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Mathematical epidemiology of HIV/AIDS and tuberculosis co-infection David, Jummy Funke 2015

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Mathematical Epidemiology of HIV/AIDS andTuberculosis Co-infectionbyJummy Funke DavidB.Sc., Ladoke Akintola University of Technology, 2011M.Sc., AIMS-Stellenbosch University, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mathematics)The University of British Columbia(Vancouver)August 2015c© Jummy Funke David, 2015AbstractThe project deals with the analysis of a general dynamical model for the spread ofHIV/AIDS and tuberculosis Co-infection. We capture in the model the dynamicsof HIV/AIDS infected individuals and investigate their impacts in the progressionof tuberculosis with and without TB treatment. It is shown that TB-only model andHIV-only model have locally asymptotically stable disease-free equilibrium whenthe basic reproduction number is less than unity and a unique endemic equilibriumexists when the basic reproduction number is greater than unity. We analyze thefull HIV/AIDS-TB coinfection model and incorporate treatment strategy for theexposed and active forms of TB. The stability of equilibria is derived through theuse of Van den Driessche method of generational matrix and Routh Harwitz sta-bility criterion. Numerical simulations are provided to justify the analytical resultsand to investigate the effect of change of certain parameters on the co-infection.Sensitivity analysis shows that reducing the most sensitive parameters β1 and β2could help to lower the basic reproduction number and thereby reducing the rate ofinfection. From the study, we conclude that treating latent and active forms of TBreduce the rate of infection, reduce the rate of progression of individuals to AIDSstage and lowers co-infection.iiPrefaceThis thesis is original, unpublished, independent work by the author, J. David.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 HIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Stages of HIV infection . . . . . . . . . . . . . . . . . . 21.2 Tuberculosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 HIV and TB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.1 Impact of HIV on TB . . . . . . . . . . . . . . . . . . . . 51.3.2 Impact of TB on HIV . . . . . . . . . . . . . . . . . . . . 51.3.3 Treatment of HIV and TB . . . . . . . . . . . . . . . . . 52 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Mathematical Analysis of the Model . . . . . . . . . . . . . . . . . . 103.1 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Model assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 13iv3.3 Model analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.1 HIV sub-model . . . . . . . . . . . . . . . . . . . . . . . 193.3.5 TB sub-model . . . . . . . . . . . . . . . . . . . . . . . 273.3.9 Analysis of the full model . . . . . . . . . . . . . . . . . 374 Numerical Simulations and Sensitivity Analysis . . . . . . . . . . . . 474.1 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . 484.2 Numerical simulations for different values of β1,β2 and discussionof results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58vList of TablesTable 3.1 Model variables, parameters and their descriptions . . . . . . . 16Table 4.1 Parameter values and their sources . . . . . . . . . . . . . . . 47Table 4.2 Initial values for different values of β1 and β2. . . . . . . . . . 48Table 4.3 Sensitivity analysis of RH . . . . . . . . . . . . . . . . . . . . 49Table 4.4 Sensitivity analysis of RT . . . . . . . . . . . . . . . . . . . . 49viList of FiguresFigure 3.1 The model diagram shows how susceptible individuals are in-fected with TB and HIV. We see from the diagram the trans-mission dynamics of the two infections. The population is di-vided into nine epidemiological classes: Susceptible (yS), Ex-posed TB (yE), Infectious TB (yI), HIV-positive (ySI), HIV in-fectious and TB Exposed (yEI), Infectious with both TB andHIV (yII), AIDS individuals (yA), treated individuals exposedor infected with TB (yT ), dually infected individuals with TBtreatment only (yT I). Classes (yEI), (yII), (yT I) and (yA) repre-sent the co-infection of HIV/AIDS and TB. . . . . . . . . . . 11Figure 4.1 The left panel shows the plot of the stability analysis of the dis-ease free equilibrium at β1 = 2.5 and β2 = 0.03 while the rightpanel shows the plot of the stability analysis of the endemicequilibrium at β1 = 5.2 and β2 = 0.3 . . . . . . . . . . . . . . 50Figure 4.2 Graphs showing the susceptibles yS, exposed to TB yE , infec-tious to TB yI , the HIV positive ySI , HIV positive exposed toTB yEI , HIV positive suffering from TB yII , those sufferingfrom AIDS yA, singly infected treated of TB yT and dually in-fected treated of TB yT I . These are simulations in differentcompartments with β1 = 5.2 and β2 = 0.3 . . . . . . . . . . . 51viiFigure 4.3 The left panel shows the plot of the stability of TB free equilib-rium E∗H at β1 = 0.4 and β2 = 0.8 while the right panel showsthe plot of the stability of HIV free equilibrium E∗T at β1 = 5.2and β2 = 0.03 . . . . . . . . . . . . . . . . . . . . . . . . . . 52Figure 4.4 The behaviour of the model for the period of 15 years with andwithout TB treatment γ1 and γ2 at β1 = 7,β2 = 0.6 and withdisease induced death. The left panel shows the impact of TBtreatment (γ1 = 1 and γ2 = 2) on the model, while the rightpanel shows the behaviour of the model without TB treatment(γ1 = 0 and γ2 = 0). . . . . . . . . . . . . . . . . . . . . . . . 53Figure 4.5 Graphs showing the behaviour of the model to changes in ini-tial conditions. The plot of the stability of equilibria E0T , E0H ,E∗T and E∗H . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Figure 4.6 Graphs showing different values of the basic reproduction num-bers RT and RH at different values of β1 and β2 respectively,with and without TB treatment . . . . . . . . . . . . . . . . . 55viiiAcknowledgmentsThank you JESUS the author and the finisher of my faith. I give my praise to theAlmighty God, who has brought me this far. I would like to express my gratitudeto my able supervisor Professor Fred Brauer who never got tired of me. I appreci-ate him for being a father, for his constructive criticism, his invaluable advice andunreserved material support. My special thanks to Professor Leah Keshet, Profes-sor Daniel Coombs and Professor Cindy Greenwood for making my stay at UBCinteresting and enjoyable. I appreciate UBC mathematics department, IAM mem-bers, Mahtbio group and all lecturers in maths for adding values to my life andgiving me a different perspective of science. I would not forget to appreciate myparents, siblings and friends for their encouragements. My special thanks goes tomy mother and my only sister for being supportive in all I do. My appreciationwould not be complete without giving a special thanks to my husband, my hero,the only man who rocks my world. Thanks my TREASURE for your challengesand encouragements. I dedicate this paper to God Almighty. Thank you Jesus forbeing the source of my strength. God bless you all.ixChapter 1IntroductionThe advent of AIDS made the relationship between HIV and Tuberculosis (TB) apublic health concern and the co-infection of TB and HIV exist when individualsare HIV positive and are either exposed or active to TB. Getting infected with TBbacteria is not automatic for HIV infected individuals unless in contact with infec-tious TB individuals. Similarly, individuals infectious to TB do not automaticallyget infected with HIV unless in contact with HIV positive individuals.1.1 HIVHIV, the Human Immunodeficiency Virus is the agent responsible for the acquiredimmunodeficiency syndrome (AIDS) [11, 19]. HIV is a retrovirus that destroys thehuman immune system by infecting the CD4+T cells. HIV virus attacks the normalfunctioning of the immune system to produce more HIV viruses [9]. Populationof viruses increase as a result of decrease in the CD4+T cells count in the body.Individuals infected with HIV can stay with this infection for years and thereforemay not be ill or show symptoms of HIV infection [11].HIV is not known to be transmitted casually but through sexual intercourse[19]. HIV is secreted in the body fluid and therefore can be transmitted in a largeramount from semen, pre-seminal fluid, vaginal secretion, and breast feeding [25].The modes of transmitting HIV includes but not limited to unprotected sexual con-tacts and non-sexual contacts (injection needles for drug use, direct blood contact,1vertical transmission) [9]. HIV is capable of suppressing the immune system whennot treated, it makes individuals to be more susceptible to other infections and willat the end be diagnosed of AIDS [1, 9].1.1.1 Stages of HIV infectionResearch shows that HIV transmission rate is not consistent and therefore the trans-mission rate differs from different stages of infection. Progression of individualsfrom one stage of infection to another changes the degree of infectiousness, andtherefore the transmission rate of viruses is grouped according to the stages of pro-gression of the infection in an infected individuals [4].HIV infection is classified into four different stages: primary infection, clini-cally asymptomatic stage, symptomatic HIV infection, and progression from HIVto AIDS.