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Graphs with specified degree distributions, simple epidemics, and local vaccination strategies

Part of: Graph theory Combinatorial probability

Published online by Cambridge University Press:  01 July 2016

Tom Britton ,
Svante Janson  and
Anders Martin-Löf
Tom Britton*
Affiliation:
Stockholm University
Svante Janson*
Affiliation:
Uppsala University
Anders Martin-Löf*
Affiliation:
Uppsala University
*
∗ Postal address: Department of Mathematics, Stockholm University, SE-10691 Stockholm, Sweden.
∗∗∗ Postal address: Department of Mathematics, Uppsala University, PO Box 480, SE-75106 Uppsala, Sweden. Email address: [email protected]
∗ Postal address: Department of Mathematics, Stockholm University, SE-10691 Stockholm, Sweden.
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Abstract

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Consider a random graph, having a prespecified degree distribution F, but other than that being uniformly distributed, describing the social structure (friendship) in a large community. Suppose that one individual in the community is externally infected by an infectious disease and that the disease has its course by assuming that infected individuals infect their not yet infected friends independently with probability p. For this situation, we determine the values of R0, the basic reproduction number, and τ0, the asymptotic final size in the case of a major outbreak. Furthermore, we examine some different local vaccination strategies, where individuals are chosen randomly and vaccinated, or friends of the selected individuals are vaccinated, prior to the introduction of the disease. For the studied vaccination strategies, we determine Rv, the reproduction number, and τv, the asymptotic final proportion infected in the case of a major outbreak, after vaccinating a fraction v.

Keywords

Degree distributionepidemic modelfinal sizelimit theoremnetworksrandom graphvaccination

MSC classification

Primary: 60C05: Combinatorial probability 05C80: Random graphs 92D30: Epidemiology
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 4 , December 2007 , pp. 922 - 948
Copyright
Copyright © Applied Probability Trust 2007 

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