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## The basic and the eﬀective reproduction numbers

The basic reproduction number, noted R0, is defined as the expected number of secondary cases produced by a single infectious individual, during the entire infectious period, in a fully suscepti-ble population (Anderson and May, 1991; Heesterbeek, 2002). The eﬀective reproduction number —also called the replacement number— is defined as the expected number of secondary cases produced by an infectious individual, at a given time or in a given context; for instance, once the population is subject to interventions such as prevention and treatment (Ridenhour et al., 2018; van den Driessche and Watmough, 2008, 2002; Hethcote, 2000).

The basic and the eﬀective reproduction numbers reflect epidemic severity, and thus are use-ful to study the impact of preventive methods on epidemic dynamics: a large basic reproduction number may be interpreted as a fast epidemic spread among the susceptible population ex-posed to the infectious disease; a decrease in the eﬀective reproduction number reflects epidemic mitigation.

The basic and the eﬀective reproduction numbers can be estimated from mathematical models (Ridenhour et al., 2018). In particular, they can be computed from deterministic com-partmental models, and thus be expressed as functions of the ODE system parameters (Heﬀernan et al., 2005). Notably, there exists a relation between the reproduction numbers and the behavior of the ODE system at the equilibrium (van den Driessche and Watmough, 2002, 2008). ODE sys-tems defining classical deterministic compartmental models for disease transmission, similar to the model depicted in fig. 1.1, often have two equilibria: a disease-free state (DFS), where there are no new infections, and an endemic state (ES), where the epidemic persists (Hethcote, 2000; van den Driessche and Watmough, 2002). The reproduction number is a threshold parameter for the ODE system equilibria: there is a transcritical bifurcation (that is, an exchange in the stability between the equilibria) for the ODE system when the reproduction number equals to 1. When the reproduction number is lower than 1, the ODE system reaches the DFS and thus the epidemic is eliminated in the long run; otherwise, the ODE system will reach the ES (Hethcote, 2000; van den Driessche and Watmough, 2002)5. See fig. 1.2 for a conceptual visualization of the transcritical bifurcation.

**Using the reproduction numbers to determine the herd immunity threshold**

Prevention-induced herd or community immunity refers to the situation in which susceptible individuals are indirectly protected against infection, due to a suﬃciently large prevention cover-age (i.e., the proportion of the population adopting the preventive method against the pathogen) (Porta, 2013). As a result, disease transmission is greatly reduced and the population is said

5 Other bifurcations may arise, depending on the assumptions made in the compartmental models. For instance, backward bifurcations may be found in SIS models where contact rates are non-constant Van Den Driessche and Watmough (2000), and Hopf bifurcations may be found in SIR models considering time delays for vaccination (Bhattacharyya and Bauch, 2010), delay in the immunity oﬀered by vaccination (Khan and Greenhalgh, 1999), or limited resources for treatment (Wang and Ruan, 2004).

Figure 1.2 – Conceptual diagram of a bifurcation in the ODE system Classical systems of ordinary diﬀerential equations (ODE) describing disease transmission usually have two equilibria: the disease-free state (DFS, depicted in blue), where there are no new infections, and the endemic state (ES, depicted in red), where the epidemic persists. A change in the stability of the equilibria occurs when the reproduction number equals to 1 (depicted by the transition from a solid to a dashed line). Note that negative populations (red, dashed line), while corresponding to the solutions of the ODE system, have no biological interpretation.

to be immune as a group; the epidemic is expected to be eliminated in the long run (i.e., the DFS is reached); see Fine et al. (2011). From the modeling perspective, the prevention coverage required for the community to reach herd immunity may be obtained by identifying the pre-vention coverage yielding an eﬀective reproduction number6 lower than 1. The larger the basic reproduction number, the larger the prevention coverage required to eliminate the epidemic.

Public health authorities usually set the target for prevention interventions depending on the prevention coverage threshold that results in herd immunity. However, as discussed in section 1.2, diﬀerent public sentiments and strategies towards the prevention versus treatment dilemma may yield a sub-optimal prevention coverage (i.e., lower than the prevention coverage threshold), despite public health recommendations. Hence, to know whether the level of prevention coverage required for herd immunity can be reached voluntarily is key for the implementation of public health interventions.

6The eﬀective reproduction number depends (implicitly or explicitly, depending on the modeling choices) on the prevention coverage.

### Modeling the decision-making about prevention adoption using a game-theoretical approach

Among the epidemiological models accounting for behavioral change, game-theoretic approaches have been used to address the individual’s decision-making through the study of the prevention versus treatment dilemma; see an early example in Bauch et al. (2013), the two previously mentioned reviews about behavioral epidemiology (Verelst et al., 2016; Wang et al., 2016) and a recent review, concerning specifically game-theoretic models by Chang et al. (2020). Game theory is a mathematical framework that allows to model rational decision-making and individual’s selection of strategies through the assessment of risk and payoﬀ. The individuals’ decisions are modeled by finding the equilibrium set of strategies; that is, the strategies that individuals benefit the most from, in the long run (Manfredi and D’Onofrio, 2013). Game theory postulates that the rational resolution of the prevention versus treatment dilemma can be mathematically modeled by maximizing the individual’s expected utility.

Utility thus becomes a fundamental tool for modeling decision-making. In the framework of infectious disease prevention, individuals assess their risk of getting infected, and the expected utility for adopting or not prevention. Since the resolution of the dilemma of prevention versus treatment is considered to be rather costly for an uninfected individual, the individuals’ expected utility may be established from the perspective of the total cost: maximizing the expected utility is equivalent to minimizing the total expected cost. The total expected cost balances the individual’s perception of the cost for adopting the strategy of using prevention versus that of adopting the strategy of being treated in the case of infection. A simplified definition of the total expected cost takes the following form:

Identifying the probability of using prevention that minimizes the total cost yields the voluntary prevention coverage.

Modeling studies using a game-theoretic approach for the individual decision-making have typically used deterministic compartmental models to describe the epidemic dynamics at the population level (Bauch et al., 2003; Bauch and Earn, 2004; Breban et al., 2007; D’Onofrio et al., 2007; Vardavas et al., 2007; Galvani et al., 2007; Breban, 2011; Liu et al., 2012). Applications of compartmental models include vaccination facing a biological attack (Bauch et al., 2003), voluntary vaccination during a public scare of vaccination against childhood infectious disease (Bauch and Earn, 2004) and recurrent decision-making on preventing seasonal infections such as influenza (Breban et al., 2007; Galvani et al., 2007), among others. The risk of infection perceived by individuals has been defined, for instance, by a free parameter taking diﬀerent values (Bauch and Earn, 2004), by epidemiological indicators reflecting the current epidemiological situation (Bauch et al., 2003; D’Onofrio et al., 2007; Breban, 2011; Liu et al., 2012) or considering the past experience of individuals facing the epidemic (Breban et al., 2007; Vardavas et al., 2007; D’Onofrio et al., 2007). Prevention has been considered to oﬀer perfect immunity (Bauch et al., 2003; Bauch and Earn, 2004; D’Onofrio et al., 2007) or short-term immunity, including recurrent decision-making (Breban et al., 2007).

Most of the above-mentioned modeling studies have concluded that the level of prevention coverage achieved through selfish individual-level decisions (i.e., decisions motivated by the in-dividual’s own interest) may diﬀer from the level of prevention coverage needed to achieve herd immunity (Bauch et al., 2003; Bauch and Earn, 2004; Breban et al., 2007; Galvani et al., 2007; Breban, 2011), unless incentives are oﬀered (Vardavas et al., 2007; Liu et al., 2012) and thus, prevention programs may fail to achieve disease elimination. However, as discussed in section 1.1, mass vaccination has resulted in epidemic elimination, globally, regionally or at least temporarily, owing to vaccination campaigns facilitating vaccine adoption on nation-wide scales. Therefore, the impact of voluntary prevention on epidemic dynamics, and whether it can eliminate epidemics or not, remains to be studied and discussed.

