Contact process on a graph with communities

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.