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Spatial epidemics and local times for critical branching random walks in dimensions 2 and 3
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  • Published: 27 August 2009

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

  • Steven P. Lalley1 &
  • Xinghua Zheng2 

Probability Theory and Related Fields volume 148, pages 527–566 (2010)Cite this article

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Abstract

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.

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Authors and Affiliations

  1. Department of Statistics, The University of Chicago, Chicago, 60637, IL, USA

    Steven P. Lalley

  2. Department of ISOM, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

    Xinghua Zheng

Authors
  1. Steven P. Lalley
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  2. Xinghua Zheng
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Correspondence to Steven P. Lalley or Xinghua Zheng.

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Lalley, S.P., Zheng, X. Spatial epidemics and local times for critical branching random walks in dimensions 2 and 3. Probab. Theory Relat. Fields 148, 527–566 (2010). https://doi.org/10.1007/s00440-009-0239-1

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  • Received: 09 January 2009

  • Revised: 18 June 2009

  • Published: 27 August 2009

  • Issue Date: November 2010

  • DOI: https://doi.org/10.1007/s00440-009-0239-1

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Keywords

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

Mathematics Subject Classification (2000)

  • Primary 60H30
  • Secondary 60K35
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