zbMATH — the first resource for mathematics

Bounding basic characteristics of spatial epidemics with a new percolation model. (English) Zbl 1221.60143
Summary: 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 K. Kuulasmaa [J. Appl. Probab. 19, 745–758 (1982; Zbl 0509.60094)]. Many models in the literature are special cases of our general model.

60K35 Interacting random processes; statistical mechanics type models; percolation theory
92D30 Epidemiology
Full Text: DOI Euclid arXiv
[1] Andersson, H. (1999). Epidemic models and social networks. Math. Scientist. 24, 128-147. · Zbl 0951.92022
[2] Andersson, H. and Britton, T. (2000). Stochastic Epidemic Models and Their Statistical Analysis . (Lecture Notes Statist. 151 ), Springer, New York. · Zbl 0951.92021
[3] Balister, P., Bollobás, B. and Walters, M. (2005). Continuum percolation with steps in the square or the disk. Random Structures Algorithms 26, 392-403. · Zbl 1072.60083
[4] 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. · Zbl 0930.92016
[5] Becker, N. G. and Utev, S. (2002). Protective vaccine efficacy when vaccine response is random. Biom. J. 44, 29-42. · Zbl 0989.62059
[6] Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31, 3-122. · Zbl 1123.05083
[7] Britton, T., Deijfen, M. and Martin-Löf, A. (2006). Generating simple random graphs with prescribed degree distribution. J. Statist. Phys. 124, 1377-1397. · Zbl 1106.05086
[8] Chayes, L. and Schonmann, R. H. (2000). Mixed percolation as a bridge between site and bond percolation. Ann. Appl. Prob. 10, 1182-1196. · Zbl 1073.60539
[9] Chung, F. and Lu, L. (2002), Connected components in random graphs with given expected degree sequences. Ann. Combinatorics 6, 125-145. · Zbl 1009.05124
[10] Cox, J. T. and Durrett, R. (1988). Limit theorems for the spread of epidemics and forest fires. Stoch. Process. Appl. 30, 171-191. · Zbl 0667.92016
[11] Diekmann, O. and Heesterbeek, J. A. P. (2000). Mathematical Epidemiology of Infectious Diseases . John Wiley, Chichester. · Zbl 0997.92505
[12] Grimmett, G. (1999). Percolation, 2nd edn. Springer, Berlin. · Zbl 0926.60004
[13] Jagers, P. (1975). Branching Processes with Biological Applications . John Wiley, London. · Zbl 0356.60039
[14] Kenah, E. and Robins, J. M. (2007). Second look at the spread of epidemics on networks. Phys. Rev. E 76, 036113, 12pp.
[15] Kuulasmaa, K. (1982). The spatial general epidemic and locally dependent random graphs. J. Appl. Prob. 19, 745-758. · Zbl 0509.60094
[16] Miller, J. C. (2008). Bounding the size and probability of epidemics on networks. J. Appl. Prob. 45, 498-512. · Zbl 1145.92029
[17] Newman, M. E. J. (2002). Spread of epidemic disease on networks. Phys. Rev. E 66, 016128, 11pp.
[18] Norros, I. and Reittu, H. (2006). On a conditionally Poissonian graph process. Adv. Appl. Prob. 38, 59-75. · Zbl 1096.05047
[19] Trapman, P. (2007). On analytical approaches to epidemics on networks. Theoret. Pop. Biol. 71, 160-173. · Zbl 1118.92055
[20] Watts, C. H. and May, R. M. (1992). The influence of concurrent partnerships on the dynamics of HIV/AIDS. Math. Biosci. 108, 89-104. · Zbl 1353.92104
[21] Wierman, J. C. (1994). Substitution method critical probability bounds for the square lattice site percolation model. Combinatorics Prob. Comput. 4, 181-188. · Zbl 0835.60089
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.