches have one infective at the beginning of theoutbreak, and the density of pathogens in the air is fixed at p(0) = P(0) = 1.Figures (3.8a) and (3.8b) show the results obtained from the reduced system ofODEs (3.75) for patches 1 and 2, respectively, while (3.8c) and (3.8d) show similar97results for the coupled PDE-ODE model (3.45). For each radius r, the epidemic inpatch 2 (right column of Figure (3.8)) is more than that in patch 1 (see Table 3.2for the parameters).0 0.5 1 1.500.10.20.30.40.50.60.7(a) ODE model: Patch 10 0.5 1 1.500.10.20.30.40.50.60.7(b) ODE model: Patch 20 0.5 1 1.500.10.20.30.40.50.60.7(c) Coupled PDE-ODE model: Patch 10 0.5 1 1.500.10.20.30.40.50.60.7(d) Coupled PDE-ODE model: Patch 2Figure 3.8: The dynamics of infected I(t) for different ring radius r. (a)and (b) show the results obtained from the reduced ODE (3.75)for patches 1 and 2, with initial conditions (S1(0), I1(0),R1(0)) =(299/300,1/300,0), (S2(0), I2(0),R2(0)) = (250/250,0,0), andp(0) = 1, while (c) and (d) show similar results obtained from thecoupled PDE-ODE model (3.45) with the same initial conditions forthe ODEs in the patches and P(0) = 1 for the diffusing patheogens.The diffusion rate of pathogens is fixed at D0 = D = 5, while all otherparameters are as given Table (3.2)As the radius of the ring (on which the patches are located) increases, that is,98as the distance between the centres of the two patches increases, the size of theepidemic in patch 1 decreases, while there seems to be no significant difference inthe size of the epidemic in patch 2. Since the shedding rate of pathogens in patch2 is larger than that in patch 1, the density of pathogens in the air around patch 2 ateach point in time is higher than those around patch 1. As a result, when the twopatches are closer to each other, the pathogens shed from patch 2 can easily diffuseto patch 1, and lead to more infections in the population. This effect depends onthe proximity of the two patches, and it weakens as the patches move farther awayfrom each other. This explains why infections in patch 1 decrease as the distancein the two patches increases. This observation is more prominent in the resultsobtained from the PDE-ODE model than in the system of ODEs. This is due to thefact that the ODEs system is valid in the limit D O(ν−1), with ν =−1/ log(ε)and ε 1. In this regime where the pathogens are diffusing fast, spatial gradientsin the pathogen density are smoothed out, and as a result the proximity of patch 2to patch 1 seems to have no significant effect on the epidemic in patch 2.However, as both patches move closer to the boundary of the domain, theyreceive signals of pathogens that is of equal strength as their shedding rates fromthe boundary (since the boundary is reflecting). In this way, there is an increasinginfection in both patches as they move closer to the boundary. This observation isnoticeable in the patch 2 dynamics shown in Figures (3.8b) and (3.8d), due to itshigh shedding rate. Specifically, it is more apparent in Figure (3.8b) than in (3.8d)because the system of ODEs used to obtain (3.8b) are only valid in the limit wherethe pathogens are diffusing very fast. We observe from these simulations that theestimates and predictions of both models qualitatively agree, even though the ODEmodel is only valid in the limit D O(ν−1).3.7 DiscussionWe developed and analyzed a coupled PDE-ODE model for studying the spreadof airborne diseases with indirect transmission. This model improves previouslydeveloped epidemic models for indirect transmission [23, 50] by incorporating themovement of pathogens, which is modelled with linear diffusion. Human popula-tions are modelled as circular patches, that are small relative to the length scale of99the domain, where each patch has an SIR dynamics for the population of suscep-tible, infected, and removed individuals respectively. The diffusion of pathogensis restricted to the region outside the patches, while human movement is not con-sidered. In our model, a susceptible individual becomes infected only by comingin contact with the pathogens (indirect transmission), and the spread of infectionwithin a patch depends on the density of pathogens around the patch. In the limitD = O(ν−1) with ν = −1/ log(ε) and ε 1 (when the pathogens are diffusingfast), the coupled PDE-ODE model is reduced to a nonlinear system of ODEs.This system of ODEs was then analyzed and used to compute the basic reproduc-tion number and the final size relation. Furthermore, the full PDE-ODE modeland the reduced system of ODEs were solved numerically, and their results agreedqualitatively.The numerical simulations for both the coupled model and the reduced systemof ODEs predicted a decrease in the epidemic as the diffusion rate of pathogensincreases, and the two models agreed in the limit D,D0→ ∞. When pathogens arediffusing slowly, it takes longer for them to diffuse away from the patches aftershedding, and as a result, more infections occur. On the other hand, when thediffusion rate is high, pathogens diffuse away from the patches immediately aftershedding, which thereby reduces infections. When there are two patches, whereinfection starts from only one of the patches with the other patch being diseasefree, and with no pathogens in the air, our models predict a delay in epidemic take-off time in the second population when pathogens diffuse slowly. This occurs as aresult of the time required for pathogens to diffuse from the infected patch to theother patch. As the diffusion rate increases, the delay in epidemic take-off timedecreases. The results of our model seems consistent with other previous results[41, 91, 104, 115, 121, 176, 179] where human populations were modeled with aPDE approach. Furthermore, we studied the effect of patch location on the spreadof infection. Our models predicted more infections when the two patches are closeto each other, and less infections when the patches are farther apart.In our model, we have assumed that the amount of pathogens in a patch canbe accounted for by knowing the density of pathogens around the patch, and in-dividuals do not move between patches. This assumption may not be true for allreal-life scenarios as the amount of pathogens in a patch may be different from the100density around the patch. Also, people may move between cities and towns. Ourmodel can be extended to incorporate human mobility between patches. This canbe achieved by allowing both humans and pathogens to diffuse in the bulk region,or by using the approach of meta population dynamics, in which individuals aretransferred from one patch to another without modelling their movement explicitly,or by using Lagrangian method to keep track of individuals’ place of residence atdifferent time. In addition to this, we can allow for infections to be transmitted inthe bulk region when a susceptible individual comes in contact with the diffusingpathogens. This would lead to a reaction-diffusion type model in the bulk region.Furthermore, similar models (indirect and/or direct transmission model) can be de-veloped for other diseases such as malaria, where the mosquitoes diffuse in the bulkregion and human populations are modelled with patches. Mosquito reservoirs canalso be incorporated into the modelling framework, where an ODE system is usedto describe mosquito life cycle from egg to adult.Notwithstanding all of these limitations and assumptions, we believe that ourproposed novel approach to modelling airborne diseases, where the movement ofpathogens is explicitly modelled with linear diffusion, will significantly contributeto knowledge and may be seen as a better approach. Our analysis and full numericalcomputations suggest that disease dynamics can be adequately studied with ourmore tractable reduced ODEs model, instead of the more intricate coupled model.The presence of parameter D0 in the reduced ODE system makes it easier to studythe effect of diffusion on diseases transmission. Furthermore, the extended systemof ODEs, which includes weak spatial effects through a Green’s interaction matrix,allowed us to study the effect of patch location on disease dynamics. Includingthis spatial information encoded in the Green’s matrix allows for characterizingthe effect of the spatial distribution of patches on disease transmission betweenspatially segregated populations.101Chapter 4A co-interaction model of HIVand syphilis infection among gay,bisexual and other men who havesex with men4.1 SynopsisWe developed a mathematical model to study the interaction of HIV and syphilisinfection among gay, bisexual and other men who have sex with men (gbMSM). Wequalitatively analysed the model and established necessary conditions under whichdisease-free and endemic equilibria are asymptotically stable. We gave analyticalexpressions for the reproduction number, and showed that whenever the reproduc-tion numbers of submodels and the co-interaction model are less than unity, theepidemics die out, while epidemics persist when they greater than unity. We pre-sented numerical simulations of the full model and showed qualitative changes ofthe dynamics of the full model to changes in the transmission rates. Our numericalsimulations using a set of reasonable parameter values showed that: (a) each of thediseases die out or co-exist whenever their respective reproduction number is lessthan or exceeds unity. (b) HIV infection impacts syphilis prevalence negatively102and vice versa. (c) one possibility of lowering the co-infection of HIV and syphilisamong gbMSM is to increase both testing and treatment rates for syphilis and HIVinfection, and decrease the rate at which HIV infected individuals go off treatment.4.2 IntroductionHIV is known to be a sexually transmitted and blood-borne infection with a highlyvariable disease progression in humans [134]. People infected with HIV experi-ence immune supression as a result of continuous destruction of the CD4+T lym-phocytes, which makes immunosupressed individuals at risk of acquiring othersexually transmitted infections (such as syphilis, gonorrhea [16, 44, 134]). At theend of 2017, approximately 37 million people were living with HIV throughout theworld, and over 900,000 reported deaths were attributed to HIV infection [142]. In2016, gay, bisexual and other men who have sex with men (gbMSM) accounted forabout half of the new HIV infections in Canada [138]. Similarly, gbMSM currentlyaccounts for most new and prevalent cases of HIV in Vancouver [69] and San Fran-cisco [39]. There were about 3320 gbMSM who were newly diagnosed with HIVin the UK in 2015 [51]. The increase of antiretroviral therapy (ART) coverage toreduce and prevent HIV transmission in British Columbia (BC), Canada, made usobserve a positive impact of ART to prevent HIV transmission and decrease HIVdiagnosis per year [78].Syphilis is known to be an infection caused by the Treponema pallidum bac-teria [44, 137], and progresses from primary → secondary → latent → tertiarystage if left untreated [137]. Infectious syphilis is more frequent in males with anincreased rate among gbMSM population in BC and Canada [70, 137]. In 2017,5% or more of gbMSM in 22 of 34 reporting countries were infected with syphilis[143]. From 2011 to 2015, the rate of reported cases of syphilis per 100,000 pop-ulation in the United States rose by 58% (from 14.8 to 23.4), with the highestrate observed in San Francisco, where the rates rose by 77% (from 84.3 to 149.6)[39]. In 2016, gbMSM accounted for about 80.6% of male infectious syphilis inthe United States [66]. Similarly, in BC, the rate of reported cases of infectioussyphilis per 100,000 population in 2016, rose to 16.0 (759 cases) when comparedto 4.2 (193 cases) in 2011 [70]. The highest rate in BC was observed in Vancou-103ver and surrounding regions, where the rates rose from 19.6 (131 cases) in 2011to 63.7 (428 cases) in 2016 [70]. In 2016, gbMSM accounted for about 63.5% ofinfectious syphilis in Vancouver [70].Recent increases in sexually transmitted infections (STIs), especially amonggbMSM, brought up about the importance for characterising the co-interaction ofHIV and syphilis. Increases in the risk of HIV and STI transmission have beenattributed to sexual behaviours over the last decade [44, 51, 66]. It is estimatedthat about 43% of gbMSM in BC with syphilis diagnoses and known HIV status in2016, were HIV positive [70]. Individuals co-infected with these two diseases aremore likely to transmit HIV to their sexual partners, and as well likely to progressto serious disease stages [44, 70]. gbMSM living with HIV are about 2 times morelikely to be infected with syphilis compared to those that are HIV negative [66].This chapter considers a single class of infectious syphilis since major stages,such as primary, secondary, early latent and infectious neurosyphilis, are generallyclassified as infectious syphilis, and is of public health concern [137]. Many math-ematical models have been previously used to assess dynamics of the co-infectionof HIV and other diseases, such as Hepatitis C virus, gonorrhea, tuberculosis andsyphilis [12, 24, 25, 34, 49, 129, 132, 154], but only Nwankwo et al. [132] useda similar approach to study the dynamics of HIV and syphilis. The study differsfrom [132] as we consider the gbMSM population in a setting where treatmentof both diseases is readily available. We make simplifying assumptions about thenatural history of both diseases and incorporate some epidemiological features ofthe co-dynamics of HIV and syphilis. Using a set of parameter values from pub-lished articles, our model aim to answer the following questions: What effect doessyphilis infection have on HIV infected population and vice versa? What is theimpact of change in transmission rate on both disease dynamics? Can we test andtreat mono-infected individuals more to reduce both diseases prevalence?The chapter is organised as follows. We develop and describe the model inSection 4.3, and analyse two sub-models in Sections 4.4 and 4.5. We present theanalysis of the full co-interaction model and some numerical simulations in Sec-tions 4.6 and 4.7 respectively while Section 4.8 discusses and concludes the paper.1044.3 Model formulation and descriptionThe total gbMSM population at time t, denoted by N(t) is divided into 8 mutuallyexclusive compartments stated in Table (4.1), so thatN(t) = S(t)+ IS(t)+UH(t)+AH(t)+TH(t)+USH(t)+ASH(t)+TSH(t). (4.1)Table 4.1: Model variables and their descriptionsVariable DescriptionS SusceptiblesIS Individuals mono-infected with syphilisUH Individuals mono-infected with HIV and unawareAH Individuals mono-infected with HIV and awareTH HIV infected individuals on treatmentUSH Coinfected individuals unaware of HIV infectionASH Coinfected individuals aware of HIV infectionTSH Coinfected individuals on HIV treatmentWe assume that at time t, new individuals enter the population at a constantrate Π. Individuals die at a constant natural mortality base rate µ . HIV infectedindividuals (UH ,AH ,USH ,ASH) not on treatment have additional HIV induced deathrates dUH ,dAH ,dUSH ,dASH respectively. We assume no death from syphilis and thatHIV infected individuals on treatment do not transmit HIV infection [56, 149].Diseases co-dynamics are complicated processes, but for simplicity, we assumethat both mono and coinfected individuals can either transmit HIV or syphilis butnot both at the same time. Susceptible individuals may acquire syphilis infectionwhen in contact with individuals in IS, USH , ASH and TSH compartments, at a rateλS (the force of infection associated with syphilis infection), given byλS = βS(IS+φ1USH +φ2ASH +φ3TSH)N, (4.2)where βS denotes the transmission rate for syphilis. Parameter βS is the probabilityof syphilis transmission from one contact between individuals in S and in othersyphilis infected