Computational and Applied Mathematics

, Volume 37, Issue 3, pp 3780–3805 | Cite as

Global stability of a diffusive and delayed virus infection model with general incidence function and adaptive immune response

  • Hui Miao
  • Zhidong TengEmail author
  • Xamxinur Abdurahman
  • Zhiming Li


In this paper, the dynamical behaviors for a five-dimensional virus infection model with diffusion and two delays which describes the interactions of antibody, cytotoxic T-lymphocyte (CTL) immune responses and a general incidence function are investigated. The reproduction numbers for virus infection, antibody immune response, CTL immune response, CTL immune competition and antibody immune competition, respectively, are calculated. By using the Lyapunov functionals and linearization methods, the threshold conditions on the global stability of the equilibria for infection-free, immune-free, antibody response, CTL response and antibody and CTL responses, respectively, are established if the space is assumed as homogeneous. When the space is inhomogeneous, the effects of diffusion, intracellular delay and production delay are obtained by the numerical simulations.


Virus infection model Delay Adaptive immune response Diffusion General incidence function Global stability 

Mathematics Subject Classification

34D40 35Q92 92B05 



This work was supported by the Natural Science Foundation of Shanxi University of Finance and Economics (Starting Fund for the Shanxi University of Finance and Economics doctoral graduates research, Grant No. Z18116), the National Natural Science Foundation of China (Grant nos. 11771373 and 11661076), the Natural Science Foundation of Xinjiang (Grant no. 2016D03022) and the Doctorial Subjects Foundation of the Ministry of Education of China (Grant no. 2013651110001).


