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Bounding basic characteristics of spatial epidemics with a new percolation model

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Ronald Meester  and
Pieter Trapman
Ronald Meester*
Affiliation:
VU University Amsterdam
Pieter Trapman*
Affiliation:
VU University Amsterdam and University Medical Center Utrecht
*
∗ Postal address: Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands.
∗∗ Current address: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden. Email address: ptrapman@math.su.se
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Abstract

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We introduce a new 1-dependent percolation model to describe and analyze the spread of an epidemic on a general directed and locally finite graph. We assign a two-dimensional random weight vector to each vertex of the graph in such a way that the weights of different vertices are independent and identically distributed, but the two entries of the vector assigned to a vertex need not be independent. The probability for an edge to be open depends on the weights of its end vertices, but, conditionally on the weights, the states of the edges are independent of each other. In an epidemiological setting, the vertices of a graph represent the individuals in a (social) network and the edges represent the connections in the network. The weights assigned to an individual denote its (random) infectivity and susceptibility, respectively. We show that one can bound the percolation probability and the expected size of the cluster of vertices that can be reached by an open path starting at a given vertex from above by the corresponding quantities for independent bond percolation with a certain density; this generalizes a result of Kuulasmaa (1982). Many models in the literature are special cases of our general model.

Keywords

Dependent percolationepidemics

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 92D30: Epidemiology
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 2 , June 2011 , pp. 335 - 347
Copyright
Copyright © Applied Probability Trust 2011 

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
Balister, P., Bollobás, B. and Walters, M. (2005). Continuum percolation with steps in the square or the disk. Random Structures Algorithms 26, 392–403.CrossRefGoogle 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
Becker, N. G. and Utev, S. (2002). Protective vaccine efficacy when vaccine response is random. Biom. J. 44, 29–42.3.0.CO;2-8>CrossRefGoogle 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.CrossRefGoogle Scholar
Chayes, L. and Schonmann, R. H. (2000). Mixed percolation as a bridge between site and bond percolation. Ann. Appl. Prob. 10, 1182–1196.CrossRefGoogle Scholar
Chung, F. and Lu, L. (2002), Connected components in random graphs with given expected degree sequences. Ann. Combinatorics 6, 125–145.CrossRefGoogle Scholar
Cox, J. T. and Durrett, R. (1988). Limit theorems for the spread of epidemics and forest fires. Stoch. Process. Appl. 30, 171–191.CrossRefGoogle Scholar
Diekmann, O. and Heesterbeek, J. A. P. (2000). Mathematical Epidemiology of Infectious Diseases. John Wiley, Chichester.Google Scholar
Grimmett, G. (1999). Percolation, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, London.Google Scholar
Kenah, E. and Robins, J. M. (2007). Second look at the spread of epidemics on networks. Phys. Rev. E 76, 036113, 12 pp.CrossRefGoogle ScholarPubMed
Kuulasmaa, K. (1982). The spatial general epidemic and locally dependent random graphs. J. Appl. Prob. 19, 745–758.CrossRefGoogle Scholar
Miller, J. C. (2008). Bounding the size and probability of epidemics on networks. J. Appl. Prob. 45, 498–512.CrossRefGoogle Scholar
Newman, M. E. J. (2002). Spread of epidemic disease on networks. Phys. Rev. E 66, 016128, 11 pp.CrossRefGoogle ScholarPubMed
Norros, I. and Reittu, H. (2006). On a conditionally Poissonian graph process. Adv. Appl. Prob. 38, 59–75.CrossRefGoogle Scholar
Trapman, P. (2007). On analytical approaches to epidemics on networks. Theoret. Pop. Biol. 71, 160–173.CrossRefGoogle ScholarPubMed
Watts, C. H. and May, R. M. (1992). The influence of concurrent partnerships on the dynamics of HIV/AIDS. Math. Biosci. 108, 89–104.CrossRefGoogle ScholarPubMed
Wierman, J. C. (1994). Substitution method critical probability bounds for the square lattice site percolation model. Combinatorics Prob. Comput. 4, 181–188.CrossRefGoogle Scholar