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Error bounds for deterministic approximations to Markov processes, with applications to epidemic models

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

J. Gani  and
Sid Yakowitz
J. Gani*
Affiliation:
The Australian National University
Sid Yakowitz*
Affiliation:
University of Arizona
*
∗Postal address: Stochastic Analysis Group, School of Mathematical Sciences, The Australian National University, Canberra ACT 2000, Australia.
∗∗Postal address: Systems and Industrial Engineering Department, University of Arizona, Tucson, AZ 85721, USA.
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Abstract

The computer age and the phenomenological complexity of the AIDS/HIV epidemic have engendered a rich profusion of deterministic and stochastic time series models for the development of an epidemic. The present study examines the reliability of deterministic approximations of fundamentally random processes. Through numerical analysis and probabilistic considerations, we derive absolute and simultaneous confidence interval bounding techniques, and offer a practical procedure based on these developments. A heartening aspect of the computational study presented at the close of this paper indicates that when the population size is in the thousands, the deterministic version to the classical logistic epidemic is a good approximation.

Keywords

DYNAMIC POPULATION EPIDEMICSAIDSHIV

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60J05: Discrete-time Markov processes on general state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 4 , December 1995 , pp. 1063 - 1076
Copyright
Copyright © Applied Probability Trust 1995 

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References

Bailey, N. T. J. (1975) The Mathematical Theory of Infectious Diseases and its Applications. Hafner Press, New York.Google Scholar
Billard, L. and Zhao, Z. (1991) Three-stage stochastic epidemic model: an application to AIDS. Math. Biosci. 107, 431–449.Google Scholar
Bucklew, J. (1990) Large Deviation Techniques in Decision, Simulation, and Estimation. Wiley, New York.Google Scholar
Fusaro, R. E., Jewell, N. P., Hauck, K. 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.Google Scholar
Hoeffding, W. (1963) Probability inequalities for the sums of bounded random variables. J. Amer. Statist. Assoc. 58, 13–29.CrossRefGoogle Scholar
Jacquez, J. A. and Simon, C. P. (1993) The stochastic SI model and recruitment and deaths I. Comparison with the closed SIS model. Math. Biosci. 117, 77–125.CrossRefGoogle ScholarPubMed
Kermack, W. O. and Mckendrick, A. G. (1927) A contribution to the mathematical theory of epidemics. Proc. R. Soc. London A115, 700–721.Google Scholar
Kryscio, R. and Lefevre, C, (1989) On the extinction of the SIS stochastic logistic epidemic. J. Appl. Prob. 27, 685–694.CrossRefGoogle Scholar
Kurtz, T. G. (1970) Solutions of ordinary differential equations as limits of pure jump processes. J. Appl. Prob. 7, 49–58.Google Scholar
Kurtz, T. G. (1971) Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J. Appl. Prob. 8, 344–356.Google Scholar
Mode, C. J. (1991) A stochastic model for the development of an AIDS epidemic in a heterosexual population. Math. Biosci. 107, 491–520.Google Scholar
Mode, C. J. and Dietz, K. (1993) On some formulas in a partnership model from the perspective of a semi-Markov process. Preprint.Google Scholar
Mode, C. J. and Salsburg, M. A. (1993) On the formulation and computer implementation of an age-dependent two-sex demographic model. Math. Biosci. 118, 211–240.Google Scholar
Schwager, S., Castillo-Chavez, C. and Hethcote, H. (1989) Statistical and mathematical approaches in HIV/AIDS modelling: A review. In Mathematical and Statistical Approaches to AIDS Epidemiology, ed. Castillo-Chavez, C. Springer-Verlag, New York.Google Scholar
Tan, W. Y. (1993) The chain multinomial models of the HIV epidemiology in homosexual populations. Math. Comput. Modelling 18, 29–72.Google Scholar
Tan, W. Y. and Byers, R. H. (1993) A stochastic model of the HIV epidemic and the HIV infection distribution in a homosexual population. Math. Biosci. 113, 115–143.Google Scholar
Tan, W. Y. and Hsu, H. (1991) Stochastic model for the AIDS epidemic in a homosexual population. In Mathematical Population and Dynamics, eds. Arino, O., Axelrod, D. E. and Kimmel, M. Marcel Dekker, New York.Google Scholar
Tan, W. Y. and Hsu, H. (1989) Some stochastic models of AIDS spread. Statistics in Medicine 8, 121–136.CrossRefGoogle ScholarPubMed
Tan, W. Y. and Tang, S. C. (1993) A stochastic model for the HIV epidemic involving both sexual contact and IV drug use. Math. Comput. Modelling 17, 31–57.Google Scholar
Tan, W. Y. and Tang, S. C. (1994) A general Markov model of the HIV epidemic in populations involving both sexual contact and IV drug use. Math. Comput. Modelling 19, 83–132.Google Scholar
Yakowitz, S. (1995) Computational methods for Markov series with large state spaces, with application to AIDS modelling. Math. Biosci. 127, 99–121.CrossRefGoogle Scholar