Stochastic sir epidemic on a reed-frost random intersection graph

Let us consider a model for the spread of a stochastic SIR epidemic on a network of individuals described by a random intersection graph. The number of cliques a typical individual belongs to follows a mixed-Poisson distribution, as does the size of a typical clique. A random intersection graph is constructed by assigning independently to each vertex a subset of a given set and drawing an edge between two vertices if and only if their respective subsets intersect. A branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. Heuristically-motivated branching process approximations are described, which lead to a threshold parameter for the model and methods for calculating the probability of a major outbreak. The effects of the amount of clustering present in the overall population structure and the infectious period distribution on the outcomes of the model are also explored.