compartments (IS, USH , ASH , TSH), times the number of contactsper year per individual. Modification parameters φ1, φ2 and φ3 respectively account105for the relative infectiousness of syphilis infected individuals with undiagnosedHIV infection (USH), coinfected with HIV and aware (ASH), and coinfected withHIV and on HIV treatment (TSH), compared to individuals mono-infected withsyphilis. We assume that coinfected individuals are about two times as infectious asmono-infected individuals [66]. Since it is believed that individuals infected withsyphilis recover with temporal immunity [151], we then assume that individualsinfected with syphilis recover after treatment and return to the susceptible class ata rate σ1.Susceptible individuals acquire HIV infection from those in the UH , AH , USHand ASH compartments, at the rate λH (the force of infection associated with HIVinfection), given byλH = βH(UH +κ1AH +κ2USH +κ3ASH)N, (4.3)where βH denotes the transmission rate for HIV. Parameter βH is the probabilityof HIV transmission from one contact between individuals in S and in other HIVinfectious compartments (UH , AH , USH , ASH), times the number of contacts peryear per individual.106Π µSIS UHµ µ+dUHUSH AHµ+dUSH µ+dAHASH THµ+dASH µTSHµλSσ1γλHη1λSσ2ρ1η3λSσ4ν2η2λSσ3λHα1ρ2 ν1Figure 4.1: Diagram of the HIV/Syphilis co-interaction model107Modification parameters κ1, κ2 and κ3 respectively account for the relativeinfectiousness of individuals monoinfected with HIV and aware (AH), coinfectedwith HIV and unaware (USH), coinfected with HIV and aware (ASH), in comparisonwith individuals mono-infected with HIV.Susceptible individuals infected with HIV at rate λH enter the HIV unawareclass UH , where they progress to HIV aware class AH following testing at a rateα1, and are then placed on treatment at a rate ρ2 to enter the class TH . Individualsin the HIV infected and on treatment classes TH and TSH can go off treatment atrates ν1 and ν2 respectively. Individuals mono-infected with HIV (UH ,AH ,TH)are infected with syphilis at rates η1λS, η2λS, η3λS to enter classes USH ,ASH ,TSHrespectively, and modification parameters η1,η2,η3 > 1 account for higher risk ofsyphilis acquisition for people living with HIV.Individuals mono-infected with syphilis, IS are infected with HIV at a rate γλHto enter the class USH , where the modification parameter γ > 1 due to higher risk ofHIV acquisition for people whose immune system are affected by syphilis infec-tion. Coinfected individuals in the class ASH are placed on treatment at a rate ρ1 toenter class TSH . Coinfected individuals in the classes USH ,ASH ,TSH are tested andtreated for syphilis at rates σ2, σ3, σ4 to move back into the classes UH ,AH ,TH ,respectively. This model assumes uniform and homogeneous mixing population.The model diagram presented in Figure 4.1 is described by the following systemof nonlinear differential equations.108dSdt= Π+σ1IS− (µ+λS+λH)S,dISdt= λSS− (µ+σ1+ γλH)IS,dUHdt= λHS− (µ+dUH +α1+η1λS)UH ,dAHdt= α1UH +σ2USH +σ3ASH +ν1TH − (µ+dAH +η2λS+ρ2)AH ,dTHdt= ρ2AH +σ4TSH − (µ+η3λS+ν1)TH , (4.4)dUSHdt= γλHIS+η1λSUH − (µ+dUSH +σ2)USH ,dASHdt= η2λSAH +ν2TSH − (µ+dASH +σ3+ρ1)ASH ,dTSHdt= ρ1ASH +η3λSTH − (µ+ν2+σ4)TSH ,We will analyse different diseases separately, and then jointly understand dif-ferent components of the general model and as well adapt to different scenarios.4.4 Syphilis sub-modelWe have the model with syphilis only by setting UH = AH = TH =USH = ASH =TSH = 0 in system (4.4), and this givesdSdt= Π+σ1IS− (µ+λS)S,dISdt= λSS− (µ+σ1)IS, (4.5)where λS = βSISNS, with total population given as NS(t) = S(t)+ IS(t).The simple SIS model in (4.5) ignored syphilis-related death and was exten-sively discussed in [151] using different stages of syphilis infection to understandthe transmission dynamics, and in [10] to track syphilis dynamics in men andwomen. Hence, the dynamics of system (4.5) based on biological considerationin the region ΞS ={(S, IS) ∈ R2+ : NS ≤ Πµ}, is easy to show as being positively109invariant with respect to the model. We therefore consider model (4.5) to be epi-demiologically and mathematically well posed with all variables and parametersbeing positive for all time series as in [24, 25, 52]. Model (4.5) has a disease freeequilibrium given by E0S = (S0, I0S) =(Πµ, 0).The linear stability of E0S will be explained by the reproduction number ReSderived using the method of next generation matrix in [52, 169]. The matrix Fshowing the rate of appearance of new infections and matrix V showing the rate oftransfer of individuals in and out of the compartments are respectivelyF = βS, and V = µ+σ1.The only eigenvalue of the next generation matrix FV−1 gives the reproductionnumber for syphilis from the model in (4.5) as:ReS = ρ(FV−1) =βS(µ+σ1), (4.6)where ρ denotes the spectral radius (the dominant eigenvalue) of FV−1.ReS is the reproduction number for syphilis dynamics given by the productof the transmission rate of syphilis infection βS and the rate that an infective pro-gresses out of syphilis infectious class1(µ+σ1). The biological interpretation ofReS is the number of syphilis infections produced by one syphilis infective dur-ing the period of infectiousness when introduced in a totally syphilis susceptiblepopulation in the presence of treatment.We can establish the local stability of the disease free equilibrium (E0S) usingLemma 4.4.1 which follows from [52] and Theorem 2 of [169].Lemma 4.4.1. The DFE E0S of model (4.5) is locally asymtotically stable (LAS) ifReS < 1 and unstable ifReS > 1.The biological interpretation ofReS < 1 is that we can eliminate syphilis fromthe population if the initial sizes of the subpopulation of syphilis sub-model are inthe attraction region E0S.To ensure that elimination of the syphilis epidemic is independent of the initialsizes of the sub-populations, we establish the global stability of the DFE E0S by110claiming the result in Lemma 4.4.2.Lemma 4.4.2. For any positive solutions (S(t), IS(t)) of model system (4.5), ifReS < 1, then, the DFE E0S is a global attractor.See Appendix A for the proof of Lemma 4.4.2.4.4.1 Endemic equilibrium pointsWe can solve equation (4.5) in terms of the force of infection λS = βSISNto findthe conditions for the existence of an equilibrium (S∗, I∗) for which syphilis isendemic in a population. By equating the right-hand side of equations (4.5) to zeroand solving for S∗ and I∗, we haveS∗ =Π(µ+σ1)µ(µ+σ1+λ ∗S ), (4.7)I∗S =λ ∗S S∗(µ+σ1), (4.8)with λ ∗S = βSI∗SN∗. From equation (4.8), we haveI∗SS∗=λ ∗S(µ+σ1)=1(µ+σ1)(βSI∗SN∗),N∗S∗= ReS,ReS = 1+λ ∗S(µ+σ1),λ ∗S =(ReS−1)Ω,where Ω denoting the mean infective period is given by Ω=1(µ+σ1).When we substitute λ ∗S into the Equations in (4.7) and (4.8), we obtainS∗ =ΠΩ(µ+σ1)µ(Ω(µ+σ1)+(ReS−1)) =ΠµReS, (4.9)I∗S =Π(ReS−1)µ(Ω(µ+σ1)+(ReS−1)) =Π(ReS−1)µ(ReS). (4.10)111And the endemic equilibrium is given by E∗S = (S∗, I∗S ).The endemic equilibrium point E∗S must be positive since the model in (4.5)keeps track of human population. We have from Equations (4.9) and (4.10) thatwhen ReS > 1, E∗S is positive and the epidemic of syphilis persists in the commu-nity.We can summarize the uniqueness of the endemic equilibrium in Lemma 4.4.3.Lemma 4.4.3. The endemic equilibrium E∗S exists and is unique if and only ifReS > 1.Proof. It is enough to show that the components of E∗S are positive only ifReS > 1.We have I∗S in Equation (4.10) to be non-zero and positive only whenReS > 1. Thesame follows for S∗. QED.4.4.2 Global stability of the endemic equilibrium for syphilis-onlymodelWe claim the result in Lemma 4.4.4.Lemma 4.4.4. The endemic equilibrium of syphilis-only model 4.5 is globallyasymptotically stable in ΞS wheneverReS > 1.See Appendix B for the proof of Lemma 4.4.4.In summary, the syphilis-only model (4.5) has a globally asymptotically stabledisease-free equilibrium whenever ReS < 1, and a unique endemic equilibriumwheneverReS > 1.4.4.3 Sensitivity analysis ofReSIn this section, we investigate the effect of testing and treating syphilis on the dy-namics of syphilis by the elasticity ofReS with respect to σ1. From Equation (4.6),we use the approach in [24, 25, 42] to compute the elasticity ([37]) of ReS withrespect to σ1 as:σ1ReS∂ReS∂σ1=− σ1µ+σ1. (4.11)112Equation (4.11) is used to measure the impact of a change in σ1 on a proportionalchange in ReS. Equation (4.11) suggests that an increase in the testing and treat-ment rate of syphilis always leads to decrease of ReS, indicating a positive impacton the control of syphilis in the community.Figure 4.2 shows the effect of increasing the treatment of syphilis in the com-munity. For the set of parameters used, the figure shows that, by increasing thetesting and treatment rate to 5 or more (ReS ≤ 0.99) (i.e., test and treat all suscep-tible males for syphilis every 2.4 months or less), the reproduction number wouldbe below unity, which indicates syphilis eradication in the community.0510150.0 2.5 5.0 7.5 10.0σ1ReSFigure 4.2: Syphilis reproduction number ReS as a function of testing andtreatment rate σ1, with all parameters as in Table B.1 except βS = 5.0.The red dashed line indicates the reproduction numberReS = 14.5 HIV sub-modelWe have the model with HIV only by setting IS =USH = ASH = TSH = 0 in (4.4)given bydSdt= Π− (µ+λH)S,dUHdt= λHS− (µ+dUH +α1)UH ,dAHdt= α1UH +ν1TH − (µ+dAH +ρ2)AH , (4.12)dTHdt= ρ2AH − (µ+ν1)TH ,113λH = βH(UH +κ1AH)NH, (4.13)with the total population given as NH(t) = S(t)+UH(t)+AH(t)+TH(t).Please note that the population is not constant and the equation of NH thatdenotes the total sub-population of HIV-only model follows thatdNHdt=Π−µN−dUHUH −dAHAH ≤Π−µN, (4.14)and (4.14) implies that limt→∞supNH(t) ≤Πµ. Therefore the dynamics of system(4.12) will be studied based on biological consideration in the regionΞH ={(S,UH ,AH ,TH)∈R4+ : NH ≤ Πµ}, which is easy to show as being positively invariant with respect tothe model. We can similarly consider model (4.12) to be epidemiologically andmathematically well posed with all variables and parameters being positive for alltime series (years) as in [52].4.5.1 Disease free equilibrium pointWe have the disease free equilibrium when UH = AH = TH = 0 in model system(4.12) . This gives E0H =(Πµ , 0, 0, 0).4.5.2 Effective reproduction numberReHSimilarly, using the method of next generational matrix in [52, 169], as in ReH =ρ(FV−1), we have the reproduction number of HIV infections produced by HIVpositive cases to beReH . Note that we have three infected classes UH , AH and TH ,and the matrix showing the rate of appearance of new infections in compartment isgiven byF =λHS00.The matrix showing the rate of transfer of individuals in and out of the com-114partments i isV = V −−V + = (µ+dUH +α1)UH(µ+dAH +ρ2)AH −α1UH −ν1TH(µ+ν1)TH −ρ2AH .The jacobian matrix ofF evaluated at the disease free equilibrium point, DFE(E0H) =(Πµ,0,0,0)is given byF =∂F (E0H)∂xl=βH βHκ1 00 0 00 0 0where xl =UH ,AH ,TH for l = 1,2,3.The jacobian matrix of V evaluated at the disease free equilibrium point DFEisV =∂V (E0H)∂xl=(µ+dUH +α1) 0 0−α1 (µ+dAH +ρ2) −ν10 −ρ2 (µ+ν1) ,and FV−1 has eigenvalues 0 and ReH . The dominant eigenvalues of the nextgeneration matrix FV−1 which is the spectral radius of the matrix FV−1, gives theeffective reproduction number for HIV from model (4.12).Therefore we haveReH = ρ(FV−1) =βH ((µ+ν1)(µ+α1κ1+dAH)+µρ2)(µ+dUH +α1)((µ+ν1)(µ+dAH)+µρ2), (4.15)and we can writeReH = BU +BA, whereBU =βH(µ+dUH +α1),BA =βHα1κ1(µ+ν1)(µ+dUH +α1)((µ+ν1)(µ+dAH)+µρ2). (4.16)ReH denotes the effective reproduction number for HIV dynamics (the number of115HIV infection produced by one HIV case).Remark 4. We can epidemiologically interpret the terms for the expression ofReHin Equation (4.16). We have denoted BU as the average number of new cases ofHIV generated by individuals in the class UH , and BA as the average number ofnew cases of HIV generated by individuals in the class AH .BU is interpreted as the product of the transmission rate of HIV infected in-dividuals in the UH class (βH) and the average duration spent in the UH class( 1µ+dUH +α1).Similarly, we can interpret BA as the product of the transmission rate of HIVinfected individuals in the AH class (βHκ1), the fraction that survives the UH class( α1µ+dUH +α1)and the average duration spent in the AH class, which include theduration of the fraction that goes off treatment from class TH(1µ+dAH + ρ2µµ+ν1).Then the reproduction number ReH is the sum of the expressions for BU and BA,which is the number of HIV infections produced by one HIV infective during theperiod of infectiousness when introduced in a totally HIV susceptible populationin the presence of treatment.We can establish the local stability of the disease free equilibrium (E0H) usingLemma 4.5.1 which follows from [52] and Theorem 2 of [169].Lemma 4.5.1. The DFE E0H of model (4.12) is locally asymtotically stable (LAS)ifReH < 1 and unstable otherwise.The biological interpretation of ReH < 1 means that we can eliminate HIVfrom the population if the initial sizes of the subpopulation of HIV sub-model arein the attraction region E0H . To be sure that eventual eradication of HIV epidemicis independent of the initial sizes of the sub-populations, we will show that thedisease free equilibrium E0H is globally asymptotically stable.1164.5.3 Global stability of the disease-free for HIV-only modelWe can rewrite model (4.12) as,dUdt= F(U,V ),dVdt= G(U,V ), G(U,0) = 0, (4.17)where U = S and V = (UH ,AH ,TH), with U ∈R1+ denoting the number of suscep-tible individuals and V ∈R3+ denoting the number of infected individuals.We now denote the disease free equilibrium by,E0H = (U∗,0), where U∗ =(Πµ). (4.18)Conditions S1 and S2 in equation (4.19) must be satisfied to guarantee local asymp-totic stability.S1 :dUdt= F(U,0), U∗ is globally asymptotic stable (g.a.s)S2 : G(U,V ) = AV − Ĝ(U,V ), Ĝ(U,V )≥ 0 for (U,V ) ∈ ΞH , (4.19)where A = DV G(U∗,0) denotes the M-matrix (the off diagonal elements of A arenon-negative) and ΞH denotes the region where the model makes biological sense.Lemma 4.5.2 holds if system (4.17) satisfies the conditions in (4.19).Lemma 4.5.2. The disease free equilibrium point E0H of HIV-only model is glob-ally asymptotically stable ifReH < 1 and conditions in (4.19) are satisfied.Proof. We have from Lemma 4.5.1 that E0H is locally asymptotically stable ifReH < 1. Now considerF(U,0) = {Π−µS},G(U,V ) = AV − Ĝ(U,V ),117A =βH − (µ+dUH +α1) κ1βH 0α1 −(µ+dAH +ρ2) ν10 ρ2 −(µ+ν1) . (4.20)Ĝ(U,V ) =Ĝ1(U,V )Ĝ2(U,V )Ĝ3(U,V )=βH(1− SNH)(UH +κ1AH)00. (4.21)We have the conditions in 4.19 satisfied since Ĝ1(U,V )≥ 0 and Ĝ2(U,V )= Ĝ3(U,V )=0⇒ Ĝ(U,V )≥ 0. And therefore we can conclude that E0H is globally asymptoti-cally stable forReH < 1.4.5.4 Endemic equilibrium pointsWe can solve equation (4.12) in terms of the force of infection λH = βH(UH +κ1AH)NHto find the conditions for the existence of an equilibrium, and for which HIV is en-demic in a population.We equate the right-hand side of equations (4.12) to zero to haveΠ− (µ+λ ∗H)S∗ = 0, (4.22)λ ∗HS∗− (µ+dUH +α1)U∗H = 0, (4.23)α1U∗H +ν1T∗H − (µ+dAH +ρ2)A∗H = 0, (4.24)ρ2A∗H − (µ+ν1)T ∗H = 0, (4.25)118From Equations (4.22) to (4.25), we haveS∗ =Π(µ+λ ∗H), (4.26)U∗H =λ ∗HS∗(µ+dUH +α1), (4.27)A∗H =α1U∗H +ν1T ∗H(µ+dAH +ρ2),=α1U∗H(µ+ν1)(µ+ν1)(µ+dAH)+µρ2, (4.28)T ∗H =ρ2A∗H(µ+ν1), (4.29)And the endemic equilibrium is given by E∗H = (S∗, U∗H , A∗H , T∗H), whereλ ∗H = βH(U∗H+κ1A∗H)N∗H.From equation (4.26) and (4.29), we haveU∗HS∗=λ ∗H(µ+dUH +α1),U∗HS∗=1(µ+dUH +α1)(βH(U∗H +κ1A∗H)N∗H),N∗HS∗=βH(µ+dUH +α1)(U∗H +κ1A∗H)U∗H),N∗HS∗= ReH ,ReH = 1+λ ∗H(µ+dUH +α1)+α1λ ∗H(µ+ν1)(µ+dUH +α1)((µ+ν1)(µ+dAH)+µρ2)+α1ρ2λ ∗H(µ+dUH +α1)((µ+ν1)(µ+dAH)+µρ2),ReH −1 = λ ∗HΣ,λ ∗H =(ReH −1)Σ,where Σ denotes the mean infective period given byΣ=1(µ+dUH +α1)(1+α1(µ+ν1)((µ+ν1)(µ+dAH)+µρ2)+α1ρ2((µ+ν1)(µ+dAH)+µρ2))119When we substitute λ ∗H into the endemic