Epidemics and opinions on networks

Abstract

This thesis studies stochastic models for epidemics, random graphs, and opinion dynamics on structured populations. It is based on two published articles and three unsubmitted manuscripts. Chapters 2 and 3 investigate stochastic SIR epidemics: one examines local vaccine mandates in a household model and their effects on outbreak risk and medically exempt individuals, while the other analyses the early growth of epidemics on random intersection graphs and derives a corresponding Malthusian parameter. Chapter 4 shows that, for a configuration model with given expected degree and given initial probabilities, the size of the giant component is maximized by allocating the remaining mass of the degree distribution over at most three integers. Chapters 5 and 6 consider opinion dynamics on clustered random graphs. One develops and analyses models inspired by Schelling’s segregation model, focusing on stationary behaviour, deterministic approximations, and reversibility. The other studies the binary voter model on structured populations and proves convergence of the fraction of vertices holding either opinion to a Wright–Fisher diffusion with an explicit diffusion constant.