Explicit bounds for critical infection rates and expected extinction times of the contact process on finite random graphs

We introduce a method to prove metastability of the contact process on ErdH{o}s-Renyi graphs and on configuration model graphs. The method relies on uniformly bounding the total infection rate from below, over all sets with a fixed number of nodes. Once this bound is established, a simple comparison with a well chosen birth-and-death process will show the exponential growth of the extinction time. Our paper complements recent results on the metastability of the contact process: under a certain minimal edge density condition, we give explicit lower bounds on the infection rate needed to get metastability, and we have explicit exponentially growing lower bounds on the expected extinction time.