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On a stochastic model of an epidemic

Published online by Cambridge University Press:  14 July 2016

C. J. Ridler-Rowe
C. J. Ridler-Rowe*
Affiliation:
University College of Swansea, Wales
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Extract

The epidemic model considered here, first given by Bartlett (see for example [2]), provides for the immigration of new susceptibles and infectives, as well as describing the spread of infection to susceptibles already present and the removal of infectives. The epidemic curve, relating the numbers of susceptibles and infectives, has been studied for certain cases by Bartlett [1], Kendall [6] and others, and provides a motivation for the results given here. With the aid of criteria given by Reuter [8], [9], the main question considered is the asymptotic behaviour of the mean duration of the epidemic. The behaviour of the limits of the transition probabilities pij(t) as t → ∞ is also investigated.

Type
Research Papers
Information
Journal of Applied Probability , Volume 4 , Issue 1 , November 1967 , pp. 19 - 33
Copyright
Copyright © Applied Probability Trust 

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References

[1] Bartlett, M. S. (1956) Deterministic and stochastic models for recurrent epidemics. Proc. Third Berkeley Symposium on Mathematical Statistics and Probability, U. of California Press 4, 81–109.Google Scholar
[2] Bartlett, M. S. (1960) Some stochastic models in ecology and epidemiology. Contributions to Probability and Statistics. (Volume dedicated to Professor Harold Hotelling) Stanford U. P., Stanford 89–96.Google Scholar
[3] Chung, K.L. (1960) Markov Chains with Stationary Transition Probabilities. Springer, Berlin-Göttingen-Heidelberg.Google Scholar
[4] Feller, W. (1940) On the integro-differential equations of purely discontinuous Markoff processes. Trans. Amer. Math. Soc. 48, 488–515; ibid. (1945) 58, 474.Google Scholar
[5] Feller, W. (1957) An Introduction to Probability Theory and its Applications. Vol. 1 (Second edition) John Wiley, New York.Google Scholar
[6] Kendall, D. G. (1956) Deterministic and stochastic epidemics in closed populations. Proc. Third Berkeley Symposium on Mathematical Statistics and Probability, U. of California Press 4, 149–165.Google Scholar
[7] Kendall, D. G. and Reuter, G. E. H. (1957) The calculation of the ergodic projection for Markov chains and processes with a countable infinity of states. Acta. Math. 97. 103–144.Google Scholar
[8] Reuter, G. E. H. (1957) Denumerable Markov processes and the associated contraction semigroups on l. Acta. Math. 97, 1–46.Google Scholar
[9] Reuter, G. E. H. (1961) Competition processes. Proc. Fourth Berkeley Symposium on Mathematical Statistics and Probability, U. of California Press 2, 421–430.Google Scholar