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 $R_*$, 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.Comment: 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

Recent Progress in Complex Network Analysis: Properties of Random Intersection Graphs

“…In Ball et al (2014), Britton et al (2008) epidemic processes were studied while Blackburn et al (2012) applied random intersection graphs results in cryptology.…”

<inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-Connectivity in Random Key Graphs With Unreliable Links

“…(a) If (1) and either there exists > 0 such that p e n > holds for all n sufficiently large or lim n→∞ p e n = 0, then (10) and lim n→∞ P…”

“…R ANDOM key graphs have received significant attention recently with applications spanning key predistribution in secure wireless sensor networks (WSNs) [2], [5], [8], [9], [13], social networks [7], [18], [39], recommender systems [25], clustering and classification analysis [4], [19], cryptanalysis of hash functions [3], circuit design [33], and the modeling of epidemics [1] and "small-world" networks [36]. They belong to a larger class of random graphs known as the random intersection graphs [2]- [7], [10], [14], [33]; in fact, they are referred to as the uniform random intersection graphs by some authors [2], [3], [6], [26], [30]- [32], [43], [44].…”

Asymptotic Behaviour of Gossip Processes and Small-World Networks

“…Hence the conditional variance of V P 0 (t)/L, given the information in F s , is small, and thus the value of V P 0 (t)/L is (almost) fixed. An analogous argument is used, for instance, in Ball, Sirl & Trapman [3], where they show that, in an epidemic in a population of large size N , the proportion of individuals ever infected is close either to zero or to a non-random value in (0, 1). A by-product of our argument is to identify the solution h of a particular integral equation, that appears in Aldous [1] and also plays a substantial part in the formula given by Chatterjee & Durrett [6], in terms of the Laplace transform of the branching process limit random variable W ; their function h is just a time translation of h 2 .…”

“…Here, M (1) is used to denote the quantities (2.2) for X * P 0 , coupled as above with Y P 0 , and F s to denote its history up to s. M (2) and M (3) are related to Y P (t; 2t) and Y P ′ (t; 2t) in similar fashion, through branching processes X * P and X * P ′ , which are independent of each other and of X * P 0 . Now the union Y * P 0 (t) of the islands of X * P 0 (t) contains Y P 0 (t), and the corresponding statement is true for X * P (t) and Y P (t; 2t) and for X * P ′ (t) and Y P ′ (t; 2t).…”