Research Repository

# Epidemics on random intersection graphs

## Authors

FRANK BALL frank.ball@nottingham.ac.uk
Professor of Applied Probability

Pieter Trapman

### Abstract

In this paper we consider a model for the spread of a stochastic SIR (Susceptible → Infectious → 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∗≤1, 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

Ball, F. G., Sirl, D. J., & Trapman, P. (2014). Epidemics on random intersection graphs. Annals of Applied Probability, 24(3), https://doi.org/10.1214/13-AAP942

Journal Article Type Article May 31, 2013 Jun 1, 2014 Jun 17, 2016 Jun 17, 2016 Annals of Applied Probability 1050-5164 1050-5164 Institute of Mathematical Statistics (IMS) Peer Reviewed 24 3 https://doi.org/10.1214/13-AAP942 Epidemic process, Random intersection graphs, Multi-type branching processes, Coupling http://eprints.nottingham.ac.uk/id/eprint/34165 http://projecteuclid.org/euclid.aoap/1398258096 Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf

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Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf