Studies in branching processes - mathematical modelling of structured populations with applications to epidemics

This thesis treats of various types of branching processes, with some applications to epidemiology. Chapters 1 and 2 introduce a novel time-varying Crump-Mode-Jagers (CMJ) branching process whose governing integral equations are then relied upon in order to provide a rigorous mathematical justification to the renewal incidence equation in infectious disease modelling. Chapter 3 focuses on the asymptotic behaviour of a structured branching population where each individual in the population is characterised by a trait or a type whose dynamics follow a Markov process. The branching process is then expected to be driven by a positive triplet of first eigenvalue problem of the first moment semigroup. A strong law of large numbers for super-critical branching Markov processes is proven assuming convergence of the renormalized semigroup in weighted total variation norm. Convergence is obtained under an $L\log L $ condition which provides a new Kesten-Stigum result in infinite dimension and relaxes the uniform convergence assumption of the renormalized first moment semigroup required in the work of Asmussen and Hering in 1976. Finally Chapter 4 treats of a novel variational auto-encoder, called $\pi$-VAE, which has the property of being a proper stochastic process. As the combination of a generative model and a stochastic process that can be used as a prior, $\pi$-VAE is then used to do full Bayesian inference. We provide particular examples with geo-spatial or epidemiological data.