Studies in branching processes - mathematical modelling of structured populations with applications to epidemics
File(s)
Author(s)
Berah, Tresnia
Type
Thesis
Abstract
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.
Version
Open Access
Date Issued
2024-12-18
Date Awarded
01/03/2025
License URL
Advisor
Pakkanen, Mikko
Passaggeri, Riccardo
Sponsor
Engineering and Physical Sciences Research Council
Publisher Department
Department of Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)