The contact process on the complete graph with random vertex-dependent infection rates

We study the contact process on the complete graph on n vertices where the rate at which the infection travels along the edge connecting vertices i and j is equal to [lambda]wiwj/n for some [lambda]>0, where wi are i.i.d. vertex weights. We show that when there is a phase transition at [lambda]c>0 such that for [lambda]<[lambda]c the contact process dies out in logarithmic time, and for [lambda]>[lambda]c the contact process lives for an exponential amount of time. Moreover, we give a formula for [lambda]c and when [lambda]>[lambda]c we are able to give precise approximations for the probability that a given vertex is infected in the quasi-stationary distribution. Our results are consistent with a non-rigorous mean field analysis of the model. This is in contrast to some recent results for the contact process on power law random graphs where the mean field calculations suggested that [lambda]c>0 when in fact [lambda]c=0.