Nonstandard finite difference scheme for a diffusive within-host virus dynamics model with both virus-to-cell and cell-to-cell transmissions

Abstract In this paper, we first propose a diffusive within-host virus dynamics model with both virus-to-cell and cell-to-cell transmissions. Then, we consider the discretization of the model by using nonstandard finite difference scheme. It is then followed by the investigation of the global stability of the equilibria of the discrete model. Our study shows that if the basic reproduction number R 0 ≤ 1 , then the infection-free equilibrium is globally asymptotically stable; however if R 0 > 1 , then the infection equilibrium is globally asymptotically stable. Numerical simulations are presented to illustrate our theoretical results.

[1]  Xiaohua Ding,et al.  Lyapunov functions and global stability for a discretized multigroup SIR epidemic model , 2015 .

[2]  Xiaohua Ding,et al.  A non-standard finite difference method for a hepatitis B virus infection model with spatial diffusion , 2014 .

[3]  Rafael J. Villanueva,et al.  Nonstandard numerical methods for a mathematical model for influenza disease , 2008, Math. Comput. Simul..

[4]  Juan Carlos Cortés,et al.  Non-standard numerical method for a mathematical model of RSV epidemiological transmission , 2008, Comput. Math. Appl..

[5]  R. Mickens.  Nonstandard Finite Difference Models of Differential Equations , 1993 .

[6]  Michael Emerman,et al.  An In Vitro Rapid-Turnover Assay for Human Immunodeficiency Virus Type 1 Replication Selects for Cell-to-Cell Spread of Virus , 2000, Journal of Virology.

[7]  Kaifa Wang,et al.  Propagation of HBV with spatial dependence. , 2007, Mathematical biosciences.

[8]  Jinliang Wang,et al.  Analysis of an HIV infection model with logistic target-cell growth and cell-to-cell transmission , 2015 .

[9]  Luca Lorenzi,et al.  Heterogeneous Viral Environment in a HIV Spatial Model , 2009, 0905.2023.

[10]  Yu Yang,et al.  Global stability of a diffusive virus dynamics model with general incidence function and time delay , 2015 .

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

[12]  Rui Xu,et al.  An HBV model with diffusion and time delay. , 2009, Journal of theoretical biology.

[13]  Sergei S. Pilyugin,et al.  Global analysis of within host virus models with cell-to-cell viral transmission , 2014 .

[14]  K. Hattaf,et al.  A numerical method for a delayed viral infection model with general incidence rate , 2016 .

[15]  Khalid Hattaf,et al.  Global stability for reaction-diffusion equations in biology , 2013, Comput. Math. Appl..

[16]  Tailei Zhang,et al.  Effect of discretization on dynamical behavior of SEIR and SIR models with nonlinear incidence , 2015, Appl. Math. Lett..

[17]  Yiyi Zhang,et al.  Dynamics of a diffusive HBV model with delayed Beddington–DeAngelis response , 2014 .

[18]  Qiulin Tang,et al.  An SEI-SI avian-human influenza model with diffusion and nonlocal delay , 2014, Appl. Math. Comput..

[19]  J. M. Murray,et al.  Turing Patterns from Dynamics of Early HIV Infection , 2012, Bulletin of mathematical biology.

[20]  Yu Huang,et al.  Global dynamics of a PDE in-host viral model , 2014 .

[21]  Ron Milo,et al.  Cell-to-cell spread of HIV permits ongoing replication despite antiretroviral therapy , 2011, Nature.

[22]  C. Bangham,et al.  The immune control and cell-to-cell spread of human T-lymphotropic virus type 1. , 2003, The Journal of general virology.

[23]  T. Fujimoto,et al.  TWO CHARACTERIZATIONS OF INVERSE-POSITIVE MATRICES: THE HAWKINS-SIMON CONDITION AND THE LE CHATELIER-BRAUN PRINCIPLE ∗ , 2004 .

[24]  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.

[25]  Kaifa Wang,et al.  Dynamics of an HBV model with diffusion and delay. , 2008, Journal of theoretical biology.

[26]  R Blumenthal,et al.  Quantitation of human immunodeficiency virus type 1 infection kinetics , 1993, Journal of virology.

[27]  Yukihiko Nakata,et al.  Global dynamics of difference equations for SIR epidemic models with a class of nonlinear incidence rates , 2012 .

[28]  Xiulan Lai,et al.  Modeling HIV-1 Virus Dynamics with Both Virus-to-Cell Infection and Cell-to-Cell Transmission , 2014, SIAM J. Appl. Math..

[29]  Yukihiko Nakata,et al.  Global stability for a discrete epidemic model for disease with immunity and latency spreading in a heterogeneous host population , 2012 .

[30]  K. Hattaf,et al.  Global properties of a discrete viral infection model with general incidence rate , 2016 .

[31]  Sebastian Bonhoeffer,et al.  Spatial models of virus-immune dynamics. , 2005, Journal of theoretical biology.

[32]  Khalid Hattaf,et al.  A generalized HBV model with diffusion and two delays , 2015, Comput. Math. Appl..

[33]  G. Webb,et al.  A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay , 2003, Journal of mathematical biology.

[34]  S. Ruan,et al.  Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions. , 2015, Mathematical biosciences.

[35]  X. Zou,et al.  Modeling cell-to-cell spread of HIV-1 with logistic target cell growth , 2015 .