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.
Similar content being viewed by others
References
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.
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.
R.V. Culshaw and S. Ruan, A delay differential equation madel of HIV infection of CD4+ T cells, Math. Biosci., 165(2000), 27–39.
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.
R.A. Goldsby, T.J. Kindt, and B.A. Osborne, “Kuby Immunology”, 4th edn., W.H. Freeman & Company, New York, 2000.
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.
J.K. Hale, “Theory of Functional Differential Equations”, Springer Verlag, New York, 1977.
C.A. Janeway, P. Travers, M. Walport, and M.J. Schlomchik, “Immunobiology” 6th edn., Garland Science Publishing, 2005.
M.A. Nowak and R.M. May, “Virus Dynamics”, Oxford University Press, 2000.
M.A. Nowak and A.J. McMichael, How HIV defeats the immune system, Scientific American, August (1995), 58–65.
A.S. Perelson, D.E. Kirschner, and R. De Boer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci., 114(1993), 81–125.
A.S. Perelson and P.W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41(1999), 3–44.
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.
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.
UNAIDS/WHO AIDS epidemic update 2006, http://www.who.int/hiv/mediacentre/2006_EpiUpdate_en.pdf
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.
Author information
Authors and Affiliations
Corresponding author
Rights 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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12591-008-0006-2