equilibrium point in (4.26) to (4.29), weobtain the endemic equilibrium point in terms ofReH asS∗ =ΠΣµΣ+(ReH −1) , (4.30)U∗H =Π(ReH −1)(µ+dUH +α1)(µΣ+(ReH −1)) , (4.31)A∗H =α1Π(µ+ν1)(ReH −1)(µ+dUH +α1)(µ(µ+dAH +ρ2)+ν1(µ+dAH))(µΣ+(ReH −1)) ,(4.32)T ∗H =α1ρ2Π(ReH −1)(µ+dUH +α1)(µ(µ+dAH +ρ2)+ν1(µ+dAH))(µΣ+(ReH −1)) ,(4.33)The endemic equilibrium point E∗H must be positive since the model in (4.12) alsokeeps track of human population. We have from Equations (4.30) - (4.33) thatwhen ReH > 1, E∗H is positive and HIV is able to attack the population. That isReH > 1 shows the possibility of HIV to prevail in the community where there isno syphilis infection.We summarize the uniqueness of the endemic equilibrium in Lemma 4.5.3.Lemma 4.5.3. The endemic equilibrium E∗H of model (4.12) exists and is unique ifand only ifReH > 1.Proof. It is enough to show that the components of E∗H are positive only ifReH > 1.We have the numerator and denominator of U∗H in Equation (4.31) to be positiveonly when ReH > 1. Therefore, both the numerator and denominator of U∗H arenon-zero and positive whenReH > 1. The same follows for S∗, A∗H and T∗H .1204.5.5 Global stability of the endemic equilibrium for HIV-only modelFor the special case of when there is no HIV-related death (i.e dUH = dAH = 0), themodel in (4.12) becomesdSdt= Π− (µ+λH)S,dUHdt= λHS− (µ+α1)UH ,dAHdt= α1UH +ν1TH − (µ+ρ2)AH , (4.34)dTHdt= ρ2AH − (µ+ν1)TH .The new model (4.34) has a similar unique endemic equilibrium as model (4.12),but with dUH = dAH = 0.LetΞH0 ={(S,UH ,AH ,TH)∈Ξh :UH =AH =TH = 0}and ReH0 =ReH |dUH=dAH=0.We claim Lemma 4.5.4 with the proof in Appendix C.Lemma 4.5.4. The endemic equilibrium of HIV-only model 4.34 is globally asymp-totically stable in ΞH \ΞH0 wheneverReH0 > 1.Using a regular perturbation argument together with Liapunov function theoryas was done in [17], the proof in Lemma (4.5.4) can be shown for the case of whendUH > 0,dAH > 0 but small.In summary, the HIV-only model in (4.5) has a globally asymptotically stabledisease-free equilibrium whenever ReH < 1, and a unique endemic equilibriumwhenever ReH > 1. This unique endemic equilibrium is globally asymptoticallystable wheneverReH0 > 1 (the case of dUH = dAH = 0).4.5.6 Sensitivity analysis ofReHFirstly, we investigate the effect of treating HIV on the dynamics of HIV by theelasticity ofReH with respect to ρ2. From Equation (4.15), we use the approach in[24, 25, 42] to compute the elasticity ([37]) ofReH with respect to ρ2 as:ρ2ReH∂ReH∂ρ2=− α1κ1µρ2(µ+ν1)((µ+ν1)(µ+dAH)+µρ2)((µ+ν1)(µ+α1κ1+dAH)+µρ2) .(4.35)121Equation (4.35) is used to measure the impact of a change in ρ2 on a proportionalchange in ReH . Equation 4.35 suggests that an increase in the rate of treatment ofHIV always lead to decrease ofReH , indicating a positive impact on the control ofHIV infection in the community.Figure 4.3a shows the effect of increasing treatment of HIV in the community.The figure predicts that even though increasing the number of cases treated canpositively impact HIV epidemics by reducing the reproduction number, but elimi-nation may only be achieved with aggresive treatment (i.e., ρ2 = 50 means treat alldiagnosed cases every week). Note that based on Equation (4.16), no matter howhigh we increase ρ2, BU will not be affected, which indicates that elimination ofHIV requires more than increasing the number of cases treated, and may never beachieved by increasing ρ2 if BU > 1.Secondly, we investigate the effect of testing HIV on the dynamics of HIV bythe elasticity ofReH with respect to α1. From Equation (4.15), we use the approachin [24, 25, 42] to compute the elasticity ([37]) ofReH with respect to α1 as:α1ReH∂ReH∂α1=α1κ1(µ+ν1)(µ+dUH)−α1((µ+ν1)(µ+dAH)+µρ2)(µ+dUH +α1)((µ+ν1)(µ+α1κ1+dAH)+µρ2) . (4.36)Equation (4.36) is used to measure the impact of a change in α1 on a proportionalchange inReH . Equation (4.36) suggests that an increase in the rate of testing HIVwill have a positive impact in decreasingReH and reducing HIV burden only if thenumerator of Equation (4.36) is negative, i.e. ifκ1(µ+ν1)(µ+dUH)−((µ+ν1)(µ+dAH)+µρ2)< 0Figure 4.3b shows the effect of increasing testing of HIV in the community. Thefigure predicts that increasing the number of cases tested could positively impactHIV epidemic by reducing the reproduction number, but elimination will never beachieved with testing alone. Note that based on Equation (4.16), no matter howhigh we increase α1, there will always be an asymptote of BA for α1 → ∞. Thisindicates that elimination of HIV requires more than increasing the number of casestested, and may never be achieved by increasing α1 if the asymptote of BA > 1.122Thirdly, we investigate the effect of the rate of treatment failure on the dynam-ics of HIV by the elasticity of ReH with respect to ν1. We compute the elasticity([37]) ofReH with respect to ν1 as:ν1ReH∂ReH∂ν1=α1κ1ν1µρ2((µ+ν1)(µ+α1κ1+dAH)+µρ2)((µ+ν1)(µ+dAH)+µρ2)(4.37)12340 20 40 60ρ2ReH(a) HIV reproduction number ReH as afunction of treatment rate ρ21230 2 4 6 8α1ReH(b) HIV reproduction number ReH as afunction of testing rate α11.01.52.02.53.00.0 0.5 1.0 1.5 2.0ν1ReH(c) HIV reproduction number ReH as afunction of rate of treatment failure ν1Figure 4.3: Impact of increasing testing rate α1, treatment rate ρ2 and rateof treatment failure ν1 on HIV reproduction number ReH , with all pa-rameters as in Table B.1 except for βH = 0.4. The red line shows whenReH = 1Equation (4.37) is used to measure the impact of a change in ν1 on a pro-portional change in ReH . Equation (4.37) suggests that a decrease in the rate oftreatment failure always lead to a decrease ofReH , indicating a positive impact on123the control of HIV in the community.Figure 4.3c shows the effect of treatment failure on the dynamics of HIV in thecommunity. This Figure predicts that increasing the rate of treatment failure (timeretained on treatment) could negatively impact HIV epidemics by increasing thereproduction number and possibly increasing HIV epidemics.4.6 Analysis of the HIV-syphilis modelHaving analyzed the two sub-models, we have the full HIV-syphilis model as in(4.4). From the equation of N that denotes the total population as in Equation(4.1), it follows thatdNdt=Π−µN−dUHUH −dAHAH −dUSHUSH −dASHASH ≤Π−µN, (4.38)and (4.38) implies that limt→∞supN(t)≤Πµ. Therefore the dynamics of system (4.4)will be studied based on biological consideration in the regionΞ={(S, IS,UH ,AH ,TH ,USH ,ASH ,TSH) ∈ R8+ : N ≤Πµ},which is easy to show as being positively invariant with respect to the model. Sim-ilarly, we can consider model (4.4) to be epidemiologically and mathematicallywell posed with all variables and parameters being positive for all time series as in[52].4.6.1 Disease free equilibrium point (DFE) of the full HIV-syphilismodelWe have the disease free equilibrium when IS =UH = AH = TH =USH = ASH =TSH = 0 in model (4.4). This givesE0 =(S0, I0S,U0H ,A0H ,T0H ,U0SH ,A0SH ,T0SH)=(Πµ, 0, 0, 0, 0, 0, 0, 0).1244.6.2 Effective reproduction numberReWe have the effective reproduction number for the full model to beRe . Using thenext generation method in [52, 169], we can show that the effective reproductionnumber for the full HIV-syphilis model (4.4) is given byRe = max{βS(µ+σ1),βH ((µ+ν1)(µ+α1κ1+dAH)+µρ2)(µ+dUH +α1)((µ+ν1)(µ+dAH)+µρ2)}, (4.39)We can establish the local stability of the disease free equilibrium (E0) usingLemma 4.6.1 which follows from [52] and Theorem 2 of [169].Lemma 4.6.1. The DFE E0 of model (4.4) is locally asymptotically stable (LAS) ifRe < 1 and unstable otherwise.Biological interpretation ofRe < 1(ReS < 1 & ReH < 1)means that we caneliminate both diseases from the population if the initial sizes of the population arein the attraction region Ξ.In the section below, we show that the elimination of HIV and syphilis epi-demics is independent on the initial sizes of the populations by showing the globalstability of the DFE E0.4.6.3 Global stability of the disease-free of the full HIV-syphilismodelWe claim the result in Lemma 4.6.2 from Lemmas 4.4.2 and 4.5.2.Lemma 4.6.2. The DFE E0 of model (4.4) is globally asymptotically stable ifRe <1 and unstable otherwise.For reference, see Appendix D for the proof of Lemma 4.6.2.4.6.4 Endemic equilibrium point of the full HIV-syphilis modelThe computation of the endemic equilibrium of the full HIV-syphilis model is an-alytically complicated, and therefore the endemic equilibria of model (4.4) corre-sponds to;1. E1 = (S1, IS1,0,0,0,0,0,0), the HIV free equilibrium, where125E1 =(ΠµReS,Π(ReS−1)µReS, 0, 0, 0, 0, 0, 0). (4.40)This exists when ReS > 1. The analysis of the equilibria E1 is similar to theendemic equilibria E∗S in equations (4.9) and (4.10).2. E2 = (S2,0,UH2,AH2,TH2,0,0,0), the syphilis free equilibrium, whereS2 =ΠΣµΣ+(ReH −1) ,UH2 =Π(ReH −1)(µ+dUH +α1)(µΣ+(ReH −1)) ,AH2 =α1Π(µ+ν1)(ReH −1)(µ+dUH +α1)(µ(µ+dAH +ρ2)+ν1(µ+dAH))(µΣ+(ReH −1)) ,(4.41)TH2 =α1ρ2Π(ReH −1)(µ+dUH +α1)(µ(µ+dAH +ρ2)+ν1(µ+dAH))(µΣ+(ReH −1)) ,This exists whenReH > 1. The analysis of the equilibria E2 is similar to theendemic equilibria E∗H in equations (4.30), (4.31), (4.32) and (4.33).3. E3 =(S3, IS3,UH3,AH3,TH3,USH3,ASH3,TSH3), the HIV-syphilis co-interactionequilibrium.We summarize the existence of the disease free equilibrium points in the followingtheorem:Theorem 4.6.3. The system of equations (4.4) has the following disease free equi-librium points:1. E0S which exist whenReS < 1.2. E0H which exist whenReH < 1.3. E0 which exists whenReS < 1 andReH < 1, i.e. Re < 1.We similarly summarize the existence of the endemic equilibrium points in thefollowing theorem:Theorem 4.6.4. The system of equations in (4.4) has the following endemic equi-librium points:1261. E∗S or E1 which exist whenReS > 1.2. E∗H or E2 which exist whenReH > 1.3. E3 which exists when ReS > 1 and ReH > 1, i.e. Re > 1. A detailed ex-planation of E3 will be given in our numerical simulations. These endemicequilibria will be explored and justified using numerical simulations. Ournumerical simulations will also explore epidemiological scenarios when(a) ReH > 1 andReS < 1,(b) ReH < 1 andReS > 1.4.7 Numerical simulations of the full modelIn order to illustrate the results of the preceding analysis, the full HIV-syphilismodel (4.4) is numerically simulated using R programming language and ggplot2[170, 175]. Unfortunately, we are unable to calibrate the model to data as a re-sult of the complexity of our model and unavailability of data on HIV-syphilisco-interaction, but we make assumptions of parameters for illustrative purposes.Hence the shape of the figures or time of epidemic take-off in our simulations maychange if the model is fitted or calibrated to the data of a particular region. Wesuggest that this theoretical study be seen as a guide for future research and datacollection.Initial conditions used are:(S(0), IS(0), UH(0),AH(0), TH(0), USH(0), ASH(0), TSH(0))= (5500,6,7,5,3,4,3,2)(4.42)which indicate the presence of both diseases in the community,(S(0), IS(0), UH(0),AH(0), TH(0), USH(0), ASH(0), TSH(0))= (5500, 0, 7, 5, 3, 0, 0, 0)(4.43)which indicate the presence of only HIV infection in the community, and(S(0), IS(0), UH(0),AH(0), TH(0), USH(0), ASH(0), TSH(0))= (5500, 6, 0, 0, 0, 0, 0, 0),(4.44)127which indicate the presence of only syphilis infection in the community. Parame-ters in Table (B.1) are also used, except otherwise stated.Table 4.2: Model parameters and their interpretations.Symbol Parameter Value(yr−1) SourceΠ Recruitment rate estimated from N ≤Π/µ 100µ Natural mortality rate 0.017 corresponds tothe life expectancy of 58.8 years0.017 [45]dUH death rate due to unaware HIV infection inmono-infected individuals0.094 [153]dAH death rate due to aware HIV infection inmono-infected individuals0.094 [153]dUSH death rate due to unaware HIV infection in co-infected individuals0.094 [153]dASH death rate due to aware HIV infection in co-infected individuals0.094 [153]βS Transmission rate for syphilis infection. Thisis the product of the probability of syphilistransmission from one contact between in-dividuals in S and in other syphilis infectedcompartments (IS, USH , ASH , TSH), and thenumber of contacts per year per individualVariableβH Transmission rate for HIV infection. This isthe product of the probability of HIV trans-mission from one contact between individu-als in S and in other HIV infectious compart-ments (UH , USH , AH , ASH), and the number ofcontacts per year per individualVariableσ1 Testing and treatment rate of syphilis amongmono-infected males in the class IS. Thevalue 4year−1 means the average time for di-agnosis and treatment is 1/σ1 = 1/4 year = 3months.4 [132]128σ2,σ3,σ4 Testing and treatment rate of syphilis amongHIV infected males in classes USH ,ASH ,TSHrespectively. The value 4year−1 means theaverage time for diagnosis and treatment is1/σ4 = 1/4 year = 3 months.4,4,4 [132]ρ2 Treatment initiation rate of HIV. The value2.5year−1 means the time from HIV diagnosisto treatment initiation among mono-infectedmales in the class AH is 1/ρ2 = 1/2.5 year= 4.8 months.2.5 Assumedρ1 Treatment initiation rate of HIV. The value2.5year−1 means the time from HIV diagno-sis to treatment initiation among coinfectedmales in the class ASH is 1/ρ1 = 1/2.5 year= 4.8 months.2.5 Assumedν1,ν2 Rate of treatment failure for mono and coin-fected individuals in classes TH and TSH re-spectively. The value 0.9375year−1 meansthe time retained on HIV treatment for mono-infected and coinfected males is 1/νi =1/0.9375 year = 12.8 months for i = 1,2.That is, HIV infected males on treatmentspend at least 12.8 months before going offtreatment0.9375,0.9375 [172]η1,η2,η3 Modification parameters accounting for thehigher risk of syphilis acquisition for peopleliving with HIV in classes UH ,AH ,TH respec-tively2.237,2.237,2.237 [66]129γ Modification parameters accounting for thehigher risk of HIV acquisition for people liv-ing with syphilis in the class IS2.5 [15,55,59,64,133]φ1,φ2,φ3 Modification parameters accounting for thehigher risk of syphilis transmission for coin-fected individuals in classes USH ,ASH ,TSHrespectively, compared with individualsmonoinfected with syphilis in the class IS2.867,2.867,2.867 [92]κ1 Modification parameter accounting for therisk of HIV transmission for individualsmonoinfected with HIV and aware (AH),compared with individuals monoinfected withHIV and unaware (UH). We assume that therisk of transmitting HIV among UH is not sig-nificantly different from AH1.0 Assumedκ2,κ3 Modification parameters accounting for thehigher risk of HIV transmission for individ-uals coinfected with HIV (USH ,ASH), com-pared with individuals monoinfected withHIV (UH)2,2 [5,133]α1 Progression (testing) rate for individualsmono infected with HIV in the class UH . Thevalue 0.5year−1 means the time from HIV in-fection to diagnosis is 1/α1 = 1/0.5 year = 2years.0.5 [168]Figure 4.4 shows the HIV and syphilis epidemics with initial condition (4.42)and parameters in Table (B.1). If the reproduction number is less than unity (ReH =0.139 < 1,ReS = 0.025 < 1,Re = 0.139 < 1) due to smaller transmission rates ofHIV and syphilis (βH = 0.02, βS = 0.1), the number of individuals living withHIV and/or syphilis decreases and converges to the asymptotcally stable disease-130free equilibrium (Figure 4.4a). Biologically, both diseases go to extinction and theepidemics of HIV and syphilis die out in the community. In contrast, if the trans-mission rates are larger (βH = 0.4, βS = 5.0) andRe > 1 (ReH = 2.780 > 1,ReS =1.245 > 1,Re = 2.780 > 1), the number of infected individuals converges to theHIV-syphilis endemic equilibrium (Figure 4.4b). This biologically means that theepidemics of both HIV and syphilis persist in the community. The simulations areconsistent with Lemma 4.6.1 and Theorem 4.6.4.Furthermore, Figure 4.5 shows the HIV and syphilis epidemics with initialcondition (4.42). If the reproduction number of syphilis is greater than unity(ReH = 0.139 < 1,ReS = 1.245 > 1,Re = 1.245 > 1) due to a larger transmissionrate of syphilis (βH = 0.02, βS = 0.5), then the reproduction number of the coinfec-tion system is greater than the unity. The number of individuals mono-infected andco-infected with HIV persists for a long time and then decreases slowly to zerobecause of the long life time of people living with HIV (Figures 4.5a and 4.5b).The number of individuals mono-infected with syphilis increases (Figure 4.5c) andthen becomes stable after about 6 years (the zoomed-in plot of IS in Figure 4.5c)to converge to the asymptotcally stable syphilis endemic equilibrium showing onepossibility of Theorem 4.6.4, (3b). This biologically means that with our choiceof parameters and over a long period of time, a community with smaller transmis-sion rate of HIV and larger transmission rate of syphilis will experience syphilisepidemics, while the epidemic of HIV will die out. In this case, the maximum re-production number of the HIV-syphilis full model will be the reproduction numberof the syphilis sub-model.Figure 4.6 similarly shows the HIV and syphilis epidemics with initial con-dition (4.42). If the reproduction number of HIV is greater than unity (ReH =2.780 > 1,ReS = 0.025 < 1,Re = 2.780 > 1) due to a larger transmission rate ofHIV (βH = 0.4, βS = 0.1), then the reproduction number of the coinfection systemis greater than unity. The number of individuals mono-infected and co-infectedwith syphilis decrease to zero (Figures 4.6b and 4.6c) in less than 2 years (thezoomed-in plot of IS in Figure 4.6c) since syphilis is curable. The number of indi-viduals mono-infected with HIV infection first increase to a maximum value andthen decrease to converge to the asymptotically HIV endemic equilibrium (Figures4.6a) showing one possibility of Theorem 4.6.4, (3a). This biologically means that131with our choice of parameters, a community with larger transmission rate of HIVand smaller transmission rate of syphilis will experience the HIV epidemic, whilethe syphilis epidemic will die out. In this case, the maximum reproduction numberof the HIV-syphilis full model will be the reproduction number of HIV sub-model.Figure 4.7 shows the impact of the presence of one disease on the other ina community where either one or both diseases persist at the initial stage of theepidemic. Figure 4.7a shows the number of individuals living with HIV usinginitial conditions (4.42) (blue line) and (4.43)) (red line). It is worth noting thatthe steady state in blue line is about 5% higher than the one in red line, whichindicates that, for the same community, the presence of syphilis infection is likelyto enhance the HIV prevalence in comparison to no syphilis infection and effortstowards eradicating syphilis infection may in turn decrease HIV prevalence.05101520250 40 80 120Time (years)Infected Population CompartmentsHIV prevalenceSyphilis prevalence(a) Disease free equilibrium010002000300040000 40 80 120Time (years)Infected Population CompartmentsHIV prevalenceSyphilis prevalence(b) Endemic equilibriumFigure 4.4: Number of HIV infected individuals (green) and syphilis infectedindividuals (red) based on initial condition (4.42) and parameters inTable B.1, with different transmission rates and reproduction number:βH = 0.02,βS = 0.1,Re = 0.139 (left); βH = 0.4,βS = 5.0,Re = 2.780(right)Figure 4.7b shows the number of individuals living with syphilis using initialconditions (4.42) (blue line) and (4.44)) (red line). Similarly, it worth noting thatthe steady state in blue line is about 30% higher than the one in red line, whichindicates that, for the same community, the presence of HIV infection is likely toenhance the syphilis prevalence in comparison to no HIV infection and efforts aimat decreasing or eradicating HIV infection will in turn decrease syphilis prevalence.132051015200 40 80 120Time (years)U H+A H+T HA0.02.55.07.50 40 80 120Time (years)U SH+A SH+T SHB030060090012000.0 2.5 5.0 7.5 10.0Time (years)I S030060090012000 40 80 120Time (years)I SCFigure 4.5: Using the initial condition in (4.42) with βH = 0.02 and βS = 5.0,the figure shows dynamics of HIV mono-infected individuals (UH+AH+TH) (A), co-infected individuals (USH + ASH + TSH) (B), and syphilismono-infected individuals (IS) (C).050010001500200025000 40 80 120Time (years)U H+A H+T HA0.02.55.07.50 40 80 120Time (years)U SH+A SH+T SHB02460.0 2.5 5.0 7.5 10.0Time (years)I S02460 40 80 120Time (years)I SCFigure 4.6: Using the initial condition in (4.42) with βH = 0.4 and βS = 0.1,the figure shows dynamics of HIV mono-infected individuals (UH+AH+TH) (A), co-infected individuals (USH + ASH + TSH) (B), and syphilismono-infected individuals (IS) (C).133010002000300040000 40 80 120Time (years)HIV infected Population(a) Population of HIV positive individuals010002000300040000 40 80 120Time (years)Syphilis infected Population(b) Population of syphilis positive individ-ualsFigure 4.7: Prevalence of HIV and syphilis with βH = 0.4 and βS = 5.0(ReH = 2.780 > 1,ReS = 1.245 > 1,Re = 2.780 > 1). (a) Figure 4.7ashows the prevalence of HIV with syphilis at the initial stage of the epi-demic (initial condition (4.42), blue dashed line) and without syphilis(initial condition (4.43), red solid line). (b) Figure 4.7b shows the preva-lence of syphilis infection with HIV at the initial stage of the epidemic(initial condition (4.42), blue dashed line) and without HIV (initial con-dition (4.44), red solid line).4.8 Discussion and conclusionWe presented a mathematical model that rigorously analysed the co-interactionof HIV and syphilis infections in the presence of treatment of both diseases. Wecarried out the stability analysis of disease-free and endemic equilibra, and showedthat1. disease-free equilibra for sub-models and the full model were locally andasymptotically stable whenever their respective reproduction numbers areless than unity.2. endemic equilibra for sub-models and the full model were locally and asymp-totically stable whenever their respective reproduction numbers are greaterthan unity.3. increasing testing and treatment rate of mono-infected individuals with syphilismay bring the reproduction number of syphilis below unity, and thereby134eradicating the disease among mono-infected individuals in the community.4. increasing the testing rate, treament rate and reducing the rate of treatmentfailure for mono-infected individuals impact HIV epidemic by lowering thereproduction number of HIV, but may not be able to eradicate the disease inthe community.Despite the limitations of assuming homogeneous mixing populations and usingparameter values from published articles, our results and analyses of the reproduc-tion number indicated that1. HIV infection increases syphilis prevalence and vice versa.2. we could bring the reproduction number of syphilis below unity if syphilisis tested and treated more, but testing and treating cases of HIV alone maynot be sufficient to bring down the prevalence of HIV as this may depend onsome other factors, for example, some parameters in Equation (4.16) (lowerHIV-related death, increase time retained on treatment and so on).Great attention has not been given to the negative effect of the co-interactionof HIV and syphilis globally, and there are not many mathematical models thatconsidered synergistic interactions with treatment of both diseases among gbMSMpopulation. Even though our approach is similar to those considered in the liter-ature [12, 23, 29, 34, 49, 129, 132, 154] in terms of the joint dynamics of bothdiseases, but treatment of both HIV and syphilis infections among gbMSM popu-lation is an essential difference that none of those studies examined. Our model canbe extended to include general population, and can also be stratified into differentage group or risk level.135Chapter 5Assessing the combined impact ofinterventions on HIV and syphilisepidemics among gay, bisexualand other men who have sex withmen in British Columbia: aco-interaction model5.1 SynopsisIntroduction: The majority of HIV and infectious syphilis cases (over 80% of allinfectious syphilis cases) in British Columbia (BC) were among gay, bisexual andother men who have sex with men (gbMSM). A recent study carried out in a set-ting where the uptake of preexposure prophylaxis (PrEP) is moderate, the authorsrevealed that the risk of acquiring bacterial sexually transmitted infections (STIs)increases among gbMSM following initiation of PrEP. We therefore developed amathematical transmission model to assess the impact of different interventions,especially PrEP on HIV and syphilis infections, and show how the combination of136testing and treating syphilis, HIV treatment as prevention (TasP), condom use andPrEP uptake could eliminate both HIV and syphilis epidemics among gbMSM inBC over the next ten years (2019−2028).Methods: The model explores epidemiological aspects of the HIV and syphilisepidemics among gbMSM in BC. We divided the gbMSM population into differ-ent disease status and examined the impact of multiple interventions on severaloutcomes, specifically the World Health Organization threshold for disease elimi-nation as a public health concern (less than one new infection per 1000 susceptiblegbMSM). We focussed on the interventions that improved PrEP uptake, TasP op-timization, improved syphilis testing and treatment, and condom use. Other out-comes we examined included HIV incidence, HIV prevalence, syphilis incidenceand all-cause mortality among people living with HIV (PLWH). We carried out dif-ferent sensitivity analyses and implemented every aspect of the model in Python.Results: Of the strategies evaluated, the combination of optimizing all aspects ofTasP, improving syphilis testing and treatment, and increasing provision of PrEPreduced the HIV incidence rate more than TasP, by as much as 88% (0.13 per1000 susceptible gbMSM), and the elimination of HIV infection was possible byoptimizing TasP or combining TasP with any other interventions. Similarly, thecombination of improving syphilis testing and treatment, and increased condomuse reduced syphilis incidence rate by as much as 80% (0.85 per 1000 susceptiblegbMSM), and the elimination of the syphilis epidemic was also possible. Combin-ing TasP and PrEP with or without other interventions reduced the HIV incidencerate more than TasP alone, while combining PrEP, and improving syphilis testingand treatment increased the syphilis incidence rate more than improving syphilistesting and treatment alone.Conclusions: The combination of any interventions with PrEP decreases the HIVincidence rate more than without PrEP, and less compared to condom use. In ad-dition, the findings highlight how increasing the number of susceptible gbMSMon PrEP can create unexpected negative impact on syphilis incidence, and showthe importance of public health policies to address the co-interaction of HIV with137syphilis, and with other STIs among gbMSM in BC and in other similar settings.5.2 IntroductionHIV incidence seems to be declining in many parts of the world among gay, bi-sexual and other men who have sex with men (gbMSM), but not as fast as in othergroup [141]. Globally, gbMSM are about 19 times more likely to be living withHIV than the other groups, and this group accounts for a disproportionate burdenof HIV and syphilis infections [141, 146]. In British Columbia (BC), Canada,gbMSM continue to have the greatest number of new HIV diagnoses, constitutingabout 60% of all new HIV diagnoses in 2016 [69, 75]. The Public Health Agencyof Canada estimated that in 2016, approximately 52% of all 11,621 people livingwith HIV (PLWH) in BC, were gbMSM, and in 2017, 69.8% (127 cases) of allnew HIV diagnoses were among gbMSM [62, 135, 136, 177]. Since 2004, thenumber of new infections diagnosed each year among gbMSM in BC has beenrelatively constant between 150 and 180 cases, and at the end of 2016, it was esti-mated that 147 (range 90−260 cases) gbMSM in BC became newly infected withHIV[62, 135, 136, 139, 177].The rate of infectious syphilis in Canada is on the rise, and with a higher burdenamong gbMSM. In BC, gbMSM account for the majority of infectious syphiliscases and remain the group most at risk of contracting syphilis [70]. Even in theUS, both HIV and syphilis remain highly concentrated epidemics among gbMSM[145, 152], and primary and secondary syphilis remain the most infectious stagesof syphilis [80]. The inflammatory genital ulcers and lesions usually caused bysyphilis create entry points for the HIV virus, which in turn, increase the risk ofHIV transmission and shedding [30, 54]. In addition, syphilis complicates theclinical course of HIV by increasing viral load levels [30, 162].Studies have shown the high impact of the antiretroviral therapy (ART) up-take in decreasing the HIV transmission among people living with HIV (PLWH)[46–48]. BC adopted the HIV ”Treatment as Prevention” (TasP) in 2010 as pub-lic health policy, to prevent new HIV infections, maximize engagement amongPLWH, increase the possibility of viral suppression, decrease morbidity and mor-tality among PLWH on treatment [79, 124]. In BC, the impact of ART in de-138creasing the HIV epidemic and reducing the transmission among gbMSM is lowerwhen compared to other populations [69, 75, 125]. Also, the rate of sexuallytransmitted infections (STIs) have been rapidly increasing during the last 10 yearsamong gbMSM [70]. Of all cases of STIs at the end of 2016, infectious syphiliswas observed to have the highest prevalence (about 86% of all cases were amonggbMSM), and among gbMSM cases with known HIV status, 43% were co-infectedwith HIV [70].Oral pre-exposure prophylaxis (PrEP) for HIV prevention is the daily use ofantiretroviral (ARV) drugs by HIV-negative people to obstruct HIV acquisition.More than 10 randomized controlled trials have shown the effectiveness/efficacyof PrEP in preventing HIV transmission among serodiscordant heterosexual cou-ples (when one of the partner is HIV positive and the other is negative), gbMSM,transgender women, high-risk heterosexual couples, and people who inject drugs[38, 60, 85, 89, 118, 119, 122, 123]. The effectiveness of PrEP among gbMSM,which is mostly dependent on adherence, ranges from 42% to 99% [76]. BeforePrEP became known, people used condoms or engaged in sero-adaptive behav-iors (e.g., having sex with only people of the same HIV status) in order to preventbeing infected with HIV [14, 103, 120]. PrEP is now widely recommended to pre-vent HIV transmission, specifically among people at high risk of HIV acquisition[9, 76, 88, 144]. Since January 2018, PrEP became provincially-funded for peoplein BC who is at higher risk of HIV infection, which is made available through theHIV Drug Treatment Program (DTP) at the British Columbia Centre for Excellencein HIV/AIDS (BCCfE) [76].Currently, PrEP is totally free for elligible invididuals in BC, and reasonablysubsidized and more affordable in Canada, by the governments of Ontario andQubec [2, 67]. Syphilis is mainly treated with a single dose of antibiotics afterdiagnosis [69], but consistent and correct use of condoms is known to significantlyreduce the risk of STIs and HIV transmission[65], and the reduction in condom useis due in part to increases in the number of people on PrEP [31, 40, 113]. PrEP canreduce the risk of acquiring HIV among sexually active gbMSM, however it doesnot offer any protection against syphilis and other STIs and may in fact accidentallyincrease the risk of STIs transmission [171].Given the continued risk of syphilis transmission, its close association with139HIV infection, and the disproportionate disease burden among gbMSM in BC,there is a need to examine and understand the co-interaction of HIV and syphilisepidemics, and their trends among