
-
Previous Article
Tumor growth dynamics with nutrient limitation and cell proliferation time delay
- DCDS-B Home
- This Issue
-
Next Article
Global dynamics of an age-structured HIV infection model incorporating latency and cell-to-cell transmission
Impacts of cluster on network topology structure and epidemic spreading
1. | Department of Mathematics, North University of China, Taiyuan Shan'xi 030051, China |
2. | Complex Systems Research Center, Shanxi University, Taiyuan Shan'xi 030006, China, and Department of Computer Science and Technology, North University of China, Taiyuan Shan'xi 030051, China |
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.
References:
[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. |
show all references
References:
[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. |






[1] |
Peigen Cao, Fang Li, Siyang Liu, Jie Pan. A conjecture on cluster automorphisms of cluster algebras. Electronic Research Archive, 2019, 27: 1-6. doi: 10.3934/era.2019006 |
[2] |
Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37 |
[3] |
Xianping Wu, Xun Li, Zhongfei Li. A mean-field formulation for multi-period asset-liability mean-variance portfolio selection with probability constraints. Journal of Industrial and Management Optimization, 2018, 14 (1) : 249-265. doi: 10.3934/jimo.2017045 |
[4] |
Inês Cruz, M. Esmeralda Sousa-Dias. Reduction of cluster iteration maps. Journal of Geometric Mechanics, 2014, 6 (3) : 297-318. doi: 10.3934/jgm.2014.6.297 |
[5] |
Valentin Ovsienko, MichaeL Shapiro. Cluster algebras with Grassmann variables. Electronic Research Announcements, 2019, 26: 1-15. doi: 10.3934/era.2019.26.001 |
[6] |
Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595 |
[7] |
Nitu Kumari, Sumit Kumar, Sandeep Sharma, Fateh Singh, Rana Parshad. Basic reproduction number estimation and forecasting of COVID-19: A case study of India, Brazil and Peru. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2021170 |
[8] |
Patrick Gerard, Christophe Pallard. A mean-field toy model for resonant transport. Kinetic and Related Models, 2010, 3 (2) : 299-309. doi: 10.3934/krm.2010.3.299 |
[9] |
Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080 |
[10] |
Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic and Related Models, 2021, 14 (3) : 429-468. doi: 10.3934/krm.2021011 |
[11] |
Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics and Games, 2021, 8 (4) : 445-465. doi: 10.3934/jdg.2021006 |
[12] |
Hélène Hibon, Ying Hu, Shanjian Tang. Mean-field type quadratic BSDEs. Numerical Algebra, Control and Optimization, 2022 doi: 10.3934/naco.2022009 |
[13] |
Octav Cornea and Francois Lalonde. Cluster homology: An overview of the construction and results. Electronic Research Announcements, 2006, 12: 1-12. |
[14] |
Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455 |
[15] |
Illés Horváth, Kristóf Attila Horváth, Péter Kovács, Miklós Telek. Mean-field analysis of a scaling MAC radio protocol. Journal of Industrial and Management Optimization, 2021, 17 (1) : 279-297. doi: 10.3934/jimo.2019111 |
[16] |
Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado. A-priori estimates for stationary mean-field games. Networks and Heterogeneous Media, 2012, 7 (2) : 303-314. doi: 10.3934/nhm.2012.7.303 |
[17] |
Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Mean-field backward stochastic Volterra integral equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1929-1967. doi: 10.3934/dcdsb.2013.18.1929 |
[18] |
Gerasimenko Viktor. Heisenberg picture of quantum kinetic evolution in mean-field limit. Kinetic and Related Models, 2011, 4 (1) : 385-399. doi: 10.3934/krm.2011.4.385 |
[19] |
Franco Flandoli, Enrico Priola, Giovanni Zanco. A mean-field model with discontinuous coefficients for neurons with spatial interaction. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3037-3067. doi: 10.3934/dcds.2019126 |
[20] |
Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. Uniform stability and mean-field limit for the augmented Kuramoto model. Networks and Heterogeneous Media, 2018, 13 (2) : 297-322. doi: 10.3934/nhm.2018013 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]