Poisson approximations for epidemics with two levels of mixing

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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