Poisson approximations for epidemics with two levels of mixing

This paper is concerned with a stochastic model for the spread of an epidemic among a population of n individuals, labeled 1, 2,..., n, in which a typical infected individual, i say, makes global contacts, with individuals chosen independently and uniformly from the whole population, and local contacts, with individuals chosen independently according to the contact distribution V i n = {v i, j n; j = 1, 2,..., n}, at the points of independent Poisson processes with rates λ G n and λ L n, respectively, throughout an infectious period that follows an arbitrary but specified distribution. The population initially comprises m n infectives and n - m n susceptibles. A sufficient condition is derived for the number of individuals who survive the epidemic to converge weakly to a Poisson distribution as n → ∞. The result is specialized to the households model, in which the population is partitioned into households and local contacts are chosen uniformly within an infective's household; the overlapping groups model, in which the population is partitioned in several ways and local mixing is uniform within the elements of the partitions; and the great circle model, in which v i,j n= v (i-j)mod n n.