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Global stability of a virus dynamics model with Beddington–DeAngelis incidence rate and CTL immune response

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Abstract

In this paper, the global stability of virus dynamics model with Beddington–DeAngelis infection rate and CTL immune response is studied by constructing Lyapunov functions. We derive the basic reproduction number R 0 and the immune response reproduction number R 0 for the virus infection model, and establish that the global dynamics are completely determined by the values of R 0. We obtain the global stabilities of the disease-free equilibrium E 0, immune-free equilibrium E 1 and endemic equilibrium E ∗ when R 0≤1, R 0>1, R 0>1, respectively.

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References

  1. Perelson, A., Nelson, P.: Mathematical models of HIV dynamics in vivo. SIAM Rev. 41, 3–44 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Perelson, A., Neumann, A., Markowitz, M., Leonard, J., Ho, D.: HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science 271, 1582–1586 (1996)

    Article  Google Scholar 

  3. Nowak, M.A., May, R.M.: Virus Dynamics. Oxford University Press, New York (2000)

    MATH  Google Scholar 

  4. Arnaout, R., Nowak, M., Wodarz, D.: HIV-1 dynamics revisited: biphasic decay by cytotoxic lymphocyte killing? Proc. R. Soc. Lond. B 265, 1347–1354 (2000)

    Article  Google Scholar 

  5. Culshaw, R.V., Ruan, S.G., Spiteri, R.J.: Optimal HIV treatment by maximising immune response. J. Math. Biol. 48, 545–562 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Nowak, M., Bangham, C.: Population dynamics of immune response to persistent viruses. Science 272, 74–79 (1996)

    Article  Google Scholar 

  7. Wang, K.F., Wang, W.D., Liu, X.N.: Global stability in a viral infection model with lytic and nonlytic immune response. Comput. Math. Appl. 51, 1593–1610 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Huang, G., Ma, W.B., Takeuchi, Y.: Global properties for virus dynamics model with Beddington–DeAngelis functional response. Appl. Math. Lett. 22, 1690–1693 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Zhou, X.Y., Shi, X.Y., Zhang, Z.H., Song, X.Y.: Dynamical behavior of a virus dynamics model with CTL immune response. Appl. Math. Comput. 213, 329–347 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Beddington, J.R.: Mutual interference between parasites or predators and its effect on searching efficiency. J. Anim. Ecol. 44, 331–340 (1975)

    Article  Google Scholar 

  11. DeAngelis, D.L., Goldstein, R.A., O’Neill, R.V.: A model for trophic interaction. Ecology 56, 881–892 (1975)

    Article  Google Scholar 

  12. Hale, J.K.: Theory of Functional Differential Equations. Springer, Berlin (1977)

    Book  MATH  Google Scholar 

  13. Hale, J.K., Verduyn Lunel, S.: Introduction to Functional Differential Equations. Springer, New York (1993)

    MATH  Google Scholar 

  14. Korobeinikov, A.: Global properties of basic virus dynamics models. Bull. Math. Biol. 66, 879–883 (2004)

    Article  MathSciNet  Google Scholar 

  15. Korobeinikov, A., Maini, P.K.: A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Math. Biosci. 1, 57–60 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Korobeinikov, A., Wake, G.C.: A Lyapunov functions and global stability for SIR, SIRS and SIS epidemiological models. Appl. Math. Lett. 15, 955–961 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wang, X., Tao, Y.D.: Lyapunov function and global properties OS virus dynamics with CTL immune response. Int. J. Biomath. 4, 443–448 (2008)

    Article  MathSciNet  Google Scholar 

  18. Wang, X., Tao, Y.D., Song, X.Y.: A delayed HIV-1 infection model with Beddington–DeAngelis functional response. Nonlinear Dyn. 62, 67–72 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, Heilongjiang, P.R. China

    Xia Wang

  2. College of Mathematics and Information Science, Xinyang Normal University, Xinyang, 464000, Henan, P.R. China

    Xia Wang, Youde Tao & Xinyu Song

Authors
  1. Xia Wang
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  2. Youde Tao
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  3. Xinyu Song
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Correspondence to Youde Tao.

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Wang, X., Tao, Y. & Song, X. Global stability of a virus dynamics model with Beddington–DeAngelis incidence rate and CTL immune response. Nonlinear Dyn 66, 825–830 (2011). https://doi.org/10.1007/s11071-011-9954-0

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  • DOI: https://doi.org/10.1007/s11071-011-9954-0

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