Hostname: page-component-84b7d79bbc-2l2gl Total loading time: 0 Render date: 2024-08-01T09:14:27.151Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Extinction of the S–I–S stochastic logistic epidemic

Published online by Cambridge University Press:  14 July 2016

Richard J. Kryscio  and
Claude Lefèvre
Richard J. Kryscio*
Affiliation:
University of Kentuckyn
Claude Lefèvre*
Affiliation:
Université Libre de Bruxelles
*
∗ Postal address: Department of Statistics, College of Arts and Sciences, University of Kentucky, Lexington, KY 40506–0027, USA.
∗∗ Postal address: Université Libre de Bruxelles, Institut de Statistique, Campus Plaine, C.P. 210, Boulevard du Triomphe, B-1050 Bruxelles, Belgium.
Get access Rights & Permissions [Opens in a new window]

Abstract

We obtain an approximation to the mean time to extinction and to the quasi-stationary distribution for the standard S–I–S epidemic model introduced by Weiss and Dishon (1971). These results are a combination and extension of the results of Norden (1982) for the stochastic logistic model, Oppenheim et al. (1977) for a model on chemical reactions, Cavender (1978) for the birth-and-death processes and Bartholomew (1976) for social diffusion processes.

Keywords

S–I–S EPIDEMICEXPECTED LENGTHQUASI-STATIONARY DISTRIBUTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 4 , December 1989 , pp. 685 - 694
Copyright
Copyright © Applied Probability Trust 1989 

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

Partially supported by a NATO Grant for international collaboration. The research of RJK was partially supported by NSF EPSCoR Grant RII-861 0671.

References

Barbour, A. D. (1975) The duration of the closed stochastic epidemic. Biometrika 62, 477–482.CrossRefGoogle Scholar
Bartholomew, D. J. (1976) Continuous time diffusion models with random duration of interest. J. Math. Sociol. 4, 187–199.Google Scholar
Bartholomew, D. J. (1982) Stochastic Models for Social Processes. Wiley, New York.Google Scholar
Cavender, J. A. (1978) Quasi-stationary distributions of birth-and-death processes. Adv. Appl. Prob. 10, 570–586.Google Scholar
Hethcote, H. W., Yorke, J. A. and Nold, A. (1982) Gonorrhea modeling: a comparison of control methods. Math. Biosci. 58, 93–109.Google Scholar
Kurtz, T. G. (1971) Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J. Appl. Prob. 8, 344–356.CrossRefGoogle Scholar
Mandl, P. (1960) On the asymptotic behavior of probabilities within classes of states of a homogeneous Markov process (in Russian). Casopis Pest. Mat. 85, 448–456.CrossRefGoogle Scholar
Norden, R. H. (1982) On the distribution of the time to extinction in the stochastic logistic population model. Adv. Appl. Prob. 14, 687–708.CrossRefGoogle Scholar
Oppenheim, I., Shuler, K. E. and Weiss, G. H. (1977) Stochastic theory of nonlinear rate processes with multiple stationary states. Physica A, 191–214.Google Scholar
Picard, P. (1965) Sur les modèles stochastiques logistiques en démographie. Ann. Inst. H. Poincaré B2, 151–172.Google Scholar
Pearson, C. E. (1983) Handbook of Applied Mathematics. Van Nostrand–Reinhold, Princeton, NJ.Google Scholar
Ross, S. (1983) Stochastic Processes. Wiley, New York.Google Scholar
Sanders, J. L. (1971) Quantitative guidelines for communicable disease control programs. Biometrics 27, 883–893.Google Scholar
Weiss, G. W. and Dishon, J. (1971) On the asymptotic behavior of the stochastic and deterministic models of an epidemic. Math. Biosci. 11, 261–265.CrossRefGoogle Scholar