• Primary HIV infection: This stage carries on for a few weeks and it of-ten occur with a short flu-like illness. The immune system in the primarystage begins to react to the virus and therefore produce HIV antibodies andcytotoxic lymphocytes as a result of a large volume of HIV in the peripheralblood. This process of developing detectable antibodies due to HIV infec-tion is called seroconversion and an incomplete seroconversion may lead toa negative HIV antibody test [4].• Clinically asymptomatic stage: The stage carries on typically for up toten years, when infected individual shows no symptoms but there may beswollen glands. HIV level in the peripheral blood reduces to very low levelsand HIV antibody test becomes positive since people remain infectious. Re-search had shown that when a viral load test (normally use to measure HIVRNA) is carried out, HIV is found to be active, but very active in the lymphnodes [4].• Symptomatic HIV infection: At this stage, HIV mutates and becomes morestronger which leads to the destruction of the CD4+T cells. Immune systembecomes damaged and the body fails to keep replacing the lost CD4+T cells.We can also refer to this stage as the pre-AIDS stage where individuals begin2to show symptoms of the infection. HIV-infected individuals on treatmentmay remain clinically asymptomatic as a result of treatment, and untreatedindividuals may continue to experience a deteriorating immune system withHIV symptoms getting worse.• Progression from HIV to AIDS: At this stage, individuals develop otheropportunistic infections (for example; Tuberculosis) and they are eventuallydiagnosed of AIDS as a result of the critical damage caused to the immunesystem [4]. Individuals with AIDS or other infections will have a reducedCD4+T cell usually from around 1000mm−3 to 200mm−3 or below [15].Symptoms of full-blown AIDS appears due to an increased viral load andthis stage can be classified as the progression from HIV to AIDS stage. Themovement from HIV stage to AIDS stage usually lasts for 8−10 years dur-ing which some individuals may progress much more rapidly while othersprogress slowly [4, 15].1.2 TuberculosisTuberculosis (TB), one of the leading cause of death from a single infectious agent,is an airborne transmitted disease caused by the releasing of Mycobacterium tuber-culosis (M. tuberculosis) droplets in the air when an infectious individual coughs,sneezes [6, 12, 18] or talks [16]. TB as known to be one of the most wide spreadinfectious diseases caused by M. tuberculosis is one of the world’s leading causesof loss of life [14, 18]. Larger number of TB cases in the United states are causedby M.tuberculosis also called tubercle bacilli [8].M.tuberculosis can be found in airborne particles called droplet nuclei andthese particles can be suspended in the air for several hours depending on the envi-ronment. M.tuberculosis is of 1−5 microns in diameter [8].Individuals exposed to infectious people at all time for long period of time standa chance of being infected. According to World Health Organization (WHO), it isestimated that one third of the population of people in the world is infected withTB and as a result leads to 2−3millions death each year [6, 13], with about 8−9millions developing active TB [13]. Although, 90%− 95% percent of TB casesoccur in developing countries and about 1.8 million new cases of tuberculosis occur3per year in India [20].TB disease is a disease with slow dynamics and therefore the epidemics mustbe studied over a long window in time [6, 13, 17]. Understanding the differencebetween TB infection and the disease itself will be very important because an in-fected individual may not be infectious [9]. Infected individuals may remain in theasymptomatic stage throughout their entire lives (Exposed or Latent TB) [3, 24]since exposed periods range from months to decades depending on individuals in-fected and their immune system [6]. Exposed stage is a stage where individualsshow no symptoms of the infection and are not infectious. Individuals progressfrom exposed stage to infectious or active stage and TB disease is as a result of thisprogression [9].Although, the probability of progression towards active TB usually depends onage of infection [2]. Individuals in this active stage show symptoms of the infectionand are infectious [9].We can group Active TB (TB disease) as pulmonary and extra-pulmonarykinds. Pulmonary TB is often seen in adults and transmitted by M.tuberculosis,while extra-pulmonary is more frequent among women, children and in HIV in-fected individuals [16]. We can classified tuberculosis within the period of fiveyears after infection as primary TB, while tuberculosis after the period of five yearsfrom initial infection can be classified as secondary TB. Only about 5% of infectedindividuals develop primary TB within the period of five years if there are no otherconditions to accelerate the infection. Exogenous reinfection (exogenous reactiva-tion) which is classified under secondary TB is the aggravation of an old infection[24].Some of the factors that can affect the transmission of M.tuberculosis are thenumber, vitality, and exacerbation of organisms within sputum droplet nuclei, andmost significantly, time spent near an infectious individual. Transmission is alsoaffected by Socio-economic status, family size, crowding, malnutrition, and badhealth care. Mathematical model for tuberculosis have been a useful tool in as-sessing the spread of the infection, the epidemiological results and control of theinfection [6].41.3 HIV and TBHIV and TB accentuate the progression of each other [21]. Immunity deterioratesas HIV infection progresses and this makes infected individuals more susceptibleto any opportunistic infection. Treatments of both HIV and TB in many societieshave altered the co-infection of HIV and TB. One third of 39.5 million of peopleinfected with HIV are co-infected with TB and individuals infected with HIV areexpected to develop TB with probability of 0.5. Many TB and HIV co-infectedindividuals are at higher risk of developing active TB (30 to 50 times more) thanonly TB infected individuals [19]. There is always increase in the recurrence rateof TB in HIV infected individuals due to endogenous reactivation and exogenousre-infection [21].1.3.1 Impact of HIV on TBThe HIV epidemic has significantly impacted TB dynamic. Over the last 5 years,one third of the detected increases in active TB cases can be related to the HIVepidemic [19]. How HIV increases the incidence of new M.tuberculosis infectionshad been described by several reports. It aggravates the degree of infectiousness ofTB and re-activates latent M.tuberculosis[10].1.3.2 Impact of TB on HIVImmunocompromised HIV infected individuals are at higher risk of opportunisticinfections like TB, TB tends to increase the HIV replication rate. Replication ofHIV may lead to fast progression to AIDS [7, 19].1.3.3 Treatment of HIV and TBAlthough HIV-related TB can both be treated and prevented, co-infection is in-creasing in developing nations where resources are limited [16]. The combinedused of highly active antiretroviral therapy (HAART) and antituberculosis treat-ment is the present drug regimen to treat HIV-TB co-infection [9]. Interaction be-tween both drug regimens can sometimes cause complications, especially betweenprotease inhibitors (antituberculosis drug) and non-nucleotide reverse transcriptaseinhibitors (antiretroviral drug). Due to these complications, some protocols have5been made to treat HIV-TB co-infection: Treatment of TB comes before HIV in-fection treatment, HAART if already in use must be adjusted to implement the useof TB drug treatment, and treatment timing is important if an infected individualhas not started HAART treatment. Drug treatment for adverse drug reactions needto be tracked. Carrying out TB test on HIV infected individuals to know a goodtime to start prophylactic treatment is also a preventive measure [9]. Starting pro-phylactic treatment can reduce the risk of progression to active TB in HIV infectedand TB exposed individuals. Some developing countries still have no access tothese drugs and treatments. However, mathematical modelling of the transmissiondynamics of the coinfection has been in place due to public health concern [21].In this project, we look at the possible future effect of treating TB on the co-infection of HIV and TB using a mathematical model. Chapter 1 summarizes thebasic background information on HIV and TB, the rest of the thesis is thereforeorganized as follows:In chapter 2, we present our motivation and review some studies done onHIV/AIDS and TB and these will be used as a basis for the formulation of ourmodel. In chapter 3, a model for HIV/AIDS and TB co-infection is developed andallows the incorporation of both infections and TB treatment. Two sub-models ofthe full model are analysed and the full model is also analysed. Computation oftheir reproduction numbers and analysis are done. Chapter 4 highlights analyticalresults using selected numerical simulations. Sensitivity is conducted to identifythe most sensitive parameter(s).6Chapter 2Literature ReviewThree selected studies done either on TB, HIV, and HIV/TB would be reviewed inthis chapter and use them as basis for our study on HIV/AIDS-TB co-infection.Roeger et al. [19] formulated a mathematical model and considered a simpledeterministic model that includes the co-infection of HIV/TB and TB treatment.The basic reproduction numbers of each of the diseases R1 (TB), R2 (HIV) andboth diseasesR = max{R1,R2} were found and the model was qualitatively anal-ysed. They obtained limited analytical results where they found the disease freeequilibrium for the full