Epidemics on random graphs

In recent years, various models of random graphs beyond the standard G(n,p) random graph have been extensively studied. In particular, there has been increasing interest in modelling epidemics on random graphs.

In this work, we study the susceptible-infective-recovered (SIR) epidemic on a random graph chosen uniformly subject to having given vertex degrees. In the SIR model, infective vertices infect each of their susceptible neighbours, and recover, at a constant rate.

Suppose that initially there are only a few infective vertices. We prove there is a threshold for a parameter involving the rates and vertex degrees below which only few infections occur. Above the threshold a large outbreak may occur. Conditional on that, the evolutions of certain quantities of interest, such as the fraction of infective vertices, converge to deterministic functions of time.

This is joint work with Svante Janson and Peter Windridge.