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Acquaintance Vaccination in an Epidemic on a Random Graph with Specified Degree Distribution

Part of: Graph theory Markov processes

Published online by Cambridge University Press:  30 January 2018

Frank Ball  and
David Sirl
Frank Ball*
Affiliation:
University of Nottingham
David Sirl*
Affiliation:
Loughborough University
*
∗ Postal address: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK. Email address: frank.ball@nottingham.ac.uk
∗∗ Postal address: Mathematics Education Centre, Loughborough University, Loughborough LE11 3TU, UK. Email address: d.sirl@lboro.ac.uk
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Abstract

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We consider a stochastic SIR (susceptible → infective → removed) epidemic on a random graph with specified degree distribution, constructed using the configuration model, and investigate the ‘acquaintance vaccination’ method for targeting individuals of high degree for vaccination. Branching process approximations are developed which yield a post-vaccination threshold parameter, and the asymptotic (large population) probability and final size of a major outbreak. We find that introducing an imperfect vaccine response into the present model for acquaintance vaccination leads to sibling dependence in the approximating branching processes, which may then require infinite type spaces for their analysis and are generally not amenable to numerical calculation. Thus, we propose and analyse an alternative model for acquaintance vaccination, which avoids these difficulties. The theory is illustrated by a brief numerical study, which suggests that the two models for acquaintance vaccination yield quantitatively very similar disease properties.

Keywords

Branching processepidemic processfinal sizenetworkrandom graphthreshold behaviourvaccination

MSC classification

Primary: 92D30: Epidemiology
Secondary: 05C80: Random graphs 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013 , pp. 1147 - 1168
Copyright
© Applied Probability Trust 

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