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

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Abstract

This paper is concerned with a stochastic model for the spread of an epidemic among a population of n individuals, labeled $1,2,\ldots,n$, 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 ${V_{i}^{n} = \{ v_{i,j}^{n} ; j=1,2, \ldots, n \}}$, at the points of independent Poisson processes with rates $\lambda_G^{n}$ and $\lambda_L^{n}$, 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 $n \to \infty$. 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 $v_{i,j}^{n} = v_{(i-j)_{\mod n}}^{n}$.

Item Type: Article
Uncontrolled Keywords: Epidemic models; local and global mixing; Poisson convergence; random graph; positively related; coupling
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 60 Probability theory and stochastic processes
MSC 2010, the AMS's Mathematics Subject Classification > 92 Biology and other natural sciences
Depositing User: Ms Lucy van Russelt
Date Deposited: 18 Aug 2006
Last Modified: 20 Oct 2017 14:12
URI: http://eprints.maths.manchester.ac.uk/id/eprint/554

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