Global stability of a delayed humoral immunity virus dynamics model with nonlinear incidence and infected cells removal rates



In this paper, we investigate the dynamical behavior of a nonlinear model for viral infection with humoral immune response and two discrete delays. The model is a four-dimensional system which describes the interactions of the virus, uninfected target cells, infected cells and B cells. We assume that the incidence rate and removal rate of infected cells are given by general nonlinear functions. We derive two threshold parameters, the basic reproduction number \(R_{0}\) and the humoral immunity number \(R_{1}\). Utilizing Lyapunov functionals and LaSalle’s invariance principle, the global asymptotic stability of the steady states of the model has been studied. We have established a set of sufficient conditions which guarantee the global stability of the model. We have proven that (1) if \(R_{0}\le 1\), then the infection-free steady state is globally asymptotically stable (GAS), (2) if \(R_{1}\le 1<R_{0}\), then the chronic-infection steady state without humoral immune response is GAS, (3) if \(R_{1}>1\), then the chronic-infection steady state with humoral immune response is GAS. We introduce an example and conduct some numerical simulations to confirm our theoretical results.


Virus dynamics Intracellular delay Global stability Humoral immune response Lyapunov functional 


  1. 1.
    Shu H, Wang L, Watmough J (2013) Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL imune responses. SIAM J Appl Math 73(3):1280–1302MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Elaiw AM, Xia X (2009) HIV dynamics: analysis and robust multirate MPC-based treatment schedules. J Math Anal Appl 356:285–301MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Li MY, Wang L (2014) Backward bifurcation in a mathematical model for HIV infection in vivo with anti-retroviral treatment. Nonlinear Anal Real World Appl 17:147–160MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Elaiw AM, Azoz SA (2013) Global properties of a class of HIV infection models with Beddington–DeAngelis functional response. Math Method Appl Sci 36:383–394MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Elaiw AM, Hassanien IA, Azoz SA (2012) Global stability of HIV infection models with intracellular delays. J. Korean Math Soc 49:779–794MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Elaiw AM (2010) Global properties of a class of HIV models. Nonlinear Anal Real World Appl 11:2253–2263MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Elaiw AM, Abukwaik RM, Alzahrani EO (2014) Global properties of a cell mediated immunity in HIV infection model with two classes of target cells and distributed delays. Int J Biomath 7(5):1450055 25 pMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Elaiw AM (2012) Global properties of a class of virus infection models with multitarget cells. Nonlinear Dyn 69:423–435MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Culshaw RV, Ruan S (2000) A delay-differential equation model of HIV infection of CD4\(^{+}\) T-cells. Math Biosci 165:27–39CrossRefMATHGoogle Scholar
  10. 10.
    Mittler J, Sulzer B, Neumann A, Perelson A (1998) Influence of delayed virus production on viral dynamics in HIV-1 infected patients. Math Biosci 152:143–163CrossRefMATHGoogle Scholar
  11. 11.
    Dixit NM, Perelson AS (2004) Complex patterns of viral load decay under antiretroviral therapy: influence of pharmacokinetics and intracellular delay. J Theoret Biol 226:95–109MathSciNetCrossRefGoogle Scholar
  12. 12.
    Wang L, Li MY (2006) Mathematical analysis of the global dynamics of a model for HIV infection of CD4\(^{+}\) T cells. Math Biosci 200(1):44–57MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hattaf K, Yousfi N (2015) A class of delayed viral infection models with general incidence rate and adaptive immune response. Int J Dyn Control. doi: 10.1007/s40435-015-0158-1
  14. 14.
    Yousfi N, Hattaf K, Tridane A (2011) Modeling the adaptative immune response in HBV infection. J Math Biol 63:933–957MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Qesmi R, ElSaadany S, Heffernan JM, Wu J (2011) A hepatitis B and C virus model with age since infection that exhibit backward bifurcation. SIAM J Appl Math 71(4):1509–1530MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Li MY, Shu H (2012) Global dynamics of a mathematical model for HTLV-I infection of CD4+ T cells with delayed CTL response. Nonlinear Anal Real World Appl 13:1080–1092MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Korobeinikov A (2004) Global properties of basic virus dynamics models. Bull Math Biol 66:879–883MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Korobeinikov A (2007) Global properties of infectious disease models with nonlinear incidence. Bull Math Biol 69:1871–1886MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Huang G, Ma W, Takeuchi Y (2011) Global analysis for delay virus dynamics model with Beddington–DeAngelis functional response. Appl Math 24(7):1199–1203Google Scholar
  20. 20.
    Li MY, Shu H (2010) Global dynamics of an in-host viral model with intracellular delay. Bull Math Biol 72:1492–1505MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Li MY, Shu H (2010) Impact of intracellular delays and target cell dynamics on in vivo viral infections. SIAM J Appl Math 70(7):2434–2448MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Li D, Ma W (2007) Asymptotic properties of a HIV-1 infection model with time delay. J Math Anal Appl 335:683–691MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Bairagi N, Adak D (2014) Global analysis of HIV-1 dynamics with Hill type infection rate and intracellular delay. Appl Math Model 38(21–22):5047–5066MathSciNetCrossRefGoogle Scholar
  24. 24.
    Yuan Z, Zou X (2013) Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays. Math Biosci Eng 10(2):483–498MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Song X, Cha X (2005) Global dynamics of viral model with saturated loss of infected cells. J Xinyang Norm Univ 18:262–267Google Scholar
  26. 26.
    Huang G, Takeuchi Y, Ma W (2010) Lyapunov functionals for delay differential equations model of viral infections. SIAM J Appl Math 70(7):2693–2708MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Nowak MA, May RM (2000) Virus dynamics: mathematical principles of immunology and virology. Oxford University, OxfordMATHGoogle Scholar
  28. 28.
    Deans JA, Cohen S (1983) Immunology of malaria. Ann Rev Microbiol 37:25–49CrossRefGoogle Scholar
  29. 29.
    Murase A, Sasaki T, Kajiwara T (2005) Stability analysis of pathogen–immune interaction dynamics. J Math Biol 51:247–267MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Wang S, Zou D (2012) Global stability of in host viral models with humoral immunity and intracellular delays. J Appl Math Mod 36:1313–1322MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Wang T, Hu Z, Liao F (2014) Stability and Hopf bifurcation for a virus infection model with delayed humoral immunity response. J Math Anal Appl 411:63–74MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Elaiw AM, AlShamrani NH (2015) Global properties of nonlinear humoral immunity viral infection models. Int J Biomath 8(5):1550058 (53 pages)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Elaiw AM, AlShamrani NH (2014) Global analysis for a delay-distributed viral infection model with antibodies and general nonlinear incidence rate. J Korean Soc Ind Appl Math 18(4):317–335MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Elaiw AM, AlShamrani NH (2015) Dynamics of viral infection models with antibodies and general nonlinear incidence and neutralize rates. Int J Dyn Control. doi: 10.1007/s40435-015-0181-2
  35. 35.
    Wang T, Hu Z, Liao F, Ma W (2013) Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity. Math Comput Simul 89:13–22MathSciNetCrossRefGoogle Scholar
  36. 36.
    Larson R, Edwards BH (2010) Calculus of a single variable. Cengage Learning Inc, USAGoogle Scholar
  37. 37.
    Hale JK, Verduyn Lunel S (1993) Introduction to functional differential equations. Springer, New YorkCrossRefMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia

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