  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.
