Susceptible-Infected Epidemics on Evolving Graphs at Critical Infection Rate

TL;DR

This paper analyzes the critical behavior of an epidemic model on evolving graphs, revealing how a key parameter determines whether outbreaks are likely to be large or small at the epidemic threshold.

Contribution

It establishes the asymptotic probability of large outbreaks in the critical evoSI model based on the sign of a specific parameter, extending understanding of phase transitions in epidemic processes.

Findings

For elta>0, probability of major outbreak is n^{-1/3}

For elta<0, probability is o(n^{-1/3})

Results depend on the sign of a parameter elta related to degree distribution moments

Abstract

Consider an SI process on a graph $G$ where each S--I connection becomes I--I at rate $\lambda$. Here S and I stand for ``susceptible'' and ``infected'' respectively. The evoSI model is a modification of the SI model in which S--I edges are broken at rate $\rho$ and the ``S'' connects to a randomly chosen vertex. It is proven in Durrett and Yao [2022, Electron. J. Probab.] that, for the supercritical evoSI process on the configuration model, there exists a quantity $\Delta$ depending on the first three moments of the degree distribution such that the sign of $\Delta$ governs the continuity of the phase transition of the final epidemic size near the critical infection rate $\lambda_c$. In this paper, we consider the critical evoSI model on the configuration model, i.e., $\lambda=\lambda_c$. We show that, if $\Delta>0$, then the probability of a major outbreak starting from a single infected individual is $Cn^{-1/3}(1+o(1))$ for some explicit constant $C>0$, where $n$ is the size of the graph. On the contrary, if $\Delta<0$, then this probability is $o(n^{-1/3})$. The case $\Delta<0$ is reminiscent of the critical {\ER} graphs, where the probability for the size of the largest component to be of order $n$ decays exponentially in $n$.