Susceptible-Infected Epidemics on Evolving Graphs at Critical Infection Rate

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 $n$.