Search Advanced

Mathematical Biosciences and Engineering

2014, Volume 11, Issue 6: 1295-1317. doi: 10.3934/mbe.2014.11.1295
Previous Article Next Article

Epidemic models for complex networks with demographics

  • 1.
    Complex Systems Research Center, Shanxi University, Taiyuan, Shanxi 030051
  • 2.
    LAMPS and CDM, Department of Mathematics and Statistics, York University, Toronto, ON, M3J1P3
  • Received: 01 March 2014 Accepted: 29 June 2018 Published: 01 September 2014

  • MSC : Primary: 58F15, 58F17; Secondary: 53C35.

  • In this paper, we propose and study network epidemic models withdemographics for disease transmission. We obtain the formula of thebasic reproduction number $R_{0}$ of infection for an SIS model withbirths or recruitment and death rate. We prove that if $R_{0}\leq1$,infection-free equilibrium of SIS model is globally asymptoticallystable; if $R_{0}>1$, there exists a unique endemic equilibrium whichis globally asymptotically stable. It is also found thatdemographics has great effect on basic reproduction number $R_{0}$.Furthermore, the degree distribution of population varies with timebefore it reaches the stationary state.

    Citation: Zhen Jin, Guiquan Sun, Huaiping Zhu. Epidemic models for complex networks with demographics[J]. Mathematical Biosciences and Engineering, 2014, 11(6): 1295-1317. doi: 10.3934/mbe.2014.11.1295

    Related Papers:

    [1] Juan Wang, Chunyang Qin, Yuming Chen, Xia Wang . Hopf bifurcation in a CTL-inclusive HIV-1 infection model with two time delays. Mathematical Biosciences and Engineering, 2019, 16(4): 2587-2612. doi: 10.3934/mbe.2019130
    [2] Bing Li, Yuming Chen, Xuejuan Lu, Shengqiang Liu . A delayed HIV-1 model with virus waning term. Mathematical Biosciences and Engineering, 2016, 13(1): 135-157. doi: 10.3934/mbe.2016.13.135
    [3] Xiaohong Tian, Rui Xu, Jiazhe Lin . Mathematical analysis of an age-structured HIV-1 infection model with CTL immune response. Mathematical Biosciences and Engineering, 2019, 16(6): 7850-7882. doi: 10.3934/mbe.2019395
    [4] A. M. Elaiw, N. H. AlShamrani . Stability of HTLV/HIV dual infection model with mitosis and latency. Mathematical Biosciences and Engineering, 2021, 18(2): 1077-1120. doi: 10.3934/mbe.2021059
    [5] Jinhu Xu, Yicang Zhou . Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay. Mathematical Biosciences and Engineering, 2016, 13(2): 343-367. doi: 10.3934/mbe.2015006
    [6] Haitao Song, Weihua Jiang, Shengqiang Liu . Virus dynamics model with intracellular delays and immune response. Mathematical Biosciences and Engineering, 2015, 12(1): 185-208. doi: 10.3934/mbe.2015.12.185
    [7] Patrick W. Nelson, Michael A. Gilchrist, Daniel Coombs, James M. Hyman, Alan S. Perelson . An Age-Structured Model of HIV Infection that Allows for Variations in the Production Rate of Viral Particles and the Death Rate of Productively Infected Cells. Mathematical Biosciences and Engineering, 2004, 1(2): 267-288. doi: 10.3934/mbe.2004.1.267
    [8] Shingo Iwami, Shinji Nakaoka, Yasuhiro Takeuchi . Mathematical analysis of a HIV model with frequency dependence and viral diversity. Mathematical Biosciences and Engineering, 2008, 5(3): 457-476. doi: 10.3934/mbe.2008.5.457
    [9] Yu Yang, Gang Huang, Yueping Dong . Stability and Hopf bifurcation of an HIV infection model with two time delays. Mathematical Biosciences and Engineering, 2023, 20(2): 1938-1959. doi: 10.3934/mbe.2023089
    [10] Urszula Foryś, Jan Poleszczuk . A delay-differential equation model of HIV related cancer--immune system dynamics. Mathematical Biosciences and Engineering, 2011, 8(2): 627-641. doi: 10.3934/mbe.2011.8.627
  • In this paper, we propose and study network epidemic models withdemographics for disease transmission. We obtain the formula of thebasic reproduction number $R_{0}$ of infection for an SIS model withbirths or recruitment and death rate. We prove that if $R_{0}\leq1$,infection-free equilibrium of SIS model is globally asymptoticallystable; if $R_{0}>1$, there exists a unique endemic equilibrium whichis globally asymptotically stable. It is also found thatdemographics has great effect on basic reproduction number $R_{0}$.Furthermore, the degree distribution of population varies with timebefore it reaches the stationary state.


