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Threshold phenomena in epidemic theory


Barbour, A D (1994). Threshold phenomena in epidemic theory. In: Kelly, F P. Probability, statistics and optimisation. Chichester: Wiley, 101-116.

Abstract

The threshold theorem for deterministic epidemics in mixing populations can usually be rewritten in such a form that a large epidemic results from trace infection if and only if $R_0>1$, where $R_0$ can be interpreted as a basic reproduction ratio for an associated population model. The Whittle stochastic threshold theorem replaces certainty with probability: if $R_0\leq 1$, a large epidemic is highly unlikely to result from the introduction of one or two infectives, whereas, if $R_0>1$, the probability of having a significant epidemic is no longer trivial. In this paper, the Whittle approximation to a model for parasitic infection in a mixing population is analysed. A feature of the model is that $R_0$ is well defined, but for certain parameter values the threshold is not at $R_0=1$. Thus to have $R_0=1$ as threshold for epidemics in mixing populations is by no means a universal rule. A related birth and death process with drift is also investigated.

Abstract

The threshold theorem for deterministic epidemics in mixing populations can usually be rewritten in such a form that a large epidemic results from trace infection if and only if $R_0>1$, where $R_0$ can be interpreted as a basic reproduction ratio for an associated population model. The Whittle stochastic threshold theorem replaces certainty with probability: if $R_0\leq 1$, a large epidemic is highly unlikely to result from the introduction of one or two infectives, whereas, if $R_0>1$, the probability of having a significant epidemic is no longer trivial. In this paper, the Whittle approximation to a model for parasitic infection in a mixing population is analysed. A feature of the model is that $R_0$ is well defined, but for certain parameter values the threshold is not at $R_0=1$. Thus to have $R_0=1$ as threshold for epidemics in mixing populations is by no means a universal rule. A related birth and death process with drift is also investigated.

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Additional indexing

Item Type:Book Section, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:1994
Deposited On:09 Apr 2010 09:25
Last Modified:29 Jul 2020 19:50
Publisher:Wiley
Series Name:Wiley Series in Probability and Mathematical Statistics
ISBN:0-471-94829-2
Additional Information:This is a preprint of an article published in [Barbour, A. D. Threshold phenomena in epidemic theory. Probability, statistics and optimisation, 101--116], Wiley Series in Probability and Mathematical Statistics Copyright © 1994
OA Status:Green
Related URLs:http://www.ams.org/mathscinet-getitem?mr=1320745

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