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Threshold limit theorems for some epidemic processes

Published online by Cambridge University Press:  01 July 2016

Bengt Von Bahr  and
Anders Martin-Löf
Bengt Von Bahr*
Affiliation:
University of Stockholm
Anders Martin-Löf*
Affiliation:
University of Stockholm
*
∗Postal address: Institutet för Försäkringsmatematik och Matematisk Statistik vid Stockholms Universitet, Hagagatan 23, Box 6701, 113 85 Stockholm, Sweden.
∗Postal address: Institutet för Försäkringsmatematik och Matematisk Statistik vid Stockholms Universitet, Hagagatan 23, Box 6701, 113 85 Stockholm, Sweden.
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Abstract

The Reed–Frost model for the spread of an infection is considered and limit theorems for the total size, T, of the epidemic are proved in the limit when n, the initial number of healthy persons, is large and the probability of an encounter between a healthy and an infected person per time unit, p, is λ/n. It is shown that there is a critical threshold λ = 1 in the following sense, when the initial number of infected persons, m, is finite: If λ ≦ 1, T remains finite and has a limit distribution which can be described. If λ > 1 this is still true with a probability σm < 1, and with probability 1 – σmT is close to n(1 – σ) and has an approximately Gaussian distribution around this value. When m → ∞ also, only the Gaussian part of the limit distribution is obtained. A randomized version of the Reed–Frost model is also considered, and this allows the same result to be proved for the Kermack–McKendrick model. It is also shown that the limit theorem can be used to study the number of connected components in a random graph, which can be considered as a crude description of a polymerization process. In this case polymerization takes place when λ > 1 and not when λ ≦ 1.

Keywords

EPIDEMIC MODELLIMIT THEOREMCRITICAL THRESHOLDBRANCHING PROCESSRANDOM GRAPHPOLYMERIZATIONMARTINGALELAPLACE APPROXIMATION
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 2 , June 1980 , pp. 319 - 349
Copyright
Copyright © Applied Probability Trust 1980 

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References

Bailey, N. T. J. (1975) The Mathematical Theory of Infectious Diseases and its Applications. Griffin, London.Google Scholar
Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, London.Google Scholar
Kendall, D. (1956) Deterministic and stochastic epidemics in closed populations. Proc. 3rd Berkeley Symp. Math. Statist. Prob. 4, 149–165.Google Scholar
Ludwig, D. (1974) Stochastic Population Theories. Lecture Notes in Biomathematics 3, Springer–Verlag, Berlin.CrossRefGoogle Scholar
Martin-Löf, A. (1973) Mixing properties, differentiability of the free energy and the central limit theorem for a pure phase in the Ising model at low temperature. Commun. Math. Phys. 32, 75–92.CrossRefGoogle 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.CrossRefGoogle Scholar
Stepanov, V. E. (1970) On the probability of connectedness of a random graph. Theory Prob. Appl. 15, 55–67.CrossRefGoogle Scholar