The Annals of Probability

A phase transition for measure-valued SIR epidemic processes

Steven P. Lalley, Edwin A. Perkins, and Xinghua Zheng

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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.

Article information

Ann. Probab., Volume 42, Number 1 (2014), 237-310.

First available in Project Euclid: 9 January 2014

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Zentralblatt MATH identifier

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60H15: Stochastic partial differential equations [See also 35R60]

Spatial epidemic Dawson–Watanabe process phase transition local extinction


Lalley, Steven P.; Perkins, Edwin A.; Zheng, Xinghua. A phase transition for measure-valued SIR epidemic processes. Ann. Probab. 42 (2014), no. 1, 237--310. doi:10.1214/13-AOP846.

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