Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Spatial epidemics: critical behavior in one dimension
Download PDF
Download PDF
  • Published: 08 April 2008

Spatial epidemics: critical behavior in one dimension

  • Steven P. Lalley1 

Probability Theory and Related Fields volume 144, pages 429–469 (2009)Cite this article

  • 243 Accesses

  • 13 Citations

  • Metrics details

Abstract

In the simple mean-field SIS and SIR epidemic models, infection is transmitted from infectious to susceptible members of a finite population by independent p-coin tosses. Spatial variants of these models are considered, in which finite populations of size N are situated at the sites of a lattice and infectious contacts are limited to individuals at neighboring sites. Scaling laws for these models are given when the infection parameter p is such that the epidemics are critical. It is shown that in all cases there is a critical threshold for the numbers initially infected: below the threshold, the epidemic evolves in essentially the same manner as its branching envelope, but at the threshold evolves like a branching process with a size-dependent drift. The corresponding scaling limits are super-Brownian motions and Dawson–Watanabe processes with killing, respectively.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Aldous, D.: Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25(2), 812–854 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Athreya, K.B., Ney, P.E.: Branching Processes. Springer, New York (1972). Die Grundlehren der mathematischen Wissenschaften, Band 196

    MATH  Google Scholar 

  3. Belhadji, L., Lanchier, N.: Individual versus cluster recoveries within a spatially structured population. Ann. Appl. Probab. 16(1), 403–422 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Billingsley, P.: Convergence of Probability Measures. Wiley Series in Probability and Statistics: Probability and Statistics, 2nd edn. Wiley-Interscience, New York (1999)

  5. Cox, J.T., Durrett, R., Perkins, E.A.: Rescaled voter models converge to super-Brownian motion. Ann. Probab. 28(1), 185–234 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dawson, D.A.: Geostochastic calculus. Canad. J. Statist. 6(2), 143–168 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dawson, D.A., Hochberg, K.J.: The carrying dimension of a stochastic measure diffusion. Ann. Probab. 7(4), 693–703 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dawson, D.A., Perkins, E.A.: Measure-valued processes and renormalization of branching particle systems. In: Stochastic Partial Differential Equations: six perspectives, Math. Surveys Monogr., vol. 64, pp. 45–106. American Mathematical Society, Providence, RI (1999)

  9. Dolgoarshinnykh, R., Lalley, S.P.: Critical scaling for the sis stochastic epidemic. J. Appl. Probab. 43, 892–898 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Durrett, R.: Stochastic spatial models. SIAM Rev. 41(4), 677–718 (1999) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  11. Durrett, R., Mytnik, L., Perkins, E.: Competing super-Brownian motions as limits of interacting particle systems. Electron. J. Probab. 10(35), 1147–1220 (2005) (electronic)

    MathSciNet  Google Scholar 

  12. Durrett, R., Perkins, E.A.: Rescaled contact processes converge to super-Brownian motion in two or more dimensions. Probab. Theory Related Fields 114(3), 309–399 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Etheridge, A.M.: An introduction to superprocesses. University Lecture Series, vol. 20. American Mathematical Society, Providence, RI (2000)

  14. Evans, S.N., Perkins, E.A.: Explicit stochastic integral representations for historical functionals. Ann. Probab. 23(4), 1772–1815 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  15. Fleischmann, K.: Critical behavior of some measure-valued processes. Math. Nachr. 135, 131–147 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  16. Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus. Graduate Texts in Mathematics, vol. 113. Springer, New York (1991)

    Google Scholar 

  17. Kesten, H.: Branching random walk with a critical branching part. J. Theoret. Probab. 8(4), 921–962 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  18. Konno, N., Shiga, T.: Stochastic partial differential equations for some measure-valued diffusions. Probab. Theory Related Fields 79(2), 201–225 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  19. 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)

    Article  MATH  MathSciNet  Google Scholar 

  20. Mollison, D.: Spatial contact models for ecological and epidemic spread. J. R. Statist. Soc. Ser. B 39(3), 283–326 (1977)

    MATH  MathSciNet  Google Scholar 

  21. Müller, C., Tribe, R.: Stochastic p.d.e.’s arising from the long range contact and long range voter processes. Probab. Theory Related Fields 102(4), 519–545 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  22. Schinazi, R.: On the role of social clusters in the transmission of infectious diseases. Theor. Pop. Biol. 61, 163–169 (2002)

    Article  MATH  Google Scholar 

  23. Shiga, T.: Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Can. J. Math. 46(2), 415–437 (1994)

    MATH  MathSciNet  Google Scholar 

  24. Sugitani, S.: Some properties for the measure-valued branching diffusion processes. J. Math. Soc. Jpn 41(3), 437–462 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  25. Walsh, J.B.: An introduction to stochastic partial differential equations. In: École d’été de probabilités de Saint-Flour, XIV—1984. Lecture Notes in Mathematics, vol. 1180, pp. 265–439. Springer, Berlin (1986)

  26. Watanabe, S.: A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ. 8, 141–167 (1968)

    MATH  MathSciNet  Google Scholar 

  27. Whittaker, E.T., Watson, G.N.: A course of modern analysis. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions, Reprint of the fourth edition (1927)

  28. Zheng, X.: Spatial epidemics in higher dimensions. Ph.D. thesis, University of Chicago (in progress)

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, University of Chicago, Eckhart 118, 5734 S. University Avenue, Chicago, IL, 60637, USA

    Steven P. Lalley

Authors
  1. Steven P. Lalley
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Steven P. Lalley.

Additional information

This work was supported by NSF grant DMS-0405102.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lalley, S.P. Spatial epidemics: critical behavior in one dimension. Probab. Theory Relat. Fields 144, 429–469 (2009). https://doi.org/10.1007/s00440-008-0151-0

Download citation

  • Received: 25 July 2007

  • Revised: 14 February 2008

  • Published: 08 April 2008

  • Issue Date: July 2009

  • DOI: https://doi.org/10.1007/s00440-008-0151-0

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

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

Mathematics Subject Classification (2000)

  • Primary: 60H30
  • Secondary: 60K35
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature