Contact process on a graph with communities

TL;DR

This paper analyzes how an epidemic spreads between two highly connected communities modeled as Erdos-Renyi graphs with bridge edges, showing conditions for long-term survival and infection timing distribution.

Contribution

It provides a mathematical analysis of epidemic spread on a multi-community network with probabilistic connectivity, extending previous models to include multiple communities and bridge edges.

Findings

Epidemic survives exponentially long when npλ > 1.

Time to infect the second community scales with the number of bridge edges.

Infection time converges to an exponential distribution under certain conditions.

Abstract

We are interested in the spread of an epidemic between two communities that have higher connectivity within than between them. We model the two communities as independent Erdos-Renyi random graphs, each with n vertices and edge probability p = n^{a-1} (0<a<1), then add a small set of bridge edges, B, between the communities. We model the epidemic on this network as a contact process (Susceptible-Infected-Susceptible infection) with infection rate \lambda and recovery rate 1. If np\lambda = b > 1 then the contact process on the Erdos-Renyi random graph is supercritical, and we show that it survives for exponentially long. Further, let \tau be the time to infect a positive fraction of vertices in the second community when the infection starts from a single vertex in the first community. We show that on the event that the contact process survives exponentially long, \tau |B|/(np) converges in distribution to an exponential random variable with a specified rate. These results generalize to a graph with N communities.