Browse > Article
http://dx.doi.org/10.12941/jksiam.2019.23.039

GLOBAL STABILITY OF VIRUS DYNAMICS MODEL WITH IMMUNE RESPONSE, CELLULAR INFECTION AND HOLLING TYPE-II  

ELAIW, A.M. (DEPARTMENT OF MATHEMATICS, KING ABDULAZIZ UNIVERSITY)
GHALEB, SH.A. (DEPARTMENT OF MATHEMATICS, KING ABDULAZIZ UNIVERSITY)
Publication Information
Journal of the Korean Society for Industrial and Applied Mathematics / v.23, no.1, 2019 , pp. 39-63 More about this Journal
Abstract
In this paper, we study the effect of Cytotoxic T Lymphocyte (CTL) and antibody immune responses on the virus dynamics with both virus-to-cell and cell-to-cell transmissions. The infection rate is given by Holling type-II. We first show that the model is biologically acceptable by showing that the solutions of the model are nonnegative and bounded. We find the equilibria of the model and investigate their global stability analysis. We derive five threshold parameters which fully determine the existence and stability of the five equilibria of the model. The global stability of all equilibria of the model is proven using Lyapunov method and applying LaSalle's invariance principle. To support our theoretical results we have performed some numerical simulations for the model. The results show the CTL and antibody immune response can control the disease progression.
Keywords
Pathogen infection; Holling-type incidence; global stability; adaptive immune response; Lyapunov function;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 L. Gibelli, A. Elaiw, M.A. Alghamdi and A.M. Althiabi, Heterogeneous population dynamics of active particles: Progression, mutations, and selection dynamics, Math. Models Meth. Appl. Sci. 27 (2017), 617-640.   DOI
2 M. A. Nowak, R. Anderson, M. Boerlijst, S. Bonhoe er, R. May, and A. McMichael, HIV-1 evolution and disease progression, Science, 274 (1996), 1008-1010.   DOI
3 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.   DOI
4 C. Lv, L. Huang, and Z. Yuan, Global stability for an HIV-1 infection model with Beddington-DeAngelis incidence rate and CTL immune response, Commun. Nonlinear Sci. Numer. Simul. 19 (2014) 121-127.   DOI
5 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).
6 X. Li and S. Fu, Global stability of a virus dynamics model with intracellular delay and CTL immune response, Math. Meth. Appl. Sci. 38 (2015), 420-430.   DOI
7 D. Huang, X. Zhang, Y. Guo, and H. Wang, Analysis of an HIV infection model with treatments and delayed immune response, Appl. Math. Model. 40 (2016), 3081-3089.   DOI
8 Y. Zhao and Z. Xu, Global dynamics for a delyed hepatitis C virus,infection model, Electron. J. Differ. Equ. 2014 (2014), 1-18.   DOI
9 A. Murase, T. Sasaki, and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics, J. Math. Biol. 51 (2005), 247-267.   DOI
10 A. M. Elaiw, E. Kh. Elnahary, E. Kh, A. M. Shehata and M. Abul-Ez, Stability of delay-distributed HIV infection models with multiple viral producer cells, J. Korean Soc. Ind. Appl. Math. 22 (2018), 29-62.   DOI
11 A. M. Elaiw, E. Kh. Elnahary and A. A. Raezah, Effect of cellular reservoirs and delays on the global dynamics of HIV, Adv. Differ. Equ. (2018).
12 S.Wang, and D. Zou, Global stability of in host viral models with humoral immunity and intracellular delays, Appl. Math. Model. 36 (2012), 1313-1322.   DOI
13 A. M. Elaiw, S. A. Ghaleb and A. Hobiny, Effect of time delay and antibodies on HCV dynamics with cure rate and two routes of infection. Journal of Applied Mathematics and Physics, 6 (2018), 1120.   DOI
14 D.S. Callaway, and A.S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol. 64 (2002), 29-64.   DOI
15 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.   DOI
16 N. E. Tarfulea, A mathematical model for CTL effect on a latently infected cell inclusive HIV dynamics and treatment, AIP Conference Proceedings, 1895 (2017).
17 A. M. Elaiw and N. H. AlShamrani, Stability of a general delay-distributed virus dynamics model with multistaged infected progression and immune response, Math. Meth. Appl. Sci. 40 (2017), 699-719.   DOI
18 Elaiw, A. M., A. A. Raezah, and A. S. Alofi, Effect of humoral immunity on HIV-1 dynamics with virus-totarget and infected-to-target infections, AIP Adv. 6 (2016).
19 L. Li and R. Xu, Global dynamics of an age-structured in-host viral infection model with humoral immunity, Adv. Differ. Equ. (2016).
20 T. Wang, Z. Hu, F. Liao and W. Ma, Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity, Math. Comput. Simul. 89 (2013), 13-22.   DOI
21 J. Xu, Y. Zhou, Y. Li and Y. Yang, Global dynamics of a intracellular infection model with delays and humoral immunity, Math. Meth. Appl. Sci. 39 (2016), 5427-5435.   DOI
22 H. Miao, Z. Teng, C. Kang, A. Muhammadhaji, Stability analysis of a virus infection model with humoral immunity response and two time delays, Math. Meth. Appl. Sci. 39 (2016), 3434-3449.   DOI
23 D. Wodarz, Killer cell dynamics: mathematical and computational approaches to immunology. Springer Verlag, New York, (2007).
