Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-22T10:25:29.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Strong approximations for some open population epidemic models

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Philip O'Neill
Philip O'Neill*
Affiliation:
University of Bradford
*
∗Postal address: Department of Mathematics, University of Bradford, Bradford, BD7 1DP, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper considers a class of epidemic models in which susceptibles may enter or leave the population according to a general continuous time density dependent Markov chain. A sequence of such epidemics indexed by N, the initial number of susceptibles, is constructed on the same probability space as a time-inhomogeneous birth-and-death process. A coupling argument is then used to demonstrate the strong convergence of the sequence of infectives to the birth-and-death process. This result is used to provide a threshold analysis of the epidemic model in question.

Keywords

EPIDEMICSSIZE OF EPIDEMICCOUPLING METHODSBIRTH-AND-DEATH PROCESSESDENSITYDEPENDENT MARKOV CHAINS

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60K99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 2 , September 1996 , pp. 448 - 457
Copyright
Copyright © Applied Probability Trust 1996 

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

Bailey, N. T. J. (1975) The Mathematical Theory of Infectious Diseases. 2nd edn. Griffin, London.Google Scholar
Bailey, N. T. J. (1991) The use of operational modeling of HIV/AIDS in a systems approach to public health decision making. Math. Biosci. 107, 413–430.CrossRefGoogle Scholar
Blythe, S. P. and Anderson, R. M. (1988) Distributed incubation and infectious periods in models of the transmission dynamics of the Human Immunodeficiency Virus (HIV). IMA J. Math. Appl. Med. Biol. 5, 1–19.CrossRefGoogle ScholarPubMed
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 , pp. 144–147.Google Scholar
Ball, F. G. and Donnelly, P. J. (1995) Strong approximations for epidemic models. Stoch. Proc. Appl. 55, 1–21.CrossRefGoogle Scholar
Ball, F. G. and O'Neill, P. D. (1994) Strong convergence of stochastic epidemics. Adv. Appl. Prob. 26, 629–655.CrossRefGoogle Scholar
Bartlett, M. S. (1955) An Introduction to Stochastic Processes. Cambridge University Press, Cambridge.Google Scholar
Jacquez, J. A. and O'Neill, P. D. (1991) Reproduction numbers and thresholds in stochastic epidemic models I. Homogeneous populations. Math. Biosci. 107, 161–186.CrossRefGoogle ScholarPubMed
Kendall, D. G. (1948) On the generalized ‘birth-and-death’ process. Ann. Math. Statist. 19, 1–15.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
Kurtz, T. G. (1978) Strong approximation theorems for density dependent Markov chains. Stoch. Proc. Appl. 6, 223–240.CrossRefGoogle 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
Mollison, D. (1977) Spatial contact models for ecological and epidemic spread. J. R. Statist. Soc. B 39, 283–326.Google Scholar
Stirzaker, D. R. (1975) A perturbation method for the stochastic recurrent epidemic. J. Inst. Math. Appl. 15, 135–160.CrossRefGoogle Scholar