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

  • Richard J. Kryscio (a1) and Claude Lefèvre (a2)

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.

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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.

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

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References

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Barbour, A. D. (1975) The duration of the closed stochastic epidemic. Biometrika 62, 477482.
Bartholomew, D. J. (1976) Continuous time diffusion models with random duration of interest. J. Math. Sociol. 4, 187199.
Bartholomew, D. J. (1982) Stochastic Models for Social Processes. Wiley, New York.
Cavender, J. A. (1978) Quasi-stationary distributions of birth-and-death processes. Adv. Appl. Prob. 10, 570586.
Hethcote, H. W., Yorke, J. A. and Nold, A. (1982) Gonorrhea modeling: a comparison of control methods. Math. Biosci. 58, 93109.
Kurtz, T. G. (1971) Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J. Appl. Prob. 8, 344356.
Mandl, P. (1960) On the asymptotic behavior of probabilities within classes of states of a homogeneous Markov process (in Russian). Casopis Pest. Mat. 85, 448456.
Norden, R. H. (1982) On the distribution of the time to extinction in the stochastic logistic population model. Adv. Appl. Prob. 14, 687708.
Oppenheim, I., Shuler, K. E. and Weiss, G. H. (1977) Stochastic theory of nonlinear rate processes with multiple stationary states. Physica A, 191214.
Picard, P. (1965) Sur les modèles stochastiques logistiques en démographie. Ann. Inst. H. Poincaré B2, 151172.
Pearson, C. E. (1983) Handbook of Applied Mathematics. Van Nostrand–Reinhold, Princeton, NJ.
Ross, S. (1983) Stochastic Processes. Wiley, New York.
Sanders, J. L. (1971) Quantitative guidelines for communicable disease control programs. Biometrics 27, 883893.
Weiss, G. W. and Dishon, J. (1971) On the asymptotic behavior of the stochastic and deterministic models of an epidemic. Math. Biosci. 11, 261265.

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