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January 2004 Poisson approximations for epidemics with two levels of mixing
Frank Ball, Peter Neal
Ann. Probab. 32(1B): 1168-1200 (January 2004). DOI: 10.1214/aop/1079021475


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 $v_{i,j}^{n} = v_{(i-j)_{\mod n}}^{n}$.


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Frank Ball. Peter Neal. "Poisson approximations for epidemics with two levels of mixing." Ann. Probab. 32 (1B) 1168 - 1200, January 2004.


Published: January 2004
First available in Project Euclid: 11 March 2004

zbMATH: 1060.92050
MathSciNet: MR2044677
Digital Object Identifier: 10.1214/aop/1079021475

Primary: 60F05 , 92D30
Secondary: 05C80 , 60K35

Keywords: coupling , epidemic models , local and global mixing , Poisson convergence , positively related , random graph

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 1B • January 2004
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