model to be locally asymptotically stable when R < 1and the disease free equilibrium point for TB-only model to be locally asymptoti-cally stable when R1 > 1 and R2 < 1. Analytical results showed that R1 < 1 andR2 > 1 may not give a stable HIV-only equilibrium and there is possibility of TBcoexisting with HIV when R2 > 1. Results of their numerical simulation showthat the increase in the rate at which TB progresses from latent to active form ofTB in individuals that are co-infected with both diseases contributed greatly to theprevalence of TB. Similarly, they were also able to show that increase in the rateat which HIV progresses from HIV to AIDS in co-infected individuals contributedto the prevalence of HIV and cause damped oscillations in the system. From theirsimulation, they found that it is possible to have co-infection of HIV and TB whenR1 < 1 and R2 > 1. Their model provided general insights into the effects of HIVinfection on TB and vice versa. Their numerical results suggested that investingmore in reducing the prevalence of HIV could be an effective way to reduce or7control the impact of TB. Results of their model were only based on local mathe-matical analysis, and their deterministic model needs to be adjusted to incorporatethe mode of HIV transmission and the treatment of latent and active forms of TB.We can therefore say that, the dynamics of the co-infection still need to be wellstudied mathematically and theoretically.Chowell-Puente et al. [9] developed an epidemiological model to analyse thedynamical interaction of HIV/AIDS and TB epidemic in South Africa among adultsbetween the age of 15 to 49. Their model was analysed to determine the level towhich the HIV epidemic aggravates the TB epidemic. They conducted sensitivityand uncertainty analysis to determine the effect of changing the parameter valueson the model and to know the parameter value(s) that is most sensitive to the basicreproduction (R0). Numerical simulations were also done to know the long termeffect of both epidemics. The exponential curve fit to the population data from1970 to 2005 was done and they numerically estimated the annual growth rate ofSouth Africa from curve fitting. The TB-free model was shown to have a globallystable disease free equilibrium when RHIV0 < 1 and a locally asymptotically stableendemic equilibrium whenRHIV0 > 1 while the HIV-free model was shown to havea locally stable disease free equilibrium when RT B0 < 1. The basic reproductionnumber R0 for the full model was determined by the max{RT B0 ,RHIV0 }. Theirresults showed that HIV-TB co-infection will eventually shift the declining trendof total TB cases in South Africa. Therefore, they suggested that treatment shouldbe focussed on HIV-positive individuals who are latently infected with TB becausethey are at a higher risk of progressing to active TB. Results of their model wereonly based on a particular geographical area (South Africa) and this estimationmay not be true in general.Hussaini [15] formulated a deterministic model which incorporates public healtheducation campaign as an intervention strategy for the prevention of HIV/AIDS.The model was analysed to know more about the epidemiological dynamics ofHIV/AIDS. The study investigated when the public health education program was100% effective. The global stability of the disease-free equilibrium of the modelwas done. The threshold analysis of the effective reproduction number showedthat we could have a positive, no, or harmful impact when public health educationcampaign is used depending on the value of impact factor (ϒ). Results of the nu-8merical simulations suggested that the universal strategy is more effective than anyother strategy in reducing new HIV cases and that the prospect of effective controlof HIV increases with increasing efficacy and coverage rate of the public healtheducation campaign.Ideas from these studies are used to formulate a general mathematical model toinvestigate implications of HIV/AIDS-TB co-infection and to show that increase inthe spread of TB infections have been associated with the spread of HIV infectionand that there would be a significant decrease in the co-infection cases if TB istreated.9Chapter 3Mathematical Analysis of theModel3.1 Model formulationMathematical model to study the dynamics of HIV/AIDS-TB co-infection is pre-sented in this chapter. The schematic diagram of the model is shown in figure 3.1.10Figure 3.1: The model diagram shows how susceptible individuals are in-fected with TB and HIV. We see from the diagram the transmissiondynamics of the two infections. The population is divided into nineepidemiological classes: Susceptible (yS), Exposed TB (yE), InfectiousTB (yI), HIV-positive (ySI), HIV infectious and TB Exposed (yEI), In-fectious with both TB and HIV (yII), AIDS individuals (yA), treatedindividuals exposed or infected with TB (yT ), dually infected individ-uals with TB treatment only (yT I). Classes (yEI), (yII), (yT I) and (yA)represent the co-infection of HIV/AIDS and TB.From figure 3.1, we have the susceptible class (yS) to be individuals with noinfection (no TB and no HIV infection). The model is structured such that sus-ceptible individuals can either be infected with TB by individuals in the epidemi-ological classes yI or yII and with HIV by individuals in the classes ySI , yEI , yII oryT I . Note that individuals with HIV can easily progress to yII or yA through TBinfection and reactivation of the exposed TB infection and we can say that HIVincreases the rate of progression of TB infection.The Exposed class (yE) are individuals with TB infection but not infectious11and they can be infected with HIV by individuals in the classes ySI , yEI , yII or yT I .The infectious TB class (yI) are individuals with TB disease and are infectious.They can also be infected with HIV by individuals in the classes ySI , yEI , yII oryT I . The HIV-positive class ySI are HIV-positive individuals and can be infectedwith TB infection by individuals in the classes yI and yII . The yEI class are HIV-positive and exposed TB individuals. The class yII are HIV-positive and infectiousTB individuals. The class yT are treated infected TB individuals. The class (yT I)are treated dually infected individuals. Finally, the class yA are AIDS individualswith either AIDS or TB and AIDS.Parameters pi denote the rate at which individuals are recruited into the suscep-tible class and µ denote the rate at which they can die naturally. Since TB can bespread by individuals in the classes yI and yII to susceptible individuals and HIVcan be transmitted by individuals in the classes ySI , yEI , yII or yT I to susceptibleindividuals. Thus susceptible individuals infected with TB enter the class yE atthe rate β1(yI +ρ1yII)Nwhile susceptible individuals infected with HIV enter theclass ySI at the rate β2(ySI +η1yEI +η2yII +η3yT I)Nwhere β1 and β2 are transmis-sion rates per year for TB and HIV respectively, the quantity(yI +ρ1yII)Nis theprobability of having contact with an individual infected with TB out of the to-tal population and(ySI +η1yEI +η2yII +η3yT I)Nis the risk measure involved withHIV levels in the population.Parameter ρ1 > 1 indicates that individuals with HIV and infectious TB aremore infectious to pass TB disease compared with individuals with only infectiousTB. The rates 1 ≤ η1 ≤ η2 ≤ η3 indicate that becoming infected with HIV fromindividuals with HIV-positive and TB disease or HIV-positive and exposed TB orHIV-positive and treated TB is easier than from just HIV-positive individuals.Exposed TB individuals can also move to HIV-positive class (yEI) at the rateβ2(ySI +η1yEI +η2yII +η3yT I)Nor progress to TB infectious class (yI) at the rate k,where β2 is the HIV transmission rate. Infectious TB individuals (yI) and exposedTB individuals (yE) can be treated and move to TB treated class (yT ) at the rate γ2and γ1 respectively. Infectious TB individuals can die from TB at the rate α , orenter the class (yII ) at the rate β2(ySI +η1yEI +η2yII +η3yT I)N. Individuals who12are HIV-positive in the class ySI can be infected with TB and they will enter theHIV-positive and exposed TB class (yEI ) at the rate β1(yI +ρ1yII)Nand die due toHIV at the rate ν . Otherwise, they progress to the AIDS class (yA) at the rate δ1.Individuals in the class (yEI) progress to the class (yII) at the rate kH or die dueto HIV at the rate ν . Figure 3.1 shows that Individuals in the class (yEI) can alsoprogress to AIDS class (yA) or treated class (yT I) at the rate δ2 and γ1 respectively.Individuals in the class (yII) progress to the treated class (yT I) and AIDS class (yA)at the rate γ2 and δ3 respectively. Individuals in the class (yII) die from both HIVand TB disease at the rate ν and α respectively. Individuals in the class (yA) die dueto both infections at the rate τ . Individuals in the class (yT ) can be reinfected byboth HIV and TB and enter the class (yE) and (yT I) at the rate β1(yI +ρ1yII)Nandβ2(ySI +η1yEI +η2yII +η3yT I)Nrespectively. Individuals in the class (yT I) can diedue to HIV at the rate ν or progress to the AIDS class (yA) at the rate δ4. Individualsin all the nine classes can die naturally at the rate µ .3.2 Model assumptions• We can have a model with a constant or varying population, so we assumesusceptible individuals are recruited into the population at a constant rate pi .• Since it is difficult to identify any symptoms clinically at the exposed levelof TB, we then assume that TB exposed individuals are not infectious andcan not transmit TB infection.