Non-standard finite difference method applied to an initial boundary value problem describing hepatitis B virus infection

In this paper, two non-standard finite difference (NSFD) schemes are proposed for a mathematical model of hepatitis B virus (HBV) infection with spatial dependence. The dynamic properties of the obtained discretized systems are completely analyzed. Relying on the theory of M-matrix, we prove that th...

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Veröffentlicht in:Journal of difference equations and applications Jg. 26; H. 1; S. 122 - 139
Hauptverfasser: Tadmon, Calvin, Foko, Severin
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Abingdon Taylor & Francis 02.01.2020
Taylor & Francis Ltd
Schlagworte:
Asymptotic properties
Boundary value problems
Computer simulation
diffusion
Discretization
Finite difference method
global stability
HBV infection
Hepatitis
Hepatitis B
Liapunov functions
Lyapunov function
Mathematical analysis
Mathematical models
Matrix methods
non-standard finite difference scheme
numerical simulations
positivity
Viruses
ISSN:1023-6198, 1563-5120
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Abstract In this paper, two non-standard finite difference (NSFD) schemes are proposed for a mathematical model of hepatitis B virus (HBV) infection with spatial dependence. The dynamic properties of the obtained discretized systems are completely analyzed. Relying on the theory of M-matrix, we prove that the proposed NSFD schemes is unconditionally positive. Furthermore, we establish that the NSFD method used preserves all constant steady states of the corresponding continuous initial boundary value problem (IBVP) model. We prove that the conditions for those equilibria to be asymptotically stable are consistent with the continuous IBVP model independently of the numerical grid size. The global asymptotical properties of the HBV-free equilibrium of the proposed NSFD schemes are derived via the construction of a suitable discrete Lyapunov function, and coincides with the continuous system. This confirms that the discretized models are dynamically consistent since they maintain essential properties of the corresponding continuous IBVP model. Finally, numerical simulations are performed from which it is demonstrated that the proposed NSFD method is advantageous over the standard finite difference (SFD) method.
AbstractList In this paper, two non-standard finite difference (NSFD) schemes are proposed for a mathematical model of hepatitis B virus (HBV) infection with spatial dependence. The dynamic properties of the obtained discretized systems are completely analyzed. Relying on the theory of M-matrix, we prove that the proposed NSFD schemes is unconditionally positive. Furthermore, we establish that the NSFD method used preserves all constant steady states of the corresponding continuous initial boundary value problem (IBVP) model. We prove that the conditions for those equilibria to be asymptotically stable are consistent with the continuous IBVP model independently of the numerical grid size. The global asymptotical properties of the HBV-free equilibrium of the proposed NSFD schemes are derived via the construction of a suitable discrete Lyapunov function, and coincides with the continuous system. This confirms that the discretized models