Dynamics of a PDE viral infection model incorporating cell-to-cell transmission

Abstract This paper is concerned with the global dynamics of a PDE viral infection model with cell-to-cell transmission and spatial heterogeneity. The basic reproduction number ℜ 0 , which is a threshold value that predicts whether the infection will go to extinction or not, is defined in a variational characterization. In quite a general setting in which every parameter can be spatially heterogeneous, it is shown that if ℜ 0 ≤ 1 , then the infection-free steady state is globally asymptotically stable, while if ℜ 0 > 1 , then the system is uniformly persistent and the infection steady state is globally asymptotically stable. The proof is based on the construction of the Lyapunov functions and usage of the Green's first identity. Finally, numerical simulation is performed in order to verify the validity of our theoretical results.

[1]  S. Iwami,et al.  Dynamics of HIV infection in lymphoid tissue network , 2016, Journal of mathematical biology.

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

[3]  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.

[4]  Alan S. Perelson,et al.  Decay characteristics of HIV-1-infected compartments during combination therapy , 1997, Nature.

[5]  C. Cosner,et al.  Spatial Ecology via Reaction-Diffusion Equations , 2003 .

[6]  O. Diekmann,et al.  On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations , 1990, Journal of mathematical biology.

[7]  Xiao-Qiang Zhao,et al.  Robust persistence for semidynamical systems , 2001 .

[8]  D. Dimitrov,et al.  Cell-to-cell spread of HIV-1 occurs within minutes and may not involve the participation of virus particles. , 1992, Virology.

[9]  H. Weinberger,et al.  Maximum principles in differential equations , 1967 .

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

[11]  Dominik Wodarz,et al.  Multiploid Inheritance of HIV-1 during Cell-to-Cell Infection , 2011, Journal of Virology.

[12]  Jaume Llibre,et al.  Detecting periodic orbits in some 3D chaotic quadratic polynomial differential systems , 2015 .

[13]  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..

[14]  G. Bocharov,et al.  Recombination: Multiply infected spleen cells in HIV patients , 2002, Nature.

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

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

[17]  G. Webb.  A reaction-diffusion model for a deterministic diffusive epidemic , 1981 .

[18]  Q. Sattentau,et al.  Cell-to-cell HIV-1 spread and its implications for immune evasion , 2009, Current opinion in HIV and AIDS.

[19]  Eric Ávila-Vales,et al.  Analysis of a HBV Model with Diffusion and Time Delay , 2012, J. Appl. Math..

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

[21]  G. Webb.  Theory of Nonlinear Age-Dependent Population Dynamics , 1985 .

[22]  Xiao-Qiang Zhao,et al.  A reaction–diffusion malaria model with incubation period in the vector population , 2011, Journal of mathematical biology.

[23]  Alan S Perelson,et al.  Quantifying T lymphocyte turnover. , 2013, Journal of theoretical biology.

[24]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[25]  Xiao-Qiang Zhao,et al.  Dynamical systems in population biology , 2003 .

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

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

[28]  Yuan Lou,et al.  Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model , 2008 .

[29]  Xiao-Qiang Zhao,et al.  Global Attractors and Steady States for Uniformly Persistent Dynamical Systems , 2005, SIAM J. Math. Anal..

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

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