    [1] Oxford University Press, Oxford, 1992.
    [2] Science, 286 (1999), 509-511.
    [3] Journal of Theoretical Biology, 235 (2005), 275-288.
    [4] J. Phys. A: Math. Theor., 40 (2007), 8607-8619.
    [5] e-print cond-mat/0301149, (2003).
    [6] J. Math. Biol., 28 (1990), 257-270.
    [7] in Mathematical Population Dynamics: Analysis of Heterogeneity (eds. O. Arino, D. Axelrod, M. Kimmel and M. Langlais), Theory of Epidemics, 1, Wuerz, Winnipeg, 1993, 33-50.
    [8] Applied Mathematics and Computation, 197 (2008), 345-357.
    [9] J. Math. Biol., 30 (1992), 717-731.
    [10] Proc. Natl. Acad. Sci. U.S.A., 101 (2004), 15124.
    [11] J. Math. Anal. Appl., 308 (2005), 343-364.
    [12] Phys. Rev. E, 69 (2004), 066105.
    [13] J. R. Soc. Interface, 2 (2005), 295-307.
    [14] Princeton University Press, 2007.
    [15] Proc. R. Soc. A, 115 (1927), 700-711.
    [16] Mathematical Biosciences, 203 (2006), 124-136.
    [17] Bulletin of Mathematical Biology, 71 (2009), 888-905.
    [18] Physica D, 238 (2009), 370-378.
    [19] World Scientific, 2009.
    [20] Phys. Rev. E, 64 (2001), 066112.
    [21] Eur. Phys. J. B, 26 (2002), 521-529.
    [22] Phys. Rev. E, 70 (2004), 030902.
    [23] Phys. Rev. E, 63 (2001), 066117.
    [24] Phys. Rev. Let., 86 (2001), 3200.
    [25] IMA Journal of Mathematics Applied in Medicine & Biology, 13 (1996), 245-257.
    [26] Phys. Rev. E, 77 (2008), 066101.
    [27] SIAM J. Appl. Math., 46 (1986), 368-375.
    [28] Rocky Mountain J. Math., 24 (1994), 351-380.
    [29] Mathematical Biosciences, 180 (2002), 29-48.
    [30] Siam J. Appl. Math., 68 (2008), 1495-1502.
    [31] Mathematical Biosciences, 190 (2004), 97-112.
    [32] Springer-Verlag, New York, 2003.
    [33] Canad. Appl. Math. Quart., 4 (1996), 421-444.

  • This article has been cited by:

    1. Fathalla A. Rihan, Duaa H. Abdel Rahman, Delay differential model for tumour–immune dynamics with HIV infection of CD4+T-cells, 2013, 90, 0020-7160, 594, 10.1080/00207160.2012.726354
    2. Jorge Duarte, Cristina Januário, Nuno Martins, C. Correia Ramos, Carla Rodrigues, Josep Sardanyés, Optimal homotopy analysis of a chaotic HIV-1 model incorporating AIDS-related cancer cells, 2018, 77, 1017-1398, 261, 10.1007/s11075-017-0314-0
    3. Jie Lou, Tommaso Ruggeri, A time delay model about AIDS-related cancer: equilibria, cycles and chaotic behavior, 2007, 56, 0035-5038, 195, 10.1007/s11587-007-0013-6
    4. Qingzhi Wen, Jie Lou, The global dynamics of a model about HIV-1 infection in vivo, 2009, 58, 0035-5038, 77, 10.1007/s11587-009-0048-y
    5. Parvaiz Ahmad Naik, Kolade M. Owolabi, Mehmet Yavuz, Jian Zu, Chaotic dynamics of a fractional order HIV-1 model involving AIDS-related cancer cells, 2020, 140, 09600779, 110272, 10.1016/j.chaos.2020.110272
    6. Joseph Páez Chávez, Burcu Gürbüz, Carla M.A. Pinto, The effect of aggressive chemotherapy in a model for HIV/AIDS-cancer dynamics, 2019, 75, 10075704, 109, 10.1016/j.cnsns.2019.03.021
    7. F.A. Rihan, D.H. Abdel Rahman, S. Lakshmanan, A.S. Alkhajeh, A time delay model of tumour–immune system interactions: Global dynamics, parameter estimation, sensitivity analysis, 2014, 232, 00963003, 606, 10.1016/j.amc.2014.01.111
    8. Ana R. M. Carvalho, Carla M. A. Pinto, New developments on AIDS-related cancers: The role of the delay and treatment options, 2018, 41, 01704214, 8915, 10.1002/mma.4657
    9. A delay-differential equation model of HIV related cancer--immune system dynamics, 2011, 8, 1551-0018, 627, 10.3934/mbe.2011.8.627
    10. Zulqurnain Sabir, Salem Ben Said, Qasem Al-Mdallal, Artificial intelligent solvers for the HIV-1 system including AIDS based on the cancer cells, 2023, 26673053, 200309, 10.1016/j.iswa.2023.200309
    11. José García Otero, Mariusz Bodzioch, Juan Belmonte-Beitia, On the dynamics and optimal control of a mathematical model of neuroblastoma and its treatment: Insights from a mathematical model, 2024, 34, 0218-2025, 1235, 10.1142/S0218202524500210
    12. Zulqurnain Sabir, Sahar Dirani, Sara Bou Saleh, Mohamad Khaled Mabsout, Adnène Arbi, A Novel Radial Basis and Sigmoid Neural Network Combination to Solve the Human Immunodeficiency Virus System in Cancer Patients, 2024, 12, 2227-7390, 2490, 10.3390/math12162490
    13. Haneche Nabil, Tayeb Hamaizia, A three-dimensional discrete fractional-order HIV-1 model related to cancer cells, dynamical analysis and chaos control, 2024, 4, 2791-8564, 256, 10.53391/mmnsa.1484994
    14. Beiqi Li, Li Zu, Quan Wang, Jiangdi Li, Dynamic behavioral analysis of a stochastic HIV model with cancer cells, 2025, 00160032, 107882, 10.1016/j.jfranklin.2025.107882
  • Reader Comments
  • © 2014 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematical Biosciences and Engineering
4.4

Metrics

Article views(4177) PDF downloads(799) Cited by(47)

Preview PDF
Download XML
Export Citation
Article outline
Abstract
References

Other Articles By Authors

Related pages

Tools

Copyright © AIMS Press

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog