The distribution of general final state random variables for stochastic epidemic models

In this paper we introduce the notion of general final stale random variables for generalized epidemic models. These random variables are defined as sums over all ultimately infected individuals of random quantities of interest associated with an individual; examples include final severity. By exploiting a construction originally due to Sellke (1983), exact results concerning the final size and general final state random variables are obtained in terms of Gontcharoff polynomials. In particular our approach highlights the way in which these polynomials arise via simple probabilistic arguments. For ease of exposition we focus initially upon the single-population case before extending our arguments to multi-population epidemics and other variants of our basic model.