Science, Irving K. Barber Faculty of (Okanagan)
Computer Science, Mathematics, Physics and Statistics, Department of (Okanagan)
DSpace
UBCO
Kaur, Avneet
Master of Science - MSc
University of British Columbia
Effective management of epidemics requires not only modeling the disease itself but also the fear response of the population to both the disease and the vaccine. Risk perceptions of the disease may drive fear of disease in the population, thus encouraging people to adopt prophylaxis. On the other hand, negative thoughts about vaccines may drive fear of vaccines in the population, thus making some people vaccine hesitant. Since these fear responses affect population behaviour and thus disease transmission rate, fear responses in turn influence the disease trajectory. In this thesis, we investigate an existing model of disease transmission that considers fear of disease and fear of vaccine as contagions that occur alongside the usual contagion of the disease. The mechanisms of fear acquisition and loss are rooted in human behavioural science. We analyse this model to determine the conditions under which each of the contagions (disease, fear of disease, or fear of vaccine) initially grows the fastest. We then analytically investigate a reduced version of the full model with just the fear of disease (i.e., no vaccine or fear of vaccine). We consider this reduced model in two scenarios, one where fear is transmitted at a rate slower than that of the disease, and the other where the two transmission rates are comparable. In the former scenario, we show that behavioural change has little to no effect on the disease dynamics, but in the latter case, we observe that sufficient behavioural intervention can suppress the epidemic yielding bifurcations in final epidemic size, the number of infection waves, and their relative peak size. In both scenarios, we observe that the disease-fearful population may retain fear of disease even after the outbreak is over. Finally, we propose an extension to the full model by adding a compartment for the population that is simultaneously fearful of both the disease and the vaccine. Analysis of the modified model gives us insights into the rich disease dynamics of this model. In particular, we observe that the final epidemic size remains low when a large fraction of the population either gets vaccinated or shifts to the double fear compartment.
INCORPORATING FEAR (UN)LEARNING MECHANISMS ININFECTIOUS DISEASE MODELLINGbyAvneet KaurB.Sc. (Honours) Physics, University of Delhi (India), 2021A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)April 2024© Avneet Kaur, 2024The following individuals certify that they have read, and recommendto the College of Graduate Studies for acceptance, a thesis entitled:INCORPORATING FEAR (UN)LEARNING MECHANISMS ININFECTIOUS DISEASE MODELLINGsubmitted by Avneet Kaur in partial fulfilment of the requirements of thedegree of Master of ScienceExamining Committee:Prof. Rebecca Tyson, Irving K. Barber Faculty of ScienceSupervisorDr. Iain Moyles, York UniversityCo-SupervisorDr. Bert Baumgaertner, University of IdahoSupervisory Committee MemberDr. Ifeoma Adaji, Irving K. Barber Faculty of ScienceSupervisory Committee MemberDr. Daniel Coombs, University of British Columbia - VancouverUniversity ExamineriiAbstractEffective management of epidemics requires not only modeling the dis-ease itself but also the fear response of the population to both the diseaseand the vaccine. Risk perceptions of the disease may drive fear of diseasein the population, thus encouraging people to adopt prophylaxis. On theother hand, negative thoughts about vaccines may drive fear of vaccines inthe population, thus making some people vaccine hesitant. Since these fearresponses affect population behaviour and thus disease transmission rate,fear responses in turn influence the disease trajectory.In this thesis, we investigate an existing model of disease transmissionthat considers fear of disease and fear of vaccine as contagions that occuralongside the usual contagion of the disease. The mechanisms of fear ac-quisition and loss are rooted in human behavioural science. We analysethis model to determine the conditions under which each of the contagions(disease, fear of disease, or fear of vaccine) initially grows the fastest.We then analytically investigate a reduced version of the full model withjust the fear of disease (i.e., no vaccine or fear of vaccine). We considerthis reduced model in two scenarios, one where fear is transmitted at a rateslower than that of the disease, and the other where the two transmissionrates are comparable. In the former scenario, we show that behaviouralchange has little to no effect on the disease dynamics, but in the lattercase, we observe that sufficient behavioural intervention can suppress theepidemic yielding bifurcations in final epidemic size, the number of infectionwaves, and their relative peak size. In both scenarios, we observe that thedisease-fearful population may retain fear of disease even after the outbreakis over.Finally, we propose an extension to the full model by adding a compart-ment for the population that is simultaneously fearful of both the diseaseand the vaccine. Analysis of the modified model gives us insights into therich disease dynamics of this model. In particular, we observe that the finalepidemic size remains low when a large fraction of the population either getsvaccinated or shifts to the double fear compartment.iiiLay SummaryPredicting and managing outbreaks requires a comprehensive under-standing of behaviour changes due to fear response of the population toboth the disease and the vaccine. Fear of disease promotes disease protec-tive actions such as contact reduction and vaccination, while fear of vaccineleads to vaccine hesitancy. These fear responses in turn affect the courseof an outbreak. In this thesis, we analyse models that incorporate humanbehavioural changes in traditional disease models and suggest general strate-gies to suppress an outbreak. One significant finding of our work is that foroutbreaks in which the population responds slowly, the ability to reduce thetotal infected population by enforcing and ensuring compliance to policiesis limited. In such scenarios, it may not be optimal to rely on fear as amotivator for behavioral change. On the other hand, for outbreaks of noveldiseases in which the population responds promptly, the total infected pop-ulation can be reduced if sufficient population retains fear of disease thusavoiding contact with infectious individuals.We propose a disease model where individuals can be simultaneouslyfearful of both the disease and the vaccine. The analysis of this model pro-vided a novel approach to lower the total infected population when severalindividuals refuse to get vaccinated. In such a scenario our result suggeststo make the vaccine hesitant individuals aware of the risks of becoming in-fected so that they choose mitigative behaviour to protect themselves fromdisease.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 1Chapter 2: The Triple contagion model . . . . . . . . . . . . . 62.1 Disease spread . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Acquisition of fears . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Decay of fears . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Vaccination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.6 Dominant Contagion . . . . . . . . . . . . . . . . . . . . . . . 13Chapter 3: Analysis of the Reduced triple contagion model . 223.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Model scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.1 Flu-like limit . . . . . . . . . . . . . . . . . . . . . . . 303.3.2 COVID-like limit . . . . . . . . . . . . . . . . . . . . . 373.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46vTABLE OF CONTENTSChapter 4: Modified triple contagion model . . . . . . . . . . . 494.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2 Model description . . . . . . . . . . . . . . . . . . . . . . . . . 494.3 Final Epidemic Size Analysis . . . . . . . . . . . . . . . . . . 534.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Chapter 5: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 70Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Appendix A: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88A.1 Scaling analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 88A.2 Flu-like limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 91A.3 COVID-like limit . . . . . . . . . . . . . . . . . . . . . . . . . 96viList of TablesTable 2.1 A description of parameters used in the Triple conta-gion model . . . . . . . . . . . . . . . . . . . . . . . . 7Table 2.2 A description of state variables used in the Triple con-tagion model . . . . . . . . . . . . . . . . . . . . . . . 8Table 3.1 A description of new parameters used in the modelanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . 25viiList of FiguresFigure 2.1 Compartmental diagram of the Triple contagion modelaccounting for fear of disease, vaccination and fear ofvaccine . . . . . . . . . . . . . . . . . . . . . . . . . . 7Figure 2.2 Parametric plot showing the parametric regions foreach contagion: the disease, the fear of disease, andthe fear of vaccine to dominate initially. . . . . . . . . 18Figure 2.3 Temporal plots of infectious subsystem to verify sys-tem dynamics in different parametric regions. . . . . . 20Figure 3.1 Plots of the disease-fearful and infectious fractionsover time, and of the final epidemic size for differentvalues of p for the Reduced model. . . . . . . . . . . . 23Figure 3.2 Plots of the disease-fearful and infectious fractionsover time, and of the final epidemic size for differentvalues of αf for the Reduced model. . . . . . . . . . . 24Figure 3.3 Plots of the final size of the disease-fearful and recov-ered populations as functions of rate of acquisition offear in the Flu-like limit. . . . . . . . . . . . . . . . . 35Figure 3.4 Temporal plots of the fraction of the population inthe disease-fearful, infectious and recovered compart-ments as functions of rate of acquisition of fear in theFlu-like limit. . . . . . . . . . . . . . . . . . . . . . . 36Figure 3.5 Plots of the final size of the disease-fearful and recov-ered populations as functions of rate of acquisition offear in the COVID-like limit. . . . . . . . . . . . . . . 44Figure 3.6 Temporal plots of the fraction of the population inthe disease-fearful, infectious and recovered compart-ments as functions of rate of acquisition of fear in theCOVID-like limit. . . . . . . . . . . . . . . . . . . . . 45viiiLIST OF FIGURESFigure 4.1 Compartmental diagram of the Modified triple conta-gion model accounting for fear of disease, vaccination,fear of vaccine, and double fear . . . . . . . . . . . . . 52Figure 4.2 Heatmap of the final epidemic size with respect to pand pfdv for the Modified triple contagion model. . . . 54Figure 4.3 Plots of the final epidemic size with respect to βfv fordifferent values of αf . . . . . . . . . . . . . . . . . . . 56Figure 4.4 Heatmap of the final epidemic size with respect to βfvand αf for the Modified triple contagion model. . . . 57Figure 4.5 Temporal plots of the population in each of the com-partments in region A of final epidemic size heatmap. 58Figure 4.6 Temporal plots of the population in each of the com-partments in region B of final epidemic size heatmap. 59Figure 4.7 Temporal plots of the population in each of the com-partments in the region C of final epidemic size heatmap. 60Figure 4.8 Temporal plots of the population in each of the com-partments in region D of final epidemic size heatmap. 61Figure 4.9 Temporal plots of the population in each of the com-partments in region E of final epidemic size heatmap. 63Figure 4.10 Temporal plots of the population in each of the com-partments in region F of final epidemic size heatmap. 64Figure 4.11 Heatmap of the final epidemic size with respect to βfvand αf for the Triple contagion model. . . . . . . . . 66Figure 4.12 Heatmap of the final epidemic size with respect to βfvand αf for the No-vaccination submodel. . . . . . . . 67ixAcknowledgementsFirst, I would like to extend my gratitude to my supervisor, Dr. RebeccaTyson, and my co-supervisor, Dr. Iain Moyles, for their consistent guidanceand unwavering support throughout my graduate studies. Their attention todetail, insightful feedback, and dedication to ensuring clarity and coherencetruly enhanced the quality of my work. I am grateful to Dr. Tyson foralways being encouraging and positive, and to Dr. Moyles for always beingenthusiastic and for introducing me to the realm of asymptotic analysis.Additionally, I would like to thank Dr. Bert Baumgaertner and Dr.Ifeoma Adaji, members of my supervisory committee, for dedicating theirtime and providing thoughtful suggestions on my thesis. Special thanks toDr. Baumgaertner for having insightful discussions on our work through thelens of cognitive science.I would also like to thank the Tyson Lab (Tylab) research group for theirsupport, valuable insights, and encouragement.Lastly, and most importantly, I am thankful to my family for being mybiggest supporters and source of strength.xDedicationTo my loving family,for their unconditional support and love.xiChapter 1IntroductionInfectious diseases cause worldwide destruction both socially and eco-nomically. There is a long list of infectious diseases that, throughout his-tory, have had a significant impact on human lives. For example, the plagueof Justinian between 541 and 543 killed approximately 100 million people,which led to the decline of the Byzantine Empire [20]. Cholera, a highly fataldisease, has been a global burden since the 1840s. Scientists estimated thatCholera cases between 2008 and 2012 may have affected as many as 1.3 to 4million people per year with 95,000 deaths [1]. The 20th century witnessedmultiple Influenza pandemics worldwide, with those in 1918, 1957, and 1968severely affecting infants and elderly people. The 1918 influenza pandemicalone resulted in approximately 500 million infections and 50 million deathsworldwide [38]. More recently, the ongoing COVID-19 pandemic has putsignificant strain on the healthcare systems and economies of nations acrossthe globe. A study conducted to evaluate the worldwide burden of infectiousdiseases revealed that almost one-third of total deaths (from any cause) weredue to infectious diseases [49].Human behavioural changes play a crucial role in shaping the trajectoryof infectious diseases. As the disease pathogen spreads, the associated mor-bidity and mortality give rise to population-level perceptions of the risk andseverity of the disease. As a result, people try to minimise those risks bychanging their behaviours, most commonly by adopting prophylactic actionssuch as vaccination, social distancing, wearing face masks, etc. These newbehaviours, in turn, shape the course of the epidemic. The feedback loopbetween disease and behaviour comes into play during outbreaks as observedin [25–27, 30, 32, 37, 52, 60–62, 74, 75] and hence it is essential to accountfor the coupled effects of disease and behaviour dynamics while developingmodels. This coupled approach has given rise to the field of BehaviouralEpidemiology, which emphasizes the incorporation of theories from sociol-ogy and psychology as key components of any model of the interaction ofhuman behaviour and disease dynamics [4, 32]. Many modelling efforts inBehavioural Epidemiology have demonstrated a change in disease dynamicsdue to human behavioural changes [5–8, 25–27, 30, 37, 52, 55, 60–62, 74, 75].1Knowledge of modelling human behavioural changes during outbreaks canhelp develop a comprehensive understanding of disease transmission dynam-ics and make better-informed health policy decisions.Pandemics such as the COVID-19, the Influenza, the SARS, and theEbola pandemics among others exhibited multiple infection waves each witha different peak size and time of onset (relative to the initial wave). However,the standard infectious disease model in epidemiology, the SIR model [43]yields only a single infection wave and hence fails to explain the complexmulti-peaked dynamics observed in real pandemics. The SIR model failsbecause of the assumption that transmission rates remain constant throughtime, i.e., that people do not change their behaviour as the pandemic pro-ceeds. This inconsistency between real epidemics and those produced by thestandard model reinforces the need for modelers to add meaningful changesin behaviour to their models.Literature ReviewModellers have been actively collaborating with researchers in the be-havioural sciences to address the limitation of traditional models. We reviewrecent efforts in the field of Behavioural Epidemiology to integrate diseasemodels with human behavioural science. Mathematical models which incor-porate behaviour are generally classified in three ways: prevalence-based,belief-based, and a combination of both as suggested in [30, 60]. The mod-els are categorised based on the type of information to which individualsrespond such as disease prevalence and related factors including hospitalisa-tions, mortality, etc or information not directly relating to disease prevalencesuch as knowledge of thoughts of people who are already fearful or the peoplewho got infected.In prevalence-based models, the decision-making process primarily re-lies on the size of infectious pool. The behaviour change in this scenarionecessarily requires the occurrence of an outbreak [30]. Some papers thatused a prevalence-based fear response approach to model the coupling be-tween disease and behaviour are [39, 59–62, 74]. For example, in [60], theauthors characterized models according to the type (belief- or prevalence-based) and source of information (local or global) to which the populationresponds as suggested by [30]. The behaviour response considered in thismodel is acquisition and decay of fear of disease which gives rise to twoclasses of susceptibles: regular (S) and fearful (Sf). The fearful popula-tion adopts prophylaxis hence reducing their contractibility of the disease,while the regular population is non-fearful and does not adopt prophylaxis2hence increasing their vulnerability to infection. A similar model formu-lation is considered in [26, 27]. In the local, prevalence-based model, theauthors assume that regular susceptibles become afraid of the disease onlyby interacting with sick individuals. On the other hand, in the global,prevalence-based model, the susceptible population responds to the globalprevalence information generally available via news, television, and the in-ternet. A similar approach was used in [32]. Another example of a coupleddisease and opinion dynamics model is that presented by Tyson et al [74] tostudy the effects of transient prophylaxis on the disease trajectory. They as-sume that individuals decide to change their behaviour based on informationcoming from prophylactic and non-prophylactic voices and the cost of pro-phylaxis. The influence of prophylactic voices increases as disease prevalenceincreases. Moreover, as disease prevalence decreases, the cost of prophylaxisdominates the behaviour calculation motivating people to switch to the lesscostly category of non-prophylaxis.In belief-based models, the decision-making process is mainly influencedby information not directly relating to disease prevalence [27, 48, 60] suchas knowledge of thoughts of people who are already fearful or the peoplewho got infected. The behaviour change in this case can happen even in theabsence of an outbreak. These models relate to behavioural changes frominformation that can spread on very different time scales than that of thedisease dynamics [30]. For example, it is known from fear acquisition theorythat fear of disease, which encourages prophylactic behaviour, can spreadmuch faster than the disease itself because this process does not require anactual contact with a sick or a disease-fearful individual [26]. Vaccine uptakeis another example of decision-making that can be independent of diseaseprevalence. Individuals who have negative experiences or thoughts aboutvaccines may exhibit vaccine hesitancy, choosing to forgo vaccination forthemselves and their dependents [30]. A model with disease-agnostic feartransmission is proposed in [60]. The authors also consider the possibilityof reversion to regular social behaviour on contact with the regular, non-fearful susceptibles. The rate of fear transmission and loss is assumed to beproportional to the size of the fearful and regular susceptible