The threshold behaviour of epidemic models

Abstract

We provide a method of constructing a sequence of general stochastic epidemics, indexed by the initial number of susceptibles N, from a time-homogeneous birth-and-death process. The construction is used to show strong convergence of the general stochastic epidemic to a birth-and-death process, over any finite time interval [0, t], and almost sure convergence of the total size of the general stochastic epidemic to that of a birth-and-death process. The latter result furnishes us with a new proof of the threshold theorem of Williams (1971). These methods are quite general and in the remainder of the paper we develop similar results for a wide variety of epidemics, including chain-binomial, host-vector and geographical spread models.