Research article

Stability of an adaptive immunity viral infection model with multi-stages of infected cells and two routes of infection

  • Received: 07 May 2019 Accepted: 12 October 2019 Published: 21 October 2019
  • This paper studies an (n + 4)-dimensional nonlinear viral infection model that characterizes the interactions of the viruses, susceptible host cells, n-stages of infected cells, CTL cells and B cells. Both viral and cellular infections have been incorporated into the model. The well-posedness of the model is justified. The model admits five equilibria which are determined by five threshold parameters. The global stability of each equilibrium is proven by utilizing Lyapunov function and LaSalle's invariance principle. The theoretical results are illustrated by numerical simulations.

    Citation: N. H. AlShamrani, A. M. Elaiw. Stability of an adaptive immunity viral infection model with multi-stages of infected cells and two routes of infection[J]. Mathematical Biosciences and Engineering, 2020, 17(1): 575-605. doi: 10.3934/mbe.2020030

    Related Papers:

  • This paper studies an (n + 4)-dimensional nonlinear viral infection model that characterizes the interactions of the viruses, susceptible host cells, n-stages of infected cells, CTL cells and B cells. Both viral and cellular infections have been incorporated into the model. The well-posedness of the model is justified. The model admits five equilibria which are determined by five threshold parameters. The global stability of each equilibrium is proven by utilizing Lyapunov function and LaSalle's invariance principle. The theoretical results are illustrated by numerical simulations.


