Global dynamics of delayed HIV infection models including impairment of B-cell functions

Volume 20, Issue 2, pp 161--188 http://dx.doi.org/10.22436/jmcs.020.02.08

Publication Date: November 08, 2019 Submission Date: June 18, 2019 Revision Date: August 26, 2019 Accteptance Date: September 16, 2019

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1841 Downloads
• 3924 Views

Authors

Ahmed Elaiw - Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. Safiya Alshehaiween - Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. - Department of Mathematics, Faculty of Science, Taibah University, P. O. Box 344, Medina 42353, Saudi Arabia. Aatef Hobiny - Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.

Abstract

In this paper, we construct delayed HIV dynamics models with impairment of B-cell functions. Two forms of the incidence rate have been considered, bilinear and general. Three types of infected cells and five-time delays have been incorporated into the models. The well-posedness of the models is justified. The models admit two equilibria, which are determined by the basic reproduction number \(R_{0}\). The global stability of each equilibrium is proven by utilizing the Lyapunov function and LaSalle's invariance principle. Numerical simulations illustrate the theoretical results.

Share and Cite

Keywords

• HIV dynamics
• global stability
• Lyapunov function
• B-cell impairment
• latent reservoirs
• time delay

MSC

•  34D20
•  34D23
•  37N25
•  92B05

References

• [1] S. Amu, N. Ruffin, B. Rethi, F. Chiodi, Impairment of B-cell functions during HIV-1 infection, AIDS, 27 (2013), 2323--2334
  • View Article
  • Google Scholar


• [2] N. Bellomo, Y. Tao, Stabilization in a chemotaxis model for virus infection, Discrete. Cont. Dyn Syst.-S, 13 (2020), 105--117
  • View Article
  • Google Scholar


• [3] J. N. Blankson, D. Persaud, R. F. Siliciano, The challenge of viral reservoirs in HIV-1 infection, Ann. Rev. Med., 53 (2002), 557--593
  • View Article
  • Google Scholar


• [4] F. Chiodi, G. Scarlatti, HIV-Induced damage of B cells and production of HIV neutralizing antibodies, Front. Immunol., 9 (2018), 4 pages
  • View Article
  • Google Scholar


• [5] R. V. Culshaw, S. Ruan, A delay-differential equation model of HIV infection of CD4$^{+}$\emph{T-cells}, Math. Biosci., 165 (2000), 27--39
  • View Article


• [6] 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
  • View Article


• [7] A. De Milito, B lymphocyte dysfunctions in HIV Infection, Curr. HIV. Res., 2 (2004), 11--21
  • View Article
  • Google Scholar


• [8] A. M. Elaiw, Global properties of a class of HIV models, Nonlinear Anal. Real World Appl., 11 (2010), 2253--2263
  • View Article


• [9] 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), 13 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] A. M. Elaiw, Global properties of a class of virus infection models with multitarget cells, Nonlinear Dynam., 69 (2012), 423--435
  • View Article


• [11] A. M. Elaiw, R. M. Abukwaik, E. O. Alzahrani, Global properties of a cell mediated immunity in HIV infection model with two classes of target cells and distributed delays, Int. J. Biomath., 7 (2014), 25 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] A. M. Elaiw, A. A. Almatrafi, A. D. Hobiny, K. Hattaf, Global properties of a general latent pathogen dynamics model with delayed pathogenic and cellular infections, Discrete Dyn. Nat. Soc., 2019 (2019), 18 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [13] A. M. Elaiw, 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
  • View Article


• [14] A. M. Elaiw, 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
  • View Article


• [15] A. M. Elaiw, M. A. Alshaikh, Stability analysis of a general discrete-time pathogen infection model with humoral immunity, J. Differ. Equ. Appl., 2019 (2019), 24 pages
  • View Article


• [16] A. M. Elaiw, M. A. Alshaikh, Stability of discrete-time HIV dynamics models with three categories of infected CD4$^{+}$ T-cells, Adv. Difference Equ., 2019 (2019), 24 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [17] A. M. Elaiw, N. H. AlShamrani, Global properties of nonlinear humoral immunity viral infection models, Int. J. Biomath., 8 (2015), 53 pages
  • View Article


• [18] A. M. Elaiw, 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
  • View Article


• [19] A. M. Elaiw, 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
  • View Article


• [20] A. M. Elaiw, N. H. AlShamrani, Stability of an adaptive immunity pathogen dynamics model with latency and multiple delays, Math. Methods Appl. Sci., 36 (2018), 125--142
  • View Article


• [21] A. M. Elaiw, N. H. AlShamrani, Stability of a general adaptive immunity virus dynamics model with multi-stages of infected cells and two routes of infection, Math. Methods Appl. Sci., 2019 (2019), 15 pages
  • Google Scholar


• [22] A. M. Elaiw, S. F. Alshehaiween, A. D. Hobiny, Global properties of delay-distributed HIV dynamics model including impairment of B-cell functions, Mathematics, 7 (2019), 27 pages
  • View Article
  • Google Scholar


• [23] A. M. Elaiw, 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
  • View Article


• [24] A. M. Elaiw, E. K. Elnahary, Analysis of general humoral immunity HIV dynamics model with HAART and distributed delays, Mathematics, 7 (2019), 35 pages
  • View Article
  • Google Scholar


