Abstract
Algorithms for the S-I-R epidemic with an initial population with m infectives and n susceptibles are examined. We propose efficient algorithms for the distributions of the total and the maximum size of the epidemic, and for the joint distribution of the maximum and the time of its occurrence. We also discuss the joint distribution of the sizes of the epidemic at two epochs. By studying the Markov chain describing an indefinite replication of the epidemic, we obtain new descriptors of the process of infections.
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References
N. T. J. Bailey The total size of a general stochastic epidemic. Biometrika 40:177–185, 1953.
N. T. J. Bailey The Elements of Stochastic Processes with Applications to the Natural Sciences. John Wiley & Sons, Inc., 1964.
N. T. J. Bailey The Mathematical Theory of Infectious Diseases and its Applications. Charles Griffin Si Company Ltd., 1975.
H.E. Daniels The maximum size of a closed epidemic.Ad-vances in Applied Probability 6(4):607–621, 1974.
H. E. Daniels The time of occurrence of the maximum of a closed epidemic. In J. P. Gabriel, C. Lefèvre, and P. Picard, editors Stochastic Processes in Epidemic Theory pages 129–136, Luminy, France, October 1988. Springer-Verlag 1990.
J. Gani On the general stochastic epidemic. In Proc. Fifth Berkeley Symp. Math. Statist. f& Prob. pages 271–279. University of California Press, 1966.
J. Gani and P. Purdue Matrix-geometric methods for the general stochastic epidemic. IMA Journal of Mathematics Applied in Medicine P& Biology 1(4):333–342, 1984.
D. M. Lucantoni The BMAP/G/1 queue: a tutorial. In L. Donatiello and R. Nelson, editors Performance Evaluation of Computer and Communication Systems Joint Totorial Papers of Performance ‘83 and Sigmetrics ‘83 pages 330–358. Springer-Verlag Berlin Heidelberg, 1993.
M. F. Neuts Matrix-geometric Solutions in Stochastic Models: An Algoritmic Approach. The Johns Hopkins University Press, 1981; Dover Publications, Inc., New York, 1995.
M. F. Neuts Models based on the Markovian arrival processes. IEICE Transactions On Communications E75-B(12):125–565, 1992.
M. F. Neuts Algorithmic Probability: A Collection of Problems. Chapman & Hall, New York, New York, 1995.
I. W. Saunders A model for myxomatosis. Mathematical Biosciences 48(1):1–15, 1980.
P. Whittle The outcome of a stochastic epidemic — a note on Bailey’s paper. Biometrika 42:116–122, 1955.
T. Williams An algebraic proof of the threshold theorem for the general stochastic epidemic. Advances in Applied Probability 3(2):223–224, 1971.
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© 1996 Springer Science+Business Media New York
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Neuts, M.F., Li, J.M. (1996). An Algorithmic Study of S-I-R Stochastic Epidemic Models. In: Heyde, C.C., Prohorov, Y.V., Pyke, R., Rachev, S.T. (eds) Athens Conference on Applied Probability and Time Series Analysis. Lecture Notes in Statistics, vol 114. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0749-8_21
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DOI: https://doi.org/10.1007/978-1-4612-0749-8_21
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