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

TL;DR

This paper develops a method to establish explicit bounds on infection rates and extinction times for the contact process on finite random graphs, advancing understanding of metastability in these stochastic models.

Contribution

It introduces a novel approach to bound infection rates uniformly, providing explicit thresholds and extinction time estimates for the contact process on Erdős-Rényi and configuration model graphs.

Findings

Explicit lower bounds on infection rates for metastability

Exponential lower bounds on expected extinction times

Method applicable to Erdős-Rényi and configuration model graphs

Abstract

We introduce a method to prove metastability of the contact process on Erd\H{o}s-R\'enyi 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.