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Strong Convergence of Stochastic Epidemics

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  01 July 2016

Frank Ball  and
Philip O'Neill
Frank Ball*
Affiliation:
University of Nottingham
Philip O'Neill*
Affiliation:
University of Nottingham
*
* Postal address: Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
** Present address: Department of Mathematics, University of Bradford, Bradford, West Yorkshire BD7 1DP, UK.
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Abstract

This paper is concerned with a model for the spread of an epidemic in a closed, homogeneously mixing population in which new infections occur at rate f(x, y) and removals occur at rate g(x, y), where x and y are the numbers of susceptible and infective individuals, respectively, and f and g are arbitrary but specified positive real-valued functions. Sequences of such epidemics, indexed by the initial number of susceptibles n, are considered and conditions are derived under which the epidemic processes converge almost surely to a birth and death process as n tends to infinity. Thus a threshold theorem for such an epidemic model is obtained. The results are extended to models which incorporate immigration and emigration of susceptibles. The theory is illustrated by several examples of models taken from the epidemic literature. Generalizations to multipopulation epidemics are discussed briefly.

Keywords

EPIDEMICSSIZE OF EPIDEMICTHRESHOLD THEOREMSBIRTH AND DEATH PROCESSESCOUPLINGEPIDEMICS WITH IMMIGRATION AND EMIGRATION

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60F15: Strong theorems 60J27: Continuous-time Markov processes on discrete state spaces 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 26 , Issue 3 , September 1994 , pp. 629 - 655
Copyright
Copyright © Applied Probability Trust 1994 

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References

Von Bahr, B. and Martin-Löf, A. (1980) Threshold limit theorems for some epidemic processes. Adv. Appl. Prob. 12, 319–349.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 behaviour of epidemic models. J. Appl. Prob. 20, 227–241.CrossRefGoogle Scholar
Ball, F. G. and Donnelly, P. J. (1992) Branching process approximation of epidemic models. Proc. 2nd World Congress of the Bernoulli Soc., Uppsala, 1990, 144–147.Google Scholar
Ball, F. G. and Donnelly, P. J. (1993) Strong approximations for epidemic models. Submitted.Google Scholar
Ball, F. G. and O'Neill, P. D. (1993) A modification of the general stochastic epidemic motivated by AIDS modelling. Adv. Appl. Prob. 25, 39–62.Google Scholar
Bartlett, M. S. (1955) An Introduction to Stochastic Processes. Cambridge University Press, London.Google Scholar
Bartlett, M. S. (1956) Deterministic and stochastic models for recurrent epidemics. Proc. 3rd Berkeley Symp. Math. Statist. Prob. 4, 81–109.Google Scholar
Gani, J. and Purdue, P. (1984) Matrix-geometric methods for the general stochastic epidemic. IMA J. Math. Appl. Med. Biol. 1, 333–342.CrossRefGoogle ScholarPubMed
GleißNer, W. (1988) The spread of epidemics. Appl. Math. Comput. 27, 167–171.Google Scholar
Griffiths, D. A. (1973) Multivariate birth and death processes as approximations to epidemic processes. J. Appl. Prob. 10, 15–26.Google Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, New York.Google Scholar
Kendall, D. G. (1956) Deterministic and stochastic epidemics in closed populations. Proc. 3rd Berkeley Symp. Math. Statist. Prob. 4, 149–165.Google Scholar
Kurtz, T. G. (1981) Approximation of Population Processes. CBMS-NSF Regional Conf. Ser. 36, SIAM publications.CrossRefGoogle Scholar
Lefèvre, C. and Piçard, P. (1993) An epidemic model with fatal risk. Math. Biosci. 117, 127–145.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
Mode, C. J. (1962) Some multi-dimensional birth and death processes and their application in population genetics. Biometrics 18, 543–567.CrossRefGoogle Scholar
Nerman, O. (1981) On the convergence of supercritical (C-M-J) branching processes. Z. Wahrsheinlichkeitsch. 57, 365–395.Google Scholar
Ripley, B. J. (1987) Stochastic Simulation. Wiley, New York.CrossRefGoogle Scholar
Saunders, I. W. (1980) A model for myxomatosis. Math. Biosci. 48, 1–15.Google Scholar
Stirzaker, D. R. (1975) A perturbation method for the stochastic recurrent epidemic. J. Inst. Math. Appl. 15, 135–160.Google Scholar
Whittle, P. (1955) The outcome of a stochastic epidemic–a note on Bailey's paper. Biometrika 42, 116–122.Google Scholar
Williams, D. (1991) Probability with Martingales. Cambridge University Press.Google Scholar
Williams, T. (1971) An algebraic proof of the threshold theorem for the general stochastic epidemic (abstract). Adv. Appl. Prob. 3, 223.Google Scholar