Skip to main content

Advertisement

Log in

Hopf bifurcation and periodic solutions in a dynamical model for HIV and immune response

  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

A mathematical model for the effect of immune response on the dynamics of HIV is considered and analyzed. Further, delay in the activation of CTLs (Cytotoxic T Lymphocytes) is accounted in model and existence of the Hopf bifurcation and periodic solutions is investigated by considering delay as parameter. The direction, stability and period of the bifurcating periodic solutions are found explicitly.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. S. Bonhoeffer, R.M. May, G.M. Shaw, and M. Nowak, Virus dynamics and drug therapy, Proc. Nat. Acad. Sci. USA, 94(1997), 6971–6976.

    Article  Google Scholar 

  2. M.S. Ciupe, B.L. Bivort, D.M. Bortz, and P.W. Nelson, Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models, Math. Biosci., 200(2006), 1–27.

    Article  MATH  MathSciNet  Google Scholar 

  3. R.V. Culshaw and S. Ruan, A delay differential equation madel of HIV infection of CD4+ T cells, Math. Biosci., 165(2000), 27–39.

    Article  MATH  Google Scholar 

  4. H.I. Freedman and V. Sree Hari Rao, The trade-off between mutual interference and time lags in predator-prey systems. Bull. Math. Biol., 45(1983), 991–1004.

    MATH  MathSciNet  Google Scholar 

  5. R.A. Goldsby, T.J. Kindt, and B.A. Osborne, “Kuby Immunology”, 4th edn., W.H. Freeman & Company, New York, 2000.

    Google Scholar 

  6. B.D. Hassard, N.D. Kazrinoff, and W.H. Wan, “Theory and Application of Hopf Bifurcation”, London Math Society Lect. Note Series, Vol. 41, Cambridge University Press, 1981.

  7. J.K. Hale, “Theory of Functional Differential Equations”, Springer Verlag, New York, 1977.

    MATH  Google Scholar 

  8. C.A. Janeway, P. Travers, M. Walport, and M.J. Schlomchik, “Immunobiology” 6th edn., Garland Science Publishing, 2005.

  9. M.A. Nowak and R.M. May, “Virus Dynamics”, Oxford University Press, 2000.

  10. M.A. Nowak and A.J. McMichael, How HIV defeats the immune system, Scientific American, August (1995), 58–65.

  11. A.S. Perelson, D.E. Kirschner, and R. De Boer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci., 114(1993), 81–125.

    Article  MATH  Google Scholar 

  12. A.S. Perelson and P.W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41(1999), 3–44.

    Article  MATH  MathSciNet  Google Scholar 

  13. M. Piatak Jr., M.S. Saag, L.C. Yang, S.J. Clark, J.C. Kappes, K.C. Luk et al., High levels of HIV-1 in plasma during all stages of infection determined by competitive PCR, Science, 259(1993), 17491754.

    Article  Google Scholar 

  14. X. Song, X. Zhou, and X. Shi, Analysis of stability and Hopf bifurcation for a delay differential equation model of HIV infection of CD 4+ T-cells, Chaos, Solitons and Fractals, 2007.

  15. UNAIDS/WHO AIDS epidemic update 2006, http://www.who.int/hiv/mediacentre/2006_EpiUpdate_en.pdf

  16. D. Wodarz and M.A. Nowak, Specific therapy regimes could lead to a long-term immunological control of HIV. Proc. Nat. Acad. Sci, USA, 96(1999), 14464–14469.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to P. K. Srivastava.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Srivastava, P.K., Chandra, P. Hopf bifurcation and periodic solutions in a dynamical model for HIV and immune response. Differ Equ Dyn Syst 16, 77–100 (2008). https://doi.org/10.1007/s12591-008-0006-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-008-0006-2

MSC

Keywords

Navigation