gbMSM. In addition, several studies have shownthe impact of PrEP in reducing the HIV incidence [140], but based on a recent studythat focused on the importance of frequent testing for STIs among gbMSM usingPrEP, in a setting similar to BC and where the uptake of PrEP is moderate, theauthors showed that the risk of bacterial STIs increases among gbMSM followinginitiation of PrEP [167]. Therefore, we developed a mathematical transmissionmodel to assess how the combination of TasP, PrEP, condom use, and testing andtreating syphilis, could eliminate HIV and syphilis epidemics in BC over the nextten years (2019−2028).5.3 MethodsWe designed a modeling approach in which HIV-syphilis disease progression andtransmission model were used. The model schematic is shown in Figure 5.1. In themodel, we combined the complex epidemiological dynamics of HIV and syphilisinfections by introducing the HIV-syphilis co-interaction. The model can be usedto predict epidemic trends, identify key factors that influence HIV and syphilis epi-demics among gbMSM, and as well predict the impact of different interventions.5.3.1 HIV-syphilis transmission modelWe developed a deterministic compartmental model for the co-interaction of HIV-syphilis transmission among the gbMSM population in BC, Canada. The modelassumptions and parameters (appropriate for the context of BC, Canada) and adetailed description of the model are presented in the supplementary informa-tion in Appendix B. The model has eight compartments (Fig 5.1): (1) Suscepti-ble (S)−individuals at risk of HIV and/or syphilis infection who were never ex-posed to both diseases and/or at risk of being re-infected with syphilis virus amongthose syphilis virus-experienced individuals who were cured; (2) Mono-infectedwith syphilis (IS)−individuals who were infected or re-infected with syphilis; (3)Mono-infected with HIV and Unaware (UH)−individuals who are infected withonly HIV and stay in this compartment until they are tested postive for HIV; (4)140Mono-infected with HIV and Aware (AH)−there are four sets of individuals inthis compartment: those who were tested positive from the unaware compartment(UH) and waiting for treatment; those individuals mono-infected with HIV whodropped out of treatment, individuals co-infected and unaware who recently gottested positive for HIV and have gotten tested and treated for syphilis; and in-dividuals co-infected and aware who recently got tested and treated for syphilis;(5) Mono-infected and on Treatment (TH)−there are two sets of individuals inthis compartment: co-infected individuals on HIV treatment who got tested andtreated for syphilis, and individuals who are eligible for and on HIV treatment; (6)Co-infected and Unaware (USH)−there are two sets of individuals in this compart-ment: those who were previously infected with syphilis and got infected with HIVbut unaware, and individuals who were mono-infected with HIV and unaware, andthen got infected with syphilis; (7) Co-infected and Aware (ASH)−there are twosets of individuals in this compartment: those who had been tested positive forHIV then got infected with syphilis and waiting for treatment, and co-infected in-dividuals who previously dropped out of HIV treatment; (8) Co-infected and onTreatment (TSH)−co-infected individuals who are on HIV treatment.S UH AH THIS USH ASH TSHΠ λHλSρ2ν1α1σ1σ4σ3η1λSσ2η2λS η3λSγλH ρ1ν2µ µ+ dUH µ+ dAHµ µ+ dUSH µ+ dASH µµFigure 5.1: Diagram of the HIV/Syphilis co-interaction modelWe assumed that individuals in the TH and TSH compartments cannot transmitHIV infection [56, 149] and that there are no syphilis related deaths since individ-uals infected with syphilis rarely die from this disease [151]. In addition, PLWH141can exit the model via either non-HIV or HIV-related mortality. Since it is difficultto get population-level data regarding the dynamics of the co-interaction of bothdiseases amongst gbMSM in BC, our initial conditions were chosen close to theobserved data of HIV and syphilis separately in BC. We calibrated the model byoptimizing the parameters βH ,α1,ρ2,βS,σ1,σ2 (see Appendix B for definitions),so that the numerical solution fits to: (1) the estimated number of PLWH and theestimated number of annual new HIV infections among the gbMSM population inBC from the Public Health Agency of Canada in 2011, 2014 and 2016 [62, 139];(2) the annual HIV diagnoses from the HIV cascade of care in BC (2011−2018)[71–75]; (3)the number of annual syphilis diagnoses from British Columbia Centrefor Disease Control (BCCDC) surveillance report (2012− 2017) [63, 70]. SincePrEP uptake was very low before 2017 [2, 67], we introduced PrEP in the modelin 2017 and the number of gbMSM on PrEP gradually increased to 4000 at the endof 2019 (see Section B.1.3 of the Appendix B for details).PythonTM version 2.7.6 was used for all the numerical and analytical coding,and the NUMPY and SCIPY libraries were used for the numerical simulations [97].Using the optimization package in the SCIPY library and with all other parametersfixed, we ran a simulation in a Nelder-Mead simplex algorithm to determine theoptimal values of the fitted parameters, while assuming a tolerance of 10−3 [130].We estimated final outputs by linear interpolation of the integrated solution andevaluation at yearly intervals. Details of differential equations, model parameters,calibration, and references can be found in the Appendix B.5.3.2 Modeling scenariosWith all parameters kept as those in the end of 2019 under the Status Quo sce-nario, we evaluated the impact of HIV TasP intervention (see Table 5.1 for details):(1) linearly decreasing the time from HIV infection to diagnosis for mono andco-infected individuals; (2) linearly decreasing the time from HIV diagnosis totreatment initiation; (3) linearly increasing the time retained on ART treatmentfor mono and co-infected individuals. For PrEP intervention, we focussed on en-rolling individuals on treatment with the uptake linearly increasing from 4000 in2019 to 5000 (low), 7000 (medium) and 10000 (high) in 2028 (see Table 5.1). Sim-142ilarly, we evaluated the impact of linearly decreasing time from syphilis infectionto treatment among both mono and co-infected individuals (Test & Treat syphilis)(see Table 5.1 for details). For condom intervention, we linearly increased the pro-portion of condom use from 65% in 2019 to 70% (low), 75% (medium) and 80%(high) in 2028 (see Table 5.1). Lastly, we evaluated the impact of combining: (a)PrEP and TasP; (b) TasP and Test & Treat syphilis; (c) PrEP and Test & Treatsyphilis; (d) condom use and TasP; (e) condom use and Test & Treat syphilis; ( f )PrEP, TasP and Test&Treat syphilis; and (g) condom use, TasP and Test & Treatsyphilis, according to the low, medium, and high scenarios as described in Table5.1. It is worth noting that no combination of PrEP and condom use was assessedin this study since different studies have shown decrease in condom use amongindividuals on PrEP [82, 90].5.3.3 Main outcomesWe compare the Status Quo (or baseline) scenario with different intervention sce-narios by forecasting the course of HIV and syphilis epidemics in BC. The follow-ing outcomes were estimated at the end of 2028: (1) the number of PLWH; (2)the number of cumulative HIV incident cases; (3) all-cause mortality cases amongHIV-positive gbMSM; (4) the number of cumulative syphilis incident cases; (5)HIV point prevalence; (6) all-cause mortality rate among HIV-positive gbMSM;(7) HIV incidence rate; and (8) syphilis incidence rate. The outcomes in (1)− (4)were presented in terms of the number of cases and the percent change when com-pared with the Status Quo. To evaluate which of the interventions could lead toHIV and/or syphilis elimination, the estimates of the HIV and syphilis incidencerates were compared to the World Health Organization (WHO) threshold for dis-ease elimination as a public health concern (less than one new infection per 1000susceptible gbMSM).5.3.4 Sensitivity analysisWe estimated the univariate sensitivity coefficients for the HIV and syphilis inci-dence changes under the TasP, PrEP and Test & Treat syphilis scenarios for the topten parameters with the highest coefficients at the end of 2028. Using the sensi-143tivity coefficients, we measured the relative change in the HIV and syphilis inci-dence with respect to the relative change in our model parameters [117, 155]. Wedemonstrated the occurence of positive and negative coefficients with an increaseor decrease in a parameter. Positive and negative coefficients denote positive andinverse association respectively, with the magnitude denoting how sensitive thetarget variable (HIV and syphilis incidence) is to changes in each parameter.In addition, the percent change in the number of cumulative HIV incident casesand syphilis incident cases with respect to Status Quo scenario from 2019 to 2028was estimated. Based on the available data and literature, we considered lower andhigher values for the parameters with the most uncertainty. Every aspect of thesensitivity and uncertainty analyses were performed using the scientific computinglibraries in PythonTM version 2.7.6.Table 5.1: Scenarios for the interventions examined in the studyInterventions StatusQuoLow Sce-narioMediumScenarioHighScenarioPrEP use (number of susceptiblegbMSM enrolled)4000 5000 7000 10000Condom use (%) 65 70 75 80TasPTime from HIV Infection to Diagnosis 3.37 years 2 years 1 year 6 monthsTime from Diagnosis to ART Treat-ment4.61 months 3.0 months 45 days 21 daysTime Retained on ART Treatment 2.72 years 3.5 years 4.5 years 6.0 yearsTest & Treat SyphilisTime from Syphilis Infection to Treat-ment (mono-infected individuals)3.26 years 2 years 8 months 3 monthsTime from Syphilis Infection to Treat-ment (co-infected individuals)18.6 years 10 years 5 years 3 yearsgbMSM: gay, bisexual and other men who have sex with men; ART: antiretroviraltreatment; PrEP: pre-exposure prophylaxis; TasP: treatment as prevention.1445.4 Results5.4.1 Status QuoWhen we kept 4000 gbMSM on PrEP from 2019 to 2028 (Status Quo scenario),our model predicted the cumulative number of HIV incident cases, syphilis incidentcases and all-cause mortality cases among PLWH to be 1389, 8039, and 961 re-spectively (see details on Tables B.4, B.5,B.6 and B.7 in Appendix B). In 2028, themodel estimated the HIV and syphilis incidence rate per 1000 susceptible gbMSM,and the mortality rate per 1000 HIV-positive gbMSM to be 4.01, 24.68 (Figures 5.2and 5.3) and 17.65, respectively (Table B.8). In addition, the HIV prevalence wasestimated to be 6432 at the end of 2028 (see details on Tables B.4, B.5,B.6 and B.7in Appendix B).5.4.2 TasPTables B.4 and B.7 show the impact of TasP interventions on the model outcomesat the end of 10 years. From the combination of all aspects of TasP, our modelpredicted that from low to high scenarios, the cumulative number of HIV in-cident cases, the cumulative number of syphilis incident cases and the 10 yearcumulative number of mortality cases among PLWH were between 842− 203(547− 1186 averted cases; 39%− 85% decrease from Status Quo (Figure 5.4)),7874− 7539 (165− 499 averted cases; 2%− 6% decrease from Status Quo), and741− 494 (220− 467 averted cases; 23%− 49% decrease from Status Quo (Fig-ure 5.4), respectively. The model estimated that in 2028, the HIV and syphilisincidence rates per 1000 susceptible gbMSM, and the mortality rate per 1000 HIV-positive gbMSM from low to high scenarios would be 1.97−0.2 (51%−95% de-crease from Status Quo), 23.39−21.76 (5%−12% decrease from Status Quo), and13.15−8.91 (25%−50% decrease from Status Quo), respectively (see Table B.8).In addition, after 10 years, the number of PLWH from low to high scenarios wasestimated to be between 6021− 5536 (6%− 14% decrease from Status Quo) (seeTable B.4). Of all TasP interventions, improving the time from HIV diagnosis totreatment seems to have the largest impact on the averted number of the cumula-tive HIV incident cases, HIV incidence rate and the mortality rate among PLWH145(see Table B.4). The combined time from syphilis infection to treatment, and timeto HIV diagnosis seems to have the largest impact on the averted number of thecumulative syphilis incident cases and incidence rate (see Table B.4). Of all indi-vidual combination of interventions, only TasP significantly reduced the mortalitycases among PLWH (Figures 5.4, 5.2 and 5.3).5.4.3 PrEPWe evaluated the impact of having 5000, 7000 and 10000 for low, medium and highPrEP uptake compared to 4000 uptake under the Status Quo scenario (Table B.6).When 10000 individuals are on PrEP, the model estimated the cumulative numberof HIV incident cases and the cumulative number of syphilis incident cases to be1172 (16% decrease from Status Quo), and 8403 (5% increase from Status Quo).In addition, the model estimated the HIV and syphilis incidence rate in 2028 to be3.26 (19% decrease from Status Quo) and 25.94 (5% increase from Status Quo)per 1000 susceptible gbMSM respectively. It is noticeable that enrolling 10000individuals on PrEP increased the syphilis incidence rate (see details in Figures 5.4,5.2 and 5.3, and on Table B.8).5.4.4 Condom useWe evaluated the impact of having 70%, 75% and 80% for low, medium and highcondom use compared to 65% condom use under the Status Quo scenario (TableB.6). When 80% of susceptible gbMSM use condom, the model estimated thecumulative number of HIV incident cases and the cumulative number of syphilisincident cases to be 1026 (363 averted cases; 26% decrease from Status Quo),and 5739 cases (2300 averted cases; 29% decrease from Status Quo). In addition,the model estimated the HIV and syphilis incidence rate in 2028 to be 2.63 (34%decrease from Status Quo) and 15.43 (37% decrease from Status Quo) per 1000susceptible gbMSM, respectively (see details in Figures 5.4, 5.2 and 5.3, and onTable B.8).1465.4.5 Test & Treat syphilisWe evaluated the combined effect of all syphilis interventions scenarios amongmono and co-infected individuals (see details on Table B.5) at the end of 10 years.Our model predicted that from low to high scenario, the cumulative number ofHIV incident cases, the cumulative number of syphilis incident cases and the 10year cumulative number of mortality cases among PLWH are between 1278−1010(111− 378 averted cases; 8%− 27% decrease from Status Quo), 6078− 2048(1961−5991 averted cases; 24%−75% decrease from Status Quo), and 928−858(33− 103 averted cases; 3%− 11% decrease from Status Quo), respectively. Themodel estimated that in 2028, the HIV and syphilis incidence rate per 1000 sus-ceptible gbMSM, and the mortality rate per 1000 HIV-positive gbMSM from lowto high scenario to be between 3.17− 2.04 (21%− 49% decrease from StatusQuo), 14.27− 1.25 (42%− 95% decrease from Status Quo), and 16.66− 15.26(6%−14% decrease from Status Quo), respectively (see Figures 5.4, 5.2 and 5.3,and Table B.8). Of all the combined syphilis interventions, improving the timefrom syphilis infection to treatment among co-infected individuals on ART seemsto have the largest impact on the averted number of the cumulative syphilis incidentcases and syphilis incidence rate.5.4.6 Combining two interventionsThe impact of combining two interventions ((1) TasP and PrEP, (2) TasP and Test& Treat Syphilis, (3) PrEP and Test & Treat Syphilis, (4) Condom use and Test &Treat Syphilis, and (5) TasP and Condom use) was evaluated in comparison to theStatus Quo scenario (see Table B.7). The combination of TasP and PrEP (mediumscenario), gave a higher reduction in the cumulative HIV incident cases that is 73%(HIV incidence rate as low as 0.61) when compared to TasP alone that is 71% (HIVincidence rate as low as 0.66) (Figures 5.4, 5.2). The combination of PrEP withTest & Treat Syphilis (high scenario), gave a lower reduction in the cumulativesyphilis incident cases that is 74% (syphilis