Epidemics on random intersection graphs

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 $R_{\ast }$, so that in a large population an epidemic with few initial infectives can give rise to a large outbreak if and only if $R_{\ast }>1$. A functional equation for the survival probability of the approximating infinite-type branching process is determined; if $R_{\ast }\le 1$, this equation has no nonzero solution, while if $R_{\ast }>1$, 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.

Published in at http://dx.doi.org/10.1214/13-AAP942 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Random graphs with clustering
• Number of Edges in Random Intersection Graph on Surface of a Sphere

Cited by in corpus (19)

• Networks beyond pairwise interactions: structure and dynamics
• Key Questions for Modelling COVID-19 Exit Strategies
• Stochastic epidemics in a homogeneous community
• Stochastic epidemics in a heterogeneous community (Part III of the book Stochastic Epidemic Models and Inference)
• Statistical inference for epidemic processes in a homogeneous community (Part IV of the book Stochastic Epidemic Models and Inference)
• k-Connectivity in Random Key Graphs with Unreliable Links
• On the strengths of connectivity and robustness in general random intersection graphs
• On $k$-connectivity and minimum vertex degree in random $s$-intersection graphs
• Random intersection graphs and their applications in security, wireless communication, and social networks
• Analyzing connectivity of heterogeneous secure sensor networks
• Recent Advances in Epidemic Modeling: Non-Markov Stochastic Models and their Scaling Limits
• Hyperbolicity, degeneracy, and expansion of random intersection graphs
• A statistical network analysis of the HIV/AIDS epidemics in Cuba
• On connectivity in a general random intersection graph
• The impact of degree variability on connectivity properties of large networks
• The real-time growth rate of stochastic epidemics on random intersection graphs
• Evaluation of vaccination strategies for SIR epidemics on random networks incorporating household structure
• A network with tunable clustering, degree correlation and degree distribution, and an epidemic thereon
• The duration of an $SIR$ epidemic on a configuration model