Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-22T21:16:22.220Z Has data issue: false hasContentIssue false
  • English
  • Français

A modification of the general stochastic epidemic motivated by AIDS modelling

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

Abstract

This paper considers a model for the spread of an epidemic in a closed, homogeneously mixing population in which new infections occur at rate βxy/(x + y), where x and y are the numbers of susceptible and infectious individuals, respectively, and β is an infection parameter. This contrasts with the standard general epidemic in which new infections occur at rate βxy. Both the deterministic and stochastic versions of the modified epidemic are analysed. The deterministic model is completely soluble. The time-dependent solution of the stochastic model is derived and the total size distribution is considered. Threshold theorems, analogous to those of Whittle (1955) and Williams (1971) for the general stochastic epidemic, are proved for the stochastic model. Comparisons are made between the modified and general epidemics. The effect of introducing variability in susceptibility into the modified epidemic is studied.

Keywords

EPIDEMICSSIZE OF EPIDEMICDETERMINISTIC AND STOCHASTIC MODELSTIME-DEPENDENT SOLUTIONTHRESHOLD THEOREMSEMBEDDED RANDOM WALKSVARIABILITY IN SUSCEPTIBILITY

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 1 , March 1993 , pp. 39 - 62
Copyright
Copyright © Applied Probability Trust 1993 

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.)

Footnotes

∗∗

Present address: Department of Mathematics, University of Bradford, Bradford BD7 1DP, UK.

References

Anderson, R. M. (1988) The epidemiology of HIV infection: variable incubation plus infectious periods and heterogeneity in sexual activity (with discussion). J. R. Statist. Soc. A 151, 66–98; 124–125.CrossRefGoogle Scholar
Anderson, R. M., Blythe, S. P., Gupta, S. and Konings, E. (1989) The transmission dynamics of the human immunodeficiency virus type 1 in the male homosexual community in the United Kingdom: the influence of changes in sexual behaviour. Phil. Trans. R. Soc. Lond. B 325, 45–98.Google ScholarPubMed
Von Bahr, B. and Martin-Löf, A. (1980) Threshold limit theorems for some epidemic processes. Adv. Appl. Prob. 12, 319–349.CrossRefGoogle 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. (1985) Deterministic and stochastic epidemics with several kinds of susceptibles. Adv. Appl. Prob. 17, 1–22.CrossRefGoogle Scholar
Billard, L. (1973) Factorial moments and probabilities for the general stochastic epidemic. J. Appl. Prob. 10, 277–288.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Volume 1, 3rd edn. Wiley, New York.Google Scholar
Fusaro, R. E., Jewell, N. P., Hauck, W. W., Heilbron, D. C., Kalbfleisch, J. D., Neuhaus, J. M. and Ashby, M. A. (1989) An annotated bibliography of quantitative methodology relating to the AIDS epidemic. Statist. Sci. 4, 264–281.CrossRefGoogle 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
Gart, J. J. (1968) The mathematical analysis of an epidemic with two kinds of susceptibles. Biometrics 24, 557–566.CrossRefGoogle ScholarPubMed
Gart, J. J. (1972) The statistical analysis of chain-binomial epidemic models with several kinds of susceptibles. Biometrics 28, 921–930.CrossRefGoogle Scholar
Gleibner, W. (1988) The spread of epidemics. Appl. Math. Comput. 27, 167–171.Google Scholar
Isham, V. (1988) Mathematical modelling of the transmission dynamics of HIV infection and AIDS: a review (with discussion). J. R. Statist. Soc. A 151, 5–30; 44–49.CrossRefGoogle Scholar
Jacquez, J. A. and O'Neill, P. (1991) Reproduction numbers and thresholds in stochastic epidemic models I. Homogeneous populations. Math. Biosci. 107, 161–186.CrossRefGoogle ScholarPubMed
Jacquez, J. A., Simon, C. P., Koopman, J., Sattenspiel, L. and Perry, T. (1988) Modeling and analyzing HIV transmission: The effect of contact patterns. Math. Biosci. 92, 119–199.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
Kendall, D. G. (1965) Mathematical models for the spread of infection. In Mathematics and Computer Science in Biology and Medicine. MRC, HMSO, London, pp. 213–225.Google Scholar
Kryscio, R. J. (1975) The transition probabilities of the general stochastic epidemic model. J. Appl. Prob. 12, 415–424.CrossRefGoogle Scholar
Kryscio, R. J. and Malice, M.-P. (1990) On modeling the incidence of AIDS. In Stochastic Processes in Epidemic Theory, eds. Gabriel, J.-P., Lefèvre, C. and Picard, P., pp.46–58 Lecture Notes in Biomathematics 86, Springer-Verlag, New York.CrossRefGoogle Scholar
Proschan, F. and Sethuraman, J. (1976) Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability. J. Multivariate Anal. 6, 608–616.CrossRefGoogle Scholar
Rajarshi, M. B. (1981) Simpler proofs of two threshold theorems for a general stochastic epidemic. J. Appl. Prob. 18, 721–724.CrossRefGoogle Scholar
Renyi, A. (1970) Probability Theory. North-Holland, Amsterdam.Google Scholar
Sellke, T. (1983) On the asymptotic distribution of the size of a stochastic epidemic. J. Appl. Prob. 20, 390–394.CrossRefGoogle Scholar
Whittle, P. (1955) The outcome of a stochastic epidemic—a note on Bailey's paper. Biometrika 42, 116–122.Google Scholar
Williams, T. (1971) An algebraic proof of the threshold theorem for the general stochastic epidemic (abstract). Adv. Appl. Prob. 3, 323.CrossRefGoogle Scholar