Une ballade en forêts aléatoires : Théorèmes limites pour des populations structurées et leurs généalogies, étude probabiliste et statistique de modèles SIR en épidémiologie, contributions à la géométrie aléatoire

Abstract : This thesis contains three parts. The first chapter considers individual-centered models of structured populations, where the individual dynamics (reproductions, competitions, deaths and aging) depend on the past, either through individuals' ages or through their ancestral lineages, and on herediraty convariates, called traits, that are transmitted from parent to offspring unless mutations occur. In large populations, the random measure-valued processes describing these populations can be approximated by solutions of PDEs or by diffusive processes, given the time-scales involved. In the second case, averaging phenomena are observed for the aging. Martingale methods introduced by Kurtz are used to separate the slow and fast time-scales, thus answering questions tackled among others by Dynkin, Kaj and coauthors... but unanswered in the cases studied here. To describe the evolution of the genealogies, historical births and deaths processes, à la Dawson et Perkins, are introduced. Their superprocess limits are studied. The distribution of the ancestral lineage of an individual chosen randomly at a given time is difficult to characterise. In the case without interaction, we can however answer this question by using many-to-one formulas that exhibit biases. An application to multi-level models is considered. The second part of this chapter focuses on two models of population genetics, with demography and randomness. In the first model, we study how the neutral diversity is affected by selection and adapation in an eco-evolutionary framework. This sheds a new light on the phylogenies that can be reconstructed from the genetical polymorphism observed at a given time. The second model considers populations with self-incompatible reproduction systems (such as some flower populations), leading to the study of inhomogeneous random walks on the quarter plane and absorbed at the boundary. In chapter two, probabilistic and statistical models for epidemiology are presented. The questions are motivated by the HIV-AIDS data collected between 1986 and 2006 in Cuba by contact-tracing. Statistical estimation, to compare for instance the relative efficiencies of different detection methods, is made difficult as infectious but non-detected individuals are unobserved. A Bayesian technique called ABC is well suited for our problem and for size of our database. The infection and detection graphs are also partially observed (5,389 vertices linked by 4,073 edges, with a giant component of 2,386 vertices and 3,168 edges, which is unique in the literature on AIDS). The statistical study of this graph and its community structure obtained by clustering methods are shown. From a more probabilistic point of view, limit theorems linked with disease propagating on large random graphs with fixed degree distributions are established. This provides three measure-valued equations to describe important degree distributions underlying the propagation, and gives a rigorous proof of the ODEs proposed by Volz. The third chapter describes two contributions in stochastic geometry. The first one deals with the approximation of the Vorob'ev expectation of random sets. The second one is motivated by the study of the Radial Spanning Tree (RST) and of the Directed Spanning Forest (DSF) introduced by Baccelli and Bordenave. These objects exhibit complex dependencies that prevent the use of classical techniques based on martingales, for instance. To show that the DSF is a tree without bi-infinite path or to study the geodesics and interfaces of the RST and show that it admits 1, 2, 3, 4 or 5 subtrees rooted at the origin with positive probability, arguments from geometry or percolation are used.
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Viet Chi Tran. Une ballade en forêts aléatoires : Théorèmes limites pour des populations structurées et leurs généalogies, étude probabiliste et statistique de modèles SIR en épidémiologie, contributions à la géométrie aléatoire. Probabilités [math.PR]. Université Lille 1, 2014. ⟨tel-01087229⟩

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