incidence rate given as 1.31) whencompared to Test & Treat Syphilis alone that is 75% (syphilis incidence rate givenas 1.25) (Figures 5.4, 5.3).1475.4.7 Combining three interventionsThe impact of combining three interventions ((1) TasP, Test & Treat Syphilis andPrEP (2) TasP, Test & Treat Syphilis and condom use) was evaluated in comparisonto the Status Quo scenario (see Table B.7). The combination of two interventionswith PrEP (medium scenario) produced a 74% reduction in the cumulative HIVincident cases (HIV incidence rate given as 0.46) but lower when compared to con-dom use that is 76% (HIV incidence rate as low as 0.41) (Figures 5.4, 5.2). Sim-ilarly, the combination of two interventions with PrEP (high scenario) produceda 73% reduction in the cumulative syphilis incident cases (syphilis incidence rategiven as 1.12) but lower when compared to condom use (high scenario) that is 80%(syphilis incidence rate given as 0.86) (Figures 5.4, 5.3).5.4.8 Conditions for the elimination of the HIV and syphilisepidemicsWe based the condition for the elimination of HIV and syphilis epidemics on theWHO threshold for disease elimination as a public health concern (< 1 new in-fection per 1000 susceptible gbMSM). The combination of PrEP with or withoutany other HIV interventions gave a much lower HIV incidence rate compared tointerventions without PrEP (see Figure 5.2 and Table B.8). On the contrary, thecombination of PrEP with any other syphilis interventions gave a much highersyphilis incidence rate compared to interventions without PrEP (see Figure 5.3 andTable B.8). For example, increasing the number of gbMSM on PrEP to 10000 gavethe syphilis incidence rate of 25.94 per 1000 susceptible gbMSM (5% increasefrom Status Quo), and the elimination of syphilis epidemic was not achieved.Based on WHO threshold, further optimizing TasP to at least the medium sce-nario (i.e., 1 year from infection to diagnosis, 3 months from diagnosis to treat-ment, and 4.5 years to be continually retained on treament) will lead to the HIVdisease elimination with the HIV incidence rate as low as 0.66 per 1000 susceptiblegbMSM (medium scenario). In addition, the combination of TasP (medium sce-nario) with any other interventions to at least the medium level (i.e., TasP, providePrEP to at least 7000 individuals, 75% of condom use, or improve syphilis testingand treatment at the medium level) could also achieve this threshold level for HIV148and with lower HIV incidence rate when compared to TasP alone (see Figure 5.2).Improving syphilis testing & treatment (high scenario) and 80% of condom usewill lead to syphilis disease elimination, with syphilis incidence rate as low as 0.85per 1000 susceptible gbMSM (see Table B.8). Similarly, syphilis infection couldalso be eliminated by combining TasP (high scenario), improving syphilis interven-tion (high scenario) and having 80% of condom use with syphilis incidence rate aslow as 0.86 per 1000 susceptible gbMSM (see Figure 5.3 and Table B.8. It may notbe possible to eliminate both diseases with increasing the number of gbMSM onPrEP. Simultaneous elimination of both diseases was achieved by combining TasP(high scenario), improving syphilis testing and treatment (high scenario), and 80%of condom use with HIV and syphilis incidence rate as low as 0.11 and 0.86 per1000 susceptible gbMSM respectively (see Figures 5.2 and 5.3 and Table B.8).149Figure 5.2: HIV incidence rate under different intervention scenarios in comparison to the WHO threshold for diseaseelimination as a public health concern at the end of 2028.WHO: World Health Organization; GBMSM: gay, bisexual and other men who have sex with men; TasP: treat-ment as prevention; PrEP: pre-exposure prophylaxis; Test & Treat: test and treat syphilis.150Figure 5.3: Syphilis incidence rate under different intervention scenarios in comparison to the WHO threshold for dis-ease elimination as a public health concern at the end of 2028.WHO: World Health Organization; GBMSM: gay, bisexual and other men who have sex with men; TasP: treat-ment as prevention; PrEP: pre-exposure prophylaxis; Test & Treat: test and treat syphilis.151(a)(b)Figure 5.4: Results for the reduction in HIV point prevalence, the cumula-tive number of HIV incident cases, and all-cause mortality cases amongPLWH (first row), and the cumulative number of syphilis incident cases(second row) among gbMSM living with HIV after 10 years of TasP,PrEP, condom use, and Test & Treat (syphilis) interventions1525.4.9 Sensitivity analysesFirst, we estimated the univariate sensitivity coefficients on the cumulative num-ber of HIV and syphilis incident cases under PrEP, TasP and syphilis interventionsscenarios at the end of 2028, and showed the top paramters with the highest coef-ficients in Figures 5.5, 5.6 and 5.7. For all interventions, the most sensitivity pa-rameters on the cumulative number of HIV incident cases (first row of Figures 5.5,5.6 and 5.7) were the proportion and effectiveness of condom use among gbMSM,and the HIV transmission rate. Similarly, for all interventions, the most sensitiv-ity parameters on the cumulative number of syphilis incident cases (second rowof Figures 5.5, 5.6 and 5.7) were the proportion and effectiveness of condom useamong gbMSM, the syphilis transmission rate and higher risk of syphilis transmis-sion among co-infected individuals on ART. The impact of each parameter on thecumulative number of HIV incident cases decreased as we moved from the low tothe high scenario of these interventions. Similarly, the impact of each parameteron the cumulative number of syphilis incident cases decreased as we moved fromthe low to the high scenario of Test and Treat syphilis, and TasP interventions.Conversely, the impact of each parameter on the cumulative number of syphilisincident cases increased as we moved from the low to the high scenario of PrEPintervention.In addition, we estimated the percent change in the cumulative number of HIVand syphilis incident cases at the end of 2028 (Figure 5.8), with respect to ourmodel predictions based on the Status Quo scenario, for the parameters with themost uncertainty based on the available literature and data. The proportion ofgbMSM using condoms and the transmission rate of HIV were the assumptionsthat mostly influenced both the changes in the cumulative number of HIV andsyphilis incident cases.153(a)(b)Figure 5.5: Results of the sensitivity analyses for the top ten parameters withthe highest sensitivity coefficients based on the scenarios for PrEP use.Row 1: cumulative number of HIV incident cases at the end of 2028;Row 2: cumulative number of syphilis incident cases at the end of 2028;PrEP: pre-exposure prophylaxis154(a)(b)Figure 5.6: Results of the sensitivity analyses for the top ten parameters withthe highest sensitivity coefficients based on the scenarios for TasP.Row 1: cumulative number of HIV incident cases at the end of 2028;Row 2: cumulative number of syphilis incident cases at the end of 2028;TasP: HIV treatment as prevention155(a)(b)Figure 5.7: Results of the sensitivity analyses for the top ten parameters withthe highest sensitivity coefficients based on the scenarios for Test &Treat. Row 1: cumulative number of HIV incident cases at the end of2028; Row 2: cumulative number of syphilis incident cases at the endof 2028; Test & Treat: test and treat syphilis156(a)(b)Figure 5.8: Results of the sensitivity analysis for the parameters with the mostuncertainty based on the available literature. Row 1: Percent changein the cumulative number of HIV incident cases in comparison to theStatus Quo at the end of 2028; Row 2: Percent change in the cumulativenumber of syphilis incident cases in comparison to the Status Quo at theend of 20281575.5 DiscussionWe have developed a co-interaction model of HIV and syphilis infections in agbMSM population using ordinary differential equations. Several models havebeen designed to show the transmission of HIV infection among gbMSM withoutexplicitly modeling its synergy with other diseases, especially with other sexuallytransmitted diseases like syphilis. Our study shows that the most successful strat-egy to reach HIV elimination entailed the optimization of all aspects of TasP, orcombination of TasP with any other interventions (impoving syphilis testing andtreatment, putting at least 7000 individuals on PrEP and increasing the proportionof gbMSM using condoms to at least 75%) to at least the medium level. We showthat the elimination of HIV epidemic is better when different interventions arecombined with PrEP than without PrEP. But the combination of any interventionswith condoms seems best.The most successful strategy to reach syphilis elimination entailed improvingsyphilis testing and treatment (high scenario), simultaneous increase in the propor-tion of gbMSM using condoms to 80% and/or optimizing all aspects of TasP. Weshowed that the elimination of syphilis epidemic is worse when different interven-tions are combined with PrEP, and seems better with condom use. The simultane-ous elimination of both diseases may never be achieved by putting more and morepeople on PrEP. The most successful strategy to reach simultaneous elimination ofHIV and syphilis epidemics entailed optimizing all aspects of TasP (high scenario),improving syphilis testing and treatment (high scenario), and the simultaneous in-crease in the proportion of gbMSM using condoms to 80%; reduction in HIV andsyphilis incidence rate by as much as 89% and 80% respectively.In BC, we have the highest number of new HIV and syphilis infections amonggbMSM [69, 70], with other studies in a similar settings also reporting a highincidence of both infections in this group [116, 167]. The reason for the increasein syphilis incidence when the PrEP uptake increases is unknown. This increasecould be due to several factors like decrease in condom use, since several studiessuggest that condom use is decreasing among gbMSM [58, 147, 148]Our results seem consistent with previous modeling studies on effectiveness ofPrEP, condom use, TasP and syphilis interventions [34, 93, 109, 116, 174]. But the158question still remain on how to give PrEP to susceptible gbMSM in BC and not in-crease the epidemic of syphilis. Our model showed that having 10000 PrEP uptakecould lead to 5% increase in syphilis incidence. According to a recent study in asimilar setting to ours, we expect that testing individuals on PrEP more and moreoften for syphilis and other STIs could avert this 5% increase in syphilis incidence.Hence, the reason to develop cost-effectiveness strategies for the distribution ofPrEP and to prioritize testing for STIs among gbMSM in BC.The findings in this work have important implications and are subject to somelimitations. First, we modeled the HIV and syphilis epidemics among gbMSM as aclosed system and did not explicitly model migration in BC. Instead, we assumed aconstant recruitment of gbMSM population due to unavailability of data. Second,we acknowledge that the model is susceptible to some degree of uncertainty sincemost parameters were based on published data and literature. Both of our sensi-tivity analyses showed how our outcomes could be influenced by the parameters.Third, based on availability of data and the complexity of the synergy that occursbetween the HIV and syphilis infections, we ssumed a homogeneous mixing pat-tern in the co-interaction model, whereas studying the impact of interventions inheterogeneous mixing settings would be a great addition to knowledge on differentintervention strategies among different gbMSM populations. Fourth, we did notstratify our model by risk and age (i.e., high and low risk, young and old) knownto significantly modify the risk of HIV and syphilis transmission [1, 96, 157, 166].Therefore we could not assess the effect of PrEP and different degrees of assor-tativity between different risk and age groups, and the complexities that exist inthe sexual networks of these individuals. Even though, these conditions are some-what important factors in the co-interaction of both diseases, modeling their effectswould greatly increase the complexity of the model, which is beyond the scope ofthis study, and is considered as one of the future works we intend to explore. Es-timating the effective reproduction number for each of the interventions and theirscenarios will also be a good way to extend our work in order to know the inter-ventions and scenarios that give the effective reproduction number less than one.It is worth noting that the success of all the aforementioned interventions is de-pendent on identifying gbMSM at risk of HIV and/or syphilis infection, and thosealready living with HIV but not diagnosed and on ART treatment. Across Canada,159the Momentum and Engage studies are the major two cohorts that can better under-stand the key barriers to accessing PrEP, TasP and trends in STIs among gbMSM[35, 77, 106, 107, 163].5.6 ConclusionThe use of a co-interaction model to combine the synergies between HIV andsyphilis infections enables us to evaluate the impact of HIV and syphilis interven-tions on both epidemics. Our model can be applied to other settings similar to BCand would be useful to measure how succesful interventions like the ones consid-ered in this study could impact the epidemics of both HIV and syphilis infections.Based on our work, we propose that it is important to develop some public healthpolicies to address the co-interaction of HIV and syphilis infections, in BC and inother parts of the world. Given the HIV-syphilis synergy and according to WHOthreshold for the elimination of both diseases as a public health concern, healthcareproviders should ensure to further optimize TasP, increase the provision of PrEP,improve syphilis testing and treatment particularly among gbMSM using PrEP, en-courage and promote consistent condom use particularly among those who may notbe eligible to receive PrEP, and initiate immediate treatment of gbMSM and theirsexual contacts when necessary. Successful implementation of these proceedingsis crucial to addressing the HIV and syphilis epidemics among gbMSM.160Chapter 6Conclusions and future directionsThe thesis studied the epidemic and endemic scenarios of infectious diseases trans-mission and prevention using mathematical models. Specifically, epidemic of dis-eases transmitted through indirect transmission pathways using the basic repro-duction number and the final size relation were carried out. In addition, the syn-demic dynamics of endemic diseases, particularly sexually transmitted diseasessuch as HIV and syphilis were addressed, with different intervention scenarios in-vestigated.In Chapter 1, we introduced some basic background of infectious diseases inhumans and simple appproach to modeling them mathematically. This chaper high-lights the impact of mathematical modeling, modeling used for public health andsome challenges.In Chapter 2, the main contribution was the developed indirect transmissionSIRP model which considered the effect of age of infection and variable pathogenshedding rates on the basic reproduction number and the final size relation, inan heterogeneous mixing environment. Following a Langrangian approach, wekept track of individual’s place of residence at all times, and showed how move-ment within and between patches could impact the final epidemic size. This studydemonstrated that: (1) the patches behave separately (independently) with no mo-bility; (2) the patches have the same level of disease prevalence with equal mobility(symmetric movement); and (3) patch 1 has highest disease prevalence with highmobility.161In Chapter 3, we extended the work done in Chapter 2 to incorporate how dif-fusion impacts the epidemic of airborne infections. A class of coupled PDE-ODEsystem was formulated and proposed as a novel