Abstract
In this paper, a class of virus infection models with CTLs response is considered. We incorporate three delays with immune impairment into the virus infection model. We begin model analysis with proving the positivity and boundedness of the solutions, local stability of the infection-free and infected steady states, and uniform persistence of the system. By analyzing corresponding characteristic equation, the local stability of each feasible equilibria and the existence of Hopf bifurcation at the infected equilibrium are addressed, respectively. By fixing the immune delay as a bifurcation parameter, we get a limit cycle bifurcation about the infected steady state. Using center manifold argument and normal form theory, we derive explicit formulae to determine the stability and direction of the limit cycles of the model. By developing a few Lyapunov functionals, we obtain conditions ensuring global stability of steady states at \(\tau _3<\tau _3^*\). By theoretical analysis and numerical simulations, we show that the immune impairment rate has a dramatic effect on the infected equilibrium of the DDE model.
Similar content being viewed by others
References
Ho, D.D., Neumann, A.U., Perelson, A.S., Chen, W., Leonard, J.M., Markowitz, M.: Rapid turnover of plasma virions and \(\mathit{CD}4\) lymphocytes in HIV-1 infection. Nature 373, 123–6 (1995)
Nelson, P.W., Perelson, A.S.: Mathematical analysis of delay differential equation models of HIV-1 infection. Math. Biosci. 179, 73–94 (2002)
Nelson, P.W., Murray, J., Perelson, A.S.: A model of HIV-1 pathogenesis that includes an intracellular delay. Math. Biosci. 163, 201–15 (2000)
Perelson, A.S., Kirschner, D.E., De Boer, R.: Dynamics of HIV infection of \(\mathit{CD}4^+T\) cells. Math. Biosci. 114, 81–125 (1993)
Perelson, A.S., Neumann, A.U., Markowitz, M., Leonard, J.M., Ho, D.D.: HIV-1 dynamics in vivo: virion clearence rate, infected cell life-span and viral generation time. Science 271, 1582–1586 (1996)
Perelson, A.S., Nelson, P.W.: Mathematical analysis of HIV-1 dynamics in vivo. SIAM. Rev. 41, 3–44 (1999)
Pitchaimani, M., Monica, C.: Global stability analysis of HIV-1 infection model with three time delays. J. Appl. Math. Comput. 48, 293–319 (2014)
Monica, C., Pitchaimani, M.: Analysis of stability and Hopf bifurcation for HIV-1 dynamics with PI and three intracellular delays. Nonlinear Anal. Real World. Appl. 27, 55–69 (2016)
Nowak, M.A., Bangham, C.R.M.: Population dynamics of immune responses to persistent viruses. Science 272, 74–79 (1996)
Wang, K., Wang, W., Pang, H., Liu, X.: Complex dynamic behavior in a viral model with delayed immune response. Phys. D 226, 197–208 (2007)
Song, X., Wang, S., Zhou, X.: Stability and Hopf bifurcation for a viral infection modelwith delayed non-lytic immuneresponse. J. Appl. Math. Comput. 33, 251–265 (2010)
Xie, Q., Huang, D., Zhang, S., Cao, J.: Analysis of a viral infection model with delayed immune response. Appl. Math. Model. 34, 2388–2395 (2010)
Song, X., Wang, S., Dong, J.: Stability properties and Hopf bifurcation of a delayed viral infection model with lytic immune response. J. Math. Anal. Appl. 373, 345–355 (2011)
Zhou, X., Song, X., Shi, X.: Analysis of stability and Hopf bifurcation for an HIV infection model with time delay. Appl. Math. Comput. 199(1), 23–38 (2008)
Buric, N., Mudrinic, M., Vasovic, N.: Time delay in a basic model of the immune response. Chaos Solitons Fract. 12, 483–489 (2001)
Canabarro, A.A., Glria, I.M., Lyra, M.L.: Periodic solutions and chaos in a non-linear model for the delayed cellular immune response. Phys. A 342, 234–241 (2004)
Wang, S., Song, X., Ge, Z.: Dynamics analysis of a delayed viral infection model with immune impairment. Appl. Math. Model. 35, 4877–4885 (2011)
MacDonald, N.: Biological Delay Systems: Linear Stability Theory. Cambridge University, Cambridge (1989)
Hale, J.: Theory of Functional Differential Equations. Springer, New York (1997)
Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, Boston (1993)
Ruan, S., Wei, J.: On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dyn. Cont. Discrete Impuls. Syst. Ser. A Math. Anal. 10, 863–874 (2003)
Cooke, K.L., Grossman, Z.: Discrete delay, distributed delay and stability switches. J. Math. Anal. Appl. 86, 592–627 (1982)
Hale, J., Lunel, S.V.: Introduction to Functional Differential Equations. Springer, New York (1993)
Hassard, B.D., Kazariniff, N.D., Wan, Y.H.: Theory and application of Hopf bifurcation. In: London Mathematical Society Lecture Note Series, vol. 41. Cambridge University Press, Cambridge, MA (1981)
Gantmacher, F.G.: The Theory of Matrices. Chelsea Publ. Co., New York (1959)
White, M., Zhao, X.: Threshold dynamics in a time-delayed epidemic model with dispersal. Math. Biosci. 218, 121 (2009)
Iwami, S., Miura, T., Nakaoka, S., Takeuchi, Y.: Immune impairment in HIV infection: existence of risky and immunodeficiency thresholds. J. Theor. Biol. 260, 490–501 (2009)
Iwami, S., Nakaoka, S., Takeuchi, Y., Miura, T.: Immune impairment thresholds in HIV infection. Immunol. Lett. 123, 149–154 (2009)
Wang, Z., Liu, X.: A chronic viral infection model with immune impairment. J. Theor. Biol. 249, 532–542 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Krishnapriya, P., Pitchaimani, M. Analysis of time delay in viral infection model with immune impairment. J. Appl. Math. Comput. 55, 421–453 (2017). https://doi.org/10.1007/s12190-016-1044-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-016-1044-5