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The Early Stage Behaviour of a Stochastic SIR Epidemic with Term-Time Forcing

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Tom Britton  and
Mathias Lindholm
Tom Britton*
Affiliation:
Stockholm University
Mathias Lindholm*
Affiliation:
Stockholm University
*
∗Postal address: Department of Mathematics, Stockholm University, SE-10691, Stockholm, Sweden. Email address: tomb@math.su.se
∗∗Current address: Department of Mathematics, Uppsala University, SE-75106, Uppsala, Sweden. Email address: mathias.lindholm@math.uu.se
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Abstract

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The general stochastic SIR epidemic in a closed population under the influence of a term-time forced environment is considered. An ‘environment’ in this context is any external factor that influences the contact rate between individuals in the population, but is itself unaffected by the population. Here ‘term-time forcing’ refers to discontinuous but cyclic changes in the contact rate. The inclusion of such an environment into the model is done by replacing a single contact rate λ with a cyclically alternating renewal process with k different states denoted {Λ(t)}t≥0. Threshold conditions in terms of R⋆ are obtained, such that R⋆>1 implies that π, the probability of a large outbreak, is strictly positive. Examples are given where π is evaluated numerically from which the impact of the distribution of the time periods that Λ(t) spends in its different states is clearly seen.

Keywords

Stochastic epidemicbranching process in a seasonal environmentseasonal forcingterm-time forcingthresholdconditions

MSC classification

Secondary: 92D30: Epidemiology 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 4 , December 2009 , pp. 975 - 992
Copyright
Copyright © Applied Probability Trust 2009 

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