• [25] A. M. Elaiw, E. K. Elnahary, A. A. Raezah, Effect of cellular reservoirs and delays on the global dynamics of HIV, Adv. Difference Equ., 2018 (2018), 36 pages
  • View Article


• [26] A. M. Elaiw, I. A. Hassanien, S. A. Azoz, Global stability of HIV infection models with intracellular delays, J. Korean Math. Soc., 49 (2012), 779--794
  • View Article


• [27] A. M. Elaiw, A. G. Hobiny, A. D. Al Agha, Global dynamics of reaction-diffusion oncolytic M1 virotherapy with immune response, Appl. Math. Comput., 367 (2019), 124758
  • View Article
  • MathSciNet
  • Google Scholar


• [28] A. M. Elaiw, 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
  • View Article


• [29] A. M. Elaiw, A. A. Raezah, S. A. Azoz, Stability of delayed HIV dynamics models with two latent reservoirs and immune impairment, Adv. Difference Equ., 2018 (2018), 25 pages
  • View Article


• [30] L. Gibelli, A. Elaiw, M. A. Alghamdi, A. M. Althiabi, Heterogeneous population dynamics of active particles: Progression, mutations, and selection dynamics, Math. Models Methods Appl. Sci., 27 (2017), 617--640
  • View Article


• [31] J. K. Hale, S. M. Verduyn Lunel, Introduction to functional differential equations, Springer-Verlag, New York (1993)
  • View Article


• [32] A. D. Hobiny, A. M. Elaiw, A. A. Almatrafi, Stability of delayed pathogen dynamics models with latency and two routes of infection, Adv. Difference Equ., 2018 (2018), 26 pages
  • View Article


• [33] G. Huang, Y. Takeuchi, W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010), 2693--2708
  • View Article


• [34] Y. Kuang, Delay Differential Equations with applications in population dynamics, Academic Press, San Diego (1993)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [35] B. Li, Y. M. Chen, X. J. Lu, S. Q. Liu, A delayed HIV-1 model with virus waning term, Math. Biosci. Eng., 13 (2016), 135--157
  • View Article


• [36] D. Li, W. Ma, Asymptotic properties of a HIV-1 infection model with time delay, J. Math. Anal. Appl., 335 (2007), 683--691
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [37] J. Z. Lin, R. Xu, X. H. 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
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [38] H. J. Liu, J.-F. Zhang, Dynamics of two time delays differential equation model to HIV latent infection, Phys. A,, 514 (2019), 384--395
  • View Article
  • MathSciNet
  • Google Scholar


• [39] Y. Lv, Z. X. Hu, F. C. Liao, The stability and Hopf bifurcation for an HIV model with saturated infection rate and double delays, Int. J. Biomath., 11 (2018), 43 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [40] P. Lydyard, A. Whelan, M. Fanger, BIOS Instant notes in immunology, Taylor & Francis, London (2005)
  • View Article
  • Google Scholar


• [41] H. Miao, X. Abdurahman, Z. D. Teng, L. Zhang, Dynamical analysis of a delayed reaction-diffusion virus infection model with logistic growth and humoral immune impairment, Chaos Solitons Fractals, 110 (2018), 280--291
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [42] H. Miao, Z. D. Teng, C. J. Kang, A. Muhammadhaji, Stability analysis of a virus infection model with humoral immunity response and two time delays, Math. Methods Appl. Sci., 39 (2016), 3434--3449
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [43] A. Murase, T. Sasaki, T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics, J. Math. Biol., 51 (2005), 247--267
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [44] P. W. Nelson, J. D. Murray, A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci., 163 (2000), 201--215
  • View Article


• [45] M. A. Nowak, C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74--79
  • View Article


• [46] M. A. Nowak, R. M. May, Virus dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford (2000)
  • Google Scholar
  • MATH


• [47] A. S. Perelson, P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM. Rev., 41 (1999), 3--44
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [48] S. K. Sahani, Yashi, Effects of eclipse phase and delay on the dynamics of HIV infection, J. Biol. Systems, 26 (2018), 421--454
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [49] H. Y. Shu, L. Wang, J. Watmough, Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses, SIAM J. Appl. Math., 73 (2013), 1280--1302
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [50] T. L. Wang, Z. X. Hu, F. C. Liao, Stability and Hopf bifurcation for a virus infection model with delayed humoral immunity response, J. Math. Anal. Appl., 411 (2014), 63--74
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [51] T. L. Wang, Z. X. Hu, F. C. Liao, W. B. Ma, Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity, Math. Comput. Simulation, 89 (2013), 13--22
  • View Article
  • MathSciNet
  • Google Scholar


• [52] L. C. Wang, M. Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of CD4$^{+}$\emph{ T cells}, Math. Biosci., 200 (2006), 44--57
  • View Article


• [53] S. F. Wang, D. Y. Zou, Global stability of in-host viral models with humoral immunity and intracellular delays, Appl. Math. Model., 36 (2012), 1313--1322
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [54] Y. Zhao, D. T. Dimitrov, H. Liu, Y. Kuang, Mathematical insights in evaluating state dependent effectiveness of HIV prevention interventions, Bull. Math. Biol., 75 (2013), 649--675
  • View Article