populations re-spectively. Another example of a model which considers belief-based changesin behaviour is [48]. The authors treat anti-vaccine sentiment as a pathogenthat can spread through interactions between individuals analogous to thetransmission of disease pathogen.There are a suite of models that consider both the prevalence-based andbelief-based decision making processes (cf. [26, 27, 37, 73]). For example,in [73], a simplified model of opinion dynamics during epidemics is pre-3sented through a spectrum of behaviours denoting prophylactic attitudesof varying degrees. The prophylactic strength of an individual is assumedto change through the disease prevalence or through belief-based interac-tion with other individuals via an influence function. The influence functiongives the rate at which individuals of a particular opinion strength changethe opinion of others on the spectrum. Different influence functions are con-sidered to weigh one side of the spectrum more than the other. Iwasa etal [37] propose yet another simplified model representing fearful and non-fearful populations. The switch from the non-fearful to the fearful state isdriven by prevalence information and the reverse is driven by the belief-based interactions with the non-fearful population. This model captures thedynamics of both disease-infected and fear-infected classes with a system ofjust two ordinary differential equations.The behavioural changes during outbreaks result in a change in the rateof disease transmission. Incorporating prevalence- and belief-based mecha-nisms of fear acquisition and loss in the disease models gives us a way tocapture the variability of the disease transmission rate over the course ofoutbreaks in a quantifiable mechanistic way. The models that incorporatebehaviour successfully explain multiple infection waves and varying peaksize and time of onset of subsequent waves observed in real-life pandemicssuch as the COVID-19, the Influenza, the SARS, and the Ebola pandemics.For example, [61, 74] obtained up to four waves from prevalence-based re-sponse models depending on how actively people respond to the outbreakand change their behaviour towards prophylaxis or non-prophylaxis relativeto the rate of disease spread. Moreover, the relative speed of transitionbetween fearful and non-fearful classes also affects the subsequent waves.A slower transition from the fearful to non-fearful class delays the onsetof subsequent waves as observed in [26, 37, 60, 74] and increases the peaksize of subsequent infection waves compared to the first infection wave asobserved in [26, 52, 60]. The size of subsequent waves increases when the ef-fective fraction of susceptibles increases because of individuals who becomecomplacent about the disease after the first infection wave.Another interesting result obtained from models with fear response is abifurcation in the total infected population and final size of disease-fearfulpopulation with respect to the fear-related parameters such as rates of ac-quisition and loss of fear. The disease-fearful population may remain fearfuleven after the outbreak is over. Specifically, the belief-based mechanismsresult in fearful individuals retaining fear of disease after the disease hasvanished [61]. The individuals residing in the fear compartment successfullyescape the infectious individuals hence bringing a bifurcation in the total4infected population as observed in [26, 60–62]. It is an open problem toidentify if a particular class of models can explain specific results. Modelswith different fear mechanisms can explain different characteristics of dis-ease dynamics in real-life pandemics as seen in [26, 27, 37, 52, 60, 61, 74].The authors of [26] qualitatively explained these characteristic results fromtheir prevalence- and belief-based model. In this thesis, we take a closerlook at their model [26] to explain their results analytically and then extendtheir model.The model [26] is named as the Triple contagion model as it includesthree different types of contagion within the susceptible population: spreadof (i) disease, (ii) fear of disease, and (iii) fear of vaccination. In the originalarticle [26], the authors present the assumptions underlying the processesof fear transmission, and loss in their model that have been drawn fromtheories on human behavioural adaptation in neuroscience. In chapter 2,we introduce the model and present an elaborate explanation of differentbehaviour changes from human behavioural science built on the concepts in[26]. Then we analyse the Triple contagion model to determine the contagionthat is initially dominant. Our analysis provides different parametric regimesin which each of the contagions present in the model, i.e., disease, fear ofdisease, and fear of vaccination take-off before the other contagions.In chapter 3, we provide a detailed analytical study of the system dy-namics. We investigate a reduced version of the Triple contagion model withjust the fear of disease. We carry out a stability analysis of the Reducedtriple contagion model. To further investigate the model, we scale the fearparameters in two ways: with respect to the rate of fear loss or the rate ofdisease recovery and conduct asymptotic analysis. In [26], the simulationswere run by scaling the fear parameters with respect to the rate of infection.The Triple contagion model accounts for individuals with fear of diseaseor fear of vaccine. It does not consider the possibility of coexistence of thetwo fears. In chapter 4, we present a modified version of the Triple conta-gion model by adding a double fear compartment. The model accounts forindividuals who are fearful of both the disease and vaccine. We conduct aqualitative analysis of the bifurcation in the total infected population withrespect to fear parameters such as p, pfdv, βfv, and αf . We look at temporalplots of the fraction of population in each compartment for different combi-nations of βfv and αf to understand the dynamics of each of the populationthat brings a change in the total infected population. Our analysis helpsus understand the structure of bifurcation in total infected population andmake suggestions on the ways in which total infected population may bereduced.5Chapter 2The Triple contagion modelIn this chapter, we discuss and analyse the Triple contagion model (2.1)proposed in [26]. We introduce the model and discuss the epidemiologicaland cognitive basis for movements between different compartments in theTriple contagion model (2.1). In section 2.6, we analyse the Triple contagionmodel (2.1) to find which contagion (disease, fear of disease, and fear ofvaccine) dominates initially and how the trajectory of the epidemic looks iffear leads or trails.The compartmental diagram of the Triple contagion model is shown inFigure 2.1. The model (2.1) is built upon the fundamental Susceptible-Infected-Recovered model [43]. The susceptible compartment from the SIRmodel is split into three groups: the susceptibles who are fearless (S), thesusceptibles who are fearful of the disease (Sfd), and the susceptibles whoare fearful of the vaccine (Sfv). The compartment I refers to the infectiousindividuals, Rnat refers to the individuals who recovered naturally from thedisease, Rvac refers to the individuals who gained immunity through vaccina-tion, and Avac refers to the fraction of population that experienced adversereactions from the vaccine. A summary of the model parameters and statevariables is given in Tables 2.1 and 2.2 respectively.6Figure 2.1: Compartmental diagram of the Triple contagion model account-ing for fear of disease, vaccination and fear of vaccine [26]. The compart-ments Sfv, S, and Sfd refer to the susceptibles fearful of the vaccine, the regu-lar susceptibles, and the susceptibles fearful of the disease respectively. Thecompartment I represents the infectious individuals. The compartmentsRnat, Rvac, and Avac refer to naturally recovered individuals, vaccinatedindividuals, and vaccinated individuals who experienced adverse reactionsrespectively. The rates of transitioning between different compartments aredisplayed on top of the arrows.Parameter Descriptionβ Rate of transmission of disease (time−1)βfd Rate of transmission of fear of disease (time−1)βfv Rate of transmission of fear of vaccine (time−1)αf Rate of loss of fear through contact (time−1)γ Rate of recovery from disease (time−1)γf Rate of spontaneous loss of fear (time−1)p Relative reduction in rate of disease transmissionη Fear difference scaling factorσ Rate of adverse reaction from vaccineϵ Maximum rate of vaccination (time−1)v(t) Rate of vaccinationTable 2.1: A description of parameters used in the Triple contagion model72.1. DISEASE SPREADVariable DescriptionS Fraction of fearless susceptible populationSfd Fraction of susceptible fearful of diseaseSfv Fraction of susceptible fearful of vaccineI Fraction of infectious populationRnat Fraction of naturally recovered populationRvac Fraction of vaccinated populationAvacFraction of vaccinees who had adverse reactionfrom vaccineTable 2.2: A description of state variables used in the Triple contagion model2.1 Disease spreadTwo coupled contagions happen in this model: one of the disease and theother of the fears (see Figure 2.1). The disease spreads through direct con-tact with an infectious individual. The fearless susceptible population comesinto contact with the infected population and acquires infection at a rate β.The susceptible population afraid of the vaccine does not behave differentlyfrom the fearless group and so we assume they acquire the disease at a rateβ as well. A successful contact between a susceptible and an infectious indi-vidual, in addition to transmission of disease, generates a source of furtherdisease spread. An infectious individual contributes to the disease trans-mission for an average of 1γ time units, eventually becoming permanentlyimmune by recovering naturally and going to the compartment Rnat. Thedisease-fearful susceptibles self-isolate to protect from disease. This fear-induced contact reduction scales down the contractibility of the disease forthe disease-fearful individuals by factor p. For example, a value of p = 0.70results in 30% decrease in the rate of transmission of disease.2.2 Acquisition of fearsThe second contagion considered in this model is fear, both of diseaseand vaccine. People adopt different behaviours according to their situationalcontext [79]. During a disease outbreak, the amygdala can be activated byperceived threats such as news of increasing cases or contact with someonewho is infected [26]. This activation triggers a fear response that can leadto increased anxiety which can motivate people to change their behavioursin order to protect themselves from the disease. In the basic SIR model,82.2. ACQUISITION OF FEARSit is assumed that every individual has the same disease contraction rate.However, people adapt their behaviours differently depending on the diseaseprevalence, their social circle, and their personal circumstances [26]. Somepeople get extremely afraid of the disease and adopt prophylactic actionslike going into isolation, wearing a face mask, practicing social distancing,frequently washing hands, etc which may reduce their susceptibility to dis-ease [74]. On the other hand, some people may develop fear of vaccine dueto their negative experiences or thoughts about the vaccines.The mechanisms of fear acquisition considered in this model are inspiredfrom the fear acquisition theory in neuroscience which gives a psychologi-cal explanation for how people develop fears and phobias [64]. It suggeststhat fears can be learned through different mechanisms such as classicalconditioning, vicarious learning, and information transmission [64]. Themechanism of fear acquisition considered in this model is vicarious learning,also known as social learning. It occurs when an individual learns a newbehaviour or develops a new emotional response by observing the behaviouror emotional response of others to a particular stimulus. We further considerthe two sub-categories of vicarious learning: imitative learning and obser-vational learning [40] to explain the two different ways in which individualslearn the fears.People develop fear by imitating the behaviour of others without anactual exposure to the stimulus [40]. For example, an individual may imitatetheir friend’s fear of a snake and develop their own fear without actuallyconfronting a snake. In the disease context, when a regular susceptible inS observes or interacts with a fearful individual in Sfd, they may acquirefear of disease through imitation at a rate βfdSfd. Similarly, when a regularsusceptible observes or interacts with a fearful individual in Sfv, they mayacquire fear of vaccine through imitation at a rate βfvSfv. The imitative fearlearning occurs on interacting with someone who is already fearful (Sfd andSfv).Another mechanism of fear acquisition is observational learning [26, 40].It is a way to develop a fear response to a stimulus by directly observing thenegative outcomes or aversive experiences of others, such as observing some-one experiencing pain, or harm from a particular stimulus [40]. An exampleof vicarious learning by observation is when an individual observes a snakeattacking someone and subsequently develops a fear of snakes. So, in thiscase, when a regular susceptible individual observes the negative outcomesof getting infected, they may develop fear of disease through observation ata rate βfdI. The spread of fear of disease in this case is hence caused by in-dividuals in I. Similarly, a regular susceptible may acquire fear of vaccine at92.3. DECAY OF FEARSa rate βfvAvac by observing the adverse reactions from vaccines experiencedby individuals in Avac. The observational fear learning occurs on observ-ing individuals who experienced effects of infection or experienced adversereaction of vaccine (I and Avac).2.3 Decay of fearsFear is not permanent and it decays in the absence of an aversive stim-ulus or in the presence of a positive stimulus. The movements from thecompartments Sfd and Sfv to S denote loss of fear of disease and vaccinerespectively. One of the mechanisms of fear decay is spontaneous fear de-cay in which an individual’s fear response vanishes without any deliberateintervention [26]. This can happen naturally over time as the individual isrepeatedly exposed to the stimulus or deprived of the exposure. The twopossible sub-mechanisms that may give rise to spontaneous loss of fear arestimulus deprivation [26] and extinction learning [54, 63]. When an indi-vidual is deprived of exposure to a specific stimulus for an extended periodof time, their previously learned association between a stimulus and a re-sponse weakens or vanishes completely because of stimulus deprivation. Forexample, someone who has a fear of dogs but has not been around any dogsfor many years may experience a reduction in their fear response becausethe means of fear acquisition has been cut down. Conversely, when an in-dividual is exposed to the aversive stimulus without experiencing negativeconsequences, their fear response is overwritten via extinction learning. Thisnew learning comes into play when the outcome violates expectations [66].Additional factors like fatigue from staying at home, protecting against dis-ease, or economic aspects can also manifest a spontaneously compromisedfear response. Our model cannot distinguish these fear sub-mechanisms thatmay lead to spontaneous loss of fear. Hence, we assume that spontaneousfear decay brought about by any of the listed or other factors happens at arate γf .A second mechanism of fear decay is complacency contagion [26, 61].There are two possible sub-mechanisms for complacency leading to fear de-cay: counter-conditioning [42] and cognitive restructuring [10, 17]. An ex-periment was conducted on mice by neuroscientists to understand and weighthe efficacy of different ways of unlearning a conditioned fear response [14].In this experiment, the researchers paired a harmless white light with anaversive mild shock delivered through the floor of the rat cage, until micebegan to manifest amygdala response to the white light. Then, they exposed102.4. VACCINATIONthe mice to white light at different time intervals without any electric shockto investigate how long the mice take to overcome aversion to the whitelight. Repeated exposure to a conditioned stimulus has long been knownto extinguish fear by exposing animals (including humans) to that stimuluswithout associated pain. The study revealed that through this exercise someof the mice learned to trust that white light does not come with shocks.Counter conditioning involves a fear-provoking stimulus paired with apositive stimulus so that individual is conditioned to leave their fear re-sponse. The second sub-mechanism of complacency is cognitive restructur-ing. If an individual’s fear decreases after observing that a feared stimulusis not really harmful or that its harmful effects go away with time, this canbe considered an example of fear reduction through cognitive restructuring[10, 17]. Our model cannot distinguish these fear sub-mechanisms that maylead to loss of fear due to complacency. An example of complacency leadingto fear decay is, someone who had developed a fear of dogs after being bittencould be exposed to playful, gentle dogs as a way to re-learn that most aresafe. In the disease context, an individual fearful of the disease becomes lessafraid after contacting someone who has recovered from the disease (Rnatindividuals) decreasing the risk of harm from contracting it. Similarly, anindividual fearful of vaccine may become complacent and leave fear of vac-cine after contacting individuals in Rvac compartment. The loss of fear, inthis case, takes place at a rate αf which is the effective contact rate for fearloss.2.4 VaccinationIt is assumed that the fearless susceptibles do not get vaccinated due tothe lack of motivation to protect themselves against disease. Moreover, thesusceptibles fearful of the vaccine choose not to get vaccinated due to theirnegative thoughts about vaccines. So, the individuals who are fearful of thedisease are the only group of people who choose to get vaccinated at a time-varying rate of vaccination v(t) as a part of their prophylactic behaviour.Note that the vaccinated individuals, like naturally recovered individuals,gain permanent immunity to the disease pathogen. Every vaccine comeswith side effects such as fatigue, fever, and soreness at the injection site[76]. These are generally mild to moderate varying from person to personand vanish on their own in few days [76]. However, adverse reactions from avaccine can prompt unvaccinated individuals to develop a fear of vaccination[16].112.5. MODELWe assume a proportion σ of the vaccinated individuals experience ad-verse reactions from the vaccine going to the compartment Avac and theremaining fraction of vaccinated individuals who didn’t experience any sideeffects move to the compartment Rvac. The vaccine-fearful vaccinees (Avac)spontaneously lose the fear of vaccine and stop contributing to the transmis-sion of fear of vaccine at a rate γf . Note that γf represents the rate at whichindividuals in Avac recover from adverse reactions and hence forget abouttheir adverse experience with vaccine [26]. We assume that complacencymechanisms may not play a role in fear loss here because the direct experi-ence of individuals in compartment Avac with the side effects of vaccine maymake them resistant to lose fear through complacency.It is assumed that the rate of vaccination increases if fear of diseaseexceeds the fear of vaccine and it decreases when fear of vaccine exceeds thatof disease. This is represented by the difference in the fraction of populationin the two fear compartments, Sfd and Sfv and is given by (2.1h). Here η isthe fear difference scaling factor and ϵ is the maximum rate of vaccination.So, as the rate of vaccination approaches ϵ, dvdt → 0.2.5 ModelIn the Triple contagion model (2.1), we assume that all the state vari-ables represent the fraction of the total population in each compartment.Hence, each of the state variables is a fraction: 0 < X < 1 for X ∈[S, Sfd, Sfv, I, Rnat, Rvac, Avac] and all of them sum up to one under the con-stant population constraint: S+Sfd+Sfv+ I+Rnat+Rvac+Avac = 1. Themodel (2.1) is a nonlinear ordinary differential equation model that cou-ples disease dynamics with behaviour adaptation mechanisms. The modelequations are given as,dSdt= −βIS︸ ︷︷ ︸Disease spread− (βfdSfd + βfvSfv)S︸ ︷︷ ︸Imitative fear acquisition− (βfdI + βfvAvac)S︸ ︷︷ ︸Observational fear acquisition+ γf(Sfd + Sfv)︸ ︷︷ ︸Spontaneous fear loss+αf(RnatSfd +RvacSfv)︸ ︷︷ ︸Fear loss by complacency, (2.1a)dSfddt= −pβISfd︸ ︷︷ ︸Disease spread− γfSfd︸ ︷︷ ︸Spontaneous fear loss− αfRnatSfd︸ ︷︷ ︸Fear loss by complacency+ βfdSfdS︸ ︷︷ ︸Imitative fear acquisition+ IS︸︷︷︸Observational fear acquisition− vSfd︸︷︷︸Vaccination, (2.1b)122.6. DOMINANT CONTAGIONdSfvdt= −βISfv︸ ︷︷ ︸Disease spread− γfSfv︸ ︷︷ ︸Spontaneous fear loss− αfRvacSfv︸ ︷︷ ︸Fear loss by complacency+ βfvSfvS︸ ︷︷ ︸Imitative fear acquisition+ AvacS︸ ︷︷ ︸Observational fear acquisition, (2.1c)dIdt= βIS + pβISfd + βISfv︸ ︷︷ ︸Disease spread− γI︸︷︷︸Recovery from disease, (2.1d)dRnatdt= γI︸︷︷︸Recovery from disease, (2.1e)dRvacdt= (1− σ)vSfd︸ ︷︷ ︸Vaccination without adverse reaction+ γfAvac︸ ︷︷ ︸Spontaneous fear loss, (2.1f)dAdt= σvSfd︸ ︷︷ ︸Adverse reaction from vaccine− γfAvac︸ ︷︷ ︸Spontaneous fear loss, (2.1g)dvdt= η(Sfd − Sfv)(ϵ− v)v. (2.1h)2.6 Dominant ContagionEquation (2.1h) driving the change in the rate of vaccination highlightsthe importance of the trade-off between fear of disease and fear of vaccinein determining vaccine uptake. This trade-off between the two fears mayalso affect the disease trajectory. The disease spread may be suppressed byprophylaxis arising from fear of disease in the form of contact reduction andvaccine acceptance [36, 77]. On the other hand, vaccine hesitancy due to fearof vaccine may drop the vaccine coverage hence promoting disease spread[58]. For any disease, the healthcare authorities are interested in developingstrategies to control the spread. We investigate the model (2.1) to find theconditions under which each of the contagions, disease, fear of disease, andfear of vaccine shows the most rapid exponential growth in the beginning (wecall it as the Dominant contagion). To figure out the dominant contagion,we calculate the basic reproduction numbers for each of the contagions inthis model.The basic reproduction number, generally denoted by R0, is a key con-cept in epidemiology that represents the average number of secondary casesproduced by a single infected individual in a population of susceptible indi-viduals [21]. It is a useful tool for understanding the potential of infection tospread within a population [41]. The infection has the potential to spread132.6. DOMINANT CONTAGIONif the basic reproduction number is larger than unity. We use the nextgeneration matrix (NGM) method proposed by Diekmann et al [22] andelaborated by Van den Driessche and Watmough [24] to calculate the basicreproduction number.The first step of the NGM method is to find the infectious subsystem ofthe model (2.1). In this model, the fear spread is considered as a contagion inaddition to disease. So, we also consider the fear-transmitting compartmentsin the infectious subsystem (in addition to the disease-infected individuals).The infectious subsystem hence consists of I, Sfd, Sfv, and Avac and is givenas follows:dIdt= βIS + pβISfd + βISfv − γI, (2.2a)dSfddt= −pβISfd − γfSfd − αfRnatSfd + βfd(Sfd + I)S − vSfd, (2.2b)dSfvdt= −βISfv − γfSfv − αfRvacSfv + βfv(Sfv +Avac)S, (2.2c)dAvacdt= σvSfd − γfAvac. (2.2d)The second step of the NGM method comprises finding the infection-free equilibrium of the system. Considering the infectious subsystem (2.2a),one of the infection-free equilibrium of the model (2.1) is when the wholepopulation is susceptible. The third step of the NGM method is to evaluatethe next-generation matrix at the infection-free equilibrium. To calculatethe next-generation matrix, we first calculate the Jacobians of the vectors fand v, where f represents the rates of flow from all the compartments intothe infected ones (both disease and fears), and v represents the cross-flowsbetween infectious compartments (both disease and fears). Then, the vectorrepresenting the change in infectious compartments can be represented asx′ = f − v, (2.3)where, the vector f is given as,f =f1f2f3f4 =βIS + pβISfd + βISfvβfd(Sfd + I)Sβfv(Sfv +Avac)SσvSfd , (2.4)142.6. DOMINANT CONTAGIONand the vector v is given as,v =v1v2v3v4 =γIpβISfd + γfSfd + αfRnatSfd + vSfdβISfv + γfSfv + αfRvacSfvγfAvac . (2.5)The Jacobian matrix of f is given as,F =βS + pβSfd + βSfv pβI βI 0βfdS βfdS 0 00 0 βfvSβfvS0 σv 0 0 , (2.6)and the Jacobian matrix of v is given as,V =γ 0 0 0pβSfd pβI + γf + αfRnat + v 0 0βSfv 0 βI + γf + αfRvac 00 0 0 γf . (2.7)We obtain the next-generation matrix evaluated at the disease-free equilib-rium as follows:G = FV −1 =βγ 0 0 0βfdγβfdγf+v0 00 0 βfvγf 00 σvγf+v 0 0 . (2.8)The basic reproduction number is obtained from the dominant eigenvalueof the next-generation matrix (2.8). The eigenvalues of the next-generationmatrix, G, can be obtained from the diagonal entries of G as,λ1 = 0, (2.9a)λ2 =βγ, (2.9b)λ3 =βfdγf + v, (2.9c)λ4 =βfvγf. (2.9d)152.6. DOMINANT CONTAGIONThe zero eigenvalue (2.9a) denotes the nullspace of the system and is as-sociated with the constant population constraint. If we remove one of thecontagious compartments (say Avac) assuming that the conservation of pop-ulation determines Avac, then we would obtain three non-zero equilibria.The dominant eigenvalue is determined by the maximum eigenvalue,i.e., max(λ2, λ3, λ4). Each eigenvalue corresponds to the basic reproductionnumber for one one of the three contagions. If the dominant eigenvalueexceeds one, that will be the contagion that has a first outbreak. Thispattern means that fear outbreaks can occur before the disease outbreak.The eigenvalue (2.9b) is the same as the basic reproduction number ofdisease from the standard SIR model. It is the ratio of the transmission rateof the disease and the recovery rate of the disease. Hence, the disease willtake off before the fear contagions when (2.9b) is the dominant eigenvalueand is greater than 1.The third eigenvalue (2.9c) is the ratio of the rate of acquiring fearof disease and the rate of losing fear. This eigenvalue denotes the basicreproduction number of the spread of fear of disease. Notice that the lossof the fear of disease receives contribution from both γf and v(0). Thevaccination term appears because immunization may reduce individuals’fear of disease. We assume the initial rate of vaccination is very small(v(0) = 0.0001) and hence vaccination has a negligible effect on determiningfear of disease as the dominant contagion. The fear of disease will proliferatebefore the other two contagions if (2.9c) is the dominant eigenvalue and isgreater than 1.Lastly, the eigenvalue (2.9d) represents the basic reproduction numbercorresponding to the fear of vaccine. It is the ratio of the rate of acquiringfear of vaccine and the rate of losing the same. Like the other two contagions,fear of vaccine will escalate before the other two infections if (2.9d) is thedominant eigenvalue and is greater than 1.An example of an epidemic in which fear of disease took off before thedisease itself is the 2014 Ebola outbreak in West Africa and worldwide [69].The initial panic and concern about the outbreak were disproportionate toits transmissibility. The perception of disease risk was largely driven bymedia coverage and statements by public health officials. Disease fear led towidespread school closures, travel restrictions, and other measures to containthe spread of the virus which was not even circulating to its full potential[69]. Eventually, people became fatigued of protecting themselves and wentback to normal. Once the fear subsided, the disease outbreak occurred. Thefear and panic caused by the Ebola outbreak had a significant impact onglobal society as it led to economic disruption, and led to a loss of public162.6. DOMINANT CONTAGIONconfidence in governments and public health officials [69].An example of an epidemic in which fear of the vaccine accelerated be-fore the spread of the disease and the fear of the disease is the 1976 swineflu outbreak in the US [29]. Some people caught the virus and one died in amilitary camp in New Jersey. The virus was perceived to be very deadly dueto its similarity with the virus that caused the 1918 Spanish flu. This led towidespread fear and panic encouraging the government and health author-ities to immediately initiate vaccination campaigns all across the country.Approximately 20% of Americans were vaccinated for the new strain, butthe virus didn’t spread outside the military camp [29]. Moreover, the vac-cine was not thoroughly tested and was causing neurological side effects in avery small fraction of the population. As a result, people lost trust in publichealth authorities [29]. This vaccination campaign was a huge failure dueto inappropriate planning and implementation of preventive actions [29].The eigenvalues (2.9c) and (2.9d) relating to fear contagions do not con-tain the parameter αf . Recall that the parameter αf represents the rateof losing fear via complacency contagion on contacting individuals who at-tained immunity against the disease either through recovery from infection(Rnat) or through vaccination (Rvac). In the initial phase of the disease,there are no individuals in the immunized compartments to induce loss offear and hence αf does not appear in the basic reproduction numbers whichmeasure contagion growth with a naive susceptible population.Figure 2.2 gives a visual presentation of the parametric regions in whicheach contagion dominates over the others initially. The x- and y-axes rep-resent βfd and βfv scaled with respect to β. All other parameters are keptfixed. We expect disease, fear of disease, and fear of vaccine to have the firstoutbreak in the parametric regions red, green, and blue respectively.The region of dominance shifts from red to green ifβfdβ=γf + v(0)γ, (2.10)and for the values of parameters considered here, the transition point is atβfdβ= 0.358. (2.11)Similarly, the system shifts from the red region to blue if,βfvβ=γfγ= 0.357, (2.12)172.6. DOMINANT CONTAGIONFigure 2.2: Parametric plot showing the parametric regions for initial dom-inance of each contagion: disease (red), fear of disease (green), and fear ofvaccine blue. The fixed parameters are: γ = 0.14, γf = 0.05, v = 0.0001.for the parameters considered here. Lastly, the transition from the blue tothe green region is given by,βfv =(γfγf + v(0))βfd, (2.13)which confirms the linear relationship as seen in Figure 2.2. For the casewhen there is no vaccination initially (v(0) = 0), the slope of the linearrelation between βfd and βfv is one. So, fear of disease is dominant initiallyif βfd > βfv and vice versa. On the other hand, if v(0) ̸= 0, the slope ofthe linear relation between βfd and βfv becomes less than one implying thata smaller rate of acquisition of fear of vaccine can shift the system to theparametric region in which fear of vaccine is dominant. When vaccination isintroduced right at the beginning of the outbreak, a part of the populationleaves Sfd compartment by getting vaccinated but the Sfv population buildsup hence making fear of vaccine dominant initially. This result representsa scenario in which vaccine becomes victim of its own success. A positiveimplication of this result is that the disease-fearful individuals are gainingpermanent immunity by getting vaccinated. These immunized individualscan then drive complacency in the vaccine-fearful individuals and can con-182.6. DOMINANT CONTAGIONtribute to reduce vaccine hesitancy.Figure 2.3 shows temporal plots of the fraction of population in the com-partments I, Sfd, and Sfv for values of βfd and βfv at different locations in theparametric plot 2.2 to verify the contagion with the most rapid exponentialgrowth. In Figure 2.3 (a) we take βfd = 0.2β, βfv = 0.1β (corresponding tothe red region in Figure 2.2) and confirm that the disease has the fastest ex-ponential growth. Similarly in Figure 2.3 (b) we take βfd = 0.5β, βfv = 0.2β(corresponding to the green region in Figure 2.2) and confirm that the fearof disease has the fastest exponential growth. In Figure 2.3 (c) we takeβfd = 0.2β, βfv = 2.5β (corresponding to the blue region in Figure 2.2) andconfirm that the fear of vaccine has the fastest exponential growth.In Figures 2.3 (d)-(f), we investigate the behaviour of the system forvalues of βfd and βfv at boundary regions in the parametric plot 2.2. Weobserve that the boundaries are regions of bi-stability and the system mayend up in either of the regions surrounding the boundary depending on thevalues of model parameters. In Figure 2.3 (d) we take βfd = 0.358β, βfv =0.2β (corresponding to the boundary between red and green region in Figure2.2) and observe that both disease and fear of disease have a comparablegrowth in the beginning with the disease becoming the faster contagioneventually. Similarly, In Figure 2.3 (e) we take βfd = 0.2β, βfv = 0.357β(corresponding to the boundary between red and blue region in Figure 2.2)and observe that disease has the fastest exponential growth in the beginning.Lastly, in Figure 2.3 (f) we take βfd = 0.6β, βfv = 0.59β (corresponding tothe boundary between blue and green region in Figure 2.2) and observe thatthe fear of disease has the fastest exponential growth initially.Our analysis allows us to determine the dominant contagion providesus with the conditions for each contagion corresponding to either disease orfears to exhibit the fastest exponential growth initially. Clearly, the differentscenarios of dominant contagions yield rich dynamics. The diversity in thedynamics at origin leads to question, what the overall dynamics look like?Further, different contagions being dominant initially may affect the overalldynamics in different ways thus raising questions about the implications ofdisease spread compared to fear propagation. We are therefore motivated tofurther analyse the dynamics of the model (2.1) and understand effect of fearon final size and disease burden. In this model, fear of disease encouragesboth the non-pharmaceutical interventions (NPIs) such as contact reduction,mask-wearing, increased hygiene, etc and the pharmaceutical interventions(PIs) such as vaccination [2]. To start with, we are motivated to analyse thesystem only with the fear of disease to understand the role played by non-pharmaceutical interventions in modifying the disease dynamics, recognizing192.6. DOMINANT CONTAGIONa) d)b) e)c) f)Figure 2.3: Temporal plots of each contagion: infectious (I) in red, fearof disease (Sfd) in green, and fear of vaccine (Sfv) in blue colour. Theparameters fixed across all these plots are: γ = 0.14, γf = 0.1, η = 0.8,σ = 0.2, ϵ = 0.2, p = 0.25, αf = 0.62, and β =27 . The initial conditions are:(S, Sfd, Sfv, I, Rnat, Rvac, Avac, v(0)) = (0.998, 0, 0, 0.002, 0, 0, 0, 0.0001).The values of βfd and βfv are chosen from different regions in the parametricplot (see Figure 2.2).202.6. DOMINANT CONTAGIONthat vaccines might not always be readily accessible to everyone [2]. In thenext chapter, we analyse a reduced version of (2.1) without vaccine and fearof vaccine.21Chapter 3Analysis of the Reducedtriple contagion model3.1 IntroductionConsider the Triple contagion model (2.1) but without vaccine and fearof vaccine. We are thus left with just two contagions: disease and fear ofdisease. The resulting Reduced triple contagion model isdSdt= −βIS − βfd(Sfd + I)S + γfSfd + αf(1− S − Sfd − I)Sfd, (3.1a)dSfddt= −pβISfd + βfd(Sfd + I)S − γfSfd − αf(1− S − Sfd − I)Sfd, (3.1b)dIdt= βIS + pβISfd − γI. (3.1c)where we have eliminated Rnat by using the constant population constraint,S + Sfd + I +Rnat = 1. For a description of variables and parameters usedin the model (3.1) and its analysis refer to Tables 2.1, 2.2, and 3.1.The Reduced triple contagion model was numerically studied in [26],where the authors demonstrated that the final epidemic size (i.e., the totalfraction of the population that is infected over the course of an outbreak)shows a sudden rise as either p or αf increase past a critical value (seeFigures 3.1 and 3.2, bottom plots reproduced by taking parameters as in[26]). They also recognised that as p increases, i.e., as fear-induced contactreduction becomes less effective, the infection dynamics transition from apair of waves to a single small-infection wave. A similar trend in solutionbehaviour was observed as a function of increasing αf . Note that the suddenrise in final epidemic size as p or αf increase also corresponds to the fearpopulation switching from a zero to a non-zero value at the end of theoutbreak. In other words, the final epidemic size is high when there are nodisease-fearful individuals left after the outbreak and the final epidemic sizeis low when there are still individuals left in the disease-fearful group at theend of the outbreak.223.1. INTRODUCTIONFigure 3.1: Plot of the fraction of the population fearful of disease (top) andthe infectious population (center) over time and plot of the final epidemicsize (bottom) for different values of p for the Reduced model. The fixedparameter values used in this simulation are: β = 27 , γ =17 , βfd = 1.1β,γf = 0.05, and αf = 2.2β.233.1. INTRODUCTIONFigure 3.2: Plot of the fraction of the population fearful of disease (top) andthe infectious population (center) over time, and plot of the final epidemicsize (bottom) for different values of αf for the Reduced model. The fixedparameter values used in this simulation are: β = 27 , γ =17 , βfd = 1.1β,γf = 0.05, and p = 0.25.243.1. INTRODUCTIONParameter Ratio Descriptionb βγ Disease reproduction numberbfβfdγfFear reproduction numbera αfγfRatio of rate of fear recovery through complacencyto rate of spontaneous fear recoveryδ γfγRatio of rate of fear recovery to rate of diseaserecoverybˆfβfdγRatio of rate of acquisition of fear to rate of diseaserecoveryaˆ αfγRatio of rate of fear recovery through complacency torate of disease recoveryTable 3.1: A description of new parameters used in the model analysisThe observed change in the number of epidemic waves arise as a conse-quence of the trade-off between vigilance (exhibited by the Sfd group) andcomplacency (exhibited by the S group) [26]. For smaller values of p andαf , people successfully reduce their susceptibility to the disease by reducingcontacts (smaller p), and remaining in the fear of disease state for longer(smaller αf). As a result, we observe a small infection wave. This resultshows how effective contact reduction can be in suppressing the disease. Asp and αf increase, prophylaxis decreases either due to decreased contact re-duction or due to higher fear loss. A larger fraction of the population thusabandons protective measures after the first wave has vanished, resulting ina bigger susceptible pool that gives rise to a second infection wave. As p andαf increase further, the effective fraction of susceptibles increases. Increasedsusceptibility results in a larger fraction of the population getting infectedand hence recovering from the disease. A larger recovered pool leads to morefear loss which in turn increases the susceptibility of individuals. This cycleof higher recovery from disease leading to more infection due to increasedsusceptibility shifts the second infection wave closer to the first one as p andαf increase. Consequently, contact reduction can control the size and timeof onset of the second wave.At this point, we have an intuitive understanding of the bifurcation be-253.2. STABILITY ANALYSIShaviour shown in Figures 3.1 and 3.2, but we do not know, in general, whatare the critical values of our bifurcation parameters for arbitrary values ofall of the other model parameters. So we will now turn to the mathematicalanalysis of (3.1). In Section 3.2, we carry out a stability analysis of thesystem (3.1) to obtain the conditions under which the system exhibits eachof the different final configurations.3.2 Stability analysisFinal epidemic size is a quantity of interest in epidemiology because itprovides a measure of the overall disease burden. To understand the finalconfiguration of the system we begin by finding the equilibria of (3.1) bysetting each of the derivatives to zero. We obtainE0(S∗0) = (S∗0 , 0, 0, 1− S∗0), (3.2a)E1(S∗fd) = (S¯, S∗fd, 0, 1− S¯ − S∗fd), (3.2b)where,S¯ =γf + αf(1− S∗fd)αf + βfd(3.3)and S∗0 and S∗fd are the fractions of the population in the regular susceptiblegroup (S), and the fearful susceptible group (Sfd) at the respective equilib-rium. Note that, multiple values of S∗0 , S¯ and S∗fd can satisfy the conditionsthat S∗0 < 1 and S¯ + S∗fd < 1 for these equilibria to be feasible. Therefore,the equilibria (3.2a) and (3.2b) give us information about the possible finalconfigurations of the system but do not provide the specific values of thefinal fraction of the population in each of the compartments.The equilibrium, E0 given by (3.2a) has no fearful