Probability Theory and Related Fields

, Volume 148, Issue 3–4, pp 527–566 | Cite as

Spatial epidemics and local times for critical branching random walks in dimensions 2 and 3

  • Steven P. LalleyEmail author
  • Xinghua ZhengEmail author


The behavior at criticality of spatial SIR epidemic models in dimensions two and three is investigated. In these models, finite populations of size N are situated at the sites of the integer lattice, and infectious contacts are limited to individuals at the same or at neighboring sites. Susceptible individuals, once infected, remain contagious for one unit of time and then recover, after which they are immune to further infection. It is shown that the measure-valued processes associated with these epidemics, suitably scaled, converge, in the large-N limit, either to a standard Dawson–Watanabe process (super-Brownian motion) or to a Dawson–Watanabe process with location-dependent killing, depending on the size of the the initially infected set. A key element of the argument is a proof of Adler’s 1993 conjecture that the local time processes associated with branching random walks converge to the local time density process associated with the limiting super-Brownian motion.


Spatial epidemic Branching random walk Dawson–Watanabe process Local times Critical scaling 

Mathematics Subject Classification (2000)

Primary 60H30 Secondary 60K35 


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© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of StatisticsThe University of ChicagoChicagoUSA
  2. 2.Department of ISOMHong Kong University of Science and TechnologyKowloonHong Kong

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