A PHASE TRANSITION FOR MEASURE-VALUED SIR EPIDEMIC PROCESSES

• 成果类型: Article
• 署名作者: Lalley, Steven P.; Perkins, Edwin A.; Zheng, Xinghua
• 署名单位: University of Chicago; University of British Columbia; Hong Kong University of Science & Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP846
• 发表日期: 2014
• 页码: 237-310
• 关键词: super-brownian motion; spatial epidemics; local-times; contact; superprocesses; dimensions; SPREAD

摘要

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, X-t(phi) = X-0(phi) + 1/2 integral(t)(0)Xs(Delta phi) ds + 1/2 integral(t)(0) Xs (phi) ds -integral(t)(0)Xs(L-s phi) ds + M-t(phi) Here Ls(x) is the local time density process associated with X, and Mt (co) is a martingale with quadratic variation [M-t(phi)](t) =integral(t)(0)Xs(phi(2)) ds. Such processes arise as scaling limits of SIR epidemic models. We show that there exist critical values 0c(d) E (0, oo) 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 O. 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.