Ball, Frank and Neal, Peter John
(2004)
*Poisson approximations for epidemics with two levels of mixing.*
Annals of Probability, 32 (1B).
pp. 1168-1200.
ISSN 0091-1798

## Abstract

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 n i ={v n i,j ;j=1,2,…,n} , at the points of independent Poisson processes with rates λ n G and λ n L , 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 n i,j =v n (i−j) modn .