Open Access
January 2014 A phase transition for measure-valued SIR epidemic processes
Steven P. Lalley, Edwin A. Perkins, Xinghua Zheng
Ann. Probab. 42(1): 237-310 (January 2014). DOI: 10.1214/13-AOP846


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.


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Steven P. Lalley. Edwin A. Perkins. Xinghua Zheng. "A phase transition for measure-valued SIR epidemic processes." Ann. Probab. 42 (1) 237 - 310, January 2014.


Published: January 2014
First available in Project Euclid: 9 January 2014

zbMATH: 1303.60059
MathSciNet: MR3161486
Digital Object Identifier: 10.1214/13-AOP846

Primary: 60H30 , 60K35
Secondary: 60H15

Keywords: Dawson–Watanabe process , Local extinction , phase transition , spatial epidemic

Rights: Copyright © 2014 Institute of Mathematical Statistics

Vol.42 • No. 1 • January 2014
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