GLOBAL STABILITY AND BACKWARD BIFURCATION OF A GENERAL VIRAL INFECTION MODEL WITH VIRUS-DRIVEN PROLIFERATION OF TARGET CELLS

In this paper, a general viral model with virus-driven proliferation of target cells is studied. Global stability results are established by employing the Lyapunov method and a geometric approach developed by Li and Muldowney. It is shown that under certain conditions, the model exhibits a global threshold dynamics, while if these conditions are not met, then backward bifurcation and bistability are possible. An example is presented to provide some insights on how the virus-driven proliferation of target cells influences the virus dynamics and the drug therapy strategies.

[1]  J. P. Lasalle.  The stability of dynamical systems , 1976 .

[2]  F. V. Vleck,et al.  Stability and Asymptotic Behavior of Differential Equations , 1965 .

[3]  James S. Muldowney,et al.  On Bendixson′s Criterion , 1993 .

[4]  Michael Y. Li,et al.  Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells. , 2006, Mathematical biosciences.

[5]  Denise E. Kirschner,et al.  Using Mathematics to Understand HIV Immune Dynamics , 1997 .

[6]  Jane M Heffernan,et al.  Monte Carlo estimates of natural variation in HIV infection. , 2005, Journal of theoretical biology.

[7]  Hongying Shu,et al.  Joint effects of mitosis and intracellular delay on viral dynamics: two-parameter bifurcation analysis , 2012, Journal of mathematical biology.

[8]  M. Li,et al.  Global dynamics of a SEIR model with varying total population size. , 1999, Mathematical Biosciences.

[9]  Hongying Shu,et al.  Role of CD4 + T-cell proliferation in HIV infection under antiretroviral therapy , 2012 .

[10]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[11]  G. García-Ramos,et al.  Fighting a virus with a virus: a dynamic model for HIV-1 therapy. , 2003, Mathematical biosciences.

[12]  J. Watmough,et al.  Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. , 2002, Mathematical biosciences.

[13]  A S Perelson,et al.  Drug concentration heterogeneity facilitates the evolution of drug resistance. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

[14]  A. Perelson,et al.  HIV-1 Dynamics in Vivo: Virion Clearance Rate, Infected Cell Life-Span, and Viral Generation Time , 1996, Science.

[15]  Shigui Ruan,et al.  Uniform persistence and flows near a closed positively invariant set , 1994 .

[16]  Xingfu Zou,et al.  DYNAMICS OF A HIV-1 INFECTION MODEL WITH CELL-MEDIATED IMMUNE RESPONSE AND INTRACELLULAR DELAY , 2009 .

[17]  Redouane Qesmi,et al.  Influence of backward bifurcation in a model of hepatitis B and C viruses. , 2010, Mathematical biosciences.

[18]  A. Korobeinikov.  Global properties of basic virus dynamics models , 2004, Bulletin of mathematical biology.

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

[20]  Michael Y. Li,et al.  Impact of Intracellular Delays and Target-Cell Dynamics on In Vivo Viral Infections , 2010, SIAM J. Appl. Math..

[21]  Liancheng Wang,et al.  HIV infection and CD4+ T cell dynamics , 2006 .

[22]  Paul Waltman,et al.  Persistence in dynamical systems , 1986 .

[23]  James S. Muldowney,et al.  Compound matrices and ordinary differential equations , 1990 .

[24]  N. P. Bhatia,et al.  Dynamical Systems: Stability, Theory and Applications , 1967 .

[25]  M. Hirsch,et al.  4. Monotone Dynamical Systems , 2005 .

[26]  Wendi Wang,et al.  Global stability of a five-dimensional model with immune responsesand delay , 2011 .

[27]  M. Nowak,et al.  Population Dynamics of Immune Responses to Persistent Viruses , 1996, Science.

[28]  T. Kajiwara,et al.  Stability analysis of pathogen-immune interaction dynamics , 2005, Journal of mathematical biology.

[29]  Alan S. Perelson,et al.  Mathematical Analysis of HIV-1 Dynamics in Vivo , 1999, SIAM Rev..

[30]  Hal L. Smith,et al.  Virus Dynamics: A Global Analysis , 2003, SIAM J. Appl. Math..

[31]  Paul Waltman,et al.  A brief survey of persistence in dynamical systems , 1991 .

[32]  James S. Muldowney,et al.  A Geometric Approach to Global-Stability Problems , 1996 .

[33]  A. Perelson,et al.  Complex patterns of viral load decay under antiretroviral therapy: influence of pharmacokinetics and intracellular delay. , 2004, Journal of theoretical biology.

[34]  Robert H. Martin.  Logarithmic norms and projections applied to linear differential systems , 1974 .