This paper is concerned with the growth rate of susceptible-infectious-recovered epidemics with general infectious period distribution on random intersection graphs. This type of graph is characterised by the presence of cliques (fully connected subgraphs). We study epidemics on random intersection graphs with a mixed Poisson degree distribution and show that in the limit of large population sizes the number of infected individuals grows exponentially during the early phase of the epidemic, as is generally the case for epidemics on asymptotically unclustered networks. The Malthusian parameter is shown to satisfy a variant of the classical Euler-Lotka equation. To obtain these results we construct a coupling of the epidemic process and a continuous-time multitype branching process, where the type of an individual is (essentially) given by the length of its infectious period. Asymptotic results are then obtained via an embedded single-type Crump-Mode-Jagers branching process.

We study a system of interacting urns where balls of different colour/type compete for their survival, and annihilate upon contact. For competition between two types, the underlying graph (finite and connected), determining the interaction between the urns, is known to be irrelevant for the possibility of coexistence, whereas for K ≥ 3 types the structure of the graph does affect the possibility of coexistence. We show that when the underlying graph is a cycle, competition between K ≥ 3 types almost surely has a single survivor, thus establishing a conjecture of Griffiths, Janson, Morris and the first author. Along the way, we give a detailed description of an auto-annihilative process on the cycle, which can be perceived as an expression of the geometry of a Möbius strip in a discrete setting.

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.

In this paper we consider Susceptible Infectious Recovered (SIR) epidemics on random graphs with clustering. To incorporate group structure of the underlying social network, we use a generalized version of the configuration model in which each node is a member of a specified number of triangles. SIR epidemics on this type of graph have earlier been investigated under the assumption of homogeneous infectivity and also under the assumption of Poisson transmission and recovery rates. We extend known results from literature by relaxing the assumption of homogeneous infectivity both in individual infectivity and between different kinds of neighbours. An important special case of the epidemic model analysed in this paper is epidemics in continuous time with arbitrary infectious period distribution. We use branching process approximations of the spread of the disease to provide expressions for the basic reproduction number R0, the probability of a major outbreak and the expected final size. In addition, the impact of random vaccination with a perfect vaccine on the final outcome of the epidemic is investigated. We find that, for this particular model, R0 equals the perfect vaccine-associated reproduction number. Generalizations to groups larger than three are discussed briefly.