Sub critical epidemics on random graphs

We study the contact process on random graphs with low infection rate ). For random d- regular graphs, it is known that the survival time is O (log n ) below the critical ) c . By contrast, on the Erd & odblac;s-R & eacute;nyi 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 lambda 2+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. (c) 2024 Published by Elsevier Inc.