    加载中


    [1] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.
    [2] N. Yousfi, K. Hattaf and A. Tridane, Modeling the adaptive immune response in HBV infection, J. Math. Biol., 63 (2011), 933-957.
    [3] L. Gibelli, A. Elaiw, M. A. Alghamdi, et al., Heterogeneous population dynamics of active particles: Progression, mutations, and selection dynamics, Math. Models Methods Appl. Sci., 27 (2017), 617-640.
    [4] A. M. Elaiw and M. A. Alshaikh, Stability of discrete-time HIV dynamics models with three categories of infected CD4+ T-cells, Adv. Differ. Equ., 2019 (2019), 407.
    [5] F. Zhang, J. Li, C. Zheng, et al., Dynamics of an HBV/HCV infection model with intracellular delay and cell proliferation, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 464-476.
    [6] N. Bellomo and Y. Tao, Stabilization in a chemotaxis model for virus infection, Discrete Contin. Dyn. syst. Ser. S, 13 (2020), 105-117.
    [7] H. Shu, L. Wang and J. Watmough, Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL imune responses, SIAM J. Appl. Math., 73 (2013), 1280-1302.
    [8] A. M. Elaiw, A. A. Raezah and S. A. Azoz, Stability of delayed HIV dynamics models with two latent reservoirs and immune impairment, Adv. Differ. Equ., 2018 (2018), 414.
    [9] M. Y. Li and H. Shu, Global dynamics of a mathematical model for HTLV-I infection of CD4+ T cells with delayed CTL response, Nonlinear Anal. Real World Appl., 13 (2012), 1080-1092.
    [10] J. Pang and J-An Cui, Analysis of a hepatitis B viral infection model with immune response delay, Int. J. Biomath., 10 (2017).
    [11] C. Vargas-De-Leon, Global properties for a virus dynamics model with lytic and nonlytic immune responses, and nonlinear immune attack rates, J. Biol. Syst., 22 (2014), 449-462.
    [12] A. Murase, T. Sasaki and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics, J. Math. Biol., 51 (2005), 247-267.
    [13] A. M. Elaiw and M. A. Alshaikh, Stability analysis of a general discrete-time pathogen infection model with humoral immunity, J. Difference Equ. Appl., 2019 (2019), 1-24.
    [14] A. M. Elaiw, E. Kh. Elnahary and A. A. Raezah, Effect of cellular reservoirs and delays on the global dynamics of HIV, Adv. Difference Equ., 2018 (2018), 85.
    [15] T. Wang, Z. Hu and F. Liao, Stability and Hopf bifurcation for a virus infection model with delayed humoral immunity response, J. Math. Anal. Appl., 411 (2014) 63-74.
    [16] A. M. Elaiw and N. H. AlShamrani, Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal, Nonlinear Anal. Real World Appl., 26 (2015), 161-190.
    [17] H. Miao, Z. Teng, C. Kang, et al., Stability analysis of a virus infection model with humoral immunity response and two time delays, Math. Methods Appl. Sci., 39 (2016), 3434-3449.
    [18] D. Wodarz, Killer cell dynamics: Mthematical and computational approaches to immunology, Springer Verlag, New York, 2007.
    [19] J. Wang, J. Pang, T. Kuniya, et al., Global threshold dynamics in a five-dimensional virus model with cell-mediated, humoral immune responses and distributed delays, Appl. Math. Comput., 241 (2014), 298-316.
    [20] A. Rezounenko, Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses, Discrete Contin. Dyn. Syst.Ser. B, 22 (2017), 1547-156.
    [21] Y. Yan and W. Wang, Global stability of a five-dimensional model with immune responses and delay, Discrete Contin. Dyn. Syst.Ser. B, 17 (2012), 401-416.
    [22] P. Dubey, U. S. Dubey and B. Dubey, Modeling the role of acquired immune response and antiretroviral therapy in the dynamics of HIV infection, Math. Comput. Simul., 144 (2018), 120- 137.
    [23] A. M. Elaiw and N. H. AlShamrani, Stability of an adaptive immunity pathogen dynamics model with latency and multiple delays, Math. Methods Appl. Sci., 36 (2018), 125-142.
    [24] R. V. Culshaw, S. Ruan and G. Webb, A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay, J. Math. Biol., 46 (2003), 425-444.
    [25] X. Lai and X. Zou, Modeling HIV-1 virus dynamics with both virus-to-cell infection and cell-tocell transmission, SIAM J. Appl. Math., 74 (2014), 898-917.
    [26] H. Pourbashash, S. S. Pilyugin, P. De Leenheer, et al., Global analysis of within host virus models with cell-to-cell viral transmission, Discrete Contin. Dyn. Syst.Ser. B, 10 (2014) 3341-3357.
    [27] J. Wang, J. Lang and X. Zou, Analysis of an age structured HIV infection model with virus-to-cell infection and cell-to-cell transmission, Nonlinear Anal. Real World Appl., 34 (2017), 75-96.
    [28] S. S. Chen, C. Y. Cheng and Y. Takeuchi, Stability analysis in delayed within-host viral dynamics with both viral and cellular infections, J. Math. Anal. Appl., 442 (2016), 642-672.
    [29] J. Lin, R. Xu and 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.
    [30] A. M. Elaiw, A. Almatrafi, A. D. Hobiny, et al., Global properties of a general latent pathogen dynamics model with delayed pathogenic and cellular infections, Discrete Dyn. Nat. Soc., 2019 (2019).
    [31] A. D. Hobiny, A. M. Elaiw and A. Almatrafi, Stability of delayed pathogen dynamics models with latency and two routes of infection, Adv. Difference Equ., 2018 (2018), 276.
    [32] Y. Yang, L. Zou and S. Ruan, Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions, Math. Biosci., 270 (2015), 183-191.
    [33] A. M. Elaiw and A. A. Raezah, Stability of general virus dynamics models with both cellular and viral infections and delays, Math. Methods Appl. Sci., 40 (2017), 5863-5880.
    [34] S. Pan and S. P. Chakrabarty, Threshold dynamics of HCV model with cell-to-cell transmission and a non-cytolytic cure in the presence of humoral immunity, Commun. Nonlinear Sci. Numer. Simul., 61 (2018), 180-197.
    [35] T. Guo, Z. Qiu and L. Rong, Analysis of an HIV model with immune responses and cell-to-cell transmission, Bull. Malays. Math. Sci. Soc., 2018 (2018), 1-27.
    [36] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708.
    [37] P. Georgescu and Y. H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal, SIAM J. Appl. Math., 67 (2006), 337-353.
    [38] A. M. Elaiw and N. H. AlShamrani, Stability of a general delay-distributed virus dynamics model with multi-staged infected progression and immune response, Math. Methods Appl. Sci., 40 (2017), 699-719.
    [39] A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883.
    [40] A. M. Elaiw and S. A. Azoz, Global properties of a class of HIV infection models with Beddington-DeAngelis functional response, Math. Methods Appl. Sci., 36 (2013), 383-394.
    [41] A. M. Elaiw, Global properties of a class of HIV models, Nonlinear Anal. Real World Appl., 11 (2010), 2253-2263.
    [42] A. M. Elaiw and N. A. Almuallem, Global properties of delayed-HIV dynamics models with differential drug efficacy in cocirculating target cells, Appl. Math. Comput., 265 (2015), 1067- 1089.
    [43] A. M. Elaiw and N. A. Almuallem, Global dynamics of delay-distributed HIV infection models with differential drug efficacy in cocirculating target cells, Math. Methods Appl. Sci., 39 (2016), 4-31.
    [44] A. M. Elaiw, I. A. Hassanien and S. A. Azoz, Global stability of HIV infection models with intracellular delays, J. Korean Math. Soc., 49 (2012), 779-794.
    [45] A. M. Elaiw, Global properties of a class of virus infection models with multitarget cells, Nonlinear Dynam., 69 (2012), 423-435.
    [46] J. K. Hale and S. V. Lunel, Introduction to functional differential equations, Springer-Verlag, New York, 1993.
    [47] X. Wang and 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.
    [48] M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections, SIAM J. Appl. Math., 70 (2010), 2434-2448.
    [49] A. M. Elaiw and E. Kh. Elnahary, Analysis of general humoral immunity HIV dynamics model with HAART and distributed delays, Mathematics, 7 (2019), 157.
    [50] A. M. Elaiw, S. F. Alshehaiween and A. D. Hobiny, Global properties of delay-distributed HIV dynamics model including impairment of B-cell functions, Mathematics, 7 (2019), 837.
    [51] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. Real World Appl., 11 (2010), 55-59.
    [52] C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610.
    [53] N. Dixit and A. Perelson, Multiplicity of human immunodeficiency virus infections in lymphoid tissue, J. Virol., 78 (2004), 8942-8945.
    [54] C. C. McCluskey, Delay versus age-of-infection-Global stability, Appl. Math. Comput., 217 (2010), 3046-3049.
    [55] P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140.
    [56] C. Browne, Immune response in virus model structured by cell infection-age, Math. Biosci. Eng., 13 (2016), 887-909.
    [57] X. Tian, R. Xu and J. Lin, Mathematical analysis of an age-structured HIV-1 infection model with CTL immune response, Math. Biosci. Eng., 16 (2019), 7850-7882.
    [58] A. M. Elaiw and A. D. AlAgha, Global dynamics of reaction-diffusion oncolytic M1 virotherapy with immune response, Appl. Math. Comput., 367 (2020), 124758.
    [59] A. Herz, S. Bonhoeffer, R. M. Anderson, et al., Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay, Proc.Natl. Acad. Sci., 93 (1996), 7247-7251.
  • 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(2637) PDF downloads(456) Cited by(1)

Article outline

Figures and Tables

Figures(7)  /  Tables(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog