Strong approximations for epidemic models

This paper is concerned with the approximation of early stages of epidemic processes by branching processes. A general model for an epidemic in a closed, homogeneously mixing population is presented. A construction of a sequence of such epidemics, indexed by the initial number of susceptibles N, from the limiting branching process is described. Strong convergence of the epidemic processes to the branching process is shown when the latter goes extinct. When the branching process does not go extinct, necessary and sufficient conditions on the sequence (tN) for strong convergence over the time interval [0, tN] are provided. Convergence of a wide variety of functionals of the epidemic process to corresponding functionals of the branching process is shown, and bounds are provided on the total variation distance for given N. The theory is illustrated by reference to the general stochastic epidemic. Generalisations to, for example, multipopulation epidemics are described briefly.