Analysis of a viral infection model with delayed immune response

Abstract It is well known that the immune response plays an important role in eliminating or controlling the disease after human body is infected by virus. In this paper, we investigate the dynamical behavior of a viral infection model with retarded immune response. The effect of time delay on stability of the equilibria of the system has been studied and sufficient condition for local asymptotic stability of the infected equilibrium and global asymptotic stability of the infection-free equilibrium and the immune-exhausted equilibrium are given. By numerical simulating,we observe that the stationary solution becomes unstable at some critical immune response time, while the delay time and birth rate of susceptible host cells increase, and we also discover the occurrence of stable periodic solutions and chaotic dynamical behavior. The results can be used to explain the complexity of the immune state of patients.

[1]  D. Wodarz,et al.  Persistent Virus Infection despite Chronic Cytotoxic T-Lymphocyte Activation in Gamma Interferon-Deficient Mice Infected with Lymphocytic Choriomeningitis Virus , 2000, Journal of Virology.

[2]  K. Cooke,et al.  On zeroes of some transcendental equations , 1986 .

[3]  M A Nowak,et al.  Virus dynamics and drug therapy. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

[4]  A S Perelson,et al.  Target cell limited and immune control models of HIV infection: a comparison. , 1998, Journal of theoretical biology.

[5]  Y. Kuang.  Delay Differential Equations: With Applications in Population Dynamics , 2012 .

[6]  Dominik Wodarz,et al.  The importance of lytic and nonlytic immune responses in viral infections. , 2002, Trends in immunology.

[7]  M A Nowak,et al.  Viral dynamics in hepatitis B virus infection. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[8]  Xianning Liu,et al.  Complex dynamic behavior in a viral model with delayed immune response , 2007 .

[9]  A S Perelson,et al.  Towards a general function describing T cell proliferation. , 1995, Journal of theoretical biology.

[10]  H. Hengartner,et al.  Different roles for cytotoxic T cells in the control of infections with cytopathic versus noncytopathic viruses. , 1996, Current opinion in immunology.

[11]  Xianning Liu,et al.  Global stability in a viral infection model with lytic and nonlytic immune responses , 2006, Comput. Math. Appl..

[12]  D. Montefiori,et al.  Control of viremia in simian immunodeficiency virus infection by CD8+ lymphocytes. , 1999, Science.

[13]  Nebojša Vasović,et al.  Time delay in a basic model of the immune response , 2001 .