The Annals of Applied Probability

Epidemics on random intersection graphs

Frank G. Ball, David J. Sirl, and Pieter Trapman

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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.

Article information

Ann. Appl. Probab., Volume 24, Number 3 (2014), 1081-1128.

First available in Project Euclid: 23 April 2014

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 92D30: Epidemiology 05C80: Random graphs [See also 60B20]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 91D30: Social networks

Epidemic process random intersection graphs multi-type branching processes coupling


Ball, Frank G.; Sirl, David J.; Trapman, Pieter. Epidemics on random intersection graphs. Ann. Appl. Probab. 24 (2014), no. 3, 1081--1128. doi:10.1214/13-AAP942.

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