approach, with human populationsmodeled using ODE, and with the movement and amount of pathogen in the airmodeled with PDE. Matched asymptotic analysis was used to reduce the coupledPDE-ODE system into an ODE. It was shown analyticaly and numerically how thechange in diffusion rate could increase or decrease the basic reproduction num-ber and the final epidemic size. The effect of the location of the patches was alsoexplored analytically and numerically. This study demonstrated that epidemic de-creases with increase in the diffusion rate, and with human populations confined ina region. The model suggested that, in order to reduce the complexity in using aPDE model, the proposed ODE system which approximates the PDE system in thelimit where diffusion is large, could be used to assess the effect of diffusion.In Chapter 4, the main contribution was the developed co-interaction modelof HIV and syphilis used to demonstrate how one disease influences the other.We analytically and numerically established the necessary conditions under whichdisease-free and endemic equilibria are asymptotically stable using the effectivereproduction number. This theoretical study showed that, HIV impacts syphilisepidemic negatively and vice versa. Using parameters from published articles, weshowed one possibility of diseases eradication. The results further showed theimportance of taking other STIs into consideration when trying to eradicate HIVepidemic.In Chapter 5, the model in Chapter 4 was extended to investigate the impactof interventions on HIV and syphilis epidemics, and how the combination of dif-ferent interventions could be used to reduce or eliminate both epidemics amonggay, bisexual, and other men who have sex with men in BC. Based on the WHOthreshold for disease elimination as a public health concern, our results suggestedthat both diseases could be eliminated if we further optimize TasP, syphilis testingand treatment, PrEP uptake and further promote condom use. In addition, the studydemonstrated that the synergy that exists between HIV and other STIs, particularlysyphilis should be taken into consideration in order to reach the elimination tar-gets. This result is consistent with the ongoing HIV and STIs testing and treatmentrecommendation among PrEP users.162For simplification, several aspects of the underlying problems were not mod-eled. For future directions, it may be possible to incorporate more realistic fea-tures into the models considered in this thesis. Some of the additional featuresinclude, but are not limited to, direct transmission pathways with/without saturatedincidence, heterogeneity, networking, model stratification by risk, age and gender,since all these are important factors to consider in the modeling of infectious dis-eases. By incorporating direct transmission in the epidemic models, and using alagrangian approach to explore human mobility (Chapters 2 & 3), it may be possi-ble to significantly improve the model outcomes.Similarly, in order to have a stronger impact of interventions, it may be neces-sary to stratify our models (Chapters 4 & 5) by different risk level and age group inthe future (depending on the availablity of data). By distributing PrEP by risk leveland age, we may be able to significantly reduce, avert and probably eliminate theepidemics within a shorter period of time. In addition, deriving conclusions (Chap-ter 5) based on the parameter set from a single best fit can be unreliable, hence theneed to assess model behavior and outcomes for all important parameters to anacceptable extent in the future (e.g., the use of Bayesian approach). 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COMMUNICATIONS ON PURE AND APPLIED ANALYSIS,7(3):659, 2008. → page 55183Appendix ASupporting information for theco-interactional model used inChapter 4Here we show the Lemmas and Proofs of some results presented in the text.A.1 The proof of Lemma 4.4.2In this proof, we show that the disease free equilibrium E0S for syphilis-only modelis a global attractor.Proof.Let f∞ = lim supt→∞f (t) and f∞ = lim inft→∞ f (t).It follows from I′S(t) = βSISSNS− (µ+σ1)IS, and SNS ≤ 1,ISNS≤ 1 thatI′S(t)≤ βSIS− (µ+σ1)IS ≤ (µ+σ1)(βS(µ+σ1)−1)IS (A.1)≤ (µ+σ1)(ReS−1)ISUsing the approach of [164], if we choose a sequence tn→∞, such that IS(tn)→ I∞S ,and I′S(tn)→ 0, we then have 0≤ (ReS−1)I∞S .184Since(βS(µ+σ1))< 1, we have I∞S = 0, and therefore limt→∞ IS(t) = 0.Similarly, choose a sequence t1n →∞, such that S(t1n)→ S∞. Then using IS(t)→0 as t→ ∞ in the first equation in (4.5), we have0≤Π−µS∞, (A.2)and Equation (A.2) gives S∞ = S∞ = Πµ . QED.A.2 The proof of Lemma 4.4.4We show here that the endemic equilibrium point E∗S is globally asymptoticallystable.Proof. We can similarly show as in Lemma 4.4.2 above, that the unique endemicequilibrium point E∗S is globally asymptotically stable forReS > 1.Recall that NS =Πµas t → ∞, and substituting S = NS− IS = Πµ − IS in (4.5),we haveI′S = λS(Πµ− IS)− (µ+σ1)IS. (A.3)We have from Dulac’s multiplier1ISthat∂∂ IS{βSISISΠ/µ(Πµ− IS)− (µ+σ1)}=−µβSΠ=−βSNS< 0 (A.4)Therefore, by Dulac’s criterion, we conclude that there are no periodic orbits in ΞS.Since ΞS is positively invariant, and the endemic equilibrium E∗S exists wheneverReS > 1, then we can infer from the Poincare-Bendixson Theorem in [98] thatfor all time t, all solutions of the limiting system emanating in ΞS remain in ΞS. Inaddition, the absence of periodic orbits in ΞS indicates that the endemic equilibriumof syphilis-only model is globally asymptotically stable wheneverReS > 1.185A.3 The proof of Lemma (4.5.4)In this proof, we show that the endemic equilibrium point of HIV-only model (4.34)is globally asymptotically stable in ΞH \ΞH0.Proof. We can similarly show as in Lemma 4.5.2 above, that the unique endemicequilibrium point for this simple case exits if and only ifReH0 > 1.Recall that NH =Πµas t→ ∞, and substitutingS = NH −UH −AH −TH = Πµ −UH −AH −THin (4.34), we haveU ′H = λH(Πµ−UH −AH −TH)− (µ+α1)UH .A′H = α1UH +ν1TH − (µ+ρ2)AH .T ′H = ρ2AH − (µ+ν1)TH .We have from Dulac’s multiplier1UHAHTHthat∂∂UH{βH(UH +κ1AH)UHAHTHΠ/µ(Πµ−UH −AH −TH)− (µ+α1)AHTH}+∂∂AH{α1UH +ν1TH − (µ+ρ2)AHUHAHTH}+∂∂TH{ρ2AH − (µ+ν1)THUHAHTH}= −{βHAHTHΠ/µ+βHκ1U2HTH− βHκ1AHU2HTHΠ/µ− βHκ1THU2HTHΠ/µ+α1UHUHA2HTH+ν1THUHA2HTH+ρ2AHUHAHT 2H}= −{βHAHTHΠ/µ+βHκ1U2HTH(1− AHΠ/µ− THΠ/µ)+α1UHUHA2HTH+ν1THUHA2HTH+ρ2AHUHAHT 2H}< 0 since AH +TH ≤Π/µ in ΞH .Therefore, by Dulac’s criterion, we conclude that there are no periodic orbits inΞH \ΞH0. Since ΞH is positively invariant, and the endemic equilibrium exists186whenever ReH0 > 1, we can infer from the Poincare-Bendixson Theorem in [98]that for all time t, all solutions of the limiting system emanating in ΞH remainin ΞH . In addition, the absence of periodic orbits in ΞH indicates that the endemicequilibrium of HIV-only model is globally asymptotically stable wheneverReH0 >1.A.4 The proof of Lemma (4.6.2)We show here that the disease-free equilibrium point E0 is globally asymptoticallystable.Proof. The proof is based on a Comparison Theorem in [108] and by following theapproach in [23, 29, 87, 126, 128, 159]. Equations of the infected compartments insystem (4.4) can be written asdISdtdUHdtdAHdtdTHdtdUSHdtdASHdtdTSHdt=(B−C)ISUHAHTHUSHASHTSH−(1− SN)BISUHAHTHUSHASHTSH, where B and C are given by187B=βS 0 0 0 φ1βS φ2βS φ3βS0 βH κ1βH 0 κ2βH κ3βH 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0, C=A1 0 0 0 0 0 00 A2 0 0 0 0 00 −α1 A3 −ν1 −σ2 −σ3 00 0 −ρ2 A4 0 0 −σ40 0 0 0 A5 0 00 0 0 0 0 A6 −ν20 0 0 0 0 −ρ1 A7.We have A1 = µ+σ1,A2 = µ+dUH +α1,A3 = µ+dAH +ρ2,A4 = µ+ν1,A5 = µ+dUSH +σ2,A6 = µ+dASH +σ3+ρ1,A7 = µ+ν2+σ4.188For all t ≥ 0 and since S≤ N in Ξ, we havedISdtdUHdtdAHdtdTHdtdUSHdtdASHdtdTSHdt≤ (B−C)ISUHAHTHUSHASHTSH(A.5)Since the eigenvalues of the matrix B−C all have negative real parts, then thelinearized differential inequality system (A.5) is stable whenever Re < 1. Con-sequently, (IS,UH ,AH ,TH ,USH ,ASH ,TSH) → (0,0,0,0,0,0,0) as t → ∞. It fol-lows by a Comparison Theorem as in [108] that (IS,UH ,AH ,TH ,USH ,ASH ,TSH)→(0,0,0,0,0,0,0) as t → ∞ and evaluating system (4.4) at IS = UH = AH = TH =USH = ASH = TSH = 0 gives S→ S0 forRe < 1. Therefore, the DFE E0 is GAS forRe < 1.189Appendix BSupporting information for theco-interactional model used inChapter 5B.1 Mathematical modelB.1.1 Model equationsThe full HIV-syphilis model, which represents the health states of the gbMSMpopulation in BC, consists of 8 ordinary differential equations. Each equation de-notes the following compartments: susceptible S, mono-infected with syphilis IS,mono-infected with HIV and unaware UH , mono-infected with HIV and aware AH ,mono-infected with HIV and on treatment TH , co-infected with HIV and unwareUSH , co-infected with HIV and aware ASH , co-infected with HIV and on treatmentTSH . Parameters explanation and assumptions are in this Appendix Table 1. The190differential equations used in the model are given by,dSdt= Π+σ1IS− (µ+λS+λH)S,dISdt= λSS− (µ+σ1+ γλH)IS,dUHdt= λHS− (µ+dUH +α1+η1λS)UH ,dAHdt= α1UH +σ2USH +σ3ASH +ν1TH − (µ+dAH +η2λS+ρ2)AH ,dTHdt= ρ2AH +σ4TSH − (µ+η3λS+ν1)TH , (B.1)dUSHdt= γλHIS+η1λSUH − (µ+dUSH +σ2)USH ,dASHdt= η2λSAH +ν2TSH − (µ+dASH +σ3+ρ1)ASH ,dTSHdt= ρ1ASH +η3λSTH − (µ+ν2+σ4)TSH ,The recruitment rate Π, the total population N, the syphilis force of infection λSand the HIV force of infection λH are given byN(t) = S(t)+ IS(t)+UH(t)+AH(t)+TH(t)+USH(t)+ASH(t)+TSH(t)Π = µN+dUHUH +dAHAH +dUSHUSH +dASHASHλS = βS(1− εξ )((1−ψ)+ψRP)(IS+φ1USH +φ2ASH +φ3TSH)NλH = βH(1− εξ )((1−ψ)+(1−θ)ψRP)(UH +κ1AH +κ2USH +κ3ASH)N .The parameters φ1, φ2, φ3 represent the relative infectivity of individuals in the co-infected and unaware USH , co-infected and aware ASH , and in the co-infected andon treatment TSH , respectively, in comparison to individuals mono-infected withsyphilis IS. The parameters κ1, κ2, κ3 represent the relative infectivity of individ-uals in the mono-infected and aware AH , co-infected and unaware USH , and inthe co-infected and aware ASH , respectively, in comparison to individuals mono-infected with HIV and unaware UH . The parameters ε, ξ respectively representthe proportion and effectiveness of condom use, the parameters ψ, θ respectivelyrepresent the proportion and effectiveness of PrEP, and the parameter RP repre-191sents the relative risk associated with using PrEP. It is worth mentioning that themodel assumes screening for syphilis among individuals co-infected and unawarecan help assess a person’s risk for getting HIV [54].B.1.2 Model parameters and variablesTable B.1: Model parameters and variables.Abbreviations: PrEP: Pre-Exposure Prophylasis, gbMSM: Gay, bisex-ual and other men who have sex with men, STIs: Sexually TransmittedInfections, ART: Antiretroviral TherapyParameter Explanation and Values ReferenceS Susceptible individualsIS Individuals mono-infected with syphilisUH Individuals mono-infected with HIV and un-awareAH Individuals mono-infected with HIV and awareTH Individuals mono-infected with HIV and ontreatmentUSH Individuals co-infected with HIV and unawareASH Individuals co-infected with HIV and awareTSH Individuals co-infected with HIV and on HIVtreatmentµ Natural mortality rate of gbMSM, estimated tobe 0.84 deaths per 100 person-yearsestimated from[32, 33, 45]dUH Mortality rate due to unaware HIV infectionin mono-infected individuals, estimated to be 4deaths per 100 person-yearsadjusted basedon [153]dAH Mortality rate due to aware HIV infection inmono-infected individuals, estimated to be 4deaths per 100 person-yearsadjusted basedon [153]dUSH Mortality rate due to unaware HIV infection inco-infected individuals, estimated to be 4 deathsper 100 person-yearsadjusted basedon [153]192dASH Mortality rate due to aware HIV infection in co-infected individuals, estimated to be 4 deaths per100 person-yearsadjusted basedon [153]βS Transmission rate for syphilis infection. This isproduct of the effective contact rate for syphilisinfection and the probability of syphilis transmis-sion per contact. The fitted value is 0.234FittedβH Transmission rate for HIV infection. This isproduct of the effective contact rate for HIV in-fection and probability of HIV transmission percontact, The fitted value is 0.195Fitted1/σ1 Time from syphilis infection to treatment for in-dividuals monoinfected with syphilis. The fittedvalue for Status Quo is 3.26 years. Interventionscenarios: 2 years, 8 months, and 3 monthsFitted1/σ2 Time from syphilis infection to treatment, andtime to HIV diagnosis for individuals coinfectedwith HIV and unaware. The fitted value for Sta-tus Quo is 18.6 years. Intervention scenarios: 10years, 5 years, and 3 yearsFitted1/σ3 Time from syphilis infection to treatment for in-dividuals coinfected with syphilis and aware. Es-timated to be 18.6 years for Status Quo. Inter-vention scenarios: 10 years, 5 years, and 3 yearsAssumed basedon σ21/σ4 Time from syphilis infection to treatment for in-dividuals coinfected with HIV and on HIV treat-ment. Estimated to be 18.6 years for Status Quo.Intervention scenarios: 10 years, 5 years, and 3yearsAssumed basedon σ21/α1 Time from HIV infection to HIV diagnosis. Thefitted value for Status Quo is 3.37 years. Inter-vention scenarios: 2 years, 1 year, and 6 monthsFitted1931/ρ2 Time to ART treatment for monoinfected indi-viduals. The fitted value for Status Quo is 4.61months. Intervention scenarios: 3 months, 45days, and 21 daysFitted1/ρ1 Time to ART treatment for co-infected individu-als. Estimated to be 4.61 months for Status Quo.Intervention scenarios: 3 months, 45 days, and21 daysAssumed basedon ρ21/ν1,1/ν2 Time retained on ART before dropping out formono and coinfected individuals respectively.Estimated to be 2.72,2.72 years respectively forStatus Quo. Intervention scenarios: 3.5 years,4.5 years, and 6.0 yearsadjusted basedon [172]γ Higher risk of HIV acquisition for people livingwith syphilis. Estimated to be 2.5adjusted basedon [15, 55, 59,64, 133]φ1,φ2,φ3 Higher risk of syphilis transmision for coin-fected individuals compared with individualsmonoinfected with syphilis. Estimated to be2.867, 2.867, 2.867[92]κ1 Higher risk of HIV transmision for individualsmonoinfected with HIV and aware, comparedwith individuals monoinfected with HIV and un-aware. Estimated to be 1.0Assumedκ2,κ3 Higher risk of HIV transmision for individu-als coinfected with HIV and unaware, comparedwith individuals monoinfected with HIV and un-aware. Estimated to be 2, 2adjusted basedon [5, 133]N Total number of gbMSM population. Estimatedto be 50900[62, 68, 69]η1,η2,η3 Higher risk of syphilis acquisition for people liv-ing with HIV. Estimated to be 2.237,2.237,2.237[66]194ε Proportion of susceptible gbMSM populationthat regularly use condoms. Estimated to be 65%adjusted basedon [168]ξ Effectiveness of condoms among HIV-negativegbMSM. Estimated to be 70%adjusted basedon [160]ψ Proportion of susceptible gbMSM population us-ing PrEP. At any time t, the parameter ψ was cal-culated as the ratio of the number of PrEP and thesize of the susceptible gbMSM populationSee SectionB.1.3 for detailθ Effectiveness of PrEP. Estimated to be 86%. Sen-sitivity scenarios: 92%, 96%, and 100%[119, 122, 156]RP Relative risk associated with using PrEP. Esti-mated to be 1.24adjusted basedon [167]B.1.3 Model assumptions about PrEP uptake in BCThe uptake of Pre-exposure prophylaxis (PrEP) since its approval Health Canadain 2016 was very low, and PrEP became fully subsidized in BC in January, 2018 forpeople at risk of HIV infection [61]. The Drug Treatment Program (DTP) of the BCCentre for Excellence in HIV/AIDS (BC-CfE) accounted for about 3225 suscepti-ble gbMSM on PrEP at the end of 2018, and currently close to 4000 in September2019 (unpublished data). Therefore, in our model, we assumed that during the pe-riod 2017−2018, the number of gbMSM on PrEP at any time was described by asigmoid