• TB could be spread through different means as discussed in the first chap-ter. We therefore assume in the model that TB infection is spread betweeninfectious and susceptible individuals by airborne spread only.• We have different means through which HIV could be transmitted, but themodel assumes HIV is transmitted between infectious and susceptible indi-viduals neglecting the mode of transmission.• Since it is possible for dually infectious individuals to develop or not developAIDS in reality and in the presence of TB treatment. Hence, we assume thatIndividual infectious with both diseases may or may not develop AIDS.13• Since we know that individuals infected with TB can not fully recover, thenwe assume that individuals would not completely recover from TB but wouldbe exposed.• We ignore the treatment of HIV/AIDS since it is difficult to cure or eradicateit.• Since HIV treatment is not considered in this model, we then assume that itis possible for HIV-positive individuals to die due to HIV,• We know that most people infected with HIV at the initial stage may or maynot show any symptoms of it, we therefore assume that individuals can diedue to HIV but at a very low probability when not co-infected with TB.• Susceptible individuals get infected with HIV following contact with HIVinfected individuals at a rate λH and they acquire TB infection from individ-uals with active TB only at a rate λT .• Individuals in the class (yA) die due to either AIDS or TB and AIDS at thesame rate τ and assume it is difficult to identify the cause of deaths in thisclass.• Since the mode of transmission is neglected and it is possible for individualsin the class yA to spread TB or HIV. We therefore assume that individuals inthe class (yA) are too weak to transmit any disease or infect others outsidethe class.• We assume that individuals in the class yA can either remain in this class ordie. We assume they will not recover from this class because the immunesystem will not be strong enough to fight against infections.• Since it is possible for infectious TB individuals to transmit infection, wetherefore assume that treated TB individuals (yT ) may not transmit infectionsince they are on treatment, but could be reinfected since they would notfully recover.• Co-infected individuals on TB treatment may also not transmit TB infection,but can transmit HIV infection since they are not on HIV treatment.14• We assume that κH ≥ κ since it is easier to be TB infectious when one isco-infected with HIV, i.e. we assume it is faster to move from class yEI toclass yII than from the class yE to class yI .Based on the assumptions above and the model diagram, the model representingthe dynamics of HIV/AIDS and tuberculosis is given by a system of non-linear or-dinary differential equation 3.1, and table 3.1 gives the description of the variablesand parameters in the model.y˙S = pi−λT yS−λHyS−µyS,y˙E = λT yS−λHyE +λT yT − (µ+ k+ γ1)yE ,y˙I = kyE −λHyI− (µ+α+ γ2)yI,y˙SI = λHyS−λT ySI− (µ+ν+δ1)ySI,y˙EI = λHyE +λT ySI− (µ+ν+ kH +δ2 + γ1)yEI, (3.1)y˙II = kHyEI +λHyI− (µ+ γ2 +δ3 +α+ν)yII,y˙A = δ1ySI +δ2yEI +δ3yII− (µ+ τ)yA +δ4yT I,y˙T = γ1yE + γ2yI−λT yT −λHyT −µyT ,y˙T I = γ1yEI + γ2yII +λHyT − (µ+ν+δ4)yT I.whereλT =β1N(yI +ρ1yII)andλH =β2N(ySI +η1yEI +η2yII +η3yT I)The total population N(t) is given byN(t) = yS(t)+ yE(t)+ yI(t)+ ySI(t)+ yEI(t)+ yII(t)+ yA(t)+ yT (t)+ yT I(t)and it satisfiesdNdt= pi−µN−α(yI + yII)−ν(ySI + yEI + yII + yT I)− τyA15where pi is the recruitment rate, βi, i = 1,2 is the transmission rate, µ is the naturaldeath rate, γi, i = 1,2 is the progression rate from infected stage to treated stage,δi, i = 1,2,3,4 is the progression rate from HIV infected stage to AIDS stage, αand ν are disease induced death rates for TB and HIV respectively. Table 3.1 givesthe detailed definitions.Table 3.1: Model variables, parameters and their descriptionsVariable DescriptionyS(t) Susceptible individualsyE(t) Latent TB individualsyI(t) Infectious(Active) TB individualsySI(t) HIV-positive individualsyEI(t) HIV infectious and TB latent individualsyII(t) Individuals Infectious with both TB and HIVyA(t) Individuals with AIDSyT (t) Treated individuals with TByT I(t) Dually infected individuals treated of TBParameter Descriptionpi Recruitment rate of susceptible individualsβ1 probability of TB transmission to a susceptible per contact with an infectious TBindividualβ2 probability of HIV transmission to a susceptible per contact with an HIV individualµ natural death rateτ death rate due to AIDSν death rate due to HIVα death rate due to TBk rate of progression of yE to yIkH rate of progression of yEI to yIIδ1 rate of progression of ySI to yAδ2 rate of progression of yEI to yAδ3 rate of progression of yII to yAδ4 rate of progression of yT I to yAγ1 Treatment rate of latent TB individualsγ2 Treatment rate of infectious TB individualsρ1 coefficient of infectiousness of yII to transmit TB diseaseη1 coefficient of infectiousness of yEI to transmit HIV-positive diseaseη2 coefficient of infectiousness of yII to transmit HIV-positive diseaseη3 coefficient of infectiousness of yT I to transmit HIV-positive disease16We will study the dynamics of the system 3.1 based on biological considerationin the regionΘ={(yS + yE + yI + ySI + yEI + yII + yA + yT + yT I) ∈R9+ : N ≤piµ}, (3.2)which is positively invariant with respect to the model system 3.1. We need toshow that all variables and parameters of model system 3.1 are all positive for alltime since the model is for human populations.Lemma 3.2.1. The region R9+ is positive everywhere for model 3.1 which estab-lishes that our model does not predict negative values for the state variables at anyfuture time.Proof.Let t1 = sup{t > 0 : yS≥ 0,yE ≥ 0,yI ≥ 0,ySI ≥ 0,yEI ≥ 0,yII ≥ 0,yA≥ 0,yT ≥ 0,yT I ≥ 0∈ [0, t]}.From the first equation in the model 3.1, we havey˙S = pi−λT yS−λHyS−µySwhere λT =β1N(yI +ρ1yII) and λH =β2N(ySI +η1yEI +η2yII +η3yT I).⇒ y˙S +(λT +λH)yS +µyS = pi⇒ddt(yS(t)exp{µt +∫ t0(λT (ξ )+λH(ξ ))dξ})= piexp{µt +∫ t0(λT (ξ )+λH(ξ ))dξ}.Then we have,yS(t1)exp{µt1 +∫ t10(λT (ξ )+λH(ξ ))dξ}−yS(0)=∫ t10piexp{µϕ+∫ ϕ0(λT (ρ)+λH(ρ))dρ}dϕ.Hence,yS(t1) = yS(0)exp{−(µt1 +∫ t10(λT (ξ )+λH(ξ ))dξ)}+exp{−(µt1 +∫ t10(λT (ξ )+λH(ξ ))dξ)}×∫ t10piexp{µϕ+∫ ϕ0(λT (ρ)+λH(ρ))dρ}dϕ ≥ 0.17This can also be shown for other compartments.Lemma 3.2.2. Every solutions in Θ remain in Θ for all time.Proof. We know that the rate of change of the total population N(t) gotten byadding equations in 3.1 is given bydNdt= pi−µN−α(yI + yII)−ν(ySI + yEI + yII + yT I)− τyA.Considering initial conditions in R9+ and t ≥ 0, we havedNdt≤ pi−µN⇒ ddt(Neµt)≤ pieµt⇒ N(t)eµt −N(0)≤piµ (eµt −1)≤piµ eµt .And for t ≥ 0,N(t)≤ N(0)e−µt +piµ . (3.3)If (y∗S,y∗E ,y∗I ,y∗SI,y∗EI,y∗II,y∗A,y∗T ,y∗T I) is an Θ limit point of a region inR9+, such thatthere exist a subsequence ti→ ∞ andlimt→∞(yS(ti),yE(ti),yI(ti),ySI(ti),yEI(ti),yII(ti),yA(ti),yT (ti),yT I(ti))= (y∗S,y∗E ,y∗I ,y∗SI,y∗EI,y∗II,y∗A,y∗T ,y∗T I).Therefore, limt→∞N(ti) = N∗ = y∗S,y∗E ,y∗I ,y∗SI,y∗EI,y∗II,y∗A,y∗T ,y∗T I.If we evaluate t = ti at i→∞, we have N∗≤piµ and we can say that (y∗S,y∗E ,y∗I ,y∗SI,y∗EI,y∗II,y∗A,y∗T ,y∗T I)∈Θ.Thus, for initial values (yS(0),yE(0),yI(0),ySI(0),yEI(0),yII(0),yA(0),yT (0),yT I(0))∈R9+, the trajectory lies within Θ and we consider the model to be well posed math-ematically and epidemiologically.3.3 Model analysisGaining insights into the dynamics of the models for HIV sub-model(HIV-onlymodel) and TB sub-model(TB-only model) will be a first step to understandingmore about the co-infection.183.3.1 HIV sub-modelWe have the model with HIV only by setting yE = yI = yEI = yII = yT = yT I = 0in 3.1 and it is given byy˙S = pi−λHyS−µyS,y˙SI = λHyS− (µ+ν+δ1)ySI, (3.4)y˙A = δ1ySI− (µ+ τ)yA,where λH =β2N(ySI) and now we have N = yS + ySI + yA.We can show for this model that the regionΘ1 ={(yS,ySI,yA) ∈R3+ : N ≤piµ},is positively invariant and solutions starting in Θ1 approach, enter or stay in Θ1Disease free equilibrium pointDisease-free equilibrium point is a steady state solution where there is no HIVinfection and AIDS disease in the population.When there are no diseases in the population, the non-negative population val-ues areySI = yA = 0. (3.5)Set the right hand side of the second and third equation in 3.4 to zero and applyequation 3.5, then the HIV sub-model has a disease-free equilibrium point (DFE)ofE0H =(piµ , 0, 0).Reproduction numberR0The basic reproduction numberR0 is defined as the number of secondary infectionsproduced by an infectious individual introduced during the period of infectious-ness into a totally susceptible population [24]. We can distinguish new infections19from all other changes in population so as to find R0. Let Fi be the vector ratesof appearance of new infections in each compartment i (i = 1,2), V +i (x) be thevector rates of transfer of individuals into the particular compartment of i by allother means, V −i (x) be the vector rates of transfer of individuals out of particularcompartment of i. We can find R0 = ρ(FV−1) [24]. In this case, we have the re-production numberRH as the number of HIV infections produced by HIV positivecases.Note that we have two infectious classes ySI,yA, and the matrix showing therate of appearance of new infections in compartment i is given byF =(λHyS0).The matrix showing the rate of transfer of individuals in and out of compartmentsi isV = V −−V + =((µ+ν+δ1)ySI(µ+ τ)yA−δ1ySI)where V + =(0δ1ySI)and V − =((µ+ν+δ1)ySI(µ+ τ)yA).The Jacobian matrix ofF evaluated at the disease free equilibrium point, DFE=(piµ ,0,0)is given byF =∂F (E0H)∂x j=(β2 00 0)where x j = ySI,yA for j = 1,2.The Jacobian matrix of V evaluated at the disease free equilibrium point DFEisV =∂V (E0H)∂x j=((µ+ν+δ1) 0−δ1 (µ+ τ)).The next generation matrix FV−1 is given by( β2(µ+ν+δ1) 00 0).20The dominant eigenvalues of FV−1 which is the spectral radius of the matrix FV−1gives the basic reproduction number for HIV/AIDS from the model (3.4) as;RH = ρ(FV−1) =β2(µ+ν+δ1),where•RH is the reproduction number for HIV/AIDS dynamics given by the productof the probability of HIV infection β2 for susceptible per contact with an HIVindividual and the probability that an infective progresses from HIV-positive toAIDS stage1(µ+ν+δ1).Stability analysis of disease-free equilibrium pointThe Jacobian matrix of the system of equations 3.4 is given byJ =−(µ+λH) −β2N yS 0λH −(µ+ν+δ1)+ β2N yS 00 δ1 −(µ+ τ).Theorem 3.3.2. The disease free equilibrium E0H point of HIV-only model is lo-cally asymptotically stable (LAS) if RH < 1 and unstable, if RH > 1.Proof. The Jacobian matrix J evaluated at the disease free equilibrium DFE pointis given asJ0H =−µ −β2 00 −(µ+ν+δ1)+β2 00 δ1 −(µ+ τ).To determine the stability of disease-free equilibrium point, we use |J0H −λ I|= 021to obtain eigenvalues of J0H .