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

Part of: Combinatorial probability Graph theory

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: svante.janson@math.uu.se
∗ 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 

References

Andersson, H. (1999). Epidemic models and social networks. Math. Scientist 24, 128–147.Google Scholar
Andersson, H. and Britton, T. (2000). Stochastic Epidemic Models and Their Statistical Analysis (Lecture Notes Statist. 151). Springer, New York.CrossRefGoogle Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Bollobás, B. (2001). Random Graphs, 2nd edn. Cambridge University Press.Google Scholar
Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31, 3–122.CrossRefGoogle Scholar
Britton, T., Deijfen, M. and Martin-Löf, A. (2006). Generating simple random graphs with prescribed degree distribution. J. Statist. Phys. 124, 1377–1397.Google Scholar
Cohen, R. Havlin, S. and ben-Avrahan, D. (2003). Efficient immunization strategies for computer networks and populations. Phys. Rev. Lett. 91, 247901.Google Scholar
Gut, A. (2005). Probability: A Graduate Course. Springer, New York.Google Scholar
Janson, S. (2006). The probability that a random multigraph is simple. Available at http://arxiv.org/abs/math/0609802.Google Scholar
Janson, S. (2007). Asymptotic equivalence and contiguity of some random graphs. In preparation.Google Scholar
Janson, S., Łuczak, T. and Ruciński, A. (2000). Random Graphs. John Wiley, New York.CrossRefGoogle Scholar
Janson, S., Knuth, D., Łuczak, T. and Pittel, B. (1994). The birth of the giant component. Random Structures Algorithms 4, 231–358.Google Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.Google Scholar
McKay, B. D. (1985). Asymptotics for symmetric 0–1 matrices with prescribed row sums. Ars Combin. 19, 15–25.Google Scholar
Molloy, M. and Reed, B. (1995). A critical point for random graphs with a given degree sequence. Random Structures Algorithms 6, 161–179.CrossRefGoogle Scholar
Molloy, M. and Reed, B. (1998). The size of the giant component of a random graph with a given degree sequence. Combin. Prob. Comput. 7, 295–305.CrossRefGoogle Scholar
Moore, C. and Newman, M. E. J. (2000). Epidemics and percolation in small world networks. Phys. Rev. E 61, 5678–5682.Google Scholar
Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Rev. 45, 167–256.CrossRefGoogle Scholar
Newman, M. E. J., Strogatz, S. H. and Watts, J. (2001). Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64, 026118.Google Scholar
Scott, J. (2000). Social Network Analysis, A Handbook, 2nd edn. Sage, London.Google Scholar