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The extinction time of a subcritical branching process related to the SIR epidemic on a random graph

Part of: Markov processes Graph theory

Published online by Cambridge University Press:  30 March 2016

Peter Windridge
Peter Windridge*
Affiliation:
Queen Mary University of London
*
∗Current address: HSBC, 8 Canada Square, London, E14 5HQ, UK. Email address: pete@windridge.org.uk
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Abstract

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We give an exponential tail approximation for the extinction time of a subcritical multitype branching process arising from the SIR epidemic model on a random graph with given degrees, where the type corresponds to the vertex degree. As a corollary we obtain a Gumbel limit law for the extinction time, when beginning with a large population. Our contribution is to allow countably many types (this corresponds to unbounded degrees in the random graph epidemic model, as the number of vertices tends to∞). We only require a second moment for the offspring-type distribution featuring in our model.

Keywords

Multitype branching processexponential tail approximationGumbelSIR epidemic

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 92D30: Epidemiology
Secondary: 05C80: Random graphs 60J28: Applications of continuous-time Markov processes on discrete state spaces
Type
Short Communications
Information
Journal of Applied Probability , Volume 52 , Issue 4 , December 2015 , pp. 1195 - 1201
Copyright
Copyright © Applied Probability Trust 2015 

References

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