Lyapunov Functions and Global Stability for Age-Structured HIV Infection Model

We study the basic age-structured population model describing the HIV infection process, which is defined by PDEs. The model allows the production rate of viral particles and the death rate of productively infected cells to vary and depend on the infection age. By using the direct Lyapunov method and constructing suitable Lyapunov functions, dynamical properties of the age-structured model without (or with) drug treatment are established. The results show that the global asymptotic stability of the infection-free steady state and the infected steady state depends only on the basic reproductive number determined by the burst size. Further, we establish mathematically that the typical ODE and DDE (delay differential equation) models of HIV infection are equivalent to two special cases of the above PDE models.

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

[2]  Michael Y. Li,et al.  Global Dynamics of an In-host Viral Model with Intracellular Delay , 2010, Bulletin of mathematical biology.

[3]  S. Ruan,et al.  A delay-differential equation model of HIV infection of CD4(+) T-cells. , 2000, Mathematical biosciences.

[4]  Libin Rong,et al.  The influence of anti-viral drug therapy on the evolution of HIV-1 pathogens , 2006, Disease Evolution: Models, Concepts, and Data Analyses.

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

[6]  A. Perelson,et al.  A model of HIV-1 pathogenesis that includes an intracellular delay. , 2000, Mathematical biosciences.

[7]  Tanya Kostova,et al.  Nonlinear age-dependent population dynamics with constant size , 1991 .

[8]  A. Perelson,et al.  Influence of delayed viral production on viral dynamics in HIV-1 infected patients. , 1998, Mathematical biosciences.

[9]  M A Nowak,et al.  Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[10]  A. Perelson,et al.  Optimizing within-host viral fitness: infected cell lifespan and virion production rate. , 2004, Journal of theoretical biology.

[11]  Jianhong Wu,et al.  A Hepatitis B and C Virus Model with Age since Infection that Exhibits Backward Bifurcation , 2011, SIAM J. Appl. Math..

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

[13]  Carlos Castillo-Chavez,et al.  How May Infection-Age-Dependent Infectivity Affect the Dynamics of HIV/AIDS? , 1993, SIAM J. Appl. Math..

[14]  M. Nowak,et al.  Virus dynamics: Mathematical principles of immunology and virology , 2001 .

[15]  Weinian Zhang,et al.  An Age-Structured Model for the Transmission Dynamics of Hepatitis B , 2010, SIAM J. Appl. Math..

[16]  Andrei Korobeinikov,et al.  Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility. , 2013, Mathematical biosciences and engineering : MBE.

[17]  Andrei Korobeinikov,et al.  Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence and non-linear incidence rate. , 2009, Mathematical medicine and biology : a journal of the IMA.

[18]  C. Connell McCluskey.  Delay versus age-of-infection - Global stability , 2010, Appl. Math. Comput..

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

[20]  Yasuhiro Takeuchi,et al.  Global Stability for Delay SIR and SEIR Epidemic Models with Nonlinear Incidence Rate , 2010, Bulletin of mathematical biology.

[21]  G. Webb,et al.  Lyapunov functional and global asymptotic stability for an infection-age model , 2010 .

[22]  Patrick W Nelson,et al.  Mathematical analysis of delay differential equation models of HIV-1 infection. , 2002, Mathematical biosciences.

[23]  Gang Huang,et al.  Lyapunov Functionals for Delay Differential Equations Model of Viral Infections , 2010, SIAM J. Appl. Math..

[24]  Shengqiang Liu,et al.  Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy. , 2010, Mathematical biosciences and engineering : MBE.

[25]  Daniel Coombs,et al.  An age-structured model of hiv infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells. , 2004, Mathematical biosciences and engineering : MBE.

[26]  Christian L. Althaus,et al.  Reassessing the Human Immunodeficiency Virus Type 1 Life Cycle through Age-Structured Modeling: Life Span of Infected Cells, Viral Generation Time, and Basic Reproductive Number, R0 , 2009, Journal of Virology.

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

[28]  Alan S. Perelson,et al.  Mathematical Analysis of Age-Structured HIV-1 Dynamics with Combination Antiretroviral Therapy , 2007, SIAM J. Appl. Math..

[29]  A. Haase,et al.  Determination of simian immunodeficiency virus production by infected activated and resting cells , 2007, AIDS.