function from 0 to 3225, achieving the half uptake in the middle of 2017.For the period 2018− 2019, we assumed that the number of gbMSM on PrEP atany time was also described by a sigmoid function from 3225 to 4000 individuals,achieving the half uptake in the middle of 2018. For the intervention starting fromthe end of 2019, we assumed that the ratio of the number of PrEP given was keptconstant as it was in 2019.B.1.4 Model calibrationFor better description of the HIV and syphilis epidemics among the gbMSM pop-ulation in BC, we included the estimates of the number of people living with HIV195(PLWH), the number of annual new HIV infections from Public Health Agency ofCanada (PHAC) [62, 139] (details in Table B.2), and the estimates of the annualHIV and syphilis diagnoses (Table B.3).Table B.2: Estimates of the number of PLWH and the number of annual newHIV infections from PHAC.Abbreviation: PLWH: People living with HIVVariables 2011 estimates 2014 estimates 2016 estimatesPLWH 5840 6013 6070[4940, 6750] [5080, 6950] [5130, 7010]Annual new HIV infections 142 141 147PLWH [110, 200] [100, 200] [90, 260]Table B.3: Published data on cases of HIV and syphilis infections from BC-CFE and BCCDC respectively.Abbreviation: BCCfE: British Columbia Centre for Excellence forHIV/AIDS; BCCDC: British Columbia Centre for Disease ControlVariables Years ReferencesAnnual HIV diagnoses 2011−2018 [71–75]Annual syphilis diagnoses 2012−2017 [63, 67, 70]The model calibration was based on the available data from Tables B.2 andB.3 A unique set of parameter values in Table B.1 that minimizes the differencebetween simulation and the target values was kept if the model simulation fittedto the following data: (1) the PHAC estimates of the number of PLWH in 2011,2014 and 2016; (2) the annual number of new HIV infections in 2011, 2014 and2016; (3) the annual number of HIV diagnoses during the period 2011−2018; (4)the annual number of syphilis diagnoses during the period 2012−2017. We ran asimulation inside a Nelder-Mead simplex algorithm to determine the optimal valueof unknown parameters, assuming a tolerance of 10−3 and with all other parametersfixed [130].196Figure B.1: PHAC estimates of PLWH and annual new HIV infections (blueerror bars) and model simulations (solid red line) during the period2011−2018Figure B.2: Annual HIV and syphilis diagnoses (blue points) and model sim-ulations (solid red line) during the period 2011−2018197For the implementation of the analysis, we used the optimization package in theSCIPY library in PythonTM version 2.7.6. We allow the model to run from 2011to 2028 adjusting for changes in the number of susceptible gbMSM on PrEP indifferent year (details in Section B.1.3). It is worth mentioning that throughout allsimulated scenarios, the total gbMSM population (N = 50900) was kept constant.The algorithm fitted the unkown parameters, and the solution that minimized thesum-squared relative residuals of the model’s estimates and the data (the solutionthat best fit the available data) are shown in Figures B.1 and B.2.B.2 Model outcomesThe model outcomes in 2028 under TasP, PrEP, Test and Treat syphilis, condomuse and different combinations of intervention scenarios are summarized in Tables(B.4), (B.5), (B.6), (B.8)198Table B.4: Model outcomes under TasP interventionsHIV Prevalence Cumulative HIV Incident cases Cumulative mortality cases, PLWH Cumulative syphilis Incident casesScenarios NReduction %ChangefromStatusQuoNAvertedcases%ChangefromStatusQuoNAvertedcases%ChangefromStatusQuoNAvertedcases%ChangefromStatusQuoStatus-Quoin 2028 6432 - - 1389 - - 961 - - 8039 - -Decrease Time from HIV Infection to DiagnosisLow 6358 74 -1% 1284 105 -8% 914 47 -5% 8029 10 0%Medium 6282 150 -2% 1176 213 -15% 866 96 -10% 8017 22 0%High 6236 196 -3% 1110 279 -20% 836 125 -13% 8008 31 0%Decrease Time from HIV Diagnosis to ART Initiation among mono-infected individualsLow 6332 100 -2% 1238 151 -11% 888 73 -8% 8014 25 0%Medium 6224 208 -3% 1076 313 -23% 809 152 -16% 7986 53 -1%High 6163 269 -4% 984 405 -29% 764 197 -20% 7970 69 -1%Decrease Time from HIV Diagnosis to ART Initiation among co-infected individualsLow 6328 104 -2% 1263 126 -9% 921 40 -4% 8048 -9 0%Medium 6213 219 -3% 1125 264 -19% 877 84 -9% 8058 -19 0%High 6146 286 -4% 1044 345 -25% 851 110 -11% 8064 -25 0%Increase Time Retained on ART among mono-infected individualsLow 6375 57 -1% 1303 86 -6% 919 42 -4% 8025 14 0%Medium 6328 103 -2% 1233 156 -11% 885 76 -8% 8014 25 0%High 6286 146 -2% 1170 219 -16% 854 107 -11% 8003 36 0%Increase Time Retained on ART among co-infected individualsLow 6367 64 -1% 1311 77 -6% 936 25 -3% 8045 -6 0%Medium 6315 117 -2% 1248 141 -10% 916 45 -5% 8049 -11 0%High 6267 165 -3% 1190 199 -14% 898 63 -7% 8054 -15 0%Decrease Time from syphilis Infection to treatment, test for HIV (co-infected & unaware)Low 6372 60 -1% 1323 66 -5% 942 19 -2% 7890 149 -2%199Medium 6281 150 -2% 1221 168 -12% 913 48 -5% 7659 380 -5%High 6209 223 -3% 1139 250 -18% 888 73 -8% 7470 569 -7%Combined TasP (Combination of the previous HIV interventions)Low 6021 411 -6% 842 547 -39% 741 220 -23% 7874 165 -2%Medium 5692 740 -12% 407 982 -71% 571 390 -41% 7675 364 -5%High 5536 896 -14% 203 1186 -85% 494 467 -49% 7539 499 -6%200Table B.5: Model outcomes under Test & Treat syphilis interventionsHIV Prevalence Cumulative HIV Incident cases Cumulative mortality cases, PLWH Cumulative syphilis Incident casesScenarios NReduction %ChangefromStatusQuoNAvertedcases%ChangefromStatusQuoNAvertedcases%ChangefromStatusQuoNAvertedcases%ChangefromStatusQuoStatus-Quoin 2028 6432 - - 1389 - - 961 - - 8039 - -Decrease Time from syphilis Infection to treatment among mono-infected individualsLow 6355 77 -1% 1375 14 -1% 950 11 -1% 7117 922 -11%Medium 6251 180 -3% 1342 46 -3% 933 28 -3% 5788 2251 -28%High 6217 215 -3% 1327 62 -4% 926 35 -4% 5326 2712 -34%Decrease Time from syphilis Infection to treatment, test for HIV (co-infected & unaware)Low 6372 60 -1% 1323 66 -5% 942 19 -2% 7890 149 -2%Medium 6281 150 -2% 1221 168 -12% 913 48 -5% 7659 380 -5%High 6209 223 -3% 1139 250 -18% 888 73 -8% 7470 569 -7%Decrease Time from syphilis Infection to treatment among individuals co-infected and awareLow 6423 9 0% 1382 7 -1% 960 1 0% 7919 120 -1%Medium 6405 27 0% 1368 21 -2% 958 3 0% 7679 360 -4%High 6384 47 -1% 1352 37 -3% 955 6 -1% 7393 646 -8%Decrease Time from syphilis Infection to treatment among individuals co-infected and on ARTLow 6391 41 -1% 1361 28 -2% 956 5 0% 7228 811 -10%Medium 6325 107 -2% 1314 75 -5% 948 13 -1% 5898 2141 -27%High 6267 164 -3% 1271 118 -8% 941 20 -2% 4751 3288 -41%Combined Test & Treat (Combination of the previous syphilis interventions)Low 6260 171 -3% 1278 111 -8% 928 33 -3% 6078 1961 -24%Medium 6060 372 -6% 1117 272 -20% 884 77 -8% 3400 4639 -58%High 5960 471 -7% 1010 378 -27% 858 103 -11% 2048 5991 -75%201Table B.6: Model outcomes under PrEP and condom use interventionsHIV Prevalence Cumulative HIV Incident cases Cumulative mortality cases, PLWH Cumulative syphilis Incident casesScenarios NReduction %ChangefromStatusQuoNAvertedcases%ChangefromStatusQuoNAvertedcases%ChangefromStatusQuoNAvertedcases%ChangefromStatusQuoStatus-Quoin 2028 6432 - - 1389 - - 961 - - 8039 - -Increase Condom UseLow 6291 141 -2% 1260 129 -9% 940 21 -2% 7219 820 -10%Medium 6161 271 -4% 1139 250 -18% 921 40 -4% 6454 1585 -20%High 6042 390 -6% 1026 363 -26% 903 58 -6% 5739 2300 -29%Increase PrEP UseLow 6398 34 -1% 1352 37 -3% 957 4 0% 8099 -60 1%Medium 6332 100 -2% 1279 110 -8% 948 13 -1% 8220 -181 2%High 6235 197 -3% 1172 217 -16% 935 26 -3% 8403 -364 5%202Table B.7: Model outcomes under the combination of different interventionsHIV Prevalence Cumulative HIV Incident cases Cumulative mortality cases, PLWH Cumulative syphilis Incident casesScenarios NReduction %ChangefromStatusQuoNAvertedcases%ChangefromStatusQuoNAvertedcases%ChangefromStatusQuoNAvertedcases%ChangefromStatusQuoStatus-Quoin 2028 6432 - - 1389 - - 961 - - 8039 - -TasP and Test & Treat (Combination of HIV and syphilis interventions)Low 5955 476 -7% 817 572 -41% 733 228 -24% 6065 1973 -25%Medium 5632 799 -12% 383 1006 -72% 563 398 -41% 3413 4626 -58%High 5507 925 -14% 194 1195 -86% 490 471 -49% 2063 5976 -74%Increase Combined TasP and Condom UseLow 5940 492 -8% 768 621 -45% 731 230 -24% 7082 957 -12%Medium 5622 810 -13% 343 1046 -75% 564 397 -41% 6190 1849 -23%High 5488 943 -15% 159 1230 -89% 489 472 -49% 5422 2617 -33%Increase Combined Test & Treat, and Condom UseLow 6147 285 -4% 1164 225 -16% 911 50 -5% 5474 2565 -32%Medium 5895 536 -8% 933 456 -33% 859 102 -11% 2815 5224 -65%High 5752 680 -11% 772 617 -44% 826 135 -14% 1570 6469 -80%Increase Combined TasP, Test & Treat, and Condom UseLow 5885 547 -9% 747 642 -46% 723 238 -25% 5470 2569 -32%Medium 5579 853 -13% 327 1062 -76% 558 403 -42% 2832 5207 -65%High 5470 962 -15% 154 1235 -89% 487 474 -49% 1584 6455 -80%Increase Combined TasP and PrEP UseLow 6002 430 -7% 821 568 -41% 739 222 -23% 7935 104 -1%Medium 5668 764 -12% 378 1011 -73% 569 392 -41% 7858 181 -2%High 5516 916 -14% 176 1213 -87% 492 469 -49% 7907 132 -2%Increase Combined Test & Treat, and PrEP UseLow 6233 199 -3% 1246 143 -10% 924 37 -4% 6122 1917 -24%203Medium 5997 435 -7% 1037 352 -25% 875 86 -9% 3468 4571 -57%High 5851 580 -9% 869 520 -37% 843 118 -12% 2124 5915 -74%Increase Combined TasP, Test & Treat, and PrEP UseLow 5939 493 -8% 797 592 -43% 731 230 -24% 6110 1928 -24%Medium 5614 818 -13% 359 1030 -74% 562 399 -42% 3481 4558 -57%High 5492 940 -15% 170 1219 -88% 489 472 -49% 2140 5899 -73%204Table B.8: HIV prevalence and incidence rates, syphilis incidence rates, mor-tality rate among PLWH under different interventionsScenarios HIV point pe-valence (%)HIV Incidencerate (per 100 sus-ceptible gbMSM)Mortality rate(per 1000 PLWH)Syphilis Incidencerate (per 1000 sus-ceptible gbMSM)Status-Quoin 202812.64 4.01 17.65 24.68Decrease Time from HIV Infection to DiagnosisLow 12.49 3.54 16.53 24.54Medium 12.34 3.15 15.61 24.4High 12.25 2.95 15.14 24.3Decrease Time from HIV Diagnosis to ART Initiation among mono-infected individualsLow 12.44 3.5 16.33 24.42Medium 12.23 2.98 14.91 24.15High 12.11 2.69 14.11 23.99Decrease Time from HIV Diagnosis to ART Initiation among co-infected individualsLow 12.43 3.54 16.87 24.62Medium 12.21 3.03 15.99 24.55High 12.07 2.74 15.47 24.51Increase Time Retained on ART among mono-infected individualsLow 12.52 3.72 16.89 24.53Medium 12.43 3.48 16.26 24.42High 12.35 3.27 15.69 24.31Increase Time Retained on ART among co-infected individualsLow 12.51 3.71 17.17 24.64Medium 12.41 3.47 16.76 24.61205High 12.31 3.26 16.38 24.59Decrease Time from syphilis Infection to treatment among mono-infected individualsLow 12.48 3.71 17.31 19.73Medium 12.28 3.33 16.86 14.67High 12.21 3.22 16.72 13.3Decrease Time from syphilis Infection to treatment, test for HIV (co-infected & unaware)Low 12.52 3.64 17.09 23.8Medium 12.34 3.16 16.32 22.61High 12.2 2.85 15.82 21.8Decrease Time from syphilis Infection to treatment among individuals co-infected and awareLow 12.62 3.96 17.62 23.97Medium 12.58 3.88 17.56 22.58High 12.54 3.78 17.5 20.98Decrease Time from syphilis Infection to treatment among individuals co-infected and on ARTLow 12.56 3.79 17.5 20.08Medium 12.43 3.45 17.24 13.38High 12.31 3.19 17.02 8.7Combined TasP (Combination of the previous HIV interventions)Low 11.83 1.97 13.15 23.39Medium 11.18 0.66 10.04 22.27High 10.88 0.2 8.91 21.76Combined Test & Treat (Combination of the previous syphilis interventions)Low 12.3 3.17 16.66 14.27Medium 11.91 2.35 15.66 4.47High 11.71 2.04 15.26 1.25Increase Condom Use206Low 12.36 3.49 17.29 21.2Medium 12.1 3.03 16.95 18.13High 11.87 2.63 16.64 15.43TasP and Test & Treat (Combination of HIV and syphilis interventions)Low 11.7 1.71 12.86 14.02Medium 11.07 0.49 9.87 4.43High 10.82 0.14 8.87 1.26Increase Combined TasP and Condom UseLow 11.67 1.73 12.97 20.21Medium 11.04 0.53 9.96 16.67High 10.78 0.15 8.89 14.02Increase Combined Test & Treat, and Condom UseLow 12.08 2.8 16.4 12.3Medium 11.58 1.89 15.34 3.41High 11.3 1.5 14.86 0.85Increase Combined TasP, Test & Treat, and Condom UseLow 11.56 1.52 12.73 12.14Medium 10.96 0.41 9.83 3.43High 10.75 0.11 8.86 0.86Increase PrEP UseLow 12.57 3.88 17.58 24.89Medium 12.44 3.62 17.43 25.31High 12.25 3.26 17.22 25.94Increase Combined TasP and PrEP UseLow 11.79 1.91 13.11 23.61Medium 11.14 0.61 10.02 22.98207High 10.84 0.17 8.91 23.2Increase Combined Test & Treat, and PrEP UseLow 12.24 3.07 16.6 14.39Medium 11.78 2.15 15.53 4.58High 11.5 1.71 15.03 1.31Increase Combined TasP, Test & Treat, and PrEP UseLow 11.67 1.66 12.83 14.15Medium 11.03 0.46 9.86 4.55High 10.79 0.13 8.87 1.32208
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The study of epidemic and endemic diseases using mathematical models David, Jummy Funke 2020
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Title | The study of epidemic and endemic diseases using mathematical models |
Creator |
David, Jummy Funke |
Publisher | University of British Columbia |
Date Issued | 2020 |
Description | Mathematical models used in epidemiology provides a comprehensive understanding of disease transmission channels and they provide recommendations for methods of control. This thesis uses different mathematical models (direct and indirect transmission models) to understand and analyze different infectious diseases dynamics and possible prevention and/or elimination strategies. As a first step in this research, an age of infection model with heterogeneous mixing and indirect transmission was considered. The simplest form of SIRP epidemic model was introduced and served as a basis for other models. Most mathematical results in this chapter were based on the basic reproduction number and the final size relation. The epidemic model was further extended to incorporate the effect of diffusion using a coupled PDE-ODE system. We proposed a novel approach to modelling air-transmitted diseases using a reduced ODE system, and showed how the reduced ODE system approximates the coupled PDE-ODE system. A deterministic compartmental model of the co-interaction of HIV and infectious syphilis transmission among gay, bisexual and other men who have sex with men (gbMSM) was developed and used to examine the impact of syphilis infection on the HIV epidemic, and vice versa. Analytical expressions for the reproduction number and necessary conditions under which disease-free and endemic equilibria are asymptotically stable were established. Numerical simulations were performed and used to support the analytical results. Finally, the co-interaction model was modified to assess the impact of combining different HIV and syphilis interventions on HIV incidence, HIV prevalence, syphilis incidence and all-cause mortality among gbMSM in British Columbia from 2019 to 2028. Plausible strategies for the elimination of both diseases were evaluated. According to our model predictions and based on the World Health Organization (WHO) threshold for disease elimination as a public health concern, we suggested the most effective strategies to eliminate the HIV and syphilis epidemics over a 10-year intervention period. The results of the research suggest diverse ways in which infectious diseases can be modelled, and possible ways to improve the health of individuals and reduce the overall disease burden, ultimately resulting in improved epidemic control. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2020-02-12 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0388610 |
URI | http://hdl.handle.net/2429/73523 |
Degree |
Doctor of Philosophy - PhD |
Program |
Interdisciplinary Studies |
Affiliation |
Graduate and Postdoctoral Studies |
Degree Grantor | University of British Columbia |
GraduationDate | 2020-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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