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On the asymptotic distribution of the maximum number of infectives in epidemic models by immigration

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

V. M. Abramov
V. M. Abramov*
Affiliation:
Technion — Israel Institute of Technology
*
∗ Postal address: 6/3 Agmon St., P.O. Box 17361, Nazareth-Illit, 17801, Israel.
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Abstract

This paper considers the asymptotic distribution of the maximum number of infectives in an epidemic model by showing that, as the initial number of susceptibles converges to infinity, the process of infectives converges almost surely to a birth and death process. The model studied here is more general than usual (see e.g. Bailey (1975), Bharucha-Reid (1960), Keilson (1979)) in that it incorporates immigration and the limiting birth and death process is non-linear. The main novelty of the present paper is the martingale approach used to prove the above-mentioned convergence.

Keywords

MAXIMUM NUMBER OF INFECTIVESGAMBLER'S RUIN PROBLEMMARTINGALEBIRTH AND DEATH PROCESS

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) 60G42: Martingales with discrete parameter 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 3 , September 1994 , pp. 606 - 613
Copyright
Copyright © Applied Probability Trust 1994 

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References

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