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A complete convergence theorem for an epidemic model

Published online by Cambridge University Press:  14 July 2016

Enrique Andjel  and
Rinaldo Schinazi
Enrique Andjel*
Affiliation:
Université de Provence
Rinaldo Schinazi*
Affiliation:
University of Colorado
*
∗Postal address: Laboratoire d'APT, CNRS/URA n° 225, Université de Provence, case X, 13331 Marseille Cedex 3, France.
∗∗Postal address: Department of Mathematics, University of Colorado, Colorado Springs. CO 80933, USA.
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Abstract

We use an interacting particle system on ℤ to model an epidemic. Each site of ℤ can be in either one of three states: empty, healthy or infected. An empty site x gets occupied by a healthy individual at a rate βn1(x) where n1(x) is the number of healthy nearest neighbors of x. A healthy individual at x gets infected at rate αn2(x) where n2(x) is the number of infected nearest neighbors of x. An infected individual dies at rate δ independently of everything else. We show that for all α, β and δ> 0 and all initial configurations, all the sites of a fixed finite set remain either all empty or all healthy after an almost surely finite time. Moreover, if the initial configuration has infinitely many healthy individuals then the process converges almost surely (in the sense described above) to the all healthy state. We also consider a model introduced by Durrett and Neuhauser where healthy individuals appear spontaneously at rate β > 0 and for which coexistence of 1's and 2's was proved in dimension 2 for some values of α and β. We prove that coexistence may occur in any dimension.

Keywords

INTERACTING PARTICLE SYSTEMSSPATIAL EPIDEMIC MODEL

MSC classification

Primary: 92D30: Epidemiology
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 3 , September 1996 , pp. 741 - 748
Copyright
Copyright © Applied Probability Trust 1996 

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References

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