Epidemics on Evolving Graphs

Dong Yao, Rick Durrett

Published 2020-03-19Version 1

The evoSIR model is a modification of the usual SIR process on a graph $G$ in which $S-I$ connections are broken at rate $\rho$ and the $S$ connects to a randomly chosen vertex. The evoSI model is the same as evoSI but recovery is impossible. In \cite{DOMath} the critical value for evoSIR was computed and simulations showed that when $G$ is an Erd\H os-R\'enyi graph with mean degree 5 and rewiring rate 4, the system has a discontinuous phase transition, i.e., as the infection rate $\lambda$ decreases to $\lambda_c$, the final fraction of once infected individuals does not converge to 0. In this paper we study evoSI and evoSIR dynamics on graphs generated by the configuration model. We show that for each model there is a quantity $\Delta$ determined by the first three moments of the degree distribuion, so that the transition is discontinuous if $\Delta>0$ and continuous if $\Delta<0$. We can also compute the limiting epidemic size in the supercritical regime for evoSI and evoSIR.