Epidemics on random intersection graphs

TL;DR

This paper models the spread of SIR epidemics on random intersection graphs, deriving a threshold parameter that predicts large outbreaks and providing rigorous probabilistic analysis of epidemic size and survival probability.

Contribution

It introduces a new stochastic epidemic model on random intersection graphs and rigorously establishes a threshold criterion for large outbreaks using branching process approximations.

Findings

Threshold parameter $R_*>1$ determines epidemic potential.

Functional equation characterizes survival probability.

Law of large numbers describes outbreak size.

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.