Epidemics on random intersection graphs

In this paper we consider a model for the spread of a stochastic SIR (Susceptible → Infectious → Recovered) epidemic on a network of individuals described by a ran-dom intersection graph. Individuals belong to a random number of cliques, each of random size, and infection can be transmitted between two individuals if and only if there is a clique they both belong to. Both the clique sizes and the number of cliques an individual belongs to follow mixed Poisson distributions. An infinite-type branch-ing process approximation (with type being given by the length of an individual’s infectious period) for the early stages of an epidemic is developed and made fully rig-orous by proving an associated limit theorem as the population size tends to infinity. This leads to a threshold parameter R∗, so that in a large population an epidemic with few initial infectives can give rise to a large outbreak if and only if R ∗> 1. A functional equation for the survival probability of the approximating infinite-type branching is determined; if R ∗ ≤ 1, this equation has no non-zero solution, whilst, if R ∗> 1, it is shown to have precisely one non-zero solution. A law of large numbers for the size of such a large outbreak is proved by exploiting a single-type branching process that approximates the susceptibility set of a typical individual