A phase transition for measure-valued SIR epidemic processes

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 $\phi$, \begin{eqnarray*}X_t(\varphi )=X_0(\varphi)+\frac{1}{2}\int _0^tX_s(\Delta \varphi ),ds+\theta \int_0^tX_s(\varphi ),ds\{}-\int_0^tX_s(L_s\varphi ),ds+M_t(\varphi ).\end{eqnarray*} Here $L_s(x)$ is the local time density process associated with $X$, and $M_t(\phi )$ is a martingale with quadratic variation $[M(\phi ){]}_t={\int }_0^t{X}_s({\phi }^2)ds$. Such processes arise as scaling limits of SIR epidemic models. We show that there exist critical values ${\theta }_c(d)\in (0,\mathrm{\infty })$ 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.