Considering the infection heterogeneity of different types of edges (lines and edges in the triangle in a network), we formulate and analyze an novel SIS model with cluster based mean-field approach for a network. We mainly focus on how network clustering influences network structure and the disease spreading over the network. In networks with double poisson distributions, power law-poisson distribution, poisson-power law distributions and double power law distributions, we find that cluster is positive(the clustering coefficient is increasing on the expected number of triangles) when the average degree of lines is fixed and the moment of triangles is less than some threshold. Once the moment of triangles exceeds that threshold, cluster will become negative(the clustering coefficient is decreasing on the expected number of triangles). For the disease, clustering always increases the basic reproduction number of the disease in networks with whether positive cluster or negative cluster. It is different from existing results that cluster always promotes the disease spread in the homogeneous or heterogeneous network.
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Figure 3.
Transmission in the triangle.
Figure 4.
Relations of the clustering coefficient
Figure 5.
Relations of the basic reproduction number
[1] | K. Chung, Y. Baek, D. Kim, M. Ha and H. Jeong, Generalized epidemic process on modular networks Phys. Rev. E, 89 (2014), 052811, 7pp. doi: 10.1103/PhysRevE.89.052811. |
[2] | E. Coupechoux and M. Lelarge, How clustering affects epidemics in random networks, Advances in Applied Probability, 46 (2014), 985-1008. doi: 10.1017/S0001867800007515. |
[3] | S. N. Dorogovtsev, Clustering of correlated networks Phys. Rev. E, 69 (2004), 027104, 5pp. doi: 10.1103/PhysRevE.69.027104. |
[4] | K. T. D. Eames, Modelling disease spread through random and regular contacts in clustered populations, Theor. Popul. Biol., 73 (2008), 104-111. doi: 10.1016/j.tpb.2007.09.007. |
[5] | J. P. Gleeson, S. Melnik and A. Hackett, How clustering affects the bond percolation threshold in complex networks Phys. Rev. E, 81 (2010), 066114, 10pp. doi: 10.1103/PhysRevE.81.066114. |
[6] | T. House and M. J. Keeling, Insights from unifying modern appoximations to infections on networks, J. R. Soc. Interface, 8 (2011), 67-73. doi: 10.1098/rsif.2010.0179. |
[7] | M. J. Keeling, The effects of local spatial structure on epidemiological invasions, Proc. R. Soc. London, Ser. B, 266 (1999), 859-867. doi: 10.1098/rspb.1999.0716. |
[8] | L. A. Meyers, Contact network epidemilogy: Bond percolation applied to infectious disease prediction and control, Bull. Amer. Math. Soc., 44 (2007), 63-86. doi: 10.1090/S0273-0979-06-01148-7. |
[9] | J. C. Miller, Spread of infectious disease through clustered populations, J. R. Soc. Interface, 6 (2009), 1121-1134. doi: 10.1098/rsif.2008.0524. |
[10] | J. C. Miller, Percolation and epidemics in random clustered networks Phys. Rev. E, 80 (2009), 020901, 4pp. doi: 10.1103/PhysRevE.80.020901. |
[11] | C. Molina and L. Stone, Modelling the spread of diseases in clustered networks, J. Thero. Biol., 315 (2012), 110-118. doi: 10.1016/j.jtbi.2012.08.036. |
[12] | M. E. J. Newman, Spread of epidemic disease on networks Phys. Rev. E, 66 (2002), 016128, 11pp. doi: 10.1103/PhysRevE.66.016128. |
[13] | M. E. J. Newman, The Structure and Function of Complex Networks, SIAM Rev., 45 (2013), 167-256. doi: 10.1137/S003614450342480. |
[14] | M. E. J. Newman, Clustering and preferential attachment in growing networks Phys. Rev. E, 64 (2001), 025102, 13pp. doi: 10.1103/PhysRevE.64.025102. |
[15] | M. E. J. Newman, Random graphs with clustering Phys. Rev. E, 103 (2009), 058701, 5pp. doi: 10.1103/PhysRevLett.103.058701. |
[16] | M. E. J. Newman, Properties of highly clustered networks Phys. Rev. E, 68 (2003), 026121, 7pp. doi: 10.1103/PhysRevE.68.026121. |
[17] | M. E. J. Newman, Power laws, Pareto distributions and Zipf's law, Contemp. Phys., 46 (2005), 323-351. doi: 10.1080/00107510500052444. |
[18] | R. Pastor-Satorras and A. Vespignani, Epidemic dynamics and endemic states in complex networks Phys. Rev. E, 63 (2001), 066117, 8pp. doi: 10.1103/PhysRevE.63.066117. |
[19] | R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, Phys. Rev. Lett., 86 (2001), 3200-3203. doi: 10.1103/PhysRevLett.86.3200. |
[20] | M. A. Serrano and M. Boguñá, Clustering in complex networks. I. General formalism Phys. Rev. E, 74 (2006), 056114, 9pp. doi: 10.1103/PhysRevE.74.056114. |
[21] | M. A. Serrano and M. Boguñá, Clustering in complex networks. Ⅱ. Percolation properties Phys. Rev. E, 74 (2006), 056115, 8pp. doi: 10.1103/PhysRevE.74.056115. |
[22] | B. Szendroi and G. Csányi, Polynomial epidemics and clustering in contact networks, Proc. R. Soc. London, Ser. B: Biological Sciences, 271 (2004), S364-S366. doi: 10.1098/rsbl.2004.0188. |
[23] | P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. |
[24] | E. M. Volz, J. C. Miller, A. Galvani and L. A. Meyers, Effects of heterogeneous and clustered contact patterns on infectious disease dynamics PLOS Comp. Biol. , 7 (2011), e1002042, 13pp. doi: 10.1371/journal.pcbi.1002042. |
[25] | D. J. Watts and S. H. Strogatz, Collective dynamics of "small-world" networks, Nature, 393 (1998), 440-442. doi: 10.1038/30918. |