We consider a stochastic SIR model on a random network (graph). Infected individuals infect susceptible neighbours at a fixed rate, and recover at another fixed rate.
The network is a random graph with given vertex degrees; equivalently, it is constructed by the configuration model.
The evolution of the number of infected individuals as a function of time is studied. We show (under suitable conditions) that as the size of the population tends to infinity, the number of infected converges after normalization to a deterministic function, described by a system of ordinary differential equations (originally found, heuristically, by Volz (2008)).
The model allows for starting with a substantial fraction of the population already infected, or already recovered (immune); the latter option can be used to study the effect of different vaccination strategies.
Based on joint papers with Malwina Luczak, Peter Windridge and Thomas House.