  1. Balasubramaniam P, Tamilalagan P, Prakash M (2015) Bifurcation analysis of HIV infection model with antibody and cytotoxic T-lymphocyte immune responses and Beddington–DeAngelis functional response. Math Methods Appl Sci 38:1330–1341MathSciNetCrossRefzbMATHGoogle Scholar
  2. Beddington JR (1975) Mutual interference between parasites or predators and its effect on searching efficiency. J Anim Ecol 44:331–340CrossRefGoogle Scholar
  3. Culshaw RV, Ruan S, Spiteri RJ (2004) Optimal HIV treatment by maximising immune response. J Math Biol 48:545–562MathSciNetCrossRefzbMATHGoogle Scholar
  4. DeAngelis DL, Goldstein RA, ÓNeill RV (1975) A model for trophic interaction. Ecology 56:881–892CrossRefGoogle Scholar
  5. Gourley SA, So JWH (2002) Dynamics of a food-limited population model incorporating nonlocal delays on a finite domain. J Math Biol 44:49–78MathSciNetCrossRefzbMATHGoogle Scholar
  6. Hale JK, Verduyn SML (1993) Introduction to functional differential equations. Springer, New YorkCrossRefzbMATHGoogle Scholar
  7. Hattaf K, Yousfi N (2013) Global stability for reaction–diffusion equations in biology. Comput Math Appl 66:1488–1497MathSciNetCrossRefzbMATHGoogle Scholar
  8. Hattaf K, Yousfi N (2015) A generalized HBV model with diffusion and two delays. Comput Math Appl 69:31–40MathSciNetCrossRefzbMATHGoogle Scholar
  9. Henry D (1993) Geometric theory of semilinear parabolic equations. In: Lecture notes in mathematics. Springer, BerlinGoogle Scholar
  10. Huang G, Ma W, Takeuchi Y (2011) Global analysis for delay virus dynamics model with Beddington–DeAngelis function response. Appl Math Lett 24:1199–1203MathSciNetCrossRefzbMATHGoogle Scholar
  11. Ji Y (2015) Global stability of a multiple delayed viral infection model with general incidence rate and an application to HIV infection. Math Biosci Eng 12:525–536MathSciNetCrossRefzbMATHGoogle Scholar
  12. Li D, Ma W (2007) Asymptotic properties of an HIV-1 infection model with time delay. J Math Anal Appl 335:683–691MathSciNetCrossRefzbMATHGoogle Scholar
  13. Lu X, Hui L, Liu S, Li J (2015) A mathematical model of HIV-1 infection with two time delays. Math Biosci Eng 12:431–449MathSciNetCrossRefzbMATHGoogle Scholar
  14. McCluskey CC, Yang Y (2015) Global stability of a diffusive virus dynamics model with general incidence function and time delay. Nonlinear Anal RWA 25:64–78MathSciNetCrossRefzbMATHGoogle Scholar
  15. Nelson P, Perelson JM (2000) A model of HIV-1 pathogenesis that includes an intracelluar delay. Math Biosci 163:201–215MathSciNetCrossRefzbMATHGoogle Scholar
  16. Nowak MA, Bangham CRM (1996) Population dynamics of immune response to persistent viruses. Science 272:74–79CrossRefGoogle Scholar
  17. Pawelek KA, Liu S, Pahlevani F, Rong L (2012) A model of HIV-1 infection with two time delays: mathematical analysis and comparison with patient data. Math Biosci 235:98–109MathSciNetCrossRefzbMATHGoogle Scholar
  18. Perelson AS, Kirschner DE, Boer RD (1993) Dynamics of HIV infection of CD4\(^+\) T cells. Math Biosci 114:81–125CrossRefzbMATHGoogle Scholar
  19. Protter MH, Weinberger HF (1967) Maximum principles in differential equations. Prentice Hall, Englewood CliffszbMATHGoogle Scholar
  20. Redlinger R (1984) Existence theorems for semilinear parabolic systems with functionals. Nonlinear Anal TMA 8:667–682MathSciNetCrossRefzbMATHGoogle Scholar
  21. Shu H, Wang L, Watmoughs J (2013) Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses. SIAM J Appl Math 73:1280–1302MathSciNetCrossRefzbMATHGoogle Scholar
  22. Song X, Neumann A (2007) Global stability and periodic solution of the viral dynamics. J Math Anal Appl 329:281–297MathSciNetCrossRefzbMATHGoogle Scholar
  23. Wang S, Feng X, He Y (2011) Global asymptotical properties for a diffused HBV infection model with CTL immune response and nonlinear incidence. Acta Math Sci 31:1959–1967MathSciNetCrossRefGoogle Scholar
  24. Wang X, Elaiw A, Song X (2012) Global properties of a delayed HIV infection model with CTL immune response. Appl Math Comput 218:9405–9414MathSciNetzbMATHGoogle Scholar
  25. Wang Y, Zhou Y, Brauer F, Heffernan JM (2013) Viral dynamics model with CTL immune response incorporating antiretroviral therapy. J Math Biol 67:901–934MathSciNetCrossRefzbMATHGoogle Scholar
  26. Wang F, Huang Y, Zou X (2014) Global dynamics of a PDE in-host viral model. Appl Anal 93:2312–2329MathSciNetCrossRefzbMATHGoogle Scholar
  27. Wang J, Pang J, Kuniya T, Enatsu Y (2014) Global threshold dynamics in a five-dimensional virus model with cell-mediated, humoral immune responses and distributed delays. Appl Math Comput 241:298–316MathSciNetzbMATHGoogle Scholar
  28. Wodarz D (2003) Hepatitis C virus dynamics and pathology: the role of CTL and antibody responses. J Gen Virol 84:1743–1750CrossRefGoogle Scholar
  29. Wu J (1996) Theory and applications of partial functional differential equations. Springer, NewYorkCrossRefzbMATHGoogle Scholar
  30. Xiang H, Feng L, Huo H (2013) Stability of the virus dynamics model with Beddington–DeAngelis functional response and delays. Appl Math Model 37:5414–5423MathSciNetCrossRefGoogle Scholar
  31. Xu R, Ma Z (2009) An HBV model with diffusion and time delay. J Theor Biol 257:499–509MathSciNetCrossRefGoogle Scholar
  32. Yan Y, Wang W (2012) Global stability of a five-dimensional model with immune responses and delay. Discrete Contin Dyn Syst B 17:401–416MathSciNetCrossRefzbMATHGoogle Scholar
  33. Yang Y, Xu Y (2016) Global stability of a diffusive and delayed virus dynamics model with Beddington–DeAngelis incidence function and CTL immune response. Comput Math Appl 71:922–930MathSciNetCrossRefGoogle Scholar
  34. 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 10:483–498MathSciNetCrossRefzbMATHGoogle Scholar
  35. Zhang Y, Xu Z (2014) Dynamics of a diffusive HBV model with delayed Beddington–DeAngelis response. Nonlinear Anal RWA 15:118–139MathSciNetCrossRefzbMATHGoogle Scholar
  36. Zhou X, Cui J (2011) Global stability of the viral dynamics with Crowley–Martin functional response. Bull Korean Math Soc 48:555–574MathSciNetCrossRefzbMATHGoogle Scholar
  37. Zhu H, Zou X (2009) Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay. Discrete Contin Dyn Syst Ser B 12:511–524MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017

Authors and Affiliations

  • Hui Miao
    • 1
  • Zhidong Teng
    • 2
    Email author
  • Xamxinur Abdurahman
    • 2
  • Zhiming Li
    • 2
  1. 1.School of Applied MathematicsShanxi University of Finance and EconomicsTaiyuanPeople’s Republic of China
  2. 2.College of Mathematics and System SciencesXinjiang UniversityUrumqiPeople’s Republic of China

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