**General objectives of this research**

The main objective of my doctoral research project was to build mathematical models for infec-tious disease transmission at the population-level, accounting for the individual-level decision-making on whether or not to adopt available preventive methods to avoid the infection, in a context where eﬀective treatment exists. We aimed to evaluate the impact of the voluntary adoption of prevention on the epidemic dynamics. In particular, our purpose was to determine whether and under what conditions voluntary prevention could eliminate epidemics.

Two applications were explored. The first part of my doctoral research focuses on volun-tary vaccination against treatable childhood infectious diseases; see chapter 2. The project was designed for analytical understanding of the results. We intended to apply our methods and find-ings to the epidemiology of an infectious disease preventable by vaccination allowing to assess epidemic elimination; we thus discussed our results in the context of the measles epidemiology. The second part of my thesis focuses on the voluntary use of pre-exposure prophylaxis (PrEP) by men who have sex with men (MSM) and who are at high risk of infection, in the current context of the HIV epidemic, where highly eﬀective antiretroviral therapies are available; see chapter 3. A more complex model for HIV transmission was built and the model was studied using numerical methods.

**Table of contents :**

**1 General introduction **

1.1 Prevention interventions aiming to the elimination of infectious diseases

1.1.1 Epidemic elimination

1.2 The prevention versus treatment dilemma

1.3 Mathematical and behavioral epidemiology

1.3.1 Modeling infectious disease transmission using deterministic compartmental models

1.3.2 The basic and the effective reproduction numbers

1.3.3 Modeling the decision-making about prevention adoption using a gametheoretical approach

1.4 General objectives of this research

1.5 General description of our methods

1.5.1 Conceptual framework

1.5.2 The mathematical model

**2 Voluntary vaccination against treatable childhood infectious diseases **

2.1 Introduction

2.1.1 Vaccines against childhood infectious diseases

2.1.2 Vaccine hesitancy

2.1.3 The measles epidemic: its place on the path towards elimination

2.1.4 Game-theoretic models for childhood vaccination

2.2 Publication

2.2.1 Description of the article

2.2.2 Results statements

2.2.3 Article

2.3 Additional figures

2.4 Further discussion

2.4.1 A note on mandatory vaccination

**3 Voluntary use of pre-exposure prophylaxis to prevent HIV infection among ,men who have sex with men **

3.1 Introduction

3.1.1 The HIV epidemic

3.1.2 PrEP uptake among MSM

3.1.3 The HIV epidemiology and PrEP rollout among the MSM population in France

3.1.4 Worldwide efforts to end AIDS and the path towards ending the HIV epidemic

3.1.5 Mathematical modeling of the HIV epidemic and PrEP uptake among MSM

3.2 Publication

3.2.1 Description of the article

3.2.2 Results statements

3.2.3 Article

3.2.4 Article’s supplementary material

3.3 Additional material

3.3.1 Computations and proofs of our analytical results

3.3.2 Additional results

3.4 Further discussion

3.4.1 Implementing HIV prevention programs aiming at epidemic elimination

3.4.2 Modeling limitations and perspectives

**4 General discussion **

4.1 Summary

4.1.1 Reaching epidemic elimination through voluntary adoption of prevention

4.1.2 Establishing public health policies aiming at the end of communicable diseases

4.2 Limitations and perspectives

4.2.1 The complexity of modeling human behavior

4.2.2 Determining and interpreting the relative cost of prevention versus treatment

4.2.3 Information dissemination and interpretation

4.2.4 Considering other behavioral models

4.2.5 Studying epidemics in other socio-economical settings

4.2.6 Epidemic dynamics at low prevalence

4.2.7 The impact of the COVID-19 pandemic on prevention interventions against measles and HIV

4.3 Conclusion