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Spatial epidemics with large finite range

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Mathew D. Penrose
Mathew D. Penrose*
Affiliation:
University of Durham
*
∗Postal address: Department of Mathematical Sciences, University of Durham, South Road, Durham, DH1 3LE, UK.
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Abstract

In the epidemic with removal with range r, each site z, once infected, remains so for a period of time Tz, the variables Tz being i.i.d. with mean μ. While infected, a site infects its healthy r-neighbours independently at total rate α. After infection, sites become immune. We show that the critical rate of infection αc (r), above which an epidemic starting from a single site may continue forever, converges to μ–1 as r →∞.

Keywords

EPIDEMICSFOREST FIRESPERCOLATIONMEAN-FIELD LIMIT

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K40: Other physical applications of random processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 4 , December 1996 , pp. 933 - 939
Copyright
Copyright © Applied Probability Trust 1996 

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References

[1] Bramson, M., Durrett, R. and Swindle, G. (1989) Statistical mechanics of crabgrass. Ann. Prob. 17, 444–481.Google Scholar
[2] Cox, J. T. and Durrett, R. (1988) Limit theorems for the spread of epidemics and forest fires. Stoch. Proc. Appl. 30, 171–191.CrossRefGoogle Scholar
[3] Durrett, R. (1988) Lecture Notes on Particle Systems and Percolation. Wadsworth and Brooks/Cole, Pacific Grove, CA.Google Scholar
[4] Greenberg, J. M. and Hastings, S. P. (1978) Spatial patterns for discrete models of excitable media. SIAM J. Appl. Math. 34, 515–523.Google Scholar
[5] Penrose, M. D. (1993) On the spread-out limit for bond and continuum percolation. Ann. Appl. Prob. 3, 253–276.Google Scholar
[6] Penrose, M. D. (1996) The threshold contact process: a continuum limit. Prob. Theory Rel. Fields 104, 77–95.Google Scholar
[7] Zhang, Y. (1993) A shape theorem for epidemics and forest fires with finite range interactions. Ann. Prob. 21, 1755–1781.Google Scholar