The COVID-19 pandemic has dramatically demonstrated the importance of epidemic models in understanding and predicting disease spread and in assessing the effectiveness of interventions. The overarching topic of this thesis is stochastic epidemic modelling, with the main focus on the role of the underlying social structure in infectious disease spread.

In Paper I we study the spread of stochastic SIR-epidemics on an extended version of the configuration model with group structure. We present expressions for the basic reproduction number R₀, the probability of a major outbreak and the expected final size, and investigate random vaccination with a perfect vaccine. We weaken the assumptions of earlier results for epidemics on this type of graph by allowing for heterogeneous infectivity both in individual infectivity and between different kinds of edges. An important special case of this model is the spread of a disease with arbitrary infectious period distribution in continuous time. 

Paper II concerns multi-type competition in a  variant of  Pólya's urn model with interaction, where balls of different colours/types annihilate upon contact. The model dynamics are governed by the structure of an underlying graph. In the special case of a cycle graph, this urn model is equivalent to a planar growth model with competing pathogens. It has earlier been shown that in the two-type case, indefinite coexistence has probability 0 for any (finite and connected) underlying graph, while for K ≥ 3 types the possibility of coexistence depends on the structure of this graph. We show that for K ≥ 3 types competing on a cycle graph, there is with probability 1 eventually only one remaining type.

In Paper III we study the real-time growth rate of SIR epidemics on random intersection graphs with mixed Poisson degree distribution. We show that during the early stage of the epidemic, the number of infected individuals grows exponentially and the Malthusian parameter is shown to satisfy a  version of the Euler-Lotka equation. These results are obtained via an approximating embedded single-type Crump-Mode-Jagers branching process. In addition, we provide a lower bound on the cumulative number of individuals that get infected before the branching process approximation breaks down. 

In Paper IV we consider stochastic SIR epidemics on inhomogeneous random graphs with degree-dependent contact rates. In this model, the per-neighbour contact rate of an individual decrease but its overall expected contact rate increases with its expected number of neighbours. We provide the basic reproduction number R₀, the probability of a large outbreak and the final size of an epidemic. We show that reducing heterogeneity in contact rates results in a higher value of the basic reproduction number R₀, and demonstrate that this result does not generally extend to the probability of a major outbreak and the final size.