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Theses

Epidemics models in infinite-dimension and optimal vaccination strategies

Abstract : This thesis is motivated by the mathematical modelling of heterogeneity in human contacts and the consequences on the dynamic and the control of contagious diseases.In the first part of the thesis, we introduce and study an infinite-dimensional deterministic SIS (Susceptible/Infected/Susceptible) model which takes into account the heterogeneity of contacts within a large population. Thanks to the monotonic properties of the flow of these equations, we prove a result on the long-time behavior of the proportion of infected people. The basic reproduction number R0, defined as the spectralradius of a kernel operator, determines whether there exists a stable endemic equilibrium (R0 > 1) or if all the solutions tends to the disease-free equilibrium (R0 <= 1).As an application, we formalize and study the problem of optimal allocation strategies for a vaccine that completely immunize from the disease those who received it. When we suppose that the contacts in the population are homogeneous, the threshold theorem states that the incidence of the infection will decrease if the proportion of vaccinated persons in the population is at least equal to 1-1/R0. In inhomogeneous models, this theorem remains true but with a better allocation of vaccine doses, we can hope for reaching herd immunity at lower cost. Hence, we study the problem where one tries to minimize simultaneously the cost of the vaccination, and a loss that may be either the effective reproduction number, or the overall proportion of infected individuals in the endemic state. By proving the continuity of these two loss functions, we obtain the existence of Pareto optimal strategies. We also show that vaccinating according to the profile of the endemic state is a critical allocation, in the sense that, if the initial reproduction number is larger than 1, then this vaccination strategy yields an effective reproduction number equal to 1.The last part of the thesis is a detailed study of the effective reproduction number and the bi-objective minimization problem associated. We prove a generalization of the Hill-Longini conjecture on the concavity and convexity of the effective reproduction number along with other theoretical results on this loss function. We then illustrate with multiple examples those properties. In particular, we investigate the three following questions.- Is it possible to always vaccinate optimally when the vaccine doses are given one at a time?- What is the effect of assortativity (the tendency to have more contacts with similar individuals) on the shape of optimal vaccination strategies?- What happens when every individuals have the same number of neighbors?
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Dylan Dronnier. Epidemics models in infinite-dimension and optimal vaccination strategies. General Mathematics [math.GM]. École des Ponts ParisTech, 2021. English. ⟨NNT : 2021ENPC0025⟩. ⟨tel-03541695⟩

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