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June 2014 Epidemics on random intersection graphs
Frank G. Ball, David J. Sirl, Pieter Trapman
Ann. Appl. Probab. 24(3): 1081-1128 (June 2014). DOI: 10.1214/13-AAP942

Abstract

In this paper we consider a model for the spread of a stochastic SIR (Susceptible $\to$ Infectious $\to$ Recovered) epidemic on a network of individuals described by a random intersection graph. Individuals belong to a random number of cliques, each of random size, and infection can be transmitted between two individuals if and only if there is a clique they both belong to. Both the clique sizes and the number of cliques an individual belongs to follow mixed Poisson distributions. An infinite-type branching process approximation (with type being given by the length of an individual’s infectious period) for the early stages of an epidemic is developed and made fully rigorous by proving an associated limit theorem as the population size tends to infinity. This leads to a threshold parameter $R_{*}$, so that in a large population an epidemic with few initial infectives can give rise to a large outbreak if and only if $R_{*}>1$. A functional equation for the survival probability of the approximating infinite-type branching process is determined; if $R_{*}\le1$, this equation has no nonzero solution, while if $R_{*}>1$, it is shown to have precisely one nonzero solution. A law of large numbers for the size of such a large outbreak is proved by exploiting a single-type branching process that approximates the size of the susceptibility set of a typical individual.

Citation

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Frank G. Ball. David J. Sirl. Pieter Trapman. "Epidemics on random intersection graphs." Ann. Appl. Probab. 24 (3) 1081 - 1128, June 2014. https://doi.org/10.1214/13-AAP942

Information

Published: June 2014
First available in Project Euclid: 23 April 2014

zbMATH: 1291.92098
MathSciNet: MR3199981
Digital Object Identifier: 10.1214/13-AAP942

Subjects:
Primary: 05C80, 60K35, 92D30
Secondary: 60J80, 91D30

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.24 • No. 3 • June 2014
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