∣∣∣∣∣∣∣∣∣∣∣∣−µ−λ −β2 00 −(µ+ν+δ1)+β2−λ 00 δ1 −(µ+ τ)−λ∣∣∣∣∣∣∣∣∣∣∣∣= 0, (3.6)We can factor out −(µ+ τ)−λ from 3.6 to haveλ1 =−(µ+ τ) < 0,which reduces 3.6 to∣∣∣∣∣∣∣−µ−λ −β20 −(µ+ν+δ1)+β2−λ∣∣∣∣∣∣∣= 0,whose eigenvalues are the diagonal elementsλ2 =−µ < 0 and λ3 = β2− (µ+ν+δ1).We can write λ3 = β2− (µ+ν+δ1) in terms of RH asλ3 = (RH −1)(µ+ν+δ1)The eigenvalue λ3 is negative or have negative whenRH−1 < 0 or when 1−RH >0 i.e. when RH < 1.Since λ1, λ2, λ3 are all negative or have negative real parts when RH < 1,we say the disease free equilibrium point is locally asymptotically stable whenRH < 1. This completes the proof.22We can rewrite model 3.4 as,dUdt= F(U,V ),dVdt= G(U,V ), G(U,0) = 0, (3.7)where U = yS and V = (ySI,yA), with U ∈R1+ denoting the number of susceptibleindividuals and V ∈R2+ denoting the number of infected individuals.We now denote the disease free equilibrium by,E0H = (U∗,0), where U∗ =(piµ). (3.8)Conditions S1 and S2 in equation 3.9 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 ) ∈Θ1, (3.9)where A = DV G(U∗,0) denotes the M-matrix (the off diagonal elements of A arenon-negative) and Θ1 denotes the region where the model makes biological sense.Theorem 3.3.3 holds if system 3.7 satisfies the conditions in 3.9.Theorem 3.3.3. The disease free equilibrium point E0H of HIV-only model is glob-ally asymptotically stable if RH < 1 and conditions in 3.9 are satisfied.Proof. We have from theorem 3.3.2 that E0H is locally asymptotically stable ifRH < 1. Now considerF(U,0) = [pi−µyS], (3.10)G(U,V ) = AV − Ĝ(U,V ), A =β2− (µ+ν+δ1) 0δ1 −(µ+ τ) . (3.11)Ĝ(U,V ) =Ĝ1(U,V )Ĝ2(U,V )=β2(1− 1N)(ySI)0 . (3.12)23We have the conditions in 3.9 satisfied since Ĝ1(U,V ) ≥ 0 and Ĝ2(U,V ) = 0⇒Ĝ(U,V ) ≥ 0. And therefore we can conclude that E0H is globally asymptoticallystable for RH < 1. This completes the proof.Endemic equilibrium pointsWe can solve equations in 3.4 in terms of the force of infection λ ∗H =β2N∗(y∗SI)to find the conditions for the existence of an equilibrium for which HIV/AIDS isendemic in the population.Equating the right-hand side of equations 3.4 to zero, we havepi−λ ∗Hy∗S−µyS = 0, (3.13)λ ∗Hy∗S− (µ+ν+δ1)y∗SI = 0, (3.14)δ1y∗SI− (µ+ τ)y∗A = 0. (3.15)From equation 3.13 to 3.15, we havey∗S =pi(µ+λ ∗H), (3.16)y∗SI =λ ∗Hy∗S(µ+ν+δ1), (3.17)y∗A =δ1y∗SI(µ+ τ) . (3.18)And the endemic equilibrium is given byE∗H = (y∗S, y∗SI, y∗A) ,where λ ∗H =β2y∗SIN∗.24From equation 3.17, we havey∗SIy∗S=λ ∗H(µ+ν+δ1),y∗SIy∗S=1(µ+ν+δ1)(β2y∗SIN∗),N∗y∗S=1(µ+ν+δ1)(β2y∗SIy∗SI),N∗y∗S=β2(µ+ν+δ1),N∗y∗S= RH ,RH =N∗y∗S,=y∗S + y∗SI + y∗Ay∗S,= 1+y∗SIy∗S+y∗Ay∗S,RH = 1+λ ∗H(µ+ν+δ1)+λ ∗Hδ1(µ+ τ)(µ+ν+δ1),RH −1 =λ ∗H(µ+ν+δ1)(1+δ1(µ+ τ)),RH −1 = λ ∗HΠ,λ ∗H =(RH −1)Π,where Π is denoted as the mean infective period which is given byΠ=1(µ+ν+δ1)(1+δ1(µ+ τ)).When λ ∗H is substituted into the endemic equilibrium point in 3.16 to 3.18, we will25obtain the endemic equilibrium point in terms of RH asy∗S =piΠµΠ+(RH −1),y∗SI =(RH −1)y∗SΠ(µ+ν+δ1), (3.19)y∗A =δ1(RH −1)y∗SΠ(µ+ τ)(µ+ν+δ1).Theorem 3.3.4. The endemic equilibrium E∗H point of HIV-only model is locallyasymptotically stable (LAS) if RH > 1.Proof. The Jacobian matrix J evaluated at the endemic equilibrium E∗H point isgiven asJ∗H =−(µ+λ ∗H)−β2RH0λ ∗Hβ2RH− (µ+ν+δ1) 00 δ1 −(µ+ τ),where RH =N∗y∗Sand λ ∗H =(RH −1)Π.To determine the stability of endemic equilibrium point, we use |J∗H −λ I|= 0to obtain eigenvalues of J∗H .|J∗H −λ I|=∣∣∣∣∣∣∣∣∣∣∣∣−(µ+λ ∗H)−λ−β2RH0λ ∗Hβ2RH− (µ+ν+δ1)−λ 00 δ1 −(µ+ τ)−λ∣∣∣∣∣∣∣∣∣∣∣∣= 0,λ1 =−(µ+ τ) and(−(µ+λ ∗H)−λ)( β2RH− (µ+ν+δ1)−λ)+β2λ ∗HRH,whose characteristic equation is given byAλ 2 +Bλ +C = 0.26Coefficients A B and C can be written in form of RH asA = 1,B = (µ+λ ∗H),C =β2λ ∗HRH.We use the Routh-Hurwitz stability criterion for second order polynomial so as tobe sure that all eigenvalues of J∗H are either negative or have negative real parts.The following conditions must hold for stability:A > 0, B > 0 and C > 0.Clearly, A > 0, B > 0 and C > 0 when λ ∗H > 0, i.e when RH > 1.All the Routh-Hurwitz criterion conditions are satisfied when RH > 1. HenceE∗H is asymptotically stable when RH > 1.3.3.5 TB sub-modelWe have the model with TB only by setting ySI = yEI = yII = yA = yT I = 0 inequation 3.1 and it is given byy˙S = pi−λT yS−µyS,y˙E = λT yS +λT yT − (µ+κ+ γ1)yE , (3.20)y˙I = κyE − (µ+α+ γ2)yI,y˙T = γ1yE + γ2yI−λT yT −µyT ,where λT =β1yINand now we have N = yS + yE + yI + yT .We can show for this model that the regionΘ2 ={(yS,yE ,yI,yT ) ∈R4+ : N ≤piµ},is positively invariant and solutions starting in Θ2 approach, enter or stay in Θ227Disease free equilibrium pointWhen there is no TB disease in the population, the non-negative population valuesareyE = yI = yT = 0. (3.21)Set the right hand side of equation 3.20 to zero and apply 3.21, then our model hasa disease-free equilibrium point (DFE) ofE0T =(piµ , 0, 0, 0).Reproduction numberR0In this case, we have the reproduction number RT as the number of TB infectionsproduced by infectious TB cases.Note that we have three infectious classes yE ,yI,yT , and the matrix showingthe rate of appearance of new infections in compartment i is given byF =λT yS +λT yT00 .The matrix showing the rate of transfer of individuals in and out of compartmentsi isV = V −−V + =(µ+κ+ γ1)yE(µ+α+ γ2)yI−κyEλT yT +µyT − γ1yE − γ2yIThe jacobian matrix ofF evaluated at the disease free equilibrium point, DFE=(piµ ,0,0,0)is given byF =∂F (E0T )∂x j=0 β1 00 0 00 0 0 where x j = yE ,yI,yT for j = 1,2,3.28The jacobian matrix of V evaluated at the disease free equilibrium point DFEisV =∂V (E0T )∂x j=(µ+κ+ γ1) 0 0−κ (µ+α+ γ2) 0−γ1 −γ2 µThe dominant eigenvalues of FV−1 which is the spectral of the matrix FV−1gives the basic reproduction number for TB from the model 3.20 as;RT = ρ(FV−1)=β1κ(µ+κ+ γ1)(µ+α+ γ2)=(β1(µ+α+ γ2))(κ(µ+κ+ γ1)),where• RT is the reproduction number for TB dynamics given by the product ofthe probability of TB infection β1 for susceptible per contact with an infectiousTB individual and the average time(1(µ+α+ γ2))an individual spends in aninfectious class times the product of the rate κ at which a latent TB individualbecomes infectious and the average time(1(µ+κ+ γ1))an individual spends inthe latent class.Stability analysis of disease-free equilibrium pointThe Jacobian matrix of the system of equations 3.20 is given byJ =−(µ+λT ) 0 −β1N yS 0λT −(µ+κ+ γ1) β1N (yS + yT ) λT0 κ −(µ+α+ γ2) 00 γ1 γ2− β1N yT −(µ+λT ).Theorem 3.3.6. The disease free equilibrium E0T point of TB-only model is locallyasymptotically stable (LAS) if RT < 1 and unstable, if RT > 1.Proof. The Jacobian matrix J evaluated at the disease free equilibrium DFE E0T29point is given asJ0T =−µ 0 −β1 00 −(µ+κ+ γ1) β1 00 κ −(µ+α+ γ2) 00 γ1 γ2 −µ.To determine the stability of disease-free equilibrium point, we use |J0T −λ I|= 0to obtain eigenvalues of J0T .|J0T −λ I|=∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣−µ−λ 0 −β1 00 −(µ+κ+ γ1)−λ β1 00 κ −(µ+α+ γ2)−λ 00 γ1 γ2 −µ−λ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣= 0λ1 =−µ < 0, λ2 =−µ < 0and ∣∣∣∣∣∣∣−(µ+κ+ γ1)−λ β1κ −(µ+α+ γ2)−λ∣∣∣∣∣∣∣= 0,which gives(−(µ+κ+ γ1)−λ)(−(µ+α+ γ2)−λ)−(β1κ)= 0. (3.22)From equation 3.22 we have;λ 2 +((µ+κ+ γ1)+(µ+α+ γ2))λ +((µ+κ+ γ1)(µ+α+ γ2)−β1κ)= 0,30which can be written in terms of RT asλ 2 +((µ+κ+ γ1)+(µ+α+ γ2))λ +(1−RT )(µ+κ+ γ1)(µ+α+ γ2) = 0(3.23)The eigenvalues λ3,4 of 3.23 are negative or have negative real parts when 1−RT >0 i.e. when RT < 1.Since λ1, λ2, λ3, λ4 are all negative or have negative real parts when RT < 1,we say the disease free equilibrium point is locally asymptotically stable whenRT < 1. This completes the proof.We can rewrite model 3.20 as,dUdt= F(U,V ),dVdt= G(U,V ), G(U,0) = 0, (3.24)where U = (yS,yT ) and V = (yE ,yI), with U ∈R2+ denoting the number of unin-fected individuals and V ∈R2+ denoting the number of infected individuals.We now denote the disease free equilibrium by,E0T = (U∗,0), where U∗ =(piµ ,0). (3.25)Conditions S1 and S2 in equation 3.26 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 ) = BV − Ĝ(U,V ), Ĝ(U,V )≥ 0 for (U,V ) ∈Θ2, (3.26)where B = DV G(U∗,0) denotes the M-matrix (the off diagonal elements of B arenon-negative) and Θ denotes the region where the model makes biological sense.Theorem 3.3.7 holds if system 3.24 satisfies the conditions in 3.26.Theorem 3.3.7. The disease free