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Epidemics on Random Graphs with Tunable Clustering

Part of: Graph theory

Published online by Cambridge University Press:  14 July 2016

Tom Britton ,
Maria Deijfen ,
Andreas N. Lagerås  and
Mathias Lindholm
Tom Britton*
Affiliation:
Stockholm University
Maria Deijfen*
Affiliation:
Stockholm University
Andreas N. Lagerås*
Affiliation:
Stockholm University
Mathias Lindholm*
Affiliation:
Stockholm University
*
∗Postal address: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden.
∗Postal address: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden.
∗Postal address: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden.
∗Postal address: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden.
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Abstract

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In this paper a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. We investigate how these quantities vary with the clustering in the graph and find that, as the clustering increases, the epidemic threshold decreases. The network is modeled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if there is at least one group that they are both members of.

Keywords

Epidemicsrandom graphclusteringbranching processepidemic threshold

MSC classification

Secondary: 92D30: Epidemiology 05C80: Random graphs
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 3 , September 2008 , pp. 743 - 756
Copyright
Copyright © Applied Probability Trust 2008 

References

Andersson, H. (1998). Limit theorems for a random graph epidemic model. Ann. Appl. Prob. 8, 1331–1349.Google Scholar
Andersson, H. (1999). Epidemic models and social networks. Math. Scientist 24, 128–147.Google Scholar
Andersson, H. and Britton, T. (2000). Epidemic Models and Their Statistical Analysis. Springer, Berlin.CrossRefGoogle Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.CrossRefGoogle Scholar
Ball, F. and Clancy, D. (1993). The final size and severity of a generalized stochastic multi-type epidemic model. Adv. Appl. Prob. 25, 721–736.Google Scholar
Ball, F., Mollison, D. and Scalia-Tomba, G. (1997). Epidemics with two levels of mixing. Ann. Appl. Prob. 7, 46–89.Google Scholar
Behrisch, M. (2007). Component evolution in random intersection graphs. Electron. J. Combinatorics 14, 12 pp.CrossRefGoogle Scholar
Deijfen, M. and Kets, W. (2007). Random intersection graphs with tunable degree distribution and clustering. Submitted.CrossRefGoogle Scholar
Dorogovtsev, S. and Mendes, J. (2003). Evolution of Networks. From Biological Nets to the Internet and WWW. Oxford University Press.CrossRefGoogle Scholar
Durrett, R. (2006). Random Graph Dynamics. Cambridge University Press.Google Scholar
Fill, J., Scheinerman, E. and Singer-Cohen, K. (2000). Random intersection graphs when m=ω(n): an equivalence theorem relating the evolution of the G(n,m,p) and G(n,p) models. Random Structures Algorithms 16, 156–176.Google Scholar
Godehardt, E. and Jaworski, J. (2002). Two models of random intersection graphs for classification. In Exploratory Data Analysis in Empirical Research, eds Schwaiger, M. and Opitz, O., Springer, Berlin, pp. 67–81.Google Scholar
Karoński, M., Scheinerman, E. and Singer-Cohen, K. (1999). On random intersection graphs: the subgraphs problem. Combinatorics Prob. Comput. 8, 131–159.CrossRefGoogle Scholar
Neal, P. (2004). SIR epidemics on Bernoulli graphs. J. Appl. Prob. 40, 779–782.Google Scholar
Neal, P. (2006). Multitype randomized Reed–Frost epidemics and epidemics upon graphs. Ann. Appl. Prob. 16, 1166–1189.CrossRefGoogle Scholar
Newman, M. (2003a). Properties of highly clustered networks. Phys. Rev. E 68, 026121.CrossRefGoogle ScholarPubMed
Newman, M. (2003b). The structure and function of complex networks. SIAM Rev. 45, 167–256.Google Scholar
Newman, M., Barabási, A. and Watts, D. (2006). The Structure and Dynamics of Networks. Princeton University Press.Google Scholar
Singer, K. (1995). Random intersection graphs. , Johns Hopkins University.Google Scholar
Stark, D. (2004). The vertex degree distribution of random intersection graphs. Random Structures Algorithms 24, 249–258.Google Scholar
Trapman, P. (2007). On analytical approaches to epidemics on networks. Theoret. Pop. Biol. 71, 160–173.Google Scholar
Von Bahr, B. and Martin-Löf, A. (1980). Threshold limit theorems for some epidemic processes. Adv. Appl. Prob. 12, 319–349.Google Scholar