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Dynamics of an epidemic in a closed population

Part of: Special processes Physiological, cellular and medical topics Limit theorems Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Åke Svensson
Åke Svensson*
Affiliation:
Stockholm University
*
∗Postal address; Department of Statistics, Stockholm University, 5106 91 Stockholm, Sweden.
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Abstract

A simple model for the intensity of infection during an epidemic in a closed population is studied. It is shown that the size of an epidemic (i.e. the number of persons infected) and the cumulative force of an epidemic (i.e. the amount of infectiousness that has to be avoided by a person that will stay uninfected during the entire epidemic) satisfy an equation of balance. Under general conditions, small deviances from this balance are, in large populations, asymptotically mixed normally distributed. For some special epidemic models the size of an asymptotically large epidemic is asymptotically normally distributed.

Keywords

EPIDEMIC MODELSCOUNTING PROCESSESMARTINGALE LIMIT THEOREMSSIZE DISTRIBUTION

MSC classification

Primary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Secondary: 60G44: Martingales with continuous parameter 60F05: Central limit and other weak theorems 92C99: None of the above, but in this section
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 2 , June 1993 , pp. 303 - 313
Copyright
Copyright © Applied Probability Trust 1993 

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References

Anderson, R. M. (1982) Population Dynamics of Infectious Diseases. Chapman and Hall, London.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
Bailey, N. T. J. (1975) The Mathematical Theory of Infectious Diseases and its Applications. Griffin, London.Google Scholar
Daley, D. J. and Kendall, D. G. (1965) Stochastic rumours. J. Inst. Maths. Applics. 1, 42–55.Google Scholar
Frauenthal, J. C. (1980) Mathematical Modelling in Epidemiology. Springer-Verlag, Berlin.Google Scholar
Jacod, J. and Shiryaev, A. N. (1987) Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin.Google Scholar
Kurtz, T. G. (1981) Approximation of Population Processes. SIAM, Philadelpia.Google Scholar
Lefevre, C. (1990) Stochastic epidemic models for S-I-R infectious diseases: a brief survey of the recent general theory. In Stochastic Processes in Epidemic Theory , ed. Gabriel, J.-P., Lefèvre, C., and Picard, P. Lecture Notes in Biomathematics 86, 1–12, Springer-Verlag, Berlin.Google Scholar
Maki, D. P. and Thompson, M. (1973) Mathematical Models and Applications. Prentice-Hall, Englewood Cliffs, NJ.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
Pittel, B. (1990) On a Daley–Kendall model of random rumours. J. Appl. Prob. 27, 14–27.Google Scholar
Watson, R. (1981) An application of a martingale central limit theorem to the standard epidemic model. Stoch. Proc. Appl. 11, 79–89.Google Scholar
Watson, R. (1988) On the size of a rumour. Stoch. Proc. Appl. 27, 141–149.Google Scholar