Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-14T05:17:46.918Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramov, V. M. (1991) Investigation of the Queueing System with Service Depending on Queue Length. Donish, Duschanbe (in Russian).Google Scholar
Bailey, N. T. J. (1975) The Mathematical Theory of Infectious Diseases and its Applications, 2nd edn. Griffin, London.Google Scholar
Ball, F. G. (1983) The threshold behavior of epidemic models. J. Appl. Prob. 20, 227–241.CrossRefGoogle Scholar
Bartlett, M. S. (1955) An Introduction to Stochastic Processes. Cambridge University Press.Google Scholar
Bharucha-Reid, A. T. (1960) Elements of the Theory of Markov Processes and their Applications. McGraw-Hill, New York.Google Scholar
Keilson, J. (1979) Markov Chain Models - Rarity and Exponentiality. Springer-Verlag, New York.CrossRefGoogle Scholar
Kendall, D. G. (1956) Deterministic and stochastic epidemics in closed populations. Proc 3rd Berkeley Symp. Math. Statist. Prob. 4, 149–165.Google Scholar
Metz, J. A. J. (1978) The epidemic in a closed population with all susceptibles equally vulnerable: some results for large susceptible populations and small initial infections. Acta Biotheoretica 27, 75–123.CrossRefGoogle Scholar
Shiryayev, A. N. (1984) Probability. Springer-Verlag, New York.CrossRefGoogle Scholar