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.

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 $K\geq 3$ types the structure of the graph does affect the possibility of coexistence. We show that when the underlying graph is a cycle, competition between $K\geq 3$ 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.

This paper is concerned with the growth rate of SIR (Susceptible-Infectious-Recovered) epidemics with general infectious period distribution on random intersection graphs. This type of graph is characterized 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 consider a stochastic SIR epidemic on an inhomogeneous random graph. In most models for epidemics on networks, the (possibly stochastic) processes that describe the contacts between individuals follow the same law for every pair of neighbours and are therefore independent of the degrees of the individuals. In this paper, we investigate the impact of this simplifying assumption. To this end, we consider a model where the contact rate between two neigbours depends on their degrees. We make assumptions that, heuristically, ensure that the per-neighbour contact rates of an individual decrease with the expected number of neighbours of that individual. We investigate the basic reproduction number $R_0$, the probability of a large outbreak, and the final size of an epidemic. We show that reducing heterogeneity in contacts (while keeping the average total contact intensity fixed) leads to an increase in the basic reproduction number $R_0$. In particular,  ignoring heterogeneity in interactions by assuming that contact rates are constant results in a higher value of $R_0$. Similar inequalities do not exist in general for the probability of a large outbreak and the final size of an epidemic.