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Scaling limits of branching and coalescing models arising in population biology

Abstract : This thesis introduces several stochastic population models, motivated by population genetics and epidemiology, and studies their large population size behavior. In Chapter 2 we consider the notion of comb, which can be seen as the planar embedding of an ultrametric measure space. We prove that any such space admits a comb representation, and that any exchangeable coalescent can be obtained by sampling from a comb. Chapter 3 studies the stationary distribution of a fragmentation-coalescence process known as Kingman's coalescent with erosion. We provide a representation of its asymptotic frequencies, and the limiting behavior of its block size distribution. Chapter 4 introduces a new model to study the impact of an Allee effect on the accumulation of deleterious mutations at the front of an expanding population. Some properties of the model are investigated using both analytical and numerical methods. Chapter 5 studies the branching approximation of a Wright-Fisher model with recombination. We provide an expression for the distribution of the block lengths and their location in a large chromosome, large population size limit. Finally, Chapter 6 and Chapter 7 are two variations of the same epidemic model, which is formulated in the framework of Crump-Mode-Jagers branching processes. In both chapters, we prove that the age of infection and compartment structure of the population converges to the solution of a partial differential equation of the McKendrick-von Foerster type. As an application of this result, Chapter 6 contains an estimation of some key epidemiological quantities of the COVID-19 epidemic in France, during the first lockdown period.
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Félix Foutel-Rodier. Scaling limits of branching and coalescing models arising in population biology. Probability [math.PR]. Sorbonne Université, 2021. English. ⟨NNT : 2021SORUS268⟩. ⟨tel-03546798⟩

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