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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 

References

Andersson, H. (1997). Epidemics in a population with social structures. Math. Biosci. 140, 79–84.CrossRefGoogle Scholar
Andersson, H. (1998). Limit theorems for a random graph epidemic model. Ann. Appl. Prob. 8, 1331–1349.CrossRefGoogle Scholar
Andersson, H. (1999). Epidemic models and social networks. Math. Scientist 24, 128–147.Google Scholar
Ball, F. and Neal, P. (2008). Network epidemic models with two levels of mixing. Math. Biosci. 212, 69–87.CrossRefGoogle ScholarPubMed
Ball, F. and Sirl, D. (2012). An SIR epidemic model on a population with random network and household structure, and several types of individuals. Adv. Appl. Prob. 44, 63–86.CrossRefGoogle Scholar
Ball, F., Sirl, D. and Trapman, P. (2009). Threshold behaviour and final outcome of an epidemic on a random network with household structure. Adv. Appl. Prob. 41, 765–796.CrossRefGoogle Scholar
Ball, F., Sirl, D. and Trapman, P. (2010). Analysis of a stochastic SIR epidemic on a random network incorporating household structure. Math. Biosci. 224, 53–73.CrossRefGoogle ScholarPubMed
Ball, F., Sirl, D. and Trapman, P. (2013). Epidemics on random intersection graphs. To appear in Ann. Appl. Prob. Google Scholar
Becker, N. G. and Starczak, D. N. (1998). The effect of random vaccine response on the vaccination coverage required to prevent epidemics. Math. Biosci. 154, 117–135.CrossRefGoogle ScholarPubMed
Britton, T., Janson, S. and Martin-Löf, A. (2007). Graphs with specified degree distributions, simple epidemics, and local vaccination strategies. Adv. Appl. Prob. 39, 922–948.CrossRefGoogle Scholar
Britton, T., Deijfen, M., Lagerås, A. N. and Lindholm, M. (2008). Epidemics on random graphs with tunable clustering. J. Appl. Prob. 45, 743–756.CrossRefGoogle Scholar
Cohen, R., Havlin, S. and ben Avraham, D. (2003). Efficient immunization strategies for computer networks and populations. Phys. Rev. Lett. 91, 247901.CrossRefGoogle ScholarPubMed
Diekmann, O., de Jong, M. C. M. and Metz, J. A. J. (1998). A deterministic epidemic model taking account of repeated contacts between the same individuals. J. Appl. Prob. 35, 448–462.CrossRefGoogle Scholar
Durrett, R. (2007). Random Graph Dynamics. Cambridge University Press.Google Scholar
Halloran, M. E., Haber, M. and Longini, I. M. Jr. (1992). Interpretation and estimation of vaccine efficacy under heterogeneity. Amer. J. Epidemiol. 136, 328–343.CrossRefGoogle ScholarPubMed
Kenah, E. and Robins, J. M. (2007). Second look at the spread of epidemics on networks. Phys. Rev. E 76, 036113.CrossRefGoogle ScholarPubMed
Kiss, I. Z., Green, D. M. and Kao, R. R. (2006). The effect of contact heterogeneity and multiple routes of transmission on final epidemic size. Math. Biosci. 203, 124–136.CrossRefGoogle ScholarPubMed
Mode, C. J. (1971). Multitype Branching Processes. Theory and Applications (Modern Analytic Comput. Meth. Sci. Math. 34). Elsevier, New York.Google Scholar
Newman, M. E. J. (2002). Spread of epidemic disease on networks. Phys. Rev. E 66, 016128.CrossRefGoogle ScholarPubMed
Newman, M. E. J., Strogatz, S. H. and Watts, D. J. (2001). Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64, 026118.CrossRefGoogle ScholarPubMed
Olofsson, P. (1996). Branching processes with local dependencies. Ann. Appl. Prob. 6, 238–268.CrossRefGoogle Scholar
Seneta, E. (1973). Non-Negative Matrices. Halsted Press, New York.Google Scholar