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UBC Theses and Dissertations

Vaccination models in infectious diseases Bai, Fan


Vaccination is the most effective method of preventing the spread of infectious diseases. In this thesis, we develop and apply mathematical models to study vaccination. The thesis consists of three main parts. Firstly, in deciding whether to be vaccinated before the outbreak of the epidemic, people need to consider the risk from vaccination and the probability of being infected. The decision of one individual is indirectly influenced by the decisions of all other individuals. Because the vaccine coverage levels are determined by all decisions of all individuals. We apply game theory to this scenario, to predict the expected vaccine coverage level. We construct and analyze four vaccination games. The novelty of this part of work is we introducing the replicator equations to describe the evolutionary processes. For all games, we are able to predict the vaccine coverage levels accurately. Secondly, In order to prove the uniqueness of Nash equilibrium in games, it is crucial to prove the attack ratios are decreasing functions of the vaccine coverage levels. We are able to obtain complete results for the cases of homogeneous mixing population. Only partial results are obtained for the case of heterogeneous mixing population. Thirdly, some insights into the dynamics of malaria infection are obtained. We propose two new malaria models. For delayed malaria transmission models, we calculate the basic reproduction number ℛ₀, the disease-free equilibrium and the possible endemic equilibrium. We also analyze the stabilities of the equilibria. For malaria vaccination model, we calculate the control reproduction number ℛc and the disease-free equilibrium. The threshold of eradication is discussed. We find that, under certain circumstances, the disease of malaria can be eradicated.

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