24 J. Wang, J. Pang, T. Kuniya and Y. Enatsu, 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.   DOI
25 A. M. Elaiw and N. H. AlShamrani, Stability of an adaptive immunity pathogen dynamics model with latency and multiple delays, Math. Meth. Appl. Sci. 36 (2018), 125-142.   DOI
26 A. M. Elaiw and N. H. AlShamrani, Stability of latent pathogen infection model with adaptive immunity and delays, J. Integr. Neurosci. 17 (2018), 547-576.   DOI
27 K. Hattaf and N. Yousfi, A class of delayed viral infection models with general incidence rate and adaptive immune response, International Journal of Dynamics and Control, 4 (2016), 254-265.   DOI
28 N. Yousfi, K. Hattaf, A. Tridane, Modeling the adaptive immune response in HBV infection, J. Math. Biol. 63 (2011), 933-957.   DOI
29 A. Rezounenko, Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses, Discrete Contin. Dyn. Syst.-Ser. B. 22 (2016).
30 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.   DOI
31 R.V. Culshaw, S. Ruan, 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.   DOI
32 X. Lai and X. Zou, Modeling HIV-1 virus dynamics with both virus-to-cell infection and cell-to-cell transmission, SIAM Journal of Applied Mathematics, 74 (2014), 898-917.   DOI
33 H. Pourbashash, S.S. Pilyugin, P. De Leenheer, C. McCluskey, Global analysis of within host virus models with cell-to-cell viral transmission, Discrete Contin. Dyn. Syst.-Ser. B. 10 (2014) 3341-3357.
34 J. Wang, J. Lang, 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.   DOI
35 S.-S. Chen, C.-Y. Cheng, Y. Takeuchi, Stability analysis in delayed within-host viral dynamics with both viral and cellular infections, J. Math. Anal. Appl. 442 (2016), 642-672.   DOI
36 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.   DOI
37 Y. Yang, L. Zou and S. Ruan, Global dynamics of a delayed within-host viral infection model with both virusto- cell and cell-to-cell transmissions, Math. Biosci. 270 (2015), 183-191.   DOI
38 A. M. Elaiw and A. A. Raezah, Stability of general virus dynamics models with both cellular and viral infections and delays, Math. Meth. Appl. Sci. 40 (2017), 5863-5880.   DOI
39 S. Pan, 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.   DOI
40 A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol. 66 (2004), 879-883.   DOI
41 A. M. Elaiw, A. A. Raezah and B. S. Alofi, Dynamics of delayed pathogen infection models with pathogenic and cellular infections and immune impairment, AIP Adv. 8 (2018).
42 A. M. Elaiw and S.A. Azoz, Global properties of a class of HIV infection models with Beddington-DeAngelis functional response, Math. Meth. Appl. Sci. 36 (2013), 383-394.   DOI
43 A. M. Elaiw, Global properties of a class of HIV models, Nonlinear Anal.-RealWorld Appl. 11 (2010), 2253-2263.   DOI
44 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 Journal of Applied Mathematics, 73 (2013), 1280-1302.   DOI
45 A. M. Elaiw, Global dynamics of an HIV infection model with two classes of target cells and distributed delays, Discrete Dyn. Nat. Soc. 2012 (2012).
46 A. M. Elaiw and N. H. AlShamrani, Stability of a general delay-distributed virus dynamics model with multistaged infected progression and immune response, Math. Meth. Appl. Sci. 40 (2017), 699-719.   DOI
47 A. M. Elaiw and N. A. Almuallem, Global dynamics of delay-distributed HIV infection models with differential drug efficacy in cocirculating target cells, Math. Meth. Appl. Sci. 39 (2016), 4-31.   DOI
48 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.   DOI
49 A. D. Hobiny, A. M. Elaiw, Almatrafi, Stability of delayed pathogen dynamics models with latency and two routes of infection, Adv. Differ. Equ. (2018).
50 A. M. Elaiw and A. A. Raezah, Stability of general virus dynamics models with both cellular and viral infections and delays, Math. Meth. Appl. Sci. 40 (2017), 5863-5880.   DOI
51 A. M. Elaiw, T. O. Alade, S. M. Alsulami, Analysis of latent CHIKV dynamics models with general incidence rate and time delays. J. Biol. Dyn. 12 (2018), 700-730.   DOI
52 A. M. Elaiw, T. O. Alade, S. M. Alsulami, Analysis of within-host CHIKV dynamics models with general incidence rate. Int. J. Biomath. 11 (2018).
53 A.M. Elaiw, Global properties of a class of virus infection models with multitarget cells, Nonlinear Dyn. 69 (2012), 423-435.   DOI
54 J.K. Hale, and S. V. Lunel, Introduction to functional differential equations, Springer-Verlag, New York, (1993).