equilibrium point E0T of HIV-only model is glob-ally asymptotically stable if RT < 1 and conditions in 3.26 are satisfied.31Proof. We have from theorem 3.3.6 that E0T is locally asymptotically stable ifRT < 1. Now considerF(U,0) =pi−µyS0 , (3.27)G(U,V ) = BV − Ĝ(U,V ), B =−(µ+κ+ γ1) β1κ −(µ+α+ γ2) . (3.28)Ĝ(U,V ) =Ĝ1(U,V )Ĝ2(U,V )=β1yI(1− yS+yTN)0 . (3.29)We have the conditions in 3.26 satisfied since Ĝ1(U,V ) ≥ 0 and Ĝ2(U,V ) = 0⇒Ĝ(U,V ) ≥ 0. And therefore we can conclude that E0T is globally asymptoticallystable for RT < 1. This completes the proof.Endemic equilibrium pointsWe can solve equations in 3.20 in terms of the force of infection λ ∗T =β1N∗(y∗I ) tofind the conditions for the existence of an equilibrium for which TB is endemic inthe population.Equating the right-hand side of equations 3.20 to zero, we havepi−λT yS−µyS = 0,λT yS +λT yT − (µ+κ+ γ1)yE = 0, (3.30)κyE − (µ+α+ γ2)yI = 0,γ1yE + γ2yI−λT yT −µyT = 0.32From (3.30), we havey∗S =pi(µ+λ ∗T ), (3.31)y∗E =(µ+α+ γ2)y∗Iκ , (3.32)y∗T =(µ+κ+ γ1)y∗Eλ ∗T− y∗S, (3.33)y∗I =λ ∗Tκ(µ+λ ∗T )y∗S(µ+α+ γ2){(µ+λ ∗T )(µ+κ+ γ1)− γ1λ ∗T}− γ2κλ ∗T. (3.34)And the endemic equilibrium is given byE∗T = (y∗S,y∗E ,y∗T ,y∗I ) ,where λ ∗T =β1y∗IN∗33From equation 3.33 we havey∗T + y∗S =(µ+κ+ γ1)y∗Eλ ∗T,y∗T + y∗Sy∗E=(µ+κ+ γ1)λ ∗T,y∗Ey∗T + y∗S=λ ∗T(µ+κ+ γ1),y∗Ey∗T + y∗S=1(µ+κ+ γ1)(β1y∗IN∗), (3.35)N∗y∗T + y∗S=1(µ+κ+ γ1)(β1y∗Iy∗E),N∗y∗T + y∗S=1(µ+κ+ γ1)(β1κ(µ+α+ γ2)),N∗y∗T + y∗S=(β1κ(µ+κ+ γ1)(µ+α+ γ2)),N∗y∗T + y∗S= RT ,RT =N∗y∗T + y∗S,=y∗S + y∗E + y∗T + y∗Iy∗T + y∗S,= 1+y∗Ey∗T + y∗S+y∗Iy∗T + y∗S,RT = 1+λ ∗T(µ+κ+ γ1)+λ ∗Tκ(µ+κ+ γ1)(µ+α+ γ2),RT −1 =λ ∗T(µ+κ+ γ1)(1+κ(µ+α+ γ2)),RT −1 = λ ∗TΩ,λ ∗T =(RT −1)Ω,where Ω is denoted as the mean infective period for TB which is given byΩ=1(µ+κ+ γ1)(1+κ(µ+α+ γ2)).34When λ ∗T is substituted into the endemic equilibrium point in 3.31 to 3.34, we willobtain the endemic equilibrium point in terms of RT asy∗S =piΩµΩ+(RT −1),y∗E =(µ+α+ γ2)y∗Iκ ,y∗T =(µ+κ+ γ1)(µ+α+ γ2)Ωy∗Iκ(RT −1)− y∗S, (3.36)y∗I =(piκ(RT −1)(µΩ+(RT −1))(µ+κ+ γ1)(µ+α+ γ2)− (RT −1)(γ1 +κγ2(µ+α+ γ2)).Theorem 3.3.8. The endemic equilibrium E∗T point of TB-only model is locallyasymptotically stable (LAS) if RT > 1.Proof. The Jacobian matrix J evaluated at the endemic equilibrium E∗T point isgiven asJ∗T =−(µ+λ ∗T ) 0−β1N∗ y∗S 0λ ∗T −(µ+κ+ γ1)β1RTλ ∗T0 κ −(µ+α+ γ2) 00 γ1 γ2− β1N∗ y∗T −(µ+λ ∗T ).where RT =N∗y∗S + y∗Tand λ ∗T =(RT −1)Ω.To determine the stability of endemic equilibrium point, we use |J∗T −λ I| = 035to obtain eigenvalues of J∗T .|J∗T−λ I|=∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣−(µ+λ ∗T )−λ 0−β1N∗ y∗S 0λ ∗T −(µ+κ+ γ1)−λβ1RTλ ∗T0 κ −(µ+α+ γ2)−λ 00 γ1 γ2− β1N∗ y∗T −(µ+λ ∗T )−λ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣= 0,We have that λ1 =−(µ+λ ∗T ) < 0 and the characteristic equation is given byλ 3 +Dλ 2 +Eλ +F = 0.Coefficients D,E and F can be written in form of RT asD = (µ+κ+ γ1)+(µ+α+ γ2)+(µ+λ ∗T ),E = (µ+λ ∗T )((µ+κ+ γ1)+(µ+α+ γ2))−λ ∗T γ1,= (µ+λ ∗T )((µ+κ)+(µ+α+ γ2))+µγ1,F = λ ∗T(β1κRT− γ1(µ+α+ γ2)−κγ2),= λ ∗T ((µ+κ)(µ+α)+µγ2) .We use the Routh-Hurwitz stability criterion for third order polynomial so as to besure that all eigenvalues of J∗T are either negative or have negative real parts. Thefollowing conditions must hold for stability:D > 0, F > 0 and DE−F > 0.Clearly, D > 0 and F > 0 when RT > 1.NowDE−F = {(µ+κ+ γ1)+(µ+λ ∗T )}{(µ+λ ∗T )[(µ+κ)+(µ+α+ γ2)]+µγ1}+(µ+α+γ2){(µ+λ ∗T )(µ+α+ γ2)+µγ1 +µ(µ+κ)}+λ ∗T γ2κ > 0, when λ ∗T > 0 i.e when RT > 1.36All the Routh-Hurwitz criterion conditions are satisfied when RT > 1. Hence E∗Tis asymptotically stable when RT > 1.3.3.9 Analysis of the full modelIn this section, we will analyse the full model 3.1.Disease free equilibrium pointDisease-free equilibrium point is a steady state solution where there is no diseasein the whole population.When there are no diseases in the population, the non-negative population val-ues areyE = yI = ySI = yEI = yII = yA = yT = yT I = 0. (3.37)Set the right hand side of (3.1) to zero and apply 3.37, then our full model has adisease-free equilibrium point (DFE) ofE0 =(piµ , 0, 0, 0, 0, 0, 0, 0, 0). (3.38)Reproduction numberR0In this case, we have the reproduction number R0 as the number of HIV/AIDSor TB infections produced by a single TB infective or single HIV/AIDS positiveindividual.Note that we have eight infectious classes yE ,yI,ySI,yEI,yII,yA,yT ,yT I , and thematrix showing the rate of appearance of new infections in compartment i is given37byF =λT (yS + yT )0λHySλHyE +λT ySIλHyI00λHyT.The matrix showing the rate of transfer of individuals in and out of compartmentsi isV = V −−V + =λHyE +(µ+κ+ γ1)yE(µ+α+ γ2)yI +λHyI−κyEλT ySI +(µ+ν+δ1)ySI(µ+ν+κH +δ2 + γ1)yEI(µ+ γ2 +δ3 +α+ν)yII− kHyEI(µ+ τ)yA−δ4yT I−δ1ySI−δ2yEI−δ3yII(µ+λT +λH)yT − γ1yE − γ2yI(µ+ν+δ4)yT I− γ1yEI− γ2yIIwhereλT =β1N(yI +ρ1yII) and λH =β2N(ySI +η1yEI +η2yII +η3yT I)The Jacobian matrix of F evaluated at the disease free equilibrium E0 point, is38given byF =∂F (E0)∂x j=0 β1 0 0 β1ρ1 0 0 00 0 0 0 0 0 0 00 0 β2 β2η1 β2η2 0 0 β2η30 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0,where x j = yE ,yI,ySI,yEI,yII,yA,yT ,yT I for j = 1, . . . ,8. Leta1 = µ+κ+ γ1,a2 = µ+α+ γ2,a3 = µ+ν+δ1,a4 = µ+ν+κH +δ2 + γ1,a5 = µ+ γ2 +δ3 +α+ν ,a6 = µ+ τ,a7 = µ+ν+δ4,so that the Jacobian matrix of V evaluated at the disease free equilibrium E0 pointisV =∂V (E0)∂x j=a1 0 0 0 0 0 0 0−κ a2 0 0 0 0 0 00 0 a3 0 0 0 0 00 0 0 a4 0 0 0 00 0 0 −κH a5 0 0 00 0 −δ1 −δ2 −δ3 a6 0 −δ4−γ1 −γ2 0 0 0 0 µ 00 0 0 −γ1 −γ2 0 0 a7.The dominant eigenvalues of FV−1 which is the spectral of the matrix FV−1 gives39the basic reproduction number for TB and HIV/AIDS from the model 3.1 as;R0 = ρ(FV−1) = max{RH ,RT},where RH =β2(µ+ν+δ1),RT =(β1(µ+α+ γ2))(κ(µ+κ+ γ1)),and these correspond to the reproduction number for HIV sub-model and TB sub-model respectively.Stability analysis of disease-free equilibrium pointTheorem 3.3.10. The disease free equilibrium E0 point of the full model is locallyasymptotically stable (LAS) if R0 < 1 and unstable, if R0 > 1.Proof. The Jacobian matrix of the system of equations (3.1) evaluated at E0 isgiven byJ =−µ 0 −β1 −β2 −β2η1 −β1ρ1−β2η2 0 0 −β2η30 −a1 β1 0 0 β1ρ1 0 0 00 κ −a2 0 0 0 0 0 00 0 0 β2−a3 β2η1 β2η2 0 0 β2η30 0 0 0 −a4 0 0 0 00 0 0 0 κH −a5 0 0 00 0 0 δ1 δ2 δ3 −a6 0 δ40 γ1 γ2 0 0 0 0 −µ 00 0 0 0 γ1 γ2 0 0 −a7.To determine the stability of disease-free equilibrium point, we use |J0−λ I| = 040to obtain eigenvalues of J0.∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣−µ−λ 0 −β1 −β2 −β2η1 −β1ρ1−β2η2 0 0 −β2η30 −a1−λ β1 0 0 β1ρ1 0 0 00 κ −a2−λ 0 0 0 0 0 00 0 0 β2−a3−λ β2η1 β2η2 0 0 β2η30 0 0 0 −a4−λ 0 0 0 00 0 0 0 κH −a5−λ 0 0 00 0 0 δ1 δ2 δ3 −a6−λ 0 δ40 γ1 γ2 0 0 0 0 −µ−λ 00 0 0 0 γ1 γ2 0 0 −a7−λ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣= 0,λ1,2 =−µ < 0,and the matrix reduces to∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣−a1−λ β1 0 0 β1ρ1 0 0κ −a2−λ 0 0 0 0 00 0 β2−a3−λ β2η1 β2η2 0 00 0 0 −a4−λ 0 0 00 0 0 κH −a5−λ 0 00 0 δ1 δ2 δ3 −a6−λ 0γ1 γ2 0 0 0 0 −µ−λ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣= 0.We have λ3 =−(µ+ν+κH +δ2 + γ1)< 0, λ4 =−(µ+ γ2 +δ3 +α+ν)< 0 andλ5 =−(µ+ν+δ4) < 0,so that either(β2− (µ+ν+δ1)−λ)(−(µ+ τ)−λ)= 0. (3.39)or (−(µ+κ+ γ1)−λ)(−(µ+α+ γ2)−λ)−β1κ = 0. (3.40)From equation (3.39) we have;λ 2 +((µ+ν+δ1)+(µ+ τ)−β2)λ +((µ+ν+δ1)(µ+ τ)−β2(µ+ τ))= 0,41which can be written in terms of RH asλ 2 +((µ+ τ)+(µ+ν+δ1)(1−RH))λ +(1−RH)(µ+ τ)(µ+ν+δ1) = 0(3.41)The eigenvalues of 3.41 are negative or have negative real parts when 1−RH > 0i.e. when RH < 1. Equation 3.40 can also be written in terms of RT as;λ 2+((µ+κ+γ1)+(µ+α+γ2))λ+(1−RT )(µ+κ+γ1)(µ+α+γ2), (3.42)The eigenvalues of (3.42) are negative or have negative real parts when 1−RT > 0i.e. when RT < 1 Since all eigenvalues are negative or have negative real partswhen RH < 1 and when RT < 1 , we say the disease free equilibrium point islocally asymptotically stable when RH < 1 and when RT < 1. And the diseasedies out. The basic reproduction number R0 = max{RH ,RT}< 1. Hence, we saythat the disease free equilibrium point E0 is locally asymptotically stable wheneverR0 < 1. This completes the proof.We can rewrite model 3.1 as,dUdt= F(U,V ),dVdt= G(U,V ), G(U,0) = 0, (3.43)where U = (yS,yT ) and V = (yE ,yI,ySI,yEI,yII,yA,yT I), with U ∈R2+ denoting thenumber of uninfected individuals and V ∈ R7+ denoting the number of infectedindividuals.We now denote the disease free equilibrium by,E0 = (U∗,0), where U∗ =(piµ ,0). (3.44)Conditions S1 and S2 in equation 3.45 must be satisfied to guarantee local asymp-42totic stability.S1 :dUdt= F(U,0), U∗ is globally asymptotic stable (g.a.s)S2 : G(U,V ) = CV − Ĝ(U,V ), Ĝ(U,V )≥ 0 for (U,V ) ∈Θ, (3.45)where C = DV G(U∗,0) denotes the M-matrix (the off diagonal elements of C arenon-negative) and Θ denotes the region where the model makes biological sense.Theorem 3.3.11 holds if system 3.43 satisfies the conditions in 3.45.Theorem 3.3.11. The disease free equilibrium point E0 of the full model is globallyasymptotically stable if R0 < 1 and conditions in 3.45 are satisfied.Proof. We have from theorem 3.3.10 that E0 is locally asymptotically stable ifR0 < 1. Now considerF(U,0) =pi−µyS−µyT , (3.46)G(U,V ) =CV − Ĝ(U,V ), (3.47)C =−a1 β1 0 0 β1ρ1 0 0κ −a2 0 0 0 0 00 0 β2−a3 β2η1 β2η2 0 β2η30 0 0 −a4 0 0 00 0 0 κH −a5 0 00 0 δ1 δ2 δ3 −a6 δ40 0 0 γ1 γ2 0 −a7. (3.48)43Ĝ(U,V )=Ĝ1(U,V )Ĝ2(U,V )Ĝ3(U,V )Ĝ4(U,V )Ĝ5(U,V )Ĝ6(U,V )Ĝ7(U,V )=β1(1− yS+yTN)(yI +ρ1yII +ρ2yA)+λHyEλHyIβ2(1− 1N)(ySI +η1yEI +η2yII +η3yT I +η4yA)+λT ySI−λHyE −λT ySI−λHyI0−λHyT.