susceptible individ-uals, no infected individuals, and a recovered population of 1 − S∗0 . WhenS∗0 ̸= 0 then this equilibrium represents a population without fear. As suchwe call this a fear-free equilibrium. When S∗0 = 1 then there is no recoveredpopulation and the equilibrium E0(1) is classically known as a disease-freeequilibrium [21] as the entire population is susceptible indicating the diseasedid not spread. The second equilibrium, E1 given by (3.2b) has a similaranalogy to E0, but with a non-zero fearful susceptible population. As suchwe call this a fear-endemic equilibrium. The existence of two equilibria, onethat is fear-free (E0(S∗0)) and the other that is fear-endemic (E1(S∗fd)), alignswith our observation in Figures 3.1 and 3.2 (top plots).To investigate the stability of the disease-free equilibrium E0(1) (DFE),E0(1) = [1, 0, 0, 0], we obtain the eigenvalues of the Jacobian around the263.2. STABILITY ANALYSISDFE. The Jacobian has a zero eigenvalue and two non-zero eigenvalues.The zero eigenvalue is related to degenerate equilibria. For some ϵ > 0the state [S∗0 , 0, 0, 1 − S∗0 ] and [S∗0 − ϵ, 0, 0, 1 − S∗0 − ϵ] are both equilibria.Therefore, a small perturbation to E0(1) will, at minimum, be unstable toa nearby equilibrium with a very small infection and recovery. We are moreinterested in E0(1) leading to solutions that are far from E0(1) indicatinga disease outbreak. Hence, we consider the non-zero eigenvalues of theJacobianλ1 = β − γ, (3.4a)λ2 = βfd − γf . (3.4b)The DFE is unstable if the eigenvalues of the corresponding Jacobian arepositive. So, for the rest of the analysis, we assumeb =βγ> 1, (3.5a)bf =βfdγf> 1. (3.5b)Note that these two quantities are, respectively, the disease and fear ofdisease reproduction numbers as defined by (2.9) in Section 2.6.To investigate the stability of the fear-free equilibrium, i.e., E0(S∗0)in (3.2a), we obtain the non-zero eigenvalues of the Jacobian around E0(S∗0):λ1 = (S∗0b− 1)γ, (3.6a)λ2 = (S∗0bf − 1)γf − (1− S∗0)αf . (3.6b)For this fear-free equilibrium to be stable, the eigenvalues of the Jacobianat E0(S∗0) must be negative. This condition is satisfied ifS∗0 < min(1b,a+ 1a+ bf), (3.7)where,a =αfγf(3.8)In the absence of fear loss due to complacency (when a = 0), condition (3.7)means that both the effective reproduction numbers, one corresponding todisease (bS∗0) and the other to fear of disease (bfS∗0), should be less than1. The Effective reproduction number is a key metric in epidemiology that273.3. MODEL SCALINGgives a measure of the expected number of new infections caused by aninfectious individual when only a fraction of the population is susceptiblewith some individuals being unsusceptible. It is equal to basic reproductionnumber times the fraction of susceptible population [33].Similarly, the non-zero eigenvalues of the Jacobian at the fear-endemicequilibrium, E1(S∗fd) from (3.2b), are:λ1 = −bfγfS∗fd, (3.9a)λ2 =[bS∗fd(p− aa+ bf)+ b(a+ 1a+ bf)− 1]γ. (3.9b)E1(S∗fd) is clearly stable if the following conditions are satisfied:p <aa+ bfand b(a+ 1a+ bf)< 1. (3.10)In the absence of fear loss due to complacency (when a = 0), the secondcondition in (3.10) becomesb < bf . (3.11)We learn from (3.11) that one of the conditions for the fear-endemic equilib-rium (E1(S∗fd)) to be stable is that the fear reproduction number (bf) shouldbe larger than the disease reproduction number (b).The stability analysis presented in this section provides insights intothe existence of the fear-free and fear-endemic equilibria observed in nu-merical simulations (Figures 3.1 and 3.2, top plots). We do not, however,obtain specific criteria regarding the observed bifurcation in final epidemicsize (sudden jump in final epidemic size) coincident with the infection dy-namics going from a pair of waves to a small single wave (Figures 3.1 and3.2, middle and bottom plots). To obtain these bifurcation conditions, weinvestigate (3.1) further using asymptotic analysis.3.3 Model scalingWe consider two different cases: one where the timescale of fear acqui-sition is comparable to the timescale of fear loss and another where it iscomparable to disease recovery time. In the former case, fear of diseasespreads on a slower time scale compared to disease recovery. This situationmay replicate in a recurring disease which does not trigger a timely fear re-sponse because people may have become complacent about the disease, forexample, the annual flu. We hence call this first case the Flu-like limit. In283.3. MODEL SCALINGthe latter case, fear mechanisms are synchronous with disease mechanisms.The fear of disease spreads at a rate proportional to disease recovery and ona faster timescale in comparison with the first case. This scenario may arisewhen a novel or deadly disease such as COVID-19 is introduced and peoplemay spontaneously change their behaviour through fear in tandem with thespread of disease [74]. We therefore call this second case the COVID-likelimit. This scaling is the one used in the numerical study of the Triple Con-tagion Model (2.1) [26]. In all cases, we assume that recovery from fear ismuch slower than recovery from disease.In each of these limits (Flu-like and COVID-like), we set an appropriateasymptotic scaling for the parameters and state variables of the model, andthen solve the order 1 and, if necessary, the order 2 equations for the statevariables of the model (3.1) (S, Sfd, and I). We then derive bifurcationconditions by determining under what conditions the solutions yield thezero or the non-zero final fear of disease population in line with steadystates (3.2a) and (3.2b).As a first step to solve for the state variables, we simplify (3.1) by scalingto a non-dimensional time t¯ = γt. All the model parameters have time−1unit, thus dividing the model equations by γ yields a non-dimensional sys-tem. We assume the initial conditions to be S(0) = S0, Sfd(0) = Sfd0, andI(0) = I0. We obtain the non-dimensionalised systemdSdt¯= −bIS + δ[−bf(Sfd + I)S + Sfd + a(1− S − Sfd − I)Sfd], (3.12a)dSfddt¯= −bpISfd − δ[−bf(Sfd + I)S + Sfd + a(1− S − Sfd − I)Sfd], (3.12b)dIdt¯= [b(S + pSfd)− 1]I, (3.12c)where,δ =γfγ, (3.13)and since we assume above that recovery from fear is much slower thanrecovery of disease then δ ≪ 1.Having defined δ we can more clearly define the Flu-like and COVID-likelimits. The Flu-like limit assumes that fear of disease spreads on a timescalesimilar to that of fear recovery. This means that the fear reproductionnumber satisfiesbf =βfdγf∼ O(1).Conversely, in the COVID-like limit we assume that fear of disease spreadson a timescale comparable to that of the disease itself. In this case, rescaling293.3. MODEL SCALINGthe fear reproduction number yieldsbf =βfdγfγγ= δ−1bˆf ∼ O(δ−1); bˆf = βfdγ. (3.14)Since everything is measured relative to the disease timescale we assumethat the disease reproduction number satisfies b ∼ O(1).3.3.1 Flu-like limitAssuming that the outbreak starts with everyone in the fearless suscepti-ble compartment (S), and a single individual in the infectious compartment(I), we find that the infection grows rapidly but the fear population remainsrelatively unchanged (see Appendix A.1). This scenario is consistent withbehaviours observed with respect to diseases like, e.g., the annual flu which,due to its regular occurrence and relatively low fatality rate may manifest areduced fear response initially. The infectious population can grow if thereare sufficient susceptibles available but the size of the fear of disease popu-lation remains small because of our assumption that fear is acquired slowly.The problem (3.12) can hence be simplified in the Flu-like limit by scalingS = x, Sfd = δy, and I = z (see Appendix A.1). We obtain the scaledproblem with the initial conditions x(0) = S0, y(0) = S¯fd0, and z(0) = I0,x˙ = −bxz − δbfxz, (3.15a)y˙ = −bpyz + bfxz − δ(y − bfxy + ay − axy − ayz), (3.15b)z˙ = (bx− 1)z + δbpyz. (3.15c)The set of ordinary differential equations (3.15) is a nonlinear system thatcannot be solved explicitly. To find an approximate solution, we express thevariables x, y, and z as the sum of a leading order and a δ order term.x = x0 + δx1 +O(δ2), (3.16a)y = y0 + δy1 +O(δ2), (3.16b)z = z0 + δz1 +O(δ2). (3.16c)Plugging the expansions (3.16) into the scaled equations (3.15), and collect-ing terms of leading order, we obtain the leading order problemx˙0 = −bx0z0, (3.17a)y˙0 = −bpy0z0 + bfx0z0, (3.17b)z˙0 = (bx0 − 1)z0, (3.17c)303.3. MODEL SCALINGwith the initial conditions x0(0) = S0, y0(0) = S¯fd0, and z0(0) = I0.Note that in the system of equations (3.17), x0 and z0 are decoupled fromy0. This result is expected because in the Flu-like limit the timescale of fearacquisition and fear loss are comparable and slower than the timescale ofdisease recovery. We know, however, that the populations are intrinsicallycoupled because population fractions in all of the compartments must sum to1. Therefore, this leading order decoupling will present consistency problemslater on.We can write (3.17a) and (3.17c) as a separable differential equation andsolve in terms of an integral (see Appendix A.2). The remaining equation fory0 is linear and we can solve it explicitly. Having determined the expressionsfor each compartment we can determine the final sizes x∞0 ̸= 0 and y∞0 withy∞0 = −bfb(1− p)x∞0 +B0x∞p0 , (3.18)where,x∞0 = limt→∞x0, (3.19a)y∞0 = limt→∞ y0. (3.19b)Note that since x∞0 ̸= 0 then (3.18) predicts a non-zero value for the disease-fearful population at the end of the outbreak. However, the steady stateanalysis in Section 3.2 indicated the possibility of two equilibrium values ofthe disease-fearful population, including the possibility that there is no fear.We thus consider the O(δ) problem,x˙1 = −bx0z1 − bx1z0 − bfx0z0, (3.20a)y˙1 = −bpy0z1 − bpy1z0 + bfx0z1 + bfx1z0 − y0 (3.20b)+ bfx0y0 − ay0 + ax0y0 + ay0z0,z˙1 = (bx0 − 1)z1 + bx1z0 + bpy0z0, (3.20c)with the initial conditions x1(0) = 0, y1(0) = 0, and z1(0) = 0.The system (3.20) is, again, not solvable analytically, but it is possible toobtain the infinite-time solutions, i.e., the values of the O(δ) state variableslong after the outbreak. To do this calculation, we assume the limitingvalues x0 → x∞0 , y0 → y∞0 , and z0 → 0.We obtain (see Appendix A.2) the δ order solution of y in the long-timelimit as,y1(t) = C0bpy∞0 − bfx∞01− bx∞0e−(1−bx∞0 )t + y∞0 ((bf + a)x∞0 − 1− a)t+C1, (3.21)313.3. MODEL SCALINGwhere, C0 and C1 are some constants. Note from (3.21) that y1 is the sum ofan exponentially decaying term (x∞0 < b−1, see Appendix A.2), a linear termin t and a constant term. Due to the linear term, y1 blows up as t→∞. Thisbehaviour, however, breaks the constant population constraint. Clearly, ycan not grow indefinitely and so one of our original assumptions must havefailed. Since y is growing large it must be that assumption of y ∼ O(1) (orSfd ∼ O(δ)) that fails. This suggests we need to rescale time where thisassumption is no longer true.We define a new timescale τ = δt in (3.12) (see Appendix A.2). Thisnew time variable τ represents a time scale that is much larger than t, whichmeans that the infinite time limit (3.15) is the initial condition for this newtime scale. Specifically, we have S → x∞0 + δx∞1 and Sfd → δy∞0 as τ → 0(see Appendix A.2), wherex∞1 = limt→∞x1. (3.22)In the long term, the infectious population vanishes, i.e., I → 0, and so theequation for I is no longer relevant. The τ timescale equations aredSdτ= Sfd − bfSSfd + a(1− S − Sfd)Sfd, (3.23a)dSfddτ= −Sfd + bfSSfd − a(1− S − Sfd)Sfd. (3.23b)The system (3.23) shows that, even after the outbreak is over, the dynam-ics between S and Sfd are sustained. If we treat fear as a pathogen thenwe have susceptibles transitioning to disease-fearful (infecteds) and back tosusceptibles (when they lose fear). This dynamic is analogous to that of anSIS model in classic disease dynamics [12].On adding (3.23a) and (3.23b), we obtain the population conservationlaw,dSdτ+dSfddτ= 0, (3.24)which upon integrating yields,S + Sfd = x∞0 + δ(x∞1 + y∞0 ) ≡ θ. (3.25)We can therefore solve (3.23) by eliminating one of the variables using (3.25).On solving (see Appendix A.2), we obtain Sfd as a function of θ as,Sfd(τ) =y∞0 (bfθ − a(1− θ))− 1bfy∞0 + [bf(θ − y∞0 )− a(1− θ)− 1]e[−(bfθ−a(1−θ)−1)τ ]. (3.26)323.3. MODEL SCALINGThe expression for Sfd(τ) in (3.26) shows that the size of the disease-fearfulpopulation may be zero or non-zero at the end of the outbreak. Whetherthe final state of the system is the zero (3.2a) or the non-zero (3.2b) fear ofdisease steady state depends on the argument of the exponential in (3.26).The argument of exponential is negative ifbf >1 + a(1− θ)θand limτ→∞Sfd = θ −(1 + a(1− θ)bf). (3.27)This solution corresponds to the case where we have a non-zero fear of diseasesteady state (3.2b) at the end of the outbreak. Recall that the steady stateanalysis in Section 3.2 determined only the existence of a zero and a non-zeroequilibrium state for Sfd. Here, with this deeper analysis, we have obtainedan expression for the limiting value of Sfd in the non-zero equilibrium (3.2b).On the other hand, if the argument of the exponential is positive, ifbf <1 + a(1− θ)θand limτ→∞Sfd = 0. (3.28)This solution corresponds to the zero-fear of disease state (3.2a) at the endof the outbreak.In the absence of fear loss due to complacency (i.e., when a = 0), the finaloutcome is determined by bfθ, which is the effective reproduction numberfor the fear contagion process.From (3.27) and (3.28) we see that the critical value of bf isbcf =1 + a(1− θ)θ, (3.29)which depends on a and θ. We note from (3.25) that θ depends on x∞1 andy∞0 which are themselves functions of bf (see (A.39) and (3.18)). To find ananalytic approximation for the bifurcation point we expand bcf as,bcf = bcf0 + δbcf1 +O(δ2). (3.30)After some algebra, we obtain (see Appendix A.2)bcf0 =1 + a(1− x∞0 )x∞0, (3.31a)bcf1 =−[x∞1 (bcf0) + y∞0 (bcf0)](1 + a)x∞20. (3.31b)In Figure 3.3, we plot the final size of the disease-fearful population andthe final size of the recovered population as functions of βfd, for the Flu-like333.3. MODEL SCALINGlimit. The red dashed line shows the critical rate of acquisition of fear ofdisease from (3.29). The value of βcfd in this case is 0.012.The final size of the recovered population, however, shows a uniform de-cline for increasing values of βfd, i.e., there is no bifurcation in final epidemicsize. This result is explained by the leading order decoupling of the fear anddisease contagion processes (3.17). When the fear of disease reproductionnumber is O(1), the fear response is weaker than the rate of disease spread,fear decouples from disease and the weak fear response does not show anysignificant impact on the final size of the recovered group.Perhaps less expected is our result that a weak fear response can leaveindividuals with fear of disease even after the outbreak is over. This fearretention may be observed when the population experiences a recurring ornon-deadly disease for which they do not adopt prophylaxis at the beginningof the outbreak due to reduced risk perception. However, as the outbreakproceeds, the expanding infectious population may create disease-fearfulnessin the population.In Figure 3.4, we plot the fraction of the population in the disease-fearful,infectious, and recovered compartments over time for different rates of thefear acquisition as measured by βfd, in the Flu-like limit. We choose βfdvalues near βcfd = 0.012 to understand the change in the overall dynamicsaround the bifurcation point. The solutions of each of these variables arerepresented by dashed lines for values of βfd < βcfd and by solid lines forβfd > βcfd. In the Sfd plot (see Figure 3.4 (top plot)), we see three solu-tions (βfd = 0.035β, 0.0385β, and 0.042β) tending toward the zero steadystate (3.2a) and each of the remaining solutions tending toward a non-zerosteady state (3.2b).In contrast, we observe that the curves for the infectious and recov-ered populations in Figure 3.4 do not change significantly with varying fearresponse. Moreover, the single infectious wave and the rapid rise of therecovered population are consistent with the classic SIR model in which in-dividuals do not alter their behaviour at all during outbreaks. Consequently,in the Flu-like limit, the system dynamics with the fear response includedor excluded are indistinguishable.We note also that, in the Flu-like limit, the disease-fearful populationtakes much longer to reach steady state than the infectious population.These longer transients in Sfd are a consequence of the longer timescalewhile conducting analysis. Recall that the δ−1 scaling (t = δ−1τ) emergednaturally in our analysis (3.23). At this longer timescale, depending on thevalue of βfd, the disease-fearful population continues to change for some timeafter the infection has vanished.343.3. MODEL SCALINGFigure 3.3: Plot of the final size of the disease-fearful (top) and recovered(bottom) populations over time as functions of rate of acquisition of fear,βfd, in the Flu-like limit. The red dashed line shows the critical rate of fearacquisition (βcfd) at which the final state transitions from a zero to a non-zero disease-fearful steady state. The solution curves represented by dashedlines correspond to βfd < βcfd and by solid lines correspond to βfd > βcfd. Thefixed parameter values used in this simulation are: β = 27 , γ =17 , δ = 0.01,γf = δγ, p = 0.25, and αf = 1.1γf .353.3. MODEL SCALINGFigure 3.4: Plots of the fraction of the population in the disease-fearful(top), infectious (center), and recovered (bottom) compartments over timeas functions of rate of acquisition of fear, βfd, in the Flu-like limit. Thefixed parameter values used in this simulation are: β = 27 , γ =17 , δ = 0.01,γf = δγ, p = 0.25, and αf = 1.1γf . The βfd colour legend in the middle plotapplies to all three plots.363.3. MODEL SCALINGThe bifurcation in the final size of the Sfd population is driven by acombination of the model parameters 2.1. We observed from (3.31) that,to leading order, this bifurcation is determined by bf and αf , and p appearsin order (δ) correction. In Figure 3.1, however, p exhibited a significantinfluence on the bifurcation in final epidemic size. The enhanced influenceof p in Figure 3.1, indicates that the Flu-like limit is not related to the [26]results, and so we need to consider a different limit in order to explain theseresults. We note that in [26], the authors expressed the fear parameters, βfdand αf , as O(1) multiples of the rate of disease transmission, β. In the nextsection, we analyse (3.12) in a limit we call the COVID-like limit.3.3.2 COVID-like limitFrom our analysis in the Flu-like limit in Section 3.3.1, we observed abifurcation in final fear state but not final epidemic size in contrast to ourobservations in Figures 3.1 and 3.2, middle and bottom plots. It is observedby Epstein et al. that p has a significant influence on the final state ofthe epidemic [26]. However, the Flu-like limit demonstrates that the effectfrom p is subdominant. This suggests that a rescaling is required to identifythe behavioural p-driven bifurcation. The parameter values taken from [26]to generate Figures 3.1 and 3.2 were expressed as O(1) multiples of theinfection transmission rate, β. So, we analyse the model further in a limit,which we call the COVID-like limit, where we assume that fear spreads ona timescale comparable to that of disease itself (see (3.14)), i.e.bf = δ−1bˆf ,As in Section 3.3.1, we assume that the outbreak starts with everyonein the fearless susceptible compartment (S), and a single individual in theinfectious compartment (I). We find that the fear population grows rapidlybut the infectious population remains relatively unchanged (see AppendixA.1). This scenario may arise in an outbreak due to a novel or deadly virus,for example SARS-CoV-2, where the population becomes more reactive andrapidly changes its behaviour due to fear in tandem with the spread of thedisease.In the COVID-like limit, (3.12) can be simplified by assuming that theI variable is much smaller than the S and Sfd variables, and thus we set373.3. MODEL SCALINGS = x, Sfd = y, and I = δz, obtaining the scaled problemx˙ = −bˆfxy + aˆ(1− x− y)y − δ(bxz + bˆfxz − y + aˆyz), (3.32a)y˙ = bˆfxy − aˆ(1− x− y)y + δ(−bpyz + bˆfxz − y + aˆyz), (3.32b)z˙ = [b(x+ py)− 1]z, (3.32c)with the initial conditions x(0) = S0, y(0) = 1− S0 − δI0, and z(0) = I0.The set of ordinary differential equations (3.32) is a nonlinear systemthat cannot be solved explicitly. As with the Flu-like limit, we express thevariables x, y, and z as the sum of a leading order and a δ order termx = x0 + δx1 +O(δ2), (3.33a)y = y0 + δy1 +O(δ2), (3.33b)z = z0 + δz1 +O(δ2). (3.33c)Plugging the expansions (3.33) into (3.32), and collecting terms of like order,we obtain the leading order problemx˙0 = −bˆfx0y0 + aˆ(1− x0 − y0)y0, (3.34a)y˙0 = bˆfx0y0 − aˆ(1− x0 − y0)y0, (3.34b)z˙0 = [b(x0 + py0)− 1]z0, . (3.34c)with the initial conditions x0(0) = S0, y0(0) = 1− S0, and z0(0) = I0.Note that,x˙0 + y˙0 = 0, (3.35)which, upon integration using initial conditions, givesx0 + y0 = 1, (3.36)We observe from (3.36) that to leading order majority of the populationstays in the susceptible compartments (S and Sfd) by avoiding infectionthrough fear-induced prophylaxis.We can eliminate one of the variables using (3.34c) and then solve (3.34b)(see Appendix A.3). We obtainy0(t) =1− S01− S0 + S0e−bˆf t, (3.37a)z0(t) = I0(1− S0 − S0e−bˆf t)−(1−p)bbˆf e(bp−1)t. (3.37b)383.3. MODEL SCALINGIf bp − 1 > 0 (where bp represents the reduced disease reproduction num-ber due to fear-induced contact reduction), the solution (3.37b) shows thatinfection is an increasing function of t. This result shows that infectionspreads rapidly with time and that fear is not enough to suppress it. Onthe other hand, if bp − 1 < 0, infection decays over time showing that fearis successful in reducing the disease.The solution (3.37a) shows that in infinite time limit, y0 plateaus at 1meaning that the whole population shifts to the fear compartment. However,the steady-state analysis in Section 3.2 indicated that at equilibrium thefearful population can also go to 0.We thus consider the O(δ) problem for the case when bp − 1 < 0. Inthe infinite-time limit, the leading order size of disease-fearful populationplateaus at 1, i.e.,y0 = 1. (3.38)Using (3.36), we obtain the infinite-time leading order value of the regularsusceptible populationx0 = 0. (3.39)We obtain the O(δ) problem with the leading order solutions (3.38)and (3.39),x˙1 = −bˆfx1 + 1− aˆ(x1 + y1 + z0), (3.40a)y˙1 = −bpz0 + bˆfx1 − 1 + aˆ(x1 + y1 + z0), (3.40b)z˙1 = (bp− 1)z1 + b(x1 + py1)z0, (3.40c)with the initial conditions x1(0) = 0, y1(0) = −I0, and z1(0) = 0.The system (3.40) is not solvable analytically, but it is possible to obtainthe infinite-time solutions, i.e., the values of the O(δ) state variables longafter the outbreak. On solving (3.40) and considering the solutions in thelong-time limit (see Appendix A.3), we obtain the infinite-time solution ifbp− 1 < 0,y∞1 = limt→∞ y1(t) =(bˆf + aˆ)I0bp−1 − 1bˆf, (3.41)Note that, if bp−1→ 0−, then y∞1 blows up, but y∞1 represents a fractionof the population and is necessarily bounded by 1. Clearly, y cannot growindefinitely, so our solution becomes invalid when bp − 1 → 0−. We can,however, investigate the solutions near this singularity by settingp =1b+ δpˆ. (3.42)393.3. MODEL SCALINGwhere, pˆ is some constant, and considering a new timescale in which thepopulations remain constrained. The appropriate rescaling is t = τ/√δ,S =√δX, Sfd = 1 +√δY , and I = δZ (see Appendix A.3). Substitutingthese scalings into (3.12), we obtain the new scaled model√δdXdτ= −bˆfX − aˆ(X + Y ) +√δ(1− bˆfXY − aˆ(X + Y )Y − aˆZ) +O(δ),(3.43a)√δdYdτ= bˆfX + aˆ(X + Y )−√δ(1− bˆfXY − aˆ(X + Y )Y − aˆZ + Z) +O(δ),(3.43b)dZdτ= (bX + Y )Z +O(√δ), (3.43c)On comparing the leading order terms on the left and right-hand side of(3.43a) and (3.43b), we obtain that (3.43a) and (3.43b) are in quasi-steadystate, yieldingY = −(bˆf + aˆaˆ)X. (3.44)On adding Equations (3.43a) and (3.43b) we obtain,dXdτ+dYdτ= −Z +O(√δ). (3.45)We can use (3.44) to eliminate one variable and use (3.45) to obtain theleading order solutions of (3.43)X(τ) =√2µI0aˆbˆftanh(√µ2I0aˆbˆfτ), (3.46a)Y (τ) = −(bˆf + aˆaˆ)√2µI0aˆbˆftanh(√µ2I0aˆbˆfτ), (3.46b)Z(τ) = I0(1− tanh2(√µ2I0aˆbˆfτ)). (3.46c)where,µ =bˆf − (b− 1)aˆaˆ. (3.47)We know that the tan-hyperbolic function exhibits a sigmoidal shape plateau-ing at 1. When the tan-hyperbolic function approaches 1, infection goes to403.3. MODEL SCALING0 (from (3.46c)) hence indicating that we cannot obtain a second infectionwave. However, we observed from Figures 3.1 and 3.2 (middle plots) that asecond infection wave arises for certain parameter values.Note that µ changes sign whenbˆf = aˆ(b− 1). (3.48)When µ < 0, the argument in the tan-hyperbolic function becomes complex,hence becoming the tangent function. We hence obtain the solution whenµ < 0 asX(τ) =√2|µ|I0aˆbˆftan(√|µ|2I0aˆbˆfτ), (3.49a)Y (τ) = −(bˆf + aˆaˆ)√2|µ|I0aˆbˆftan(√|µ|2I0aˆbˆfτ), (3.49b)Z(τ) = I0(1 + tan2(√|µ|2I0aˆbˆfτ)), (3.49c)The solution (3.49c) indicates the emergence of a second infection wavewhich correspondingly results in an increase in final epidemic size. Hence,the final epidemic size can be kept low if µ > 0 oraˆ <bˆfb− 1 . (3.50)In our simulations, we use b = 2. The condition (3.50) then becomesaˆ < bˆf , (3.51)which says that the rate of fear recovery triggered by the recovered groupshould not be larger than the rate of fear acquisition to yield a drop in finalepidemic size. The negative sign in (3.49b) shows that the final fear popu-lation drops suddenly when the final epidemic size surges. We thus obtainthe critical value, bˆcf at which final epidemic size and final fear populationsize show a bifurcation from (3.48).Note that the solutions (3.49) blow up when,√|µ|I0aˆ2bˆfτ =π2, (3.52)413.3. MODEL SCALINGThe solutions in (3.49) break the constant population constraint. We there-fore need to consider a different time scale such that the populations re-main bounded. An interesting study left for future work is to further anal-yse (3.12) in the COVID-like limit to figure out the correct timescale inwhich the solutions remain bounded.In Figure 3.5, we plot the final size of the disease-fearful population andfinal epidemic size respectively for different values of βfd in the COVID-likelimit. The red-dashed line shows the critical rate of acquisition of fear of dis-ease (bˆcf ) from Equation (3.48). The critical value of βfd denotes the point ofswitch from zero to non-zero fear of disease population and a correspondingswitch from a high to low final epidemic size. In this case, we assumed thatthe fear of disease spreads on a timescale comparable to disease recovery.Therefore, we observe that the bifurcations in final epidemic size and finalfear population size are synchronous. The risk perceptions of such outbreakmay prompt individuals to become vigilant, actively engaging in preventivemeasures and public health protocols as the outbreak unfolds. We observedfrom our analysis that fast adoption of vigilance through fear of disease canpotentially mitigate the spread of the disease.In Figure 3.6, we plot the fraction of the population in the disease-fearful,infectious, and recovered compartments over time for different rates of fearacquisition as measured by βfd, in the COVID-like limit. The critical value,βcfd = 0.342. The solutions of each of these variables are represented bydashed lines for values of βfd < βcfd and by solid lines for βfd > βcfd. Fromthe infection plot, we observe that extremely fast rates of fear acquisitionlead to a small single infection wave. This result shows that if fear acquisitionis fast enough it can prevent subsequent infection waves. As βfd decreases,the loop between fear acquisition and fear recovery through complacencybecomes important and leads to the rise of a second infection wave. Recallthat we obtained the emergence of a second infection wave in our analysiswhen the tan-hyperbolic function in (3.46) changes to the tangent functionin (3.49). As βfd decreases even further, we observe a second bifurcationin infection dynamics in which the system goes from two infection waves toa big single infection wave. Explaining this second bifurcation in infectiondynamics remains a future project.From the top plot in Figure 3.6, we observe that, corresponding to thetransition from a small single infection wave to a pair of infection waves,the final disease-fearful population goes from a non-zero to a zero fractionrespectively. When the fear acquisition is fast, more individuals recruit intothe disease-fearful population. The synchronous disease and fear dynamicsmean that fearful individuals adopt prophylaxis when needed. If, at the423.3. MODEL SCALINGsame time, fear loss is sufficiently slow, these fearful individuals adopt pro-phylactic measures for so long that the infectious population vanishes. Thissituation results in part of the population getting stuck in the fear compart-ment and the system ending up with a lower final epidemic size as seen inbottom plot in Figure 3.6. As βfd decreases further, a smaller fraction of thepopulation acquires fear of disease, which increases the effective fraction ofsusceptible individuals that are exposed to the disease and leads to a largerfinal epidemic size.433.3. MODEL SCALINGFigure 3.5: Plots of the final size the disease-fearful (top) and recoveredpopulations (bottom) as functions of rate of acquisition of fear, βfd, in theCOVID-like limit. The red dashed line shows the critical rate of fear acqui-sition denoting the transition from zero to non-zero fear steady state. Thefixed parameter values used in this simulation are: β = 27 , γ =17 , αf = 2.2β,γf = 0.05, δ =γγf, and p = 0.25.443.3. MODEL SCALINGFigure 3.6: Plots of the fraction of the population in the disease-fearful(top), infectious (center), and recovered (bottom) compartments over timeas functions of rate of acquisition of fear, βfd, in the COVID-like limit. Thefixed parameter values used in this simulation are: β = 27 , γ =17 , αf = 2.2β,γf = 0.05, δ =γγf, and p = 0.25.453.4. DISCUSSION3.4 DiscussionIn this chapter, we conducted a stability analysis of the system (3.1)to understand the final configuration of each of the compartments. Thestability analysis showed that the system (3.1) has two equilibria, one isfear-free and the other is fear-endemic. To obtain the conditions for theswitch between the two equilibria (3.2a) and (3.2b), and to find an expressionfor the final size of the fear population in the fear-endemic equilibrium, wecarried out an asymptotic analysis of the system. We considered the modelin two regimes: the Flu-like and the COVID-like limits. In both of thesescalings, we assumed that rate of fear loss is less than rate of recovery fromdisease.In the Flu-like limit, we assumed that the timescale of fear acquisitionis comparable to the timescale of fear loss. In this limit, both the diseaseand fear reproduction numbers are larger than 1, but fear spreads relativelyslowly compared to the disease. Through our analysis, we found that thesystem behaves like an SIS system with S and I being analogous to Sand Sfd in the Reduced model (3.12). This result is expected because thetimescales of fear acquisition and fear loss are slower than the timescale ofdisease spread and recovery. A weak fear response results in the decouplingof disease and fear spread. This decoupling leads to continued dynamicsbetween S and Sfd even after the outbreak is over. A characteristic result ofthe SIS model is that the disease either vanishes or becomes endemic [12].This endemicity of infection (analogous to fear in our case) implies that aweak fear response can result in people retaining fear even after the outbreakis over. This result indicates that the population does not take appropriateactions when needed and sticks to protective actions when they are notneeded. With regard to COVID-19, it has been observed that, even whenthe spread is minimal, a part of the population continues mask-wearing andsocial distancing [45].On the other hand, the population that retains fear of the disease canpotentially serve as the initial fearful population for future outbreaks. Ithas been shown [60] that the population with fear retention reduces thelikelihood of resurgence of the disease. If a second outbreak occurs, theinitial fearful population reduces the disease reproduction number by rapidlyengaging in prophylaxis. Note that fear is not permanent [26] and so thefear-retaining population eventually loses this fear. We thus expect fearretention to help with disease outcomes only if the second outbreak occursbefore whole population loses fear.Our results further showed that, in the Flu-like limit, the fear response463.4. DISCUSSIONhas a negligible influence on final epidemic size. Moreover, the dynamicsare consistent with the classic SIR model in which individuals do not al-ter their behaviour at all during outbreaks. Consequently, in the Flu-likelimit, the system dynamics with the fear response included or excluded areindistinguishable. From a policy perspective, this reduced capacity to influ-ence the final epidemic size poses a considerable challenge to governments,as the efficacy of public health measures often depends on the population’swillingness to comply with recommended guidelines [32]. Unfortunately, insituations where fear is not substantially influential, it may not be optimalto rely on fear as a motivator for behavioral change.In the Flu-like limit, we observed a bifurcation in the final disease-fearfulpopulation, but, in contrast with our observations in Figures 3.1 and 3.2(middle and bottom plots), the final epidemic size and infection dynamicsdid not change with varying fear response. So, we analysed the model furtherin the COVID-like limit.In the COVID-like limit, we assumed that the timescale of fear spreadis comparable to the timescale of disease recovery. Through our analysis,we demonstrated the conditions under which a second epidemic wave couldemerge. These conditions correspond to a bifurcation in final epidemic size.Our results show that when bp− 1 > 0, fear response is not enough to sup-press the disease. The dynamics are therefore dominated by disease spreadrather than fear spread and hence we do not see a bifurcation in final epi-demic size in this case. On the other hand, when bp−1→ 0−, we found thata fast fear response is effective in lowering the final epidemic size if fear lossdue to complacency is slower than fear acquisition. This result is intuitive:if fear loss is slower than fear acquisition, a large fraction of people will stayin the fear compartment and protect themselves from disease. This bifurca-tion in final epidemic size was also observed in the qualitative analysis of aslightly different model proposed in [61]. This model [61] included imitativefear loss, which corresponds to the fearful population (Sfd) losing fear uponinteraction with regular susceptibles (S), in place of the spontaneous fearloss mechanism that is implemented in our model.An analysis similar to bp−1→ 0− can be conducted to obtain bifurcationconditions for any value of p satisfying bp − 1 < 0. [26] obtained thesebifurcations (see Figures 3.1 and 3.2) for bp = 0.5 < 1.From our analysis in this chapter, we identify that fear learning and fearloss are complex mechanisms. The relative timescales of fear acquisitionand disease spread determine whether fear-induced protection affects thedisease dynamics or not. Model (2.1) considered population groups holdinga single fear: fear of disease or fear of vaccine. It is possible, however, for an473.4. DISCUSSIONindividual to be simultaneously fearful of both the disease and the vaccine[68]. These individuals with simultaneous fear of disease and fear of vaccinemay reduce their contacts in order to reduce their susceptibility but mayalso avoid vaccination. In the next chapter, we add a compartment to theoriginal model (2.1) to account for such individuals.48Chapter 4Modified triple contagionmodel4.1 IntroductionDuring outbreaks, side effects of both the infection and the vaccine arerelevant to people’s assessment of risk. Several studies have been conductedto understand the relationship between the fear of disease and the fear ofvaccines. Human behaviour during an outbreak depends on the perceivedrisk of the disease, the perceived risks and benefits of health-protective be-haviours, and the costs involved [68]. In a COVID-19 vaccine hesitancy sur-vey, researchers found that, during disease outbreaks, people exhibit dualdefense responses, i.e., they avoid death or discomfort not only from diseasebut also from disease-protecting tools [11, 68]. To reduce their risk of death,humans not only adopt prophylaxis to reduce their susceptibility to the dis-ease but also refrain from getting vaccinated in case of negative thoughtsabout the vaccine [11, 68]. The Triple contagion model (2.1) in Chapter 2does not consider the individuals who are fearful of both the disease and thevaccine. We call these individuals “doubly afraid”. We add a compartmentSfdv to the existing model (2.1) to account for the doubly afraid susceptibleindividuals.4.2 Model descriptionIn the Modified triple contagion model, we make some assumptions re-garding the acquisition and loss of double fear. We assume that individualsacquire a single fear before becoming doubly afraid and lose one fear or bothfear simultaneously. In the epidemiological context, consider individuals whohave become afraid of the disease and have entered the compartment Sfd.As vaccinations increase, adverse events may drive them to also be afraid ofthe vaccine. These individuals become doubly afraid and enter the compart-ment Sfdv. Similarly, individuals who are afraid of only the vaccine enter494.2. MODEL DESCRIPTIONthe compartment Sfv. These individuals may become afraid of the diseasedue to high case numbers hence becoming doubly afraid and entering Sfdv.In addition to the underlying mechanisms of fear acquisition and fearloss in the model (2.1), we assume that the doubly afraid individuals alsocontribute to the rate of acquisition of single fears and double fear. Theregular fearless susceptibles, S, can acquire either the fear of disease at arate βfd or the fear of vaccine at a rate βfv on interacting with the doublyafraid individuals. The individuals with just the fear of disease can becomedoubly afraid at a rate βfv through interactions with vaccine-fear transmit-ting individuals, which include the Sfv, Avac, and Sfdv individuals. On theother hand, the individuals with just the fear of vaccine may transform intodoubly afraid individuals through interactions with the disease-fear trans-mitting individuals, which include Sfd, I, and Sfdv, at a rate βfd.The doubly afraid individuals may lose one of the fears through com-placency. The individuals who have naturally recovered from the diseasecan make the doubly afraid individuals complacent about the disease en-couraging them to leave the fear of disease hence leaving them with just thefear of vaccine. Similarly, the vaccinees in the Rvac compartment can makethe doubly afraid individuals complacent about the vaccines hence leavingthem with just the fear of disease. The rate of loss of double fear throughcomplacency is assumed to be αf . To simplify the model, we assume thatthe per capita rate of transition from double fear to single fear is equal tothat from single fear to no fear.Additionally, we assume that the doubly afraid susceptibles can spon-taneously lose both fears simultaneously hence going to the regular suscep-tible group, S. A spontaneous loss of double fear may take place due to,e.g., pandemic fatigue. Constant vigilance and the need to abide by restric-tions/policies may exhaust people. As a result, people may not exerciseprophylaxis as enthusiastically as at the beginning of the outbreak. Forexample, during the COVID-19 pandemic, the increasing number of cases,uncertainty about the nature of the virus, and the experience of multiple in-fection waves resulted in pandemic fatigue [34]. In addition, long lockdownsand restrictions brought emotional and economic burdens. All these factorsled people to abandon protective actions altogether which posed significanthealth threat to the public [34].The doubly afraid individuals contract the disease at a rate of diseasetransmission β scaled by the fear-induced contact reduction parameter pfdv.The fear-induced contact reduction parameter for the doubly afraid individ-uals (pfdv) may be smaller or larger than the fear-induced contact reductionparameter for the individuals with just the fear of disease (p). The fear re-504.2. MODEL DESCRIPTIONsponse of doubly afraid individuals may vary with characteristic behaviourof individuals. Doubly afraid individuals might be more anxious and reducecontacts more (hence pfdv < p) but they may also be stubborn (hence theirdouble fear) and actually act as if nothing is wrong (hence pfdv > p). Inour analysis, we assume that pfdv < p to account for individuals who beingaware of the risks of disease and their higher susceptibility to the disease asunvaccinated individuals may exhibit heightened anxiety.It is assumed that only the individuals with fear of disease get vacci-nated. The