Since the first cases of HIV/AIDS disease were recognised in the early 1980s, a large number of mathematical models have been proposed. However, the mobility of people among regions, which has an obvious impact on the spread of the disease, has not been much considered in the modelling studies. One of the main reasons is that the models for the spread of the disease in multiple populations are very complex and, as a consequence, they can easily become intractable. In this thesis we provide various new results pertaining to the spread of the disease in mobile populations, including epidemic intervention in multiple populations. We first develop stochastic models for the spread of the disease in a single heterosexual population, considering both constant and varying population sizes. In particular, we consider a class of continuous-time Markov chains (CTMCs). We establish deterministic and Gaussian diffusion analogues of these stochastic processes by applying the theory of density dependent processes. A range of numerical experiments are provided to show how well the deterministic and Gaussian counterparts approximate the dynamic behaviour of the processes. We derive threshold parameters, known as basic reproduction numbers, for both cases above the threshold which the disease is uniformly persistent and below the threshold which disease-free equilibrium is locally attractive. We find that the threshold conditions for both constant and varying population sizes have the same form. In order to take into account the mobility of people among regions, we extend the stochastic models to multiple populations. Various stochastic models for multiple populations are formulated as CTMCs. The deterministic and Gaussian diffusion counterparts of the corresponding stochastic processes for the multiple populations are also established. Threshold parameters for the persistence of the disease in the multiple population models are derived by applying the concept of next generation matrices. The results of this study can serve as a basic framework how to formulate and analyse a more realistic stochastic model for the spread of HIV in mobile heterogeneous populations—classifying all individuals by age, risk, and level of infectivities, and at the same time considering different modes of the disease transmission. Assuming an accurate mathematical model for the spread of HIV/AIDS disease, another question that we address in this thesis is how to control the spread of the disease in a mobile population. Most previous studies for the spread of the disease focus on identifying the most significant parameters in a model. In contrast, we study these problems as optimal epidemic intervention problems. The study is mostly motivated by the fact that more and more local governments allocate budgets over a certain period of time to combat the disease in their areas. The question is how to allocate this limited budget to minimise the number of new HIV cases, say on a country level, over a finite time horizon as people move among regions. The mathematical models developed in the first part of this thesis are used as dynamic constraints of the optimal control problems. In this thesis, we also introduce a novel approach to solve quite general optimal control problems using the Cross-Entropy (CE) method. The effectiveness of the CE method is demonstrated through several illustrative examples in optimal control. The main application is the optimal epidemic intervention problems discussed above. These are highly non-linear and multidimensional problems. Many existing numerical techniques for solving such optimal control problems suffer from the curse of dimensionality. However, we find that the CE technique is very efficient in solving such problems. The numerical results of the optimal epidemic strategies obtained via the CE method suggest that the structure of the optimal trajectories are highly synchronised among patches but the trajectories do not depend much on the structure of the models. Instead, the parameters of the models (such as the time horizon, the amount of available budget, infection rates) much affect the form of the solution.

Since the first cases of HIV/AIDS disease were recognised in the early 1980s, a large number of mathematical models have been proposed. However, the mobility of people among regions, which has an obvious impact on the spread of the disease, has not been much considered in the modelling studies. One of the main reasons is that the models for the spread of the disease in multiple populations are very complex and, as a consequence, they can easily become intractable. In this thesis we provide various new results pertaining to the spread of the disease in mobile populations, including epidemic intervention in multiple populations. We first develop stochastic models for the spread of the disease in a single heterosexual population, considering both constant and varying population sizes. In particular, we consider a class of continuous-time Markov chains (CTMCs). We establish deterministic and Gaussian diffusion analogues of these stochastic processes by applying the theory of density dependent processes. A range of numerical experiments are provided to show how well the deterministic and Gaussian counterparts approximate the dynamic behaviour of the processes. We derive threshold parameters, known as basic reproduction numbers, for both cases above the threshold which the disease is uniformly persistent and below the threshold which disease-free equilibrium is locally attractive. We find that the threshold conditions for both constant and varying population sizes have the same form. In order to take into account the mobility of people among regions, we extend the stochastic models to multiple populations. Various stochastic models for multiple populations are formulated as CTMCs. The deterministic and Gaussian diffusion counterparts of the corresponding stochastic processes for the multiple populations are also established. Threshold parameters for the persistence of the disease in the multiple population models are derived by applying the concept of next generation matrices. The results of this study can serve as a basic framework how to formulate and analyse a more realistic stochastic model for the spread of HIV in mobile heterogeneous populations—classifying all individuals by age, risk, and level of infectivities, and at the same time considering different modes of the disease transmission. Assuming an accurate mathematical model for the spread of HIV/AIDS disease, another question that we address in this thesis is how to control the spread of the disease in a mobile population. Most previous studies for the spread of the disease focus on identifying the most significant parameters in a model. In contrast, we study these problems as optimal epidemic intervention problems. The study is mostly motivated by the fact that more and more local governments allocate budgets over a certain period of time to combat the disease in their areas. The question is how to allocate this limited budget to minimise the number of new HIV cases, say on a country level, over a finite time horizon as people move among regions. The mathematical models developed in the first part of this thesis are used as dynamic constraints of the optimal control problems. In this thesis, we also introduce a novel approach to solve quite general optimal control problems using the Cross-Entropy (CE) method. The effectiveness of the CE method is demonstrated through several illustrative examples in optimal control. The main application is the optimal epidemic intervention problems discussed above. These are highly non-linear and multidimensional problems. Many existing numerical techniques for solving such optimal control problems suffer from the curse of dimensionality. However, we find that the CE technique is very efficient in solving such problems. The numerical results of the optimal epidemic strategies obtained via the CE method suggest that the structure of the optimal trajectories are highly synchronised among patches but the trajectories do not depend much on the structure of the models. Instead, the parameters of the models (such as the time horizon, the amount of available budget, infection rates) much affect the form of the solution.

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Cette thèse propose une étude mathématique des stratégies de vaccination.La partie I présente le cadre mathématique, notamment le modèle à compartiments Susceptible - Infected – Recovered.La partie II aborde les techniques mathématiques de type contrôle optimal employées afin de trouver une stratégie optimale de vaccination au niveau de la société. Ceci se fait en minimisant le coût de la société. Nous montrons que la fonction valeur associée peut avoir une régularité plus faible que celle attendue dans la littérature. Enfin, nous appliquons les résultats à la vaccination contre la coqueluche.La partie III présente un modèle où le coût est défini au niveau de l'individu. Nous reformulons le problème comme un équilibre de Nash et comparons le coût obtenu avec celui de la stratégie sociétale. Une application à la grippe A(H1N1) indique la présence de perceptions différentes liées à la vaccination.La partie IV propose une implémentation numérique directe des stratégies présentées. / This thesis propose a mathematical analysis of the vaccination strategies.The first part introduces the mathematical framework, in particular the Susceptible – Infected – Recovered compartmental model.The second part introduces the optimal control tools used to find an optimal vaccination strategy from the societal point of view, which is a minimizer of the societal cost. We show that the associated value function can have a less regularity than what was assumed in the literature. These results are then applied to the vaccination against the whooping cough.The third part defines a model where the cost is defined at the level of the individual. We rephrase this problem as a Nash equilibrium and compare this results with the societal strategy. An application to the Influenza A(H1N1) 2009-10 indicates the presence of inhomogeneous perceptions concerning the vaccination risks.The fourth and last part proposes a direct numerical implementation of the different strategies.