Subcritical epidemics on random graphs

We study the contact process on random graphs with low infection rate λ. For random d-regular graphs, it is known that the survival time is O(log⁡n) below the critical λ_c. By contrast, on the Erdős-Rényi random graphs G(n,d/n), rare high-degree vertices result in much longer survival times. We show that the survival time is governed by high-density local configurations. In particular, we show that there is a long string of high-degree vertices on which the infection lasts for time n^λ2+o(1). To establish a matching upper bound, we introduce a modified version of the contact process which ignores infections that do not lead to further infections and allows for a sharper recursive analysis on branching process trees, the local-weak limit of the graph. Our methods, moreover, generalize to random graphs with given degree distributions that have exponential moments.