(3.49)We have the condition (S2) in 3.45 not satisfied since Ĝ4(U,V )< 0, Ĝ5(U,V )< 0and Ĝ7(U,V ) < 0. And therefore we can conclude that E0 may not be globallyasymptotically stable for R0 < 1. This completes the proof.Endemic equilibrium pointsThe computation of the endemic equilibrium of the full model (co-infection model)is difficult analytically, and therefore the model 3.1 endemic equilibria correspondsto,1. E1 = (yS1,0,0,ySI1,0,0,yA1,0,0)(yS1,0,0,ySI1,0,0,yA1,0,0)=(piΠµΠ+(RH −1),0,0,(RH −1)yS1Π(µ+ν+δ1),0,0,δ1(RH −1)yS1Π(µ+ τ)(µ+ν+δ1)),(3.50)the TB free equilibrium. This exists when RH > 1. The analysis of theequilibria E1 is similar to the endemic equilibria E∗H in equation 3.19.442. E2 = (yS2,yE2,yI2,0,0,0,0,yT 2,0), the HIV free equilibrium, whereyS2 =piΩµΩ+(RT −1),yE2 =(µ+α+ γ2)yI2κ ,yT 2 =(µ+κ+ γ1)(µ+α+ γ2)ΩyI2κ(RT −1)− yS2, (3.51)yI2 =piκ(RT −1)(µΩ+(RT −1))(µ+κ+ γ1)(µ+α+ γ2)− (RT −1)(γ1 +κγ2(µ+α+ γ2)).This exists when RT > 1. The analysis of the equilibria E2 is similar to theendemic equilibria E∗T in equation 3.36.3. E3 =(yS3,yE3,yI3,ySI3,yEI3,yII3,yA3,yT 3,yT I3), the HIV-TB co-infection equi-librium. This exists when each component of E3 is positive.We summarize the existence of the disease free equilibrium points in the followingtheorem:Theorem 3.3.12. The system of equations 3.1 has the following disease free equi-librium points:1. E0H which exist when RH < 1.2. E0T which exist when RT < 1.3. E0 which exists when RH < 1 and RT < 1, i.e. R0 < 1.And summarize the existence of the endemic equilibrium points in the follow-ing theorem:Theorem 3.3.13. The system of equations in 3.1 has the following endemic equi-librium points:1. E∗H or E1 which exist when RH > 1.2. E∗T or E2 which exist when RT > 1.453. E3 which exists when RH > 1 and RT > 1, i.e. R0 > 1. We will give adetailed explanation of E3 in our numerical simulations.Remarks:The model revealed the following scenarios regarding the effects of HIV/AIDSand Tuberculosis in an endemic section:1. A scenario where we have population of individuals infected with only TB(TB sub-model).2. A scenario where we have population of individuals infected with only HIV(HIV sub-model).3. A scenario where there are individuals with both infection (co-infection model).We shall also explore the impact of these scenarios on the progression of HIV andTB infection using numerical simulations.In summary, we have been able to show the mathematical analysis for TB sub-model, HIV sub-model and the full co-infection model. We show that the basicreproduction numberR0 determines the dynamics of the model. Disease free equi-librium point E0 is also shown to be locally asymptotically stable whenR0 < 1 andunstable if R0 > 1, we therefore have that solutions converge to E0 and diseasesdie out. We show that the Endemic equilibrium point E∗ is locally asymptoticallystable if R0 > 1 and unstable if R0 < 1, we then have that solutions converge toE∗ and any initial epidemics of TB and HIV/AIDS will become endemic in thepopulation. Our analytical results will be justified by our numerical simulations.46Chapter 4Numerical Simulations andSensitivity AnalysisResults of the numerical simulation are given in this section and the set of parame-ters used are given in table 4.1. Initial values used for different values of β1 and β2is also given in table 4.2.Table 4.1: Parameter values and their sourcesSymbol Value References Symbol Value ReferencesN(0) 500,000 δ1 0.1 yr−1 [22, 23]pi 7142 δ2 0.102 yr−1 [5]µ 1/70 yr−1 [9, 22, 23] δ3 0.25 yr−1 [5]β1 Variable δ4 0.125 yr−1 [21]β2 Variable γ1 1 yr−1 [22]τ 0.33 yr−1 [5, 23] γ2 2 yr−1 [22]ν 0.01 yr−1 [21] ρ1 100 [20]α 0.02 yr−1 [21] η1 1κ 1 yr−1 [22] η2 1κH 1.3κ [22] η3 147Table 4.2: Initial values for different values of β1 and β2.β1 β2 yS(0) yE(0) yI(0) ySI(0) yEI(0) yII(0) yA(0) yT (0) yT I(0)2.5 0.03 60N10015N1002N1004N10014N1004N100N100 0 05.2 0.3 60N10015N1002N1004N10014N1004N100N100 0 05.2 0.03 60N10015N1002N1004N10014N1004N100N100 0 00.4 0.8 60N10015N1002N1004N10014N1004N100N100 0 07 0.6 60N10015N1002N1004N10014N1004N100N100 0 05.2 0.03 99N100N100 0 0 0 0 0 0 02.5 0.03 0 N 0 0 0 0 0 0 00.4 0.8 99N100 0 0N100 0 0 0 0 02.5 0.03 0 0 0 N 0 0 0 0 04.1 Sensitivity analysisWe are able to know how important each parameter is to the spread of the diseasethrough sensitivity indices of R0 to all different parameters. This is helpful inassigning the correct and appropriate parameter for making an endemic scenario[20].Sensitivity analysis in this section describes the effect of changes in the param-eter values on the model. Now we will let ε be any of the non-negative parametersthat make up R0 in the model. A small perturbation in ε by ∆ε will also cause aperturbation in R0 by ∆R0. We define the normalized sensitivity index by ϒε (theratio of the corresponding normalized changes[9]). Therefore, the sensitivity indexϒε is computed by using the normalized forward sensitivity index method:ϒε =∆R0R0/∆εε =εR0·∂R0∂ε . (4.1)We can calculate the sensitivity index in terms ofRH andRT sinceR0 = max{RH ,RT}.The sensitivity indices in terms of RH =β2(µ+ν+δ1)is given asϒβ2 = 1, ϒµ =−µ(µ+ν+δ1), ϒν =−ν(µ+ν+δ1), ϒδ1 =−δ1(µ+ν+δ1).The sensitivity indices of RT =(β1(µ+α+ γ2))(κ(µ+κ+ γ1))is given asϒβ1 = 1, ϒγ2 =−γ2(µ+α+ γ2), ϒα =−α(µ+α+ γ2), ϒγ1 =−γ1(µ+κ+ γ1), ϒκ =µ+ γ1(µ+κ+ γ1),48ϒµ =−µ(µ+α+ γ2)−µ(µ+κ+ γ1)Most parameter values used were gotten from previous HIV-TB model manuscripts[5, 9, 20–23]. Using the parameter values in table 4.1, the sensitivity indexes arecomputed in table 4.3 and 4.4 asTable 4.3: Sensitivity analysis of RH .Sensitivity index Valueϒβ2 1ϒµ −0.0141ϒν −0.0805ϒδ1 −0.8046Table 4.4: Sensitivity analysis of RT .Sensitivity index Valueϒβ1 1ϒγ2 −0.9831ϒα −0.0098ϒγ1 −0.4965ϒκ 0.5035ϒµ −0.0141The sign in front of each of the values in tables 4.3 and 4.4 shows what willhappen to R0 if the parameter is increased or decreased. R0(RH or RT ) increaseswhen sensitivity indeces with positive signs increase, while R0(RH or RT ) de-creases when sensitivity indeces with negative signs increase and vice versa.The most sensitive parameters toRH andRT are found to be β2 and β1 respec-tively. Sensitivity indeces ϒβ1 = 1 and ϒβ2 = 1 mean thatRH orRT approximatelydecreases by 1% when either β1 or β2 is decreased by 1%.Since decrease in β1 and β2 is the possible intervention strategy for the reduc-tion of R0. We will consider changes of parameters β1 and β2 and see their effectson RT and RH .494.2 Numerical simulations for different values of β1,β2and discussion of resultsAll parameter values used in the simulations are given in table 4.1. Table 4.2 showsdifferent initial conditions for different values of β1 and β2. It shows the effect ofβ1 on RT and the effect of β2 on RH .In general, 500 years from our figures may not be a reasonable timescale fromthe model to be predictive, but we have decided to use it as an illustration to showthat TB infection persists for decades.Figure 4.1: The left panel shows the plot of the stability analysis of the dis-ease free equilibrium at β1 = 2.5 and β2 = 0.03 while the right panelshows the plot of the stability analysis of the endemic equilibrium atβ1 = 5.2 and β2 = 0.3From the left panel figure of figure 4.1, we considered initial conditions fromtable 4.2 and R0 < 1 (RT < 1 and RH < 1) to establish the stability of the diseasefree equilibrium E0 given by 3.38. It is shown numerically that yS converges to Nas t → ∞, and every other disease in the population dies out. From the right panelfigure of figure 4.1, we considered initial conditions from table 4.2 and R0 > 1(RT > 1 or RH > 1) to establish the stability of the endemic equilibrium E∗. Wehave shown numerically that for R0 > 1 and as t→ ∞, the state variables convergeto E∗ and the endemic equilibrium exist.50Figure 4.2: Graphs showing the susceptibles yS, exposed to TB yE , infectiousto TB yI , the HIV positive ySI , HIV positive exposed to TB yEI , HIVpositive suffering from TB yII , those suffering from AIDS yA, singlyinfected treated of TB yT and dually infected treated of TB yT I . Theseare simulations in different compartments with β1 = 5.2 and β2 = 0.3Figure 4.2 represents the behaviour of individuals in various stages of the co-infection of HIV/AIDS and TB over a period of 500 years in which treatment oflatent and active TB treatment is incorporated.Left panel figure on the first row shows that treatment of TB reduces the suscep-tible population to a stable state and remains constant. It shows that when t → ∞,susceptible does not go to zero due to TB treatment.Middle panel figure on the third row shows the behaviour of individuals in-fected with TB and on TB. Individuals on TB treatment increase with respect todecrease in all other TB infected individuals and we can see from middle panel51figure on the first row that untreated TB infection leads to increase in the numberof infected individuals.Middle and right panel figures on the first row, and, left and right panel figureson the third row show that treatment of TB result in their decrease to low levels.These imply that the population of TB individuals decrease up to a certain stage andbecome constant as t→ ∞ and does not go to zero due to TB treatment. Left panelfigure on the second row shows that the population drops and starts increasing dueto some AIDS individuals recovering from TB for the dually infected individuals.Middle and right panel figures on the second row show a decrease in the numberof dually infected individuals to almost zero where they remain constant due to thenumber of those entering the AIDS class, those on TB