change in the rate of vaccination with time is governed by Equa-tion (4.1i). It is assumed that the rate of vaccination increases if the pool ofindividuals choosing to vaccinate exceeds the pool of individuals who refuseto get vaccinated and vice-versa. This is represented by the difference inthe fraction of the population in the vaccine-accepting (disease-fearful, Sfd)and vaccine-avoiding (vaccine fearful, Sfv and doubly afraid, Sfdv) groups.In Equation (4.1i), ϵ is the maximum rate of vaccination. As the rate ofvaccination approaches ϵ, dvdt → 0.The compartmental diagram of the Modified triple contagion model isgiven in Figure 4.1. This model is built on top of the Triple Contagionmodel (2.1) by adding the double fear compartment, Sfdv. Since each ofthe state variables is a fraction and these sum up to one under the constantpopulation constraint: S + Sfd + Sfv + Sfdv + I +Rnat +Rvac +Avac = 1.The new set of ordinary differential equations is,dSdt= −βIS − βfd(Sfd + I + Sfdv)S − βfv(Sfv +Avac + Sfdv)S+ γf(Sfd + Sfv + Sfdv) + αf(RnatSfd +RvacSfv), (4.1a)dSfddt= −pβISfd + βfd(Sfd + I + Sfdv)S − βfv(Sfv +Avac + Sfdv)Sfd− γfSfd + αf(RvacSfdv −RnatSfd)− vSfd, (4.1b)dSfvdt= −βISfv + βfv(Sfv +Avac + Sfdv)S − βfd(Sfd + I + Sfdv)Sfv− γfSfv + αf(RnatSfdv −RvacSfv), (4.1c)dSfdvdt= −pfdvβISfdv + βfd(Sfd + I + Sfdv)Sfv+ βfv(Sfv +Avac + Sfdv)Sfd − αf(RvacSfdv +RnatSfdv)− γfSfdv, (4.1d)dIdt= βIS + pβISfd + βISfv + pfdvβISfdv − γI, (4.1e)dRnatdt= γI, (4.1f)514.2. MODEL DESCRIPTIONFigure 4.1: Compartmental diagram of the Modified triple contagion modelaccounting for fear of disease, vaccination, fear of vaccine, and double fear.The compartments Sfv, S, Sfd, and Sfdv refer to the susceptibles fearful of thevaccine, the regular susceptibles, the susceptibles fearful of the disease, andthe susceptibles fearful of both the disease and vaccine respectively. Thecompartment I represents the infectious individuals. The compartmentsRnat, Rvac, and Avac refer to naturally recovered individuals, vaccinatedindividuals, and vaccinated individuals who experienced adverse reactionsrespectively. The rates of transitioning between different compartments aredisplayed on top of the arrows.524.3. FINAL EPIDEMIC SIZE ANALYSISdRvacdt= (1− σ)vSfd + γfAvac, (4.1g)dAdt= σvSfd − γfAvac, (4.1h)dvdt= η(Sfd − Sfv − Sfdv)(ϵ− v)v. (4.1i)4.3 Final Epidemic Size AnalysisIn Chapter 3, for the case of fast fear acquisition (bf ∼ O(δ−1)), weexplained how the final epidemic size shows a sudden drop as the rate ofacquisition of fear of disease increases and the infection dynamics corre-spondingly goes from a pair of waves to a small single wave. We observedthat this drop in the final epidemic size also corresponds to the final fearpopulation going from zero to non-zero as the rate of acquisition of fear ofdisease increases. Further, the analysis showed that the change in the finalstate of the system largely depends on the trade-off between vigilance (ex-hibited by the Sfd group) and complacency (exhibited by the S group). Inthis chapter, we conduct a qualitative investigation of the final epidemic sizewith respect to βfv, the rate of acquisition of fear of vaccine and the doublefear, and αf , the rate of loss of fears. Both Sfd and Sfdv individuals reducecontacts to protect from disease. We plot a heatmap of final epidemic sizewith respect to p and pfdv to examine the relative effect of contact reductionof the two groups, Sfd and Sfdv, on the final epidemic size.Figure 4.2 represents the heatmap of the final epidemic size with respectto p and pfdv. We observe that if p = 0, the case of perfect isolation by theSfd group such that they become non-susceptible to the disease, the criticalvalue of pfdv to show a bifurcation in final epidemic size is pcfdv = 0.35approximately with all other model parameters fixed. And if pfdv = 0,the case when the Sfdv group isolates perfectly, then the critical value ofp to show a bifurcation in final epidemic size is pc = 0.52 approximatelywith all other model parameters fixed. The critical value, pcfdv is less than pcpossibly because the doubly afraid individuals rely only on contact reductionto protect from disease. We might expect the critical values pc and pcfdv to bethe same when one of the groups is perfectly isolating. However, we observefrom the heatmap 4.2 that p and pfdv have an unsymmetrical effect on thebifurcation in final epidemic size. The boundary between the parametricregion corresponding to low and high final epidemic size is elliptical. Asa result, the critical values of pfdv are bigger than p for low values of p,but as p increases the critical values of pfdv are less than p. For low values534.3. FINAL EPIDEMIC SIZE ANALYSISFigure 4.2: Heatmap of the final epidemic size with respect to p and pfdvfor the Modified triple contagion model. The fixed parameter values usedare: β = 27 , βfd = 1.1β, βfv = 1.6β, αf = 1.2β γf = 0.05, η = 0.8, σ = 0.2,ϵ = 0.2, γ = β2 .of p, Sfd people protect well enough, so Sfdv people do not need to put inmuch effort in reducing contacts. On the other hand, for large values of p,Sfdv people compensate for the reduced protection of Sfd by reducing theircontacts more efficiently. This unsymmetric influence of p and pfdv on finalepidemic size shows that one of the groups has to invest relatively moreefforts in reducing contacts to suppress the disease.From analysis in Chapter 3, we observed that complacency plays a cru-cial role in deciding the disease trajectory. In Figure 4.3, we plot the fi-nal epidemic size, i.e., the total infected population with respect to βfv forthree different values of αf with all other model parameters kept fixed. Forαf = 0.3, the final epidemic size decreases slightly and becomes constantas βfv increases. For αf = 0.49, we observe that the final epidemic sizeincreases as βfv increases, stays constant for a range of βfv and then againdecreases for higher values of βfv. And lastly for αf = 0.8, the final epidemicsize increases slightly and then plateaus for increasing values of βfv. These544.3. FINAL EPIDEMIC SIZE ANALYSIStrends in the final epidemic size are unintuitive. We might expect that thefinal epidemic size increases as βfv increases due to the growing anti-vaccinesentiment and thus decreased vaccination levels. However, we observe thatthe final epidemic size may increase or decrease for different values of αf asβfv increases.In Figure 4.4, we plot a heatmap of the final epidemic size with respectto βfv and αf keeping all other model parameters fixed to verify the trendsobserved in Figure 4.3. We label different regions of this heatmap withletters A for black-maroon region (αf < 0.4), B for maroon region (0.4 <αf < 0.57), C for red region (0.57 < αf < 0.62), D for bright yellow region(0.41 < αf < 0.8), E for light yellow region between the bright yellow andwhite regions, and F for white region towards the top right of the heatmap4.4. We observe that for αf < 0.4, the final epidemic size decreases slightly(transition vertically from maroon to black region within region A) as βfvincreases. For αf approximately in the range (0.4, 0.57), we observe anincrease and then a decrease in the final epidemic size as βfv increases.Lastly for larger values of αf , the final epidemic size increases slightly andstays constant as βfv increases. Note that the range of values of αf hereare dependent on other model parameters and therefore may change as themodel parameters change.We explore the βfv−αf phase space to understand the system dynamicsthat results in the trends observed in the final epidemic size in Figure 4.4.Figure 4.5 shows the fraction of population in each of the compartmentsfor parameters in region A of final epidemic size heatmap 4.4. We observethat a majority of the susceptible population ends up in the double fearcompartment. We get a small infection wave and hence the final epidemicsize is the lowest in this parametric region of the heatmap 4.4. The curve forRvac shows that only around 2.5% of the population gets vaccinated. Thepopulation that gets vaccinated is low because a majority of the popula-tion shifts into the double fear compartment. A larger population avoidingvaccines compared to the population accepting vaccines leads to the rateof vaccination grow to only 1.25% of the maximum vaccinating capacity, asshown in the curve of rate of vaccination.Figure 4.6 shows the fraction of population in each of the compartmentsfor parameters in region B of final epidemic size heatmap 4.4. In this case,a majority of the susceptible population ends up in the regular susceptiblegroup, S, with the rest of the susceptible compartments going to 0. We geta single infection wave slightly higher than the wave in Figure 4.5 becauseexcept the people who choose to vaccinate, rest of the susceptible populationresides in the regular susceptible compartment and not in the protected554.3. FINAL EPIDEMIC SIZE ANALYSISFigure 4.3: Plots of the final epidemic size with respect to βfv for differentvalues of αf . The fixed parameter values used are: β =27 , βfd = 1.1β,γf = 0.05, p = 0.25, pfdv = 0.15, η = 0.8, σ = 0.2, ϵ = 0.2, γ =β2 .564.3. FINAL EPIDEMIC SIZE ANALYSISFigure 4.4: Heatmap of the final epidemic size with respect to βfv and αffor the Modified triple contagion model. The fixed parameter values usedare: β = 27 , βfd = 1.1β, γf = 0.05, p = 0.25, pfdv = 0.15, η = 0.8, σ = 0.2,ϵ = 0.2, γ = β2 .574.3. FINAL EPIDEMIC SIZE ANALYSISFigure 4.5: Temporal plots of the population in each of the compartmentsfor the parameter values in region A of final epidemic size heatmap 4.4.584.3. FINAL EPIDEMIC SIZE ANALYSISFigure 4.6: Temporal plots of the population in each of the compartmentsin region B of final epidemic size heatmap 4.4.compartments such as Sfd and Sfdv. On the positive side, more than 40%of the population gets vaccinated hence gaining permanent immunity to thedisease. The curve of compartment A in this case is slightly higher than inFigure 4.5 because high vaccine coverage results in a larger fraction of thepopulation experiencing adverse reactions from the vaccine.Figure 4.7 shows the fraction of population in each of the compartmentsfor parameters in region C of final epidemic size heatmap 4.4. The finalepidemic size in region C is slightly higher than final epidemic size in regionB. We observe a big infection wave with an emerging secondary infectionwave. This emerging secondary infection wave becomes distinct in Figure4.8 that shows the fraction of population in each of the compartments forparameters in region D of the final epidemic size heatmap 4.4. The creation594.3. FINAL EPIDEMIC SIZE ANALYSISFigure 4.7: Temporal plots of the population in each of the compartmentsin region C of final epidemic size heatmap 4.4.of a second infection wave when the final epidemic size increases in this caseis conceptually similar to the multi-peak transition obtained in the COVID-like limit in Chapter 3. Note that the second infection wave is smaller thanthe first one. In region D, the final epidemic size is almost double the finalepidemic size in the region C and the total vaccinated population in theregion D is a tenth of that in the region C. This result indicates that evenslight changes in fear response can lead to significant rise in the total infectedpopulation. Moreover, going from region D to region C, the disease spreadis suppressed by vaccination.Figure 4.9 depicts the fraction of population in each of the compart-ments for parameters in region E of final epidemic size heatmap 4.4. Thefinal epidemic size in region E increases further compared to region D with604.3. FINAL EPIDEMIC SIZE ANALYSISFigure 4.8: Temporal plots of the population in each of the compartmentsin region D of final epidemic size heatmap 4.4.614.3. FINAL EPIDEMIC SIZE ANALYSISthe second infection wave becoming bigger than the first one. Note that theSfd peak in region E is very narrow showing that there is less time to getvaccinated before people become complacent about the disease and hencethe full vaccination potential is not achieved. The total vaccinated popu-lation in region E is a tenth of the total vaccinated population in regionD. Moreover, a majority of the susceptible population fears vaccine at theend of the outbreak. This result is counter-intuitive because we expect thatlesser vaccine coverage leads to less adverse reactions from the vaccine hencecutting-off the vaccine fear stimulus. This is a scenario where we see vac-cination driving anti-vaccine sentiments. From Figure 4.9, we observe thatthe transition from Sfdv to Sfd is negligible because of small Rvac popula-tion. Further, Sfv and I increase as Sfdv decreases. The increase in the Sfvis brought by the naturally recovered individuals (Rnat) who drive doublyafraid individuals to the Sfv compartment by becoming complacent aboutthe disease. This result tells us that the influence of Rvac individuals mayreduce relative to Rnat individuals due to low vaccine coverage hence driv-ing anti-vaccine sentiments. A majority of vaccine-fearful population at theend of an outbreak is not a desired scenario for an outbreak and should beavoided. The population that retains fear of vaccine after the outbreak hasvanished serves as the initial fearful population for the future outbreaks [61]and this can possibly reduce the initial rate of vaccination.In region F of final epidemic size heatmap 4.4, we observe that the twoinfection waves from region E merge to give a big infection wave. The finalepidemic size is the highest in this region because a negligible fraction of thepopulation gets vaccinated with the majority susceptible population endingup in the fear of vaccine compartment.From the temporal plots in different parametric regions of final epidemicsize heatmap 4.4, we hypothesize that the final epidemic size is low eitherdue to high vaccination or due to high contact reduction of the doubly afraidindividuals. For αf = 0.49 in Figure 4.3, the final size is low for smaller valuesof βfv due to high vaccination. We observed that vaccination can lower thesecond infection wave (shift from region E to D) or completely vanish thesecond infection wave (shift from region D to C). As βfv increases, fear ofvaccine increases and vaccine coverage decreases which results in a spikein the final epidemic size. As βfv increases further, the rate of acquisitionof double fear dominates the rate of double fear loss. As a result, a largerfraction of population resides in the double fear compartment, extinguishingthe infectious population, hence reducing the final epidemic size. This showsthat the double fear also provides sufficient protection despite doubly afraidindividuals choosing not to get vaccinated. A strategy to lower the final624.3. FINAL EPIDEMIC SIZE ANALYSISFigure 4.9: Temporal plots of the population in each of the compartmentsin region E of final epidemic size heatmap 4.4.634.3. FINAL EPIDEMIC SIZE ANALYSISFigure 4.10: Temporal plots of the population in each of the compartmentsin region F of final epidemic size heatmap 4.4.644.3. FINAL EPIDEMIC SIZE ANALYSISepidemic size when a fraction of the population denies to get vaccinated isby demonstrating the risks of becoming infected so that these individualschoose mitigative behaviour and enter the double-fear compartment.To verify our hypothesis that the final epidemic is low either due to highvaccination or due to a larger doubly afraid population, we consider twosubmodels of the model 4.1. We observed that final epidemic size remainslow for smaller values of βfv due to high vaccine coverage. The double fearcompartment does not play a crucial role for smaller values of βfv becauserate of loss of double fear is dominant over rate of acquisition of doublefear and hence the double fear population drifts out of the compartment.So, we consider the submodel without the double fear compartment. Thissubmodel is same as the Triple Contagion model 2.1. Figure 4.11 showsthe final epidemic size heatmap obtained from the Triple contagion model.We observe that for a fixed αf , as βfv increases, the final epidemic sizerises suddenly hence supporting our hypothesis that the bifurcation in finalepidemic size for smaller values of βfv is due to high vaccination and doublefear does not play a crucial role in this parametric regime.Further, we observed that the final epidemic size also remains low forlarger values of βfv because larger fraction of the population stays in thedouble fear compartment. For larger values of βfv, the double fear compart-ment plays a crucial role and vaccination is not very significant in the dropof final epidemic size. We consider a no-vaccination submodel with the rateof vaccination assumed to be 0 at all times and starting with a non-zero Sfvpopulation. Figure 4.12 represents the final epidemic size heatmap for theno-vaccination submodel. We observe that for a given αf , as βfv increasesthe final epidemic size shows a sudden drop hence supporting our hypothesisthat the sudden drop in the final epidemic size in Figure 4.3 is due to theprotection from double fear. Note that in figure 4.12, βfv and αf exhibita constant relationship for smaller values of βfv followed by a non-linearrelationship as βfv increases, followed by a close to constant relationshipfor larger values of βfv. Exploring the bifurcation relationship between βfvand αf in a similar fashion to the Reduced model (3.1) in Chapter 3 is aninteresting avenue of future work.654.3. FINAL EPIDEMIC SIZE ANALYSISFigure 4.11: Heatmap of the final epidemic size with respect to βfv and αffor the Triple contagion model. The parameter values used are same as inFigure 4.4664.3. FINAL EPIDEMIC SIZE ANALYSISFigure 4.12: Heatmap of the final epidemic size with respect to βfv and αffor the no-vaccination submodel. The parameter values used are same as inFigure 4.4674.4. DISCUSSION4.4 DiscussionTo summarise this chapter, we introduced a modification of model (2.1)by adding a double fear compartment, Sfdv. Sfdv accounts for individualsfearful of both the disease and the vaccine simultaneously such that theyreduce contacts but refuse to get vaccinated. We conducted a qualitativeanalysis of the final epidemic size with respect to p and pfdv which showedthat the bifurcation relation between p and pfdv is unsymmetric and thatone of the groups (Sfd or Sfdv) has to put more efforts in reducing contactswhen the other group is not protecting enough to reduce the final epidemicsize.We discussed in Chapter 3 that the final state of the system is largely de-pendent on the trade-off between vigilance (exhibited by the Sfd group) andcomplacency (exhibited by the S group). In this chapter, we investigatedthe final epidemic size with respect to both βfv and αf through a heatmap,and then we temporally explored different parametric regimes of the finalepidemic size heatmap. The results showed that the final epidemic size maydecrease, increase and then decrease or increase and plateau depending onthe relative values of βfv and αf .Vaccination is an effective tool to suppress the disease [36, 77]. We ob-served from the infection dynamics in going from region E to D to C in finalepidemic size heatmap 4.4 that vaccination could lower the second infectionwave (shift from region E to D) or completely vanish the second infectionwave (shift from region D to C). However, for most combinations of βfv andαf , full vaccination potential could not be achieved because of the limitedtime for which people retain fear of disease before becoming complacent. Amaximum of 40% of the population got vaccinated in a very small paramet-ric regime (βfv < 0.2 and 0.4 < αf < 0.6 approximately). Moreover, thetemporal plots in region E of final epidemic size heatmap 4.4 depicted howvaccination can drive anti-vaccine sentiments. Lower willingness to get vac-cinated and easy propagation of anti-vaccine sentiments highlight the needof more efforts in advertising the benefits and reduced risks of vaccine todiscourage anti-vaccine sentiment driven from vaccination.Our analysis showed that contact reduction due to double fear can alsobe effective in reducing the final epidemic size even lower than vaccina-tion. If several individuals refuse to get vaccinated, our analysis suggeststo make these individuals aware of the risks of becoming infected so thatthey choose mitigative behaviour and enter the double-fear compartment toprotect themselves from disease.The temporal plots of the fraction of infectious population showed that684.4. DISCUSSIONwe may get a small infection peak that may start to split into two infectionwaves, giving a distinct second wave with a higher or lower peak in differentparametric regimes of the final epidemic