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On the asymptotic size and duration of a class of epidemic models

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

Published online by Cambridge University Press:  14 July 2016

Åke Svensson
Åke Svensson*
Affiliation:
Stockholm University
*
∗Postal address: Department of Mathematical Statistics, Stockholm University, S-106 91 Stockholm, Sweden.
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Abstract

Models for epidemic spread of infections are formulated by defining intensities for relevant counting processes. It is assumed that an infected individual passes through k stages of infectivity. The times spent in the different stages are random. Many well-known models for the spread of infections can be described in this way. The models can also be applied to describe other processes of epidemic character (such as models for rumour spreading). Asymptotic results are derived both for the size and for the duration of the epidemic.

Keywords

COUNTING PROCESSESMARTINGALE LIMIT THEOREMSSIZE DISTRIBUTIONASYMPTOTIC DURATION

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 Papers
Information
Journal of Applied Probability , Volume 32 , Issue 1 , March 1995 , pp. 11 - 24
Copyright
Copyright © Applied Probability Trust 1995 

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References

Anderson, D. and Watson, R. (1980) On the spread of a disease with gamma distributed latent and infectious periods. Biometrika 67, 191–198.Google Scholar
Anderson, R. M. (1982) Population Dynamics of Infectious Diseases. Chapman and Hall, London.Google Scholar
Anderson, R. M. and May, R. M. (1992) Infectious Diseases of Humans. Oxford University Press.Google Scholar
Von Bahr, B. and Martin-Löf, A. (1980) Threshold limit theorems for some epidemic processes. Adv. Appl. Prob. 12, 319–349.CrossRefGoogle Scholar
Bailey, N. T. J. (1975) The Mathematical Theory of Infectious Diseases and its Applications. Griffin, London.Google Scholar
Barbour, A. D. (1975) The duration of the closed stochastic epidemic. Biometrika 62, 477–482.CrossRefGoogle Scholar
Becker, N. G. (1989) Analysis of Infectious Disease Data. Chapman and Hall, London.Google Scholar
Frauenthal, J. C. (1980) Mathematical Modeling in Epidemiology. Springer-Verlag, Berlin.Google Scholar
Jacod, J. and Shiryaev, A. N. (1987) Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin.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, pp. 1–12, Springer-Verlag, Berlin.Google Scholar
Liptser, R. Sh. and Shiryaev, A. N. (1989) Theory of Martingales. Kluwer Academic Publishers, Dordrecht.Google Scholar
Svensson, Å. (1993a) On the duration of a Maki-Thompson epidemic. Math. Biosci. 117, 211–220.Google Scholar
Svensson, Å. (1993b) Dynamics of an epidemic in a closed population. Adv. Appl. Prob. 25, 303–313.Google Scholar
Watson, R. (1980) On the size distribution for some epidemic models. J. Appl. Prob. 17, 912–921.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.CrossRefGoogle Scholar