treatment and due to deaths(natural and disease deaths).Figure 4.3: The left panel shows the plot of the stability of TB free equilib-rium E∗H at β1 = 0.4 and β2 = 0.8 while the right panel shows the plotof the stability of HIV free equilibrium E∗T at β1 = 5.2 and β2 = 0.03The left panel figure of figure 4.3 shows the stability of TB free equilibriumE∗H . It shows that HIV/AIDS persist in the society while other disease dies out. Thepopulation has a higher number of susceptible individuals exposing the populationto a slower progression towards AIDS due to TB treatment. The model respondsto changing β1 and β2 to 0.4 and 0.8 respectively. It means TB free equilibriumoccurs when β1 is very low and β2 is very high i.e when RT < 1 and RH > 1and this represents TB-only model. The right panel figure of figure 4.3 shows the52stability of HIV free equilibrium E∗T . This shows that TB persists in the societywhile the other disease dies out. The model responds to changing β1 and β2 to5.2 and 0.03 respectively. It means HIV free equilibrium occurs when β1 is veryhigh and β2 is very low i.e whenRT > 1 andRH < 1 and this represents HIV-onlymodel.Figure 4.4 shows the effect and impact of treating and not treating TB on themodel. Comparing figure on the right and left panel where TB is treated and nottreated respectively, we see that latent TB yE decreases faster in the right panelmaking TB infectious yI and HIV positive with TB disease yII increase and expos-ing the population to a faster progression towards the AIDS class within the periodof 15 years. While this is the other way in left panel due to TB treatment. TBtreatment lowers the rate of progression of the exposed individuals and this leadsto increase in dually infected individuals on TB treatment. We can say that TBtreatment at the exposed and infectious stage may prevent or reduce co-infection.From the right panel of figure 4.4, we observe that after 3 years, individuals in-fected with TB disease and HIV reduce than in the left panel. This happens becauseTB is treated in the left panel and untreated in the right panel. The decrease in thepopulation of yII in the right panel is because co-infection triggers the symptomsof AIDS and therefore decrease in yII would lead to increase in yA.Figure 4.4: The behaviour of the model for the period of 15 years with andwithout TB treatment γ1 and γ2 at β1 = 7,β2 = 0.6 and with diseaseinduced death. The left panel shows the impact of TB treatment (γ1 = 1and γ2 = 2) on the model, while the right panel shows the behaviour ofthe model without TB treatment (γ1 = 0 and γ2 = 0).53Figure 4.5: Graphs showing the behaviour of the model to changes in initialconditions. The plot of the stability of equilibria E0T , E0H , E∗T and E∗HFigure 4.5 shows the behaviour and response of the model to different initialconditions. The left panel figure on the first row is the disease free E0T for TB-submodel and it shows that TB infection dies out while susceptible goes to N as t→∞.The left panel figure on the second row is the disease free E0H for HIV-sub modeland it shows that HIV infection dies out while susceptible goes to N as t → ∞.Figures on the right panel of the first and second row are respectively similar tofigure 4.3 and their descriptions follow from figure 4.3.54Figure 4.6: Graphs showing different values of the basic reproduction num-bers RT and RH at different values of β1 and β2 respectively, with andwithout TB treatmentFigure 4.6 can be explained from the analytical solution that R0 from the in-teraction of two diseases (HIV/AIDS and TB) is illustrated as max{RH ,RT} andthis is shown in figure 4.6. We have the left panel figure in 4.6 by varying β1 andβ2 from 0 to 6 with no TB treatment while we have the right panel figure in 4.6 byvarying β1 and β2 from 0 to 6 with TB treatment . The disease threshold is deter-mined by the value of our parameters. Varying β1 and β2 with TB treatment andusing the parameter values from table 4.1, we find from the right panel of figure4.6 that RH is greater than RT and therefore RH is the epidemic threshold valuewhen TB is treated. Also,varying β1 and β2 without TB treatment and using theparameter values from table 4.1, we find from the left panel of figure 4.6 that RTis greater than RH and therefore RT is the epidemic threshold value when TB isnot treated. These mean that it is possible for the disease threshold to change if weincrease or decrease β1 or β2 and with or without TB treatment, e.g. γ1 = 0,γ2 = 0as in the left panel changes R0 to RT while γ1 = 1,γ2 = 2 as in the right panelchanges R0 to RH .4.3 ConclusionWe considered a general mathematical model of nine nonlinear differential equa-tions on HIV/AIDS and TB co-infection. We denoted the population of susceptibleindividuals by yS, the population of latent TB individuals by yE , the population ofinfectious (active) TB individuals by yI , the population of HIV-positive individ-uals by ySI , the population of HIV-positive and latent TB individuals by yEI , the55population of HIV-positive and infectious TB individuals by yII , the population ofAIDS individuals by yA, the population of treated individuals with TB by yT andthe population of dually infected individuals treated of TB by yT I .The threshold parameter R0 was calculated and used to determine the con-ditions under which the HIV/AIDS and TB could be transmitted and remainedendemic in the population. We analyzed the model to know the level at which theHIV/AIDS epidemic aggravates the spread of Tuberculosis (TB) and vice versa.We thus showed that three disease-free equilibrium points E0H ,E0T ,E0 respectivelyfor HIV-sub model, TB-sub model and the full model are locally asymptoticallystable when R0 < 1 i.e. RT < 1 and RH < 1. We also showed that the populationwith both HIV and TB infection have three endemic equilibrium points E∗H ,E∗T ,E∗respectively for HIV-sub model, TB-sub model and the full model which were lo-cally asymptotically stable when R0 > 1 i.e. RT > 1 or RH > 1. Global stabilityanalysis of the three disease-free equilibrium points was established. We foundthe most sensitive parameters to be β1 and β2 and showed how changes to theseparameters with or without TB treatment affect the basic reproduction number.Numerical simulations were used to compare the endemic scenarios revealedby analytical results. Simulations were purely hypothetical since the data used arenot for a particular community but the qualitative features that revealed the impactof each of the scenarios on HIV/AIDS and TB transmission were shown. Figure 4.6gave a linear relationship between the two reproduction number, and this showedthat RT gave the epidemic threshold value when TB was not treated, while RHgave the epidemic threshold value with TB treatment. Our results suggested thatthe better scenarios were where some of the individuals (the right panel of figure??) have lower infection levels i.e. when TB was treated, and the worst scenarioswere where there are co-infection of both HIV/AIDS and TB (the left panel offigure ??) without TB treatment.Thus, we can interpret the situation in an epidemiological manner that a societywith some individuals infected with TB and without TB treatment is at the worstrisk of being co-infected with HIV which in turn creates socio-economic effects ifno intervention is implemented in time for either or both HIV/AIDS and TB infec-tion. We conclude that TB treatment for individuals with TB infections results in asignificant reduction (as in the left panel of figure 4.4) of the number of individuals56progressing to active TB, reduction of the co-infected individuals and reduction ofthe disease induced death. Also, effective treatment of TB for the co-infected indi-viduals also reduced the number of individuals that progress to AIDS class. Thusin a situation where treatment of HIV is not readily available, we can thereforeadvise public health authorities that treating both the exposed and active form ofTB in both singly and dually TB infected individuals could be a good public healthmeasure to improve life for HIV-positive individuals.As part of future work to improve the model in question, restructuring of themodel to include HIV/AIDS treatment for only HIV/AIDS individuals (HIV-submodel) and co-infected individuals could be a better approach to studying the dy-namics of HIV/AIDS and TB, and could be the best measure to reduce R0 andco-infection. We also wish to find the global stability of the endemic equilibriumpoints in the future work and a nonlinear relationship between RT and RH . De-spite all its limitations, the model provided useful information and insights into thepotential impact of treating Tuberculosis on the dynamics of HIV/AIDS and TBco-infection.57Bibliography[1] AIDSG. What are the stages of HIV infection.http://www.aids.gov/hiv-aids-basics/just-diagnosed-with-hiv-aids/hiv-in-your-body/stages-of-hiv/, Accessed November 2014. → pages 2[2] J. P. Aparicio, A. F. Capurro, and C. Castillo-Chavez. Transmission anddynamics of tuberculosis on generalized households. J. theor. Biol., 206:327–341, 2000. → pages 4[3] J. P. Aparicio, A. F. Capurro, and C. Castillo-Chavez. Markers of diseaseevolution: The case of tuberculosis. Journal of theoretical Biology, 215:227–237, 2002. → pages 4[4] AVT. Stages of HIV infection.http://www.avert.org/stages-hiv-infection.htm, Accessed December 2014. →pages 2, 3[5] C. P. Bhunu, W. Garira, and Z. Mukandavire. Modelling HIV/AIDS andtuberculosis coinfection. 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