Strong Convergence of Stochastic Epidemics

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Abstract
This paper is concerned with a model for the spread of an epidemic in a closed, homogeneously mixing population in which new infections occur at ratef(x,y) and removals occur at rateg(x,y), wherexand y are the numbers of susceptible and infective individuals, respectively, andfandgare arbitrary but specified positive real-valued functions. Sequences of such epidemics, indexed by the initial number of susceptiblesn,are considered and conditions are derived under which the epidemic processes converge almost surely to a birth and death process asntends to infinity. Thus a threshold theorem for such an epidemic model is obtained. The results are extended to models which incorporate immigration and emigration of susceptibles. The theory is illustrated by several examples of models taken from the epidemic literature. Generalizations to multipopulation epidemics are discussed briefly.

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