size heatmap 4.4. These wavesmay then merge to give a big infection wave in region F of final epidemicsize heatmap 4.4. Generally, we expect a higher second infection peak toarise when models consider waning immunity because of the replenishmentof the susceptible pool if immunity wanes fast enough [15, 31, 51]. Weobtained it based on complacency mechanisms without considering waningimmunity. Models in [26, 61, 74] also observed a higher second infectionpeak without accounting for waning immunity. The conceptual reasoningfor a higher second infection peak in [26, 61, 74] is that complacency fromthe first infection wave gives rise to a false perception of the risk of thedisease in the population. As a result, people stick to normal behaviour anddo not take protective actions when needed hence significantly increasingthe infected individuals.Lastly, we observed that in regions D, E, and F of final epidemic sizeheatmap 4.4 corresponding to higher values of final epidemic size, the fearof vaccine group has the highest proportion at the end of the outbreak.In regions A, B, and C of final epidemic size heatmap 4.4, on the otherhand, either double fear susceptibles or regular susceptibles make a majorityat the end of the outbreak. It is an interesting future work to determinethe conditions for which each of the susceptible group forms the largestproportion at the end of the outbreak.69Chapter 5ConclusionHuman behavioural changes are known to play a crucial role in shapingthe trajectory of infectious diseases [25–27, 30, 32, 37, 52, 60–62, 74, 75].Disease prevalence alters perceptions of disease risk that encourage peo-ple to change their behaviour towards prophylaxis [51, 74]. The behaviourdynamics in turn shape the disease dynamics. Models in the field of Be-havioural Epidemiology account for this feedback loop between disease andbehaviour dynamics by incorporating changes of behaviour in the equations.Such models are classified as: (1) prevalence-based, (2) belief-based, or (3)a combination of both as suggested in [30, 60]. In this thesis, we analysedthree different prevalence- and belief-based models, starting with the Triplecontagion model [26]. This model considers three contagions: disease, fear ofdisease, and fear of vaccine. Then, we analysed a Reduced triple contagionmodel that comprises only the fear of disease with no vaccine-related terms.Lastly, we analysed the Modified triple contagion model which includes anadditional fear compartment representing people holding fear of disease andfear of vaccine simultaneously.Our analysis in Chapter 2, showed that any of the contagions (disease,fear of disease, or fear of vaccine) can initially show a growth faster thanthe other contagions in different parametric regimes. We observed thatfear of disease can show rapid growth before the disease itself. This result isconsistent with earlier work [27] showing that fear can become the dominantcontagion even in the absence of a driver (disease in this case) becausefear transmission does not require direct contact. Knowing the initiallydominant contagion can help governments decide the appropriate course ofaction in the initial phase of outbreak. For example, [53] recommends tospread information as early as possible, when the infection spreads rapidlyin diseases with short-term outbreaks, in order to reduce disease severity.Moreover, a quick and effective implementation of policies in the initial phaseof outbreak is crucial to avoid delayed and unsuccessful execution of policies[47].The loss of fear through complacency does not play an important rolein driving the disease dynamics in the initial phase of the outbreak. We70observed that αf does not appear in the expressions of basic reproductionnumbers corresponding to fear of disease and fear of vaccine. In the initialphase of the disease, there are no individuals in the immunized compart-ments to induce loss of fear, and therefore complacency contagion cannotalter the disease transmission. This observation was also made in [26] whileinvestigating the effect of αf on disease trajectory. The authors found thatthe infection waves corresponding to different values of αf overlap initiallyand show different dynamics only after the first infection peak is reached.Fear response to disease and emotions of other individuals can result ina bifurcation in final epidemic size. In Chapter 3, our goal was to theoret-ically determine the conditions leading to bifurcations observed in earlierstudies of this and similar models [26, 37, 61]. Numerical analysis of theReduced triple contagion model in [26] demonstrated a bifurcation in finalepidemic size (shift from high to low), the infection dynamics (shift froma pair of infection waves to a small single infection wave), and the finaldisease-fearful population (shift from a zero to a non-zero fraction) with re-spect to behaviour related parameters (p, βfd, and αf). These bifurcationswere also observed in [61] with a slightly different model that included im-itative fear loss (Sfd → S on interaction between Sfd and S individuals).The authors discussed that fear retention in the population even after theoutbreak is over is due to belief-based fear acquisition because it occursindependently of disease-prevalence. Another model that obtained these bi-furcations is [37], which considered observational fear acquisition and fearloss due to complacency and imitation of the non-compliant behaviour asthe mechanisms that drive dynamics between the regular susceptible anddisease-fearful populations.The effectiveness of contact reduction (p) and the rate of fear loss (αf)can significantly affect the disease trajectory. Our numerical results showedthat smaller values of p and αf yield a small infection wave, demonstratingthe effectiveness of contact reduction in suppressing the disease spread asalso observed in [26, 44]. We also observed that as p and αf increase further,the infection dynamics shift to a pair of infection waves with the secondinfection wave emerging ever sooner after the first one, thus demonstratingthat contact reduction can control the time of onset of the second infectionwave as also observed in [18, 26, 62]. An interesting study left for futurework is to mathematically explain the shift of the second infection wavetoward the first one.The relative speeds of disease and fear spread can affect the final fearstate. The stability analysis showed the existence of two equilibria, onewhich is fear-free and another which is fear-endemic. We found that one71of the conditions for the fear-endemic equilibrium to be stable is that thefear reproduction number should exceed the disease reproduction number.Recall, that the fear-endemic equilibrium corresponds to a low final epidemicsize. A similar condition was also obtained in [62], where the authors foundthat final epidemic size is low if fear spread is fast compared to disease spreadand fear loss is not too quick. This comparison of the rates of fear spreadand disease spread was also discussed in [61]. The authors observed that fora given value of the disease reproduction number, as the fear reproductionnumber crosses a threshold, the final epidemic size drops to 0.The stability analysis did not provide information on bifurcations in finalepidemic size and infection dynamics, or the final size of each subpopulation.We thus conducted an asymptotic analysis of the Reduced triple contagionmodel (3.1). We considered two different cases: the Flu-like and the COVID-like limits. In both cases, we assumed that recovery from fear is muchslower than recovery from disease. This assumption is consistent with studiesof fear loss [3, 50]. Research on overcoming fears suggests that recoveryfrom fear requires the help of therapies and external interventions [50]. Therequirement of intensive therapy to overcome fear supports our assumptionthat a natural fear-loss, considered in our model via γf , would take extremelylong.In the Flu-like limit, we assume that the timescale of fear acquisition iscomparable to the timescale of fear loss. Therefore, fear of disease spreadson a slower time scale compared to that of disease. The initial behaviour ofthe system in this limit showed that infectious population grows rapidly butthe fear population remains relatively unchanged. This decoupling of fearfrom disease dynamics was also observed in [25]. This scenario may manifestin outbreaks of diseases that recur or are associated with low perceptionsof risk [51]. Meta-analyses of the relationship between risk perceptions anddisease-preventive behaviour [13, 67], show that perceptions of risk play animportant role in shaping health-related behaviours.In the COVID-like limit, we assume that fear spreads on a faster timescalethan in the first case. The initial behaviors of the system in this limit showedthat fear population grows rapidly but the infectious population remains rel-atively unchanged. This scenario may arise in an outbreak due to a novel ordeadly virus which is associated with high risk perceptions that encouragehealth-protective behaviour [13]. A study conducted on the popular percep-tions regarding COVID-19 showed that people perceive novel diseases likeCOVID-19 as significantly threatening compared to seasonal flu [56]. Suchdiseases hence show an elevated fear response [56]. The results in this limitshowed that the system dynamics may yield a single infection wave or a72pair of infection waves. We observed that if fear acquisition is fast enough,it may successfully prevent subsequent infection waves, as also observed in[61, 62].In both the Flu-like and COVID-like limits, we observed that final epi-demic size is high for smaller values of βfd. Slow cross-transition betweenthe regular susceptible and disease-fearful population results in polarization(population divides between prophylactic and non-prophylactic behaviours)[74]. On investigating the effects of polarization on disease dynamics, theauthors found that polarization results in a larger final epidemic size as weobserved in our analysis.The Triple contagion model (2.1) considered population groups holdinga single fear: fear of disease or fear of vaccine. It is possible, however,for an individual to be simultaneously fearful of both the disease and thevaccine such that they have traits of both type of individuals [68]. Theseindividuals with simultaneous fear of disease and fear of vaccine may reducetheir contacts in order to reduce their susceptibility but may also avoidvaccination.In Chapter 4, we analysed the Modified triple contagion model, a modifi-cation to the Triple contagion model by adding the double fear compartmentSfdv. To the best of our knowledge, no previous studies have examined the ef-fects of simultaneous fear of disease and fear of vaccine on disease dynamics.In this model, we assumed that being aware of the disease risks, the doublyafraid individuals reduce their contact more efficiently to protect from dis-ease since they do not get vaccinated. The doubly afraid individuals have ahigher risk perception compared to the disease-fearful individuals, therefore,these individuals put more effort to reduce their susceptibility [13]. More-over, we obtained that if one of the groups out of Sfd and Sfdv do not reducecontacts efficiently, the other group increases contact reduction to compen-sate for the reduced protection of the other group. This highlights the needto take a collective approach towards compliance to disease-mitigative ac-tions in contrast to visualising benefits on individual-level [65].A faster fear acquisition may not always reduce disease spread. In Figure4.3 (middle plot), we observed that the final epidemic size first increaseswith increasing value of βfv. An increase in βfv represents, in addition to anincrease in rate of acquisition of fear of vaccine, an increase in transition fromdisease-fearful to double fear population. As double fear increases, a largerfraction of population exhibits more contact reduction which should reducethe final epidemic size. This counter-intuitive increase in final epidemic sizewas also observed in [74], where the authors found that if the populationproactively responds to the disease dynamics (i.e., acquires and loses fear73as disease prevalence increases or decreases), a fast fear response results inan increase in the final epidemic size and multiple subsequent waves withdecreasing peak size.We observed from different regions in the final epidemic size heatmap4.4 that as the final epidemic size increases with respect to βfv and αf , theinfection dynamics goes from a small infection wave to a pair of infectionwaves, to a big single infection wave. The creation of second wave whenfinal epidemic size increases is similar to the multi-peak transition obtainedin the COVID-like limit in Chapter 3. We could get up to two waves in ouranalysis. Authors of [61, 74] obtained up to four waves from their models byconsidering a fear response that is based only on disease prevalence. Sucha fear response can increase the duration of the outbreak and also producemultiple infection waves [55]. Future research on our model could exploremechanisms that may give rise to more than two waves. One possible ex-tension is to include waning immunity. [51] highlight how including waningimmunity helped them reproduce the third infection wave of COVID-19 inautumn 2021.Our results showed that the second infection wave can be smaller orlarger than the first infection wave. Conceptually, we get a larger secondinfection wave when people decide not to take protective actions when re-quired as observed in 1918 Influenza [26, 38]. The amplification in fear lossdue to complacency contagion increases the fraction of susceptibles, henceincreasing the size of the second infection wave [26]. Further, [74] discussedthat a higher second infection wave can arise when the population reactsto disease dynamics proactively. On the other hand, the second infectionwave may be smaller than the first infection wave when people continueactive vigilance after the disease vanishes at the end of the first infectionwave [26]. An important next step of this work is to find the mathematicalconditions under which we get a lower or higher second infection peak.Vaccination is an effective tool to suppress the disease [36, 51, 77]. Ourresults showed that vaccination can mitigate the second infection wave. Asimilar result was obtained in [26]. The authors found that subsequentinfection waves can be avoided or reduced in size by vaccinating people.They found that if the rate of spread of fear of vaccine is slow and rate ofspread of fear of disease is fast enough, vaccination can successfully decreasethe fraction of population getting infected.We observed that limited vaccine coverage can increase vaccine hesitancy.A lower vaccine coverage led to lesser people giving positive feedback aboutthe vaccine and more people recovering naturally from disease hence givinga sense that vaccines are not too useful. This resulted in majority of popu-74lation becoming complacent about the disease and shifting from the doublefear compartment to the fear of vaccine compartment. We also observedthis scenario in Chapter 2 in the parametric region where fear of vaccine isdominant. The World Health Organization listed vaccine hesitancy in thetop ten threats to global health [58]. The surge in vaccine refusal due toits fear can result in the rebirth of diseases that were controlled by vaccineslike measles, pertussis, and polio [26]. Lesser willingness to get vaccinatedand easy propagation of anti-vaccine sentiments [57] highlights the need ofmore efforts in publicising the benefits of vaccines.Our results showed that the final epidemic size remains low when alarge pool of the population either gets vaccinated or shifts to the doublefear compartment. Contact reduction can be very effective in suppressingthe disease [26, 44] and hence developing fear of disease in the populationcan serve as an effective disease mitigation strategy. Moreover, the WorldHealth Organization has emphasized that non-pharmaceutical interventionssuch as contact reduction play an important role in reducing disease severity[9]. This result provides us a strategy to reduce the final epidemic sizewhen several individuals refuse to get vaccinated. Our analysis suggestedto make these vaccine-fearful individuals aware of the risks of becominginfected so that they choose mitigative behaviour and enter the double-fearcompartment to protect themselves from disease. The idea of developingdouble fear in the population to reduce the final epidemic size is a novelapproach.Our model consisted of many behaviour-related parameters and vari-ables. Throughout our analysis, we did not focus on individual parametervalues, instead, we looked at ratios of parameters. It is challenging to mapthe simulation parameters to real life because of the lack of data. To cal-ibrate these models, future efforts should be focused on collecting relevantdata. In future, we would like to fit our model to data for specific diseases toeither determine or get a sense of the order of some of the behaviour-relatedparameters.From the lens of behavioural science, our model can be extended by incor-porating information transmission via mass media [19, 61] as a means of fearspread. During a disease outbreak, constant exposure to information aboutthe disease through mass media can elevate fear response, giving rise to fearof disease even in individuals who have never come into contact with aninfected or disease-fearful person [61]. This elevated fear response can miti-gate the disease [19] or create panic [61]. Moreover, unverified information onmass media can facilitate misinformation [74] about the efficacy and safetyof vaccines which can increase vaccine hesitancy [46]. In the disease context,75[78] shows that information spread through media can increase perceptionsof the risk and severity of disease. Moreover, exaggerated media reportscan result in polarization in the population, with some people exhibitingheightened anxiety while others not reacting at all [70]. Including the effectof information transmission via media becomes important when people arenot well connected and information spread majorly happens through media.For example, in the 2003 SARS outbreak, low incidence rate of the outbreakmade it less likely that the general public encounters an infectious individualand thus gains awareness about the disease information. The informationabout the disease primarily spread through media reports [28].Non-pharmaceutical interventions such as social distancing require con-sistent reminders for effective compliance as individuals reevaluate their de-cision to adhere to social distancing policies based on the disease prevalence[32]. Effects of adaptive contact reduction has been studied in [32], wherethe decision to protect from disease is driven by disease prevalence and thecosts involved in reducing contacts. The authors obtained oscillatory dy-namics in social distancing behaviour and infection spread. Further, theirresults showed that a highly rational and adaptive population can worsenthe disease by yielding multiple small infection waves hence resulting in alarger final epidemic size. It would be interesting to incorporate this ideainto our model to examine the impact of adaptive behaviour on disease out-come in the presence of pharmaceutical interventions, for example, vaccine.Further, adaptive fear response comes into play in the context of habitu-ation. If the virus circulates for too long, it may not elicit the same fearresponse because people get used to it [71]. This reduced fear response dueto habituation comes into action on a timescale longer than the scale weare considering in our analysis. Habituation can be incorporated into ourmodel by defining fear-induced contact reduction parameters, p and pfdv, asincreasing functions of time.Changes in human behaviour are complex, especially when driven by adynamically changing risk factor like infection [26, 74]. Our model includedaspects of human behaviour rooted in neuroscientific theories of human be-haviour adaptation and fear response [26, 68]. In order to fully understanddisease dynamics within a human population, it is necessary that math-ematical models included terms founded in behavioural science. There isa wealth of literature in cognitive science and public health that studythe impact of disease dynamics on human behavioural changes and viceversa [30, 75]. Knowledge of these theories can help in developing informeddisease-mitigation strategies [30, 32]. Several studies have considered fearresponse to disease and vaccine independently [23, 26, 51, 72]. 