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 $\lambda w_i w_j / n$ for some $\lambda >0$, where $w_i$ are i.i.d. vertex weights. We show that when $E[w_1^2] < \infty$ there is a phase transition at $\lambda_c > 0$ so 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 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 $\lambda_c = 0$.