Advertisement

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
Article

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.

Keywords

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

Mathematics Subject Classification (2000)

Primary 60H30 Secondary 60K35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adler, R.J.: Superprocess local and intersection local times and their corresponding particle pictures. In: Seminar on Stochastic Processes, 1992 (Seattle, WA, 1992), vol 33 of Progr. Probab., pp. 1–42. Birkhäuser Boston (1993)Google Scholar
  2. 2.
    Aldous D.: Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25(2), 812–854 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bramson M., Durrett R., Swindle G.: Statistical mechanics of crabgrass. Ann. Probab. 17(2), 444–481 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Dawson D.A.: Geostochastic calculus. Can. J. Stat. 6(2), 143–168 (1978)zbMATHCrossRefGoogle Scholar
  5. 5.
    Dawson D.A., Hochberg K.J.: The carrying dimension of a stochastic measure diffusion. Ann. Probab. 7(4), 693–703 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dawson, D.A., Perkins, E.A.: Measure-valued processes and renormalization of branching particle systems. In: Stochastic Partial Differential Equations: Six Perspectives, vol 64 of Math. Surveys Monogr., pp. 45–106. Amer. Math. Soc., Providence (1999)Google Scholar
  7. 7.
    Dolgoarshinnykh R.G., Lalley S.P.: Critical scaling for the SIS stochastic epidemic. J. Appl. Probab. 43(3), 892–898 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Durrett R., Perkins E.A.: Rescaled contact processes converge to super-Brownian motion in two or more dimensions. Probab. Theory Relat. Fields 114(3), 309–399 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Etheridge, A.M.: An introduction to superprocesses, volume 20 of University Lecture Series. American Mathematical Society, Providence, RI (2000)Google Scholar
  10. 10.
    Evans S.N., Perkins E.A.: Explicit stochastic integral representations for historical functionals. Ann. Probab. 23(4), 1772–1815 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Fleischmann K.: Critical behavior of some measure-valued processes. Math. Nachr. 135, 131–147 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hoeffding W.: Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc. 58, 13–30 (1963)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Iscoe I.: Ergodic theory and a local occupation time for measure-valued critical branching Brownian motion. Stochastics 18(3–4), 197–243 (1986)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Karatzas I., Shreve S.E.: Brownian Motion and Stochastic Calculus, vol. 113 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (1991)Google Scholar
  15. 15.
    Konno N., Shiga T.: Stochastic partial differential equations for some measure-valued diffusions. Probab. Theory Relat. Fields 79(2), 201–225 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Lalley S.P.: Spatial epidemics: critical behavior in one dimension. Probab. Theory Relat. Fields 144(3–4), 429–469 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Lalley, S.P., Perkins, E., Zheng, X.: A phase transition in spatial sir epidemics (in preparation, 2009)Google Scholar
  18. 18.
    Lalley, S.P., Zheng, X.: Occupation statistics of critical branching random walks. Ann. Probab., arXiv:0707.3829v2 (in revision 2007)Google Scholar
  19. 19.
    Lawler, G.F., Limic, V.: Random Walk: A Modern Introduction. http://www.math.uchicago.edu/~lawler/srwbook.pdf (2007)
  20. 20.
    Martin-Löf A.: The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier. J. Appl. Probab. 35(3), 671–682 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Perkins, E., Zheng, X.: Spatial epidemics and percolations in dimensions four and higher (in preparation, 2009)Google Scholar
  22. 22.
    Reimers M.: One-dimensional stochastic partial differential equations and the branching measure diffusion. Probab. Theory Relat. Fields 81(3), 319–340 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Spitzer, F.: Principles of Random Walks, 2nd edn. Graduate Texts in Mathematics, vol. 34. Springer, New York (1976)Google Scholar
  24. 24.
    Sugitani S.: Some properties for the measure-valued branching diffusion processes. J. Math. Soc. Jpn 41(3), 437–462 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Walsh, J.B.: An introduction to stochastic partial differential equations. In: École d’été de probabilités de Saint-Flour, XIV—1984, volume 1180 of Lecture Notes in Math., pp. 265–439. Springer, Berlin (1986)Google Scholar
  26. 26.
    Watanabe S.: A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ. 8, 141–167 (1968)zbMATHMathSciNetGoogle Scholar

Copyright information

© 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

Personalised recommendations