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Lin, Adaptability and life satisfaction: The moder-ating role of social support, Frontiers in Psychology, 7 (2016). → pages886Appendix87Appendix AA.1 Scaling analysisAs a first step to solve for the state variables, we simplify (3.1) by scalingto a non-dimensional time t¯ = γt,dSdt¯= −bIS + δ[−bf(Sfd + I)S + Sfd + a(1− S − Sfd − I)Sfd], (A.1a)dSfddt¯= −bpISfd − δ[−bf(Sfd + I)S + Sfd + a(1− S − Sfd − I)Sfd], (A.1b)dIdt¯= [b(S + pSfd)− 1]I. (A.1c)with S(0) = S0, Sfd(0) = Sfd0, and I(0) = I0.We assume that the outbreak starts with everyone in the fearless suscep-tible compartment (S), and an infectious individual. To satisfy these initialconditions, the state variables in (A.1) are scaled asS = 1 + δx, (A.2a)Sfd = δy, (A.2b)I = δz. (A.2c)The system of equations (A.1) then translate to,δx˙ = −bδz(1 + δx)− δ2bf(y + z)(1 + δx) + δ2y+ δ2a(−δ(x+ y + z))y, (A.3a)δy˙ = −pbδ2yz + δ2bf(y + z)(1 + δx)− δ2y− δ2a(−δ(x+ y + z))y, (A.3b)z˙ = b(1 + δx)z + δbpyz − z. (A.3c)There are two possible scaling, one where the rate of fear acquisition iscomparable to the rate of fear loss and another where it is comparable tothe rate of disease recovery.88A.1. Scaling analysisCase 1: βfd ∼ γfIn this scaling, the rate of fear acquisition is assumed to be comparableto the rate of fear lossβfd ∼ γf << γ. (A.4)where, the rate of fear loss is assumed to be smaller than the rate of diseaserecovery.On dividing (A.4) by γf , we obtainbf ∼ O(1). (A.5)Ignoring order δ and higher order terms, we obtain the leading orderproblem of the scaled model (A.3) as,x˙ = −bz, (A.6a)y˙ = 0, (A.6b)z˙ = (b− 1)z. (A.6c)On solving (A.6c), we obtain,z(t) = I0e(b−1)t. (A.7)Equations (A.7) and (A.6b) respectively tell us that the infection growsrapidly (using assumption (3.5a)) but the fear population is relatively un-changed. The infectious population grows because there are sufficient sus-ceptibles but the fear of disease population remains small because of ourassumption of slow fear acquisition. Therefore, after some transience, weexpect S ∼ O(1), Sfd ∼ O(δ), and I ∼ O(1). This scenario is evident in thecontext of, e.g., the annual flu season which due to its regular occurrenceand a relatively lower fatality rate compared to more severe diseases maymanifest a reduced fear response initially. We hence call this first case theFlu-like limit. We investigate the system (A.1) by assuming the system be-haviour after some transience as the initial state. This assumption allowsus to solve for the system dynamics starting from a later time. Hence, letS = x, Sfd = δy, and I = z with the initial conditions being x(0) = S0,y(0) = S¯fd0, z(0) = I0.Case 2: βfd ∼ γIn this scaling, the rate of fear acquisition is assumed to be comparableto the rate of disease recoveryβfd ∼ γ. (A.8)89A.1. Scaling analysisOn dividing (A.8) by γf , we obtainbf ∼ O(δ−1), (A.9)This gives,bf = bˆfδ−1, (A.10)where, bˆf is proportionality constant.We further assume thata = aˆδ−1. (A.11)Assuming that the outbreak starts with everyone in the fearless suscepti-ble compartment (S), and a single individual in the infectious compartment(I). We substitute (A.10) and (A.11) in (A.3). We obtainx˙ = −bz(1 + δx)− bˆf(y + z)(1 + δx) + δy− δaˆ(x+ y + z))y, (A.12a)y˙ = −pbδyz + bˆf(y + z)(1 + δx)− δy+ δaˆ(x+ y + z)y, (A.12b)z˙ = b(1 + δx)z + δbpyz − z. (A.12c)To leading order, we havez(t) = I0e(b−1)t. (A.13)The leading order solution of z(t) is same in the two scalings (A.7) and(A.13). The leading order equations for x and y are,x˙ = −bz − bˆf(y + z), (A.14a)y˙ = bˆf(y + z). (A.14b)with the initial conditions x(0) = −I0 and y(0) = 0. On solving (A.14b),we obtain,y(t) =I0bˆf1− b+ bˆf(ebˆf t − e(b−1)t). (A.15)If bˆf ≥ (b− 1), the fear exponential ebˆf t grows faster than the disease expo-nential e(b−1)t and y loses asymptotic consistency approximately whenebˆf t ∼ δ−1. (A.16)90A.2. Flu-like limitWhen (A.16) is satisfied,I ∼ I0δ1−(b−1bˆf)<< 1. (A.17)Equations (A.13) and (A.17) respectively tell us that the fear populationgrows quickly but the disease population is relatively unchanged. Therefore,after some transience, we expect S ∼ O(1), Sfd ∼ O(1), and I ∼ O(δ). Thisscenario is evident in the context of novel or deadly diseases , e.g., COVID-19which manifest a fast fear response compared to the first scaling. We hencecall this second case the COVID-like limit. We investigate the system (A.1)by assuming the system behaviour after some transience as the initial state.This assumption allows us to solve for the system dynamics starting froma later time. Consequently, let S = x, Sfd = y, and I = δz with the initialconditions being x(0) = S0, y(0) = S¯fd0, z(0) = I0.A.2 Flu-like limitWe learned from Section A.1 that in the Flu-like limit we can solve forsystem dynamics by scaling S = x, Sfd = δy, and I = z with the initialconditions being x(0) = S0, y(0) = S¯fd0, z(0) = I0. Plugging this scalinginto (A.1), we arrive at the scaled system,x˙ = −bxz − δbfxz, (A.18a)y˙ = −bpyz + bfxz − δ(y − bfxy + ay − axy − ayz), (A.18b)z˙ = (bx− 1)z + δbpyz. (A.18c)We expand each of these state variables as a sum of a leading order andan O(δ) term. We assume that the leading order terms x0, y0, and z0 arebigger than the O(δ) terms δx1, δy1, and δz1,x = x0 + δx1 +O(δ2), (A.19a)y = y0 + δy1 +O(δ2), (A.19b)z = z0 + δz1 +O(δ2). (A.19c)The corresponding leading order system of (A.18) is,x˙0 = −bx0z0, (A.20a)y˙0 = −bpy0z0 + bfx0z0, (A.20b)z˙0 = (bx0 − 1)z0. (A.20c)91A.2. Flu-like limitwith the initial conditions being x0(0) = S0, y0(0) = S¯fd0, z0(0) = I0. Notethat in this last system of equations, x0 and z0 decouple from y0, so we cansolve for z as follows. We first divide Equations (A.20a) and (A.20c),z˙0x˙0=1bx0− 1, (A.21)which gives,z0(x0) =1blog x0 − x0 +A0, (A.22)where,A0 = I0 − 1blogS0 + S0. (A.23)Integrating (A.20a) using (A.22), we obtaint =1b∫ x0S0duu(u− 1b log u−A0) . (A.24)In (A.24), the integrand has asymptotes at u = 0 and f(u) = 0 where,f(u) = u− 1blog u−A0. (A.25)The function f(u) has a minimum at u = b−1 because f ′(b−1) = 0 andf ′′(b−1) > 0. Near the disease and fear-free equilibrium, we generally haveS0 ≈ 1 and I0 ≈ 0, which means that A0 ≈ I0 + S0 ≈ 1. Hence, for b > 1,we find that f(b−1) < 0. The minima, b−1, lies between the lower andupper roots, u− and u+, of f(u) respectively. Note that f(S0) = −I0 < 0(using Equations (A.25) and (A.23)), so f(S0) < f(u+) = 0. Since x0decreases monotonically, f(x0) < f(u+). Furthermore, limu→0 f(u) → ∞,so, 0 < u− < b−1. Therefore, the long-time limit of x0 is,x∞0 = u−. (A.26)Observe also that x˙0 + z˙0 = −z0, which, upon integration gives,x∞0 − S0 − I0 = −∫ ∞0z0dt. (A.27)We solve for y0 by integrating (A.20b) which can also be written asy˙0 − p( ˙log x0)y0 = −bfbx˙0. (A.28)92A.2. Flu-like limitOn integrating this linear equation, we obtain,y0 = − bfb(1− p)x0 +B0xp0, (A.29)where,B0 =[S¯10 +bfS0b(1− p)]S−p0 . (A.30)(A.29) gives an estimate of the final size of the fear of disease compart-ment as,y∞0 = −bfb(1− p)x∞0 +B0x∞p0 , (A.31)Note that this result predicts only a non-zero fear-population at the endof the outbreak. Based on equilibria (3.2a) of the model (3.1), however,the fear population at the end of the outbreak can also go to 0. So, weinvestigate the problem (A.18) at O(δ).The set of equations at O(δ) arex˙1 = −bx0z1 − bx1z0 − bfx0z0, (A.32a)y˙1 = −bpy0z1 − bpy1z0 + bfx0z1 + bfx1z0 − y0+ bfx0y0 − ay0 + ax0y0 + ay0z0, (A.32b)z˙1 = (bx0 − 1)z1 + bx1z0 + bpy0z0. (A.32c)The initial values at O(δ) are x1(0) = 0, y1(0) = 0, and z1(0) = 0. As in theleading order problem, y1 decouples from x1 and z1 and we can write theseequations in matrix form as,[x˙1z˙1]=[ −bz0 −bx0bz0 bx0 − 1] [x1z1]+[ −bfx0z0bpy0z0 − z0]. (A.33)Note that the quantities x0 and z0 are both time-dependent variables, andso (A.33) is a linear matrix equation with time-varying coefficients. There isno general analytic solution of (A.33) because, for any s and t, C(t)C(s) ̸=C(s)C(t). We can, however, sum the two equations to investigate the O(δ)equations further. We havex˙1 + z˙1 = −z1 − y˙0, (A.34)On integrating (A.34), we obtain,x∞1 + y∞0 − S¯10 =∫ ∞0z1dt, (A.35)93A.2. Flu-like limitTo extract information from (A.35), we transform x1 using the scaling x1 =x0u, which yieldsx˙1 = x0u˙+ x˙0u= x0u˙− bx0z0u, (A.36)On equating Equations (A.32) and (A.36), we obtain,u˙ = −bz1 − bfz0, (A.37)with the initial condition u(0) = 0. On integrating this equation, we obtain,u∞ =x∞1x∞0= b(x∞1 + y∞0 − S¯10)− bf∫ ∞0z0dt= b(x∞1 + y∞0 − S¯10) + bf(x∞0 − S0 − I0). (A.38)Finally, solving for x∞1 we obtain,x∞1 = x∞0[b(y∞0 − S¯10) + bf(x∞0 − S0 − I0)(1− ax∞0 )]. (A.39)Since we are interested in the final state of each of the populations, wesimplify (A.32) by substituting for the long-time leading order variables asx0 → x∞0 , y0 → y∞0 , and z0 → 0. In the long-time limit, the system ofequations (A.32) translates tox˙1 = −bx∞0 z1, (A.40a)y˙1 = (−bpy∞0 + bfx∞0 )z1 − y∞0 + bfx∞0 y∞0 − ay∞0 + ax∞0 y∞0 , (A.40b)z˙1 = (bx∞0 − 1)z1. (A.40c)On integrating the z˙1, x˙1, and y˙1 equations respectively, we obtainz1(t) = C0e−(1−bx∞0 )t, (A.41a)x1(t) =bx∞0(1− bx∞0 )C0e−(1−bx∞0 )t + x∞1 , (A.41b)y1(t) =(bpy∞0 − bfx∞0 )C0e−(1−bx∞0 )t(1− bx∞0 )+ y∞0 (−1 + bfx∞0 − a+ ax∞0 )t+ C1.(A.41c)where, C0 and C1 are unknown constants.Note from (A.41c) that y1 is the sum of an exponentially decaying term(x∞0 < b−1), a linear term in t and a constant term. As t → ∞, y1 blows94A.2. Flu-like limitup hence breaking the constant population constraint. Clearly, y can notgrow indefinitely and so we need to look for a long-time rescaling wherefear no longer decouples from the problem. This happens when t ∼ O(δ−1).Therefore, we do a long-time rescaling of the model (A.1) by substitutingt = δ−1τ where τ represents a time scale relatively much larger than t. Thelong-time scaling of δ−1 is a distinguished limit that comes from fear acqui-sition becoming the dominant mechanism. In the long term, the infectiouspopulation vanishes, i.e., I → 0. We get the new set of equations as,dSdτ= Sfd − bfSSfd + a(1− S − Sfd)Sfd, (A.42a)dSfddτ= −Sfd + bfSSfd − a(1− S − Sfd)Sfd. (A.42b)where, the initial condition for this new time scale is determined as a limitfrom the previous one. Specifically, we have S → x∞0 + δx∞1 and Sfd → δy∞0as τ → 0. This kind of scaling is a common practice in the method ofmatched asymptotics, refer [35] for more details.On adding (A.42a) and (A.42b), we obtain the population conservationlaw,dSdτ+dSfddτ= 0, (A.43)and on integrating (A.43) we obtain,S + Sfd = x∞0 + δ(x∞1 + y∞0 ) = θ (constant). (A.44)We integrate (A.42b) and use (A.44) with the initial condition Sfd(0) = y∞0to solve for Sfd asSfd(τ) =y∞0 (bfθ − a(1− θ))− 1bfy∞0 + [bf(θ − y∞0 )− a(1− θ)− 1]e[−(bfθ−a(1−θ)−1)τ ]. (A.45)The two steady states (3.2a) and (3.2b) corresponding to zero or non-zerofear of disease population are resolved by the argument of the exponentialin (A.45).The argument of the exponential is negative if[bfθ − a(1− θ)− 1] > 0 or bf > 1 + a(1− θ)θ, (A.46)Then as τ →∞,Sfd → θ −(1 + a(1− θ)bf), (A.47a)S →(1 + a(1− θ)bf). (A.47b)95A.3. COVID-like limitThis solution corresponds to the non-zero fear of disease population at theend of the outbreak. On the other hand, the argument in the exponential ispositive if[bfθ − a(1− θ)− 1] < 0 or bf < 1 + a(1− θ)θ, (A.48)Then as τ →∞,Sfd → 0, (A.49a)S → θ. (A.49b)This solution corresponds to the zero fear of disease population at the endof the outbreak.The critical value of bf beyond which S∞fd transitions from the zero tothe non-zero steady state isbcf =1 + a(1− θ)θ. (A.50)From (A.44), note that θ comprises x∞0 and y∞1 which are a function ofbf . So, we can find the bifurcation point by expanding bcf as,bcf = bcf0 + δbcf1. (A.51)Substituting (A.44) in (A.51), we obtainbf =1 + a[1− x∞0 − δ(x∞1 + y∞0 )]x∞0 + δ(x∞1 + y∞0 ),=1 + a(1− x∞0 )x∞0+ δ[−(x∞1 + y∞0 )(1 + a)x∞20]. (A.52)On comparing (A.51) and (A.52), we obtainbcf0 =1 + a(1− x∞0 )x∞0, (A.53a)bcf1 =−[x∞1 (bcf0) + y∞0 (bcf0)](1 + a)x∞20. (A.53b)A.3 COVID-like limitWe learned from Section A.1 that in the COVID-like limit we can solvefor system dynamics by scaling S = x, Sfd = y, and I = δz with the initial96A.3. COVID-like limitconditions being x(0) = S0, y(0) = 1 − S0 − δI0, z(0) = I0. Plugging thisscaling into (A.1), we arrive at the scaled systemx˙ = −bˆfxy + aˆ(1− x− y)y − δ(bxz + bˆfxz − y + aˆyz), (A.54a)y˙ = bˆfxy − aˆ(1− x− y)y + δ(−bpyz + bˆfxz − y + aˆyz), (A.54b)z˙ = [b(x+ py)− 1]z. (A.54c)We then expand each of the state variables as a sum of a leading orderand an O(δ) term by plugging (A.19) into (A.54). We obtain the leadingorder system of (A.54) with the initial conditions x0(0) = S0, y0(0) = 1−S0,and z0(0) = I0x˙0 = −bˆfx0y0 + aˆ(1− x0 − y0)y0, (A.55a)y˙0 = bˆfx0y0 − aˆ(1− x0 − y0)y0, (A.55b)z˙0 = [b(x0 + py0)− 1]z0. (A.55c)Note that,x˙0 + y˙0 = 0 =⇒ x0 + y0 = 1. (A.56)We eliminate one variable using (A.56) and solve (A.55b) to obtainy0(t) =1− S01− S0 + S0e−bˆf t. (A.57)On solving (A.55c), we obtainz0(t) = I0(1− S0 − S0e−bˆf t)−(1−p)bbˆf e(bp−1)t. (A.58)If bp − 1 > 0 (where bp represents the reduced disease reproduction num-ber due to fear-induced contact reduction), the solution (A.58) shows thatinfection is an increasing function of t. This result shows that infectionspreads rapidly with time and that fear is not enough to suppress it. Sincethe total population is constrained, the infectious population cannot growindefinitely. On the other hand, if bp − 1 < 0, infection decays over timeshowing that fear is successful in reducing the disease.The solution (A.57) shows that in infinite time limit, y0 plateaus at 1meaning that the whole population shifts to the fear compartment. However,the steady-state analysis in Section 3.2 indicated that at equilibrium thefearful population can also go to 0.97A.3. COVID-like limitWe thus consider the O(δ) problem for the case when bp − 1 < 0. Inthe infinite-time limit, the leading order size of disease-fearful populationplateaus at 1, i.e.,y0 = 1. (A.59)Using (A.56), we obtain the infinite-time leading order value of the regularsusceptible populationx0 = 0. (A.60)Now we consider theO(δ) problem with the leading order solutions (A.60)and (A.59),x˙1 = −bˆfx1 + 1− aˆ(x1 + y1 + z0), (A.61a)y˙1 = −bpz0 + bˆfx1 − 1 + aˆ(x1 + y1 + z0), (A.61b)z˙1 = (bp− 1)z1 + b(x1 + py1)z0, (A.61c)with initial conditions x1(0) = 0, y1(0) = −I0, and z1(0) = 0. On addingEquations (A.61a) and (A.61b), we obtainx˙1 + y˙1 = −z˙0 − z0, (A.62)and on integrating (A.62), we obtainx1 + y1 = −z0 − zˆ0, (A.63)where,zˆ0 =∫ t0z0ds =I0bp− 1(e(bp−1)t − 1). (A.64)Solving for y1 using Equations (A.61b) and (A.63), we obtainy1(t) = E0e(bp−1)t + y∞1 − (E0 + y∞1 + I0)e−bˆf t, (A.65)where,E0 =−bpI0bp− 1 −aˆI0(bp− 1)(bp+ bˆf − 1). (A.66)If bp− 1 < 0, theny∞1 = limt→∞ y1(t) =(bˆf + aˆ)I0bp−1 − 1bˆf. (A.67)To solve for z1, let z1 = uz0. Then,z˙1 = z0u˙+ (bp− 1)z0u, (A.68)98A.3. COVID-like limitand (A.61c) becomes,z˙1 = (bp− 1)z0u+ b(x1 + py1)z0. (A.69)On equating Equations (A.61c) and (A.69) and using (A.63), we obtain,u˙ = −b(z0 + zˆ0 + (1− p)y1), (A.70)which has the solution,u = −b(v0e(bp−1)t + u∞t+ u0e−bˆf t − (v0 + u0)), (A.71)where,v0 =bp2I0(bp− 1)2 −aˆ(1− p)I0(bp− 1)2(bp+ bˆf − 1), (A.72a)u0 = (1− p)(− 1bˆ2f+aˆI0bˆ2f (bp+ bˆf − 1)), (A.72b)u∞ =−pI0bp− 1 −(1− p)bˆf+aˆ(1− p)I0bˆf(bp− 1). (A.72c)We can also compute,zˆ1 =∫ t0z1ds= −bI0(v02(bp− 1)e2(bp−1)t +u∞(bp− 1)2 ((bp− 1)t− 1)e(bp−1)t+u0(bp− 1− bˆf)e(bp−1−bˆf)t − (v0 + u0)(bp− 1) e(bp−1)t)+ zˆ∞1 ,(A.73)where,zˆ∞1 = −bI0(v02(bp− 1) −u0bˆf(bp− 1)(bp− 1− bˆf)+u∞(bp− 1)2). (A.74)Note that, as p → 1b−, then y∞1 , zˆ∞0 , and zˆ∞1 blow up. Clearly, y cannotgrow indefinitely, so our solution becomes invalid when p → 1b−. We can,however, investigate the solutions near this singularity by settingp =1b+ δpˆ. (A.75)99A.3. COVID-like limitThen the leading order and O(δ) solutions become,z0 ∼ I0(1 + bδpˆt), (A.76a)zˆ0 ∼ I0t+ δI02bpˆt2, (A.76b)y1 ∼ −(bˆf + aˆ)bˆfI0t+(I0aˆbˆ2f− 1bˆf− I0)−(I0aˆbˆ2f− 1bˆf)e−bˆf t, (A.76c)z1 ∼ I202bˆf((b− 1)aˆ− bˆf)t2 − I0bˆ2f(I0bˆ2f + (b− 1)aˆI0 − (b− 1)bˆf)t+ I0(b− 1)(1bˆ2f− I0abˆ3f)e−bˆf t − I0(b− 1)(1bˆ2f− I0abˆ3f). (A.76d)Note from (A.76d) that z1 is the sum of a quadratic term in t, a linear termin t, an exponentially decaying term in t, and a constant. Due to the firsttwo terms, z1 blows up as t → ∞. Similarly, zˆ0 also blows up. Since z0and z1 represent a fraction of the population, they cannot grow indefinitelyand so our result (A.76) is useful over some interval of time, but eventuallyloses asymptotic consistency. To solve this problem, we seek a new timescalein which the populations remain constrained. The appropriate rescaling ist = τ/√δ. Substituting for t in (A.76), we obtainz ∼ I202bˆf((b− 1)aˆ− bˆf)τ2 + I0, (A.77a)zˆ ∼ I0τ√δ, (A.77b)y ∼ 1−√δ(bˆf + aˆbˆf)I0τ, (A.77c)x ∼ δx1 = −δ(y1 + z0 + zˆ0) = −√δ(√δI0 −(bˆf + aˆbˆf)I0τ + I0τ).(A.77d)We thus scale the state variables as S =√δX, Sfd = 1+√δY , and I = δZ.Substituting these scalings into (A.1), we obtain the new scaled model√δdXdτ= −bˆfX − aˆ(X + Y ) +√δ(1− bˆfXY − aˆ(X + Y )Y − aZ) +O(δ),(A.78a)100A.3. COVID-like limit√δdYdτ= bˆfX + aˆ(X + Y )−√δ(1− bˆfXY − aˆ(X + Y )Y − aˆZ + Z) +O(δ),(A.78b)dZdτ= (bX + Y )Z +O(√δ). (A.78c)On comparing the leading order terms on the left and right-hand side of(A.78a) and (A.78b), we obtain that (A.78a) and (A.78b) are in quasi-steadystate, yieldingY = −(bˆf + aˆaˆ)X. (A.79)On differentiating (A.79) with respect to τ , we obtainXτ + Yτ = − bˆfaˆXτ . (A.80)and on adding Equations (A.78a) and (A.78b) we obtain,dXdτ+dYdτ= −Z +O(√δ). (A.81)Using (A.79), (A.80) and (A.81), we can simplify (A.78c) asZτ = − bˆf2aˆµ(X2)τ , (A.82)where,µ =bˆf − (b− 1)aˆaˆ. (A.83)On solving (A.82), we obtain,Z = − bˆf2aˆµ(X2 − C). (A.84)where C is some constant. On comparing Equations (A.81) and (A.84), weobtainXτ =−µ2(X2 − C), (A.85)and on solving (A.85), we obtainX(τ) =√C tanh(µ2√Cτ + C2), (A.86)where C2 is also some constant.101A.3. COVID-like limitWe observe from (A.77) that as τ → 0,X ∼ I0aˆbˆfτ, (A.87a)Y ∼ −(bˆf + aˆbˆf)I0τ, (A.87b)Z ∼ I202bˆf((b− 1)aˆ− bˆf)τ2 + I0. (A.87c)To match the solution in (A.87a) with (A.86), we take C2 = 0. For a smallτ , on expressing tan-hyperbolic function as a taylor expansion in (A.86), weobtainX(τ) ∼ Cµτ2. (A.88)On comparing (A.88) and (A.87a) we obtain,C =2µI0aˆbˆf. (A.89)This gives,X(τ) =√2µI0aˆbˆftanh(√µ2I0aˆbˆfτ). (A.90)From (A.79), we obtainY (τ) = −(bˆf + aˆaˆ)√2µI0aˆbˆftanh(√µ2I0aˆbˆfτ). (A.91)and from (A.84), we obtainZ(τ) = I0(1− tanh2(√µ2I0aˆbˆfτ)). (A.92)From (A.83), if aˆ << 1, we obtain,µ ≈ bˆfaˆ. (A.93)When aˆ is too small, µ is positive and we do not see the bifurcation in thefinal epidemic size. Further note that µ changes sign when,aˆ =bˆfb− 1 . (A.94)102A.3. COVID-like limitWhen µ becomes negative, the argument of the tan-hyperbolic func-tion becomes complex, therefore, the tanh in the solutions (A.90), (A.91),and (A.92) becomes tan. This switch from sigmoid shape of tanh functionto increasing shape of the tan function gives rise to the second wave whichalso corresponds to a jump in the final epidemic size. Let us investigatewhat the solutions look like whenµ < 0, (A.95)and hence,aˆ >bˆfb− 1 . (A.96)In our simulations, we used b = 2. From (A.96), we obtain the condition,aˆ > bˆf , (A.97)which says that the rate of fear recovery triggered by the recovered groupshould at least be equal to the rate of fear acquisition to show a bifurcationin the final epidemic size and the final fear population.We write µ = −|µ|, then (A.85) becomes,Xτ =|µ|2(X2 + C). (A.98)The constant C appears with a positive sign to make sure thatX remainspositive for τ << 1. Then, the solutions (A.90), (A.91), and (A.92) become,X(τ) =√2|µ|I0aˆbˆftan(√|µ|2I0aˆbˆfτ), (A.99a)Y (τ) = −(bˆf + aˆaˆ)√2|µ|I0aˆbˆftan(√|µ|2I0aˆbˆfτ), (A.99b)Z(τ) = I0(1 + tan2(√|µ|2I0aˆbˆfτ)). (A.99c)These solutions (A.99) blow up when,√|µ|I0aˆ2bˆfτ =π2, (A.100)103A.3. COVID-like limithence instigating the bifurcation in the final infected and recovered popula-tions.We can obtain the critical value of bˆ∗f at which final epidemic size andfinal fear population size show a bifurcation as,bˆ∗f = aˆ(b− 1) (A.101)The solutions in (A.99) break the constant population constraint. Wetherefore need to consider a different time scale such that the populationsremain bounded. It is an interesting future work to further analyse (A.1)in the COVID-like limit to figure out the correct timescale in which thesolutions remain bounded.104
Thesis/Dissertation
eng
Mathematics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Tyson, Rebecca
Moyles, Iain
Incorporating fear (un)learning mechanisms in infectious disease modelling
Text