Phase transitions of epidemics on random graphs

Abstract

This thesis focuses on the study of two mathematical models of epidemics, namely the contact process and the threshold-θ contact process. Until recently, rigorous understanding of their phase transition was mostly limited to the cases of elementary underlying networks such as lattices, homogeneous trees, and random regular graphs. The purpose of this work is to present recent advances in analyzing their behavior on general random graphs, including Galton-Watson trees, Erdos-Renyi random graphs, and the configuration model.

In the contact process, each individual is either infected or healthy. An infected individual can infect its neighbors or become healthy, with both events happening independently at certain rates. Its behavior can intuitively be understood as follows: when the infection is weak, the epidemic dies out quickly (extinction phase); on the other hand, when the infection is strong, the epidemic survives for a long time (survival phase). We propose new techniques to study the survival time of the process and establish the necessary and sufficient criteria for the existence of the extinction phase on Galton-Watson trees. Moreover, the corresponding analogs on random graphs are developed. Additionally, we extend our methods to derive the asymptotics for the value of the infection rate at the threshold.

We then consider the discrete-time threshold-θ contact process, where an individual becomes infected in the next time step with probability p, if it had at least θ neighboring infections in the current step. On (sparse) Erdos-Renyi graphs with large enough average degree, we establish a discontinuous phase transition in the emergence of metastability: (i) if p is large enough, the process starting from a high density of infections shows a long survival, keeping its infection density above a certain level; (ii) if either p is small or the initial state has a low density of infections, the process exhibits a short survival. We further present the analogs of the result on other random graphs. Finally, we discuss the contrasting result for the low-degree random regular graph on which the process only displays a short survival.