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Asymptotic analysis of the general stochastic epidemic with variable infectious periods

Part of: Limit theorems

Published online by Cambridge University Press:  14 July 2016

A. N. Startsev
A. N. Startsev*
Affiliation:
Institute of Mathematics, Uzbek Academy of Sciences
*
∗ Postal address: Institute of Mathematics, Uzbek Academy of Sciences, Ul. Khodjaeva 29, Akademgorodok, 700143 Tashkent, Uzbekistan.
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Abstract

A generalisation of the classical general stochastic epidemic within a closed, homogeneously mixing population is considered, in which the infectious periods of infectives follow i.i.d. random variables having an arbitrary but specified distribution. The asymptotic behaviour of the total size distribution for the epidemic as the initial numbers of susceptibles and infectives tend to infinity is investigated by generalising the construction of Sellke and reducing the problem to a boundary crossing problem for sums of independent random variables.

Keywords

epidemicssize of epidemicvariable infectious periodlimit theoremsnon-Markovian model

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60F17: Functional limit theorems; invariance principles
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 1 , March 2001 , pp. 18 - 35
Copyright
Copyright © by the Applied Probability Trust 2001 

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References

Bailey, N. T. J. (1975). The Mathematical Theory of Infectious Diseases and its Applications, 2nd edn. Griffin, London.Google Scholar
Ball, F. (1986). A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models. Adv. Appl. Prob. 18, 289–310.CrossRefGoogle Scholar
Ball, F. G., and Barbour, A. D. (1990). Poisson approximation for some epidemic models. J. Appl. Prob. 27, 479–490.CrossRefGoogle Scholar
Borisov, I. S., and Borovkov, A. A. (1987). Second-order approximation of random polygonal lines in the Donsker–Prokhorov invariance principle. Theory Prob. Appl. 31, 179–202.Google Scholar
Daley, D. J. (1990). The size of epidemics with variable infectious periods. Res. Rept SMS-012-90, Australian National University.Google Scholar
Daley, D. J., and Gani, J. (1999). Epidemic Modelling: an Introduction (Cambridge Studies Math. Biol. 15). Cambridge University Press.Google Scholar
Daniels, H. E. (1967). The distribution of the total size of an epidemic. In Proc. 5th. Berkeley Symp. Math. Stat. Prob. University of California Press, Berkeley, pp. 281–293.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Application, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Gani, J. (1965). On the partial differential equation of epidemic theory. I. Biometrika 52, 617–622.Google Scholar
Gani, J. (1967). On the general stochastic epidemic. In Proc. 5th Berkeley Symp. Math. Statist. Prob. University of California Press, Berkeley, pp. 271–279.Google Scholar
Lefèvre, C., and Picard, P. (1990). A non-standard family of polynomials and the final size distribution of Reed–Frost epidemic processes. Adv. Appl. Prob. 22, 25–48.CrossRefGoogle Scholar
Lefèvre, C., and Utev, S. (1995). Poisson approximation for the final state of a generalized epidemic process. Ann. Prob. 23, 1139–1162.CrossRefGoogle Scholar
Ludwig, D. (1974). Stochastic Population Theories (Lecture Notes Biomath. 3). Springer, Berlin.Google Scholar
Martin-Löf, A. (1986). Symmetric sampling procedures, general epidemic processes and their threshold limit theorems. J. Appl. Prob. 23, 265–282.CrossRefGoogle Scholar
Martin-Löf, A. (1998). The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier. J. Appl. Prob. 35, 671–682.Google Scholar
Nagaev, A. V., and Startsev, A. N. (1970). The asymptotic analysis of a stochastic model of an epidemic. Theory Prob. Appl. 15, 98–107.Google Scholar
Picard, P. and Lefèvre, C. (1990). A unified analysis of the final size and severity distribution in collective Reed–Frost epidemic processes. Adv. Appl. Prob. 22, 262–294.CrossRefGoogle Scholar
Scalia-Tomba, G. (1985). Asymptotic final-size distribution for some chain-binomial processes. Adv. Appl. Prob. 17, 477–495.CrossRefGoogle Scholar
Scalia-Tomba, G. (1990). On the asymptotic final size distribution of epidemic in heterogeneous population. In Stochastic Processes in Epidemic Theory (Lecture Notes Biomath. 86), eds Gabriel, J.-P., Lefèvre, C. and Picard, P. Springer, New York.Google Scholar
Sellke, T. (1983). On the asymptotical distribution of the size of a stochastic epidemic. J. Appl. Prob. 20, 390–394.CrossRefGoogle Scholar
Siskind, V. (1965). Solution of the general stochastic epidemic theory. Biometrika 52, 617–622.CrossRefGoogle Scholar
Startsev, A. N. (1971). Limit theorems for the epidemic size in the general stochastic model. In Random Processes and Statistical Decision. Fan, Tashkent, pp. 60–73 (in Russian).Google Scholar
Startsev, A. N. (1988). On approximation conditions of the distribution of the maximum sums of independent random variables. Ann. Acad. Sci. Fennicae, Ser. A I. Math. 13, 269–275.Google Scholar
Startsev, A. N. (1994). On distribution of the first passage time for a class of a two-dimensional Markov random walk. Uzbek Math. J. 4, 60–66.Google Scholar
Startsev, A. N. (1997). On epidemic size distribution in a non-Markovian model. Theory Prob. Appl. 41, 730–740.Google Scholar
Startsev, A. N. and Cha˘, Z. S. (1987). Limit theorems for the epidemic size in a generalized probability model. In Probability Models and Mathematical Statistics, ed. Kh. Sirazhdinov, S. Fan, Tashkent, pp. 92–105 (in Russian).Google Scholar
Von Bahr, B. and Martin-Löf, A. (1980). Threshold limit theorems for some epidemic processes. Adv. Appl. Prob. 12, 319–349.Google Scholar
Wang, J. S. (1977). Gaussian approximation of some closed stochastic epidemic models. J. Appl. Prob. 14, 221–231.CrossRefGoogle Scholar
Watson, R. (1980). On the size distribution for some epidemic models. J. Appl. Prob. 17, 912–921.CrossRefGoogle Scholar
Weiss, G. H. (1965). On the spread of epidemics by carriers. Biometrics 21, 481–490.Google Scholar
Whittle, P. (1955). The outcome of stochastic epidemic. Biometrika 42, 116–122.Google Scholar