Research article Special Issues

Global stability for a class of HIV virus-to-cell dynamical model with Beddington-DeAngelis functional response and distributed time delay

  • Received: 03 March 2020 Accepted: 16 June 2020 Published: 29 June 2020
  • A HIV virus-to-cell dynamical model with distributed delay and Beddington-DeAngelis functional response is proposed in this paper. Using the characteristic equations and analytical means, the principle reproduction number R0 on the local stability of infection-free and chronic-infection equilibria is established. Furthermore, by constructing suitable Lyapunov functionals and using LaSalle invariance principle, we show that if R0 ≤ 1 the infection-free equilibrium is globally asymptotically stable, while if R0 > 1 the chronic-infection equilibrium is globally asymptotically stable. Numerical simulations are presented to illustrate the theoretical results. Comparing the effects between discrete and distributed delays on the stability of HIV virus-to-cell dynamical models, we can see that they could be same and different even opposite.

    Citation: Xinran Zhou, Long Zhang, Tao Zheng, Hong-li Li, Zhidong Teng. Global stability for a class of HIV virus-to-cell dynamical model with Beddington-DeAngelis functional response and distributed time delay[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 4527-4543. doi: 10.3934/mbe.2020250

    Related Papers:

  • A HIV virus-to-cell dynamical model with distributed delay and Beddington-DeAngelis functional response is proposed in this paper. Using the characteristic equations and analytical means, the principle reproduction number R0 on the local stability of infection-free and chronic-infection equilibria is established. Furthermore, by constructing suitable Lyapunov functionals and using LaSalle invariance principle, we show that if R0 ≤ 1 the infection-free equilibrium is globally asymptotically stable, while if R0 > 1 the chronic-infection equilibrium is globally asymptotically stable. Numerical simulations are presented to illustrate the theoretical results. Comparing the effects between discrete and distributed delays on the stability of HIV virus-to-cell dynamical models, we can see that they could be same and different even opposite.


    加载中


    [1] D. D. Richman, D. M. Margolis, M. Delaney, W. C. Greene, D. Hazuda, R. J. Pomerantz, The challenge of finding a cure for HIV infection, Science, 323 (2009), 1304-1307.
    [2] V. Leonenko, G. Bobashev, Analyzing influenza outbreaks in Russia using an age-structured dynamic transmission model, Epidemics, 29 (2019), 100-358.
    [3] Y. Cai, K. Wang, W. Wang, Global transmission dynamics of a Zika virus model, Appl. Math. Lett., 92 (2019), 190-195.
    [4] Y. Cai, Z. Ding, B. Yang, Z. Peng, W. Wang, Transmission dynamics of Zika virus with spatial structure-A case study in Rio de Janeiro, Brazil, Physica A: Stat. Mech. Appl., 514 (2019), 729-740.
    [5] E. Grigorieva, E. Khailov, Determination of the opimal controls for an Ebola epidemic model, Disc. Cont. Dynam. Syst. S., 11 (2018), 1071-1101.
    [6] J. Huang, S. Ruan, X. Wu, X. Zhou, Seasonal transmission dynamics of measles in China, Theor. Biosci., 137 (2018), 185-195.
    [7] T. Zhang, X. Zhao, Mathematical modeling for schistosomiasis with seasonal influence: A case study in Hubei, China, SIAM J. Appl. Dyn. Syst., 19 (2020), 1438-1471.
    [8] M. A. Nowak, C. R. M. Bangham, Population dynamics of immune responses to persistent virus, Science, 272 (1996), 74-79.
    [9] R. M. Anderson, R. M. May, The population dynamics of microparasites and their invertebrate hosts, Philos. Trans. R. Soc. Lond. Ser. B, 291 (1981), 451-524.
    [10] A. S. Perelson, P. W. Nelson, Mathematical analysis of HIV-I: dynamics in vivo, SIAM Rev., 41 (1999), 3-44.
    [11] X. Lai, X. Zou, Modeling HIV-1 virus dynamics with both virus-to-cell infection and cell-to-cell transmission, SIAM J. Appl. Math., 74 (2014), 898-917.
    [12] A. L. Hill, D. I. Rosenbloom, M. A. Nowak, R. F. Siliciano, Insight into treatment of HIV infection from viral dynamics models, Immun. Rev., 285 (2018), 9-25.
    [13] P. Aavani, L. S. Allen, The role of CD4 T cells in immune system activation and viral reproduction in a simple model for HIV infection, Appl. Math. Model., 75 (2019), 210-222.
    [14] D. Olabode, L. Rong, X. Wang, Optimal control in HIV chemotherapy with termination viral load and latent reservoir, Math. Biosci. Eng., 16 (2018), 619-635.
    [15] X. Wang, L. Rong, HIV low viral load persistence under treatment: Insights from a model of cell-to-cell viral transmission, Appl. Math. Lett., 94 (2019), 44-51.
    [16] P. W. Nelson, A.S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94.
    [17] R. Xu, Global stability of an HIV-1 infection model with saturation infection and intracellular delay, J. Math. Anal. Appl., 375 (2011), 75-81.
    [18] J. Lin, R. Xu, X, Tian, Threshold dynamics of an HIV-1 virus model with both virus-to-cell and cell-to-cell transmissions, intracellular delay, and humoral immunity, Appl. Math. Comput., 315 (2017), 516-530.
    [19] G. Huang, W. Ma, Y. Takeuchi, Global properties for virus dynamics model with BeddingtonDeAngelis functional response, Appl. Math. Lett., 22 (2009), 1690-1693.
    [20] G. Huang, W. Ma, Y. Takeuchi, Global analysis for delay virus dynamics model with BeddingtonDeAngelis functional response, Appl. Math. Lett., 24 (2011), 1199-1203.
    [21] Y. Nakata, Global dynamics of a viral infection model with a latent period and beddingtonDeAngelis response, Nonlinear Anal. TMA, 74 (2011), 2929-2940.
    [22] H. Xiang, L. Feng, H. Huo, Stability of the virus dynamics model with Beddington-DeAngelis functional response and delays, Appl. Math. Model., 37 (2013), 5414-5423.
    [23] R. Xu, Global dynamics of an HIV-1 infection model with distributed intracellular delays, Comput. Math. Appl., 61 (2011), 2799-2805.
    [24] J. Wang, M. Guo, X. Liu, Z. Zhao, Threshold dynamics of HIV-1 virus model with cell-to-cell transmission, cell-mediated immune responses and distributed delay, Appl. Math. Comput., 291 (2016), 149-161.
    [25] Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays, J. Math. Anal. Appl., 375 (2011), 14-27.
  • Reader Comments
  • © 2020 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

Metrics

Article views(3251) PDF downloads(340) Cited by(2)

Article outline

Figures and Tables

Figures(2)  /  Tables(2)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog