A phase transition for measure-valued SIR epidemic processes

Abstract

We consider measure-valued processes $X=(X_{t})$ that solve the following martingale problem: for a given initial measure $X_{0}$, and for all smooth, compactly supported test functions $\varphi$,

\begin{eqnarray*}X_{t}(\varphi )&=&X_{0}(\varphi)+\frac{1}{2}\int_{0}^{t}X_{s}(\Delta\varphi )\,ds+\theta \int_{0}^{t}X_{s}(\varphi )\,ds\\&&{}-\int_{0}^{t}X_{s}(L_{s}\varphi )\,ds+M_{t}(\varphi ).\end{eqnarray*}

Here $L_{s}(x)$ is the local time density process associated with $X$, and $M_{t}(\varphi )$ is a martingale with quadratic variation $[M(\varphi )]_{t}=\int_{0}^{t}X_{s}(\varphi ^{2})\,ds$. Such processes arise as scaling limits of SIR epidemic models. We show that there exist critical values $\theta_{c}(d)\in(0,\infty)$ for dimensions $d=2,3$ such that if $\theta>\theta_{c}(d)$, then the solution survives forever with positive probability, but if $\theta<\theta_{c}(d)$, then the solution dies out in finite time with probability 1. For $d=1$ we prove that the solution dies out almost surely for all values of $\theta$. We also show that in dimensions $d=2,3$ the process dies out locally almost surely for any value of $\theta$; that is, for any compact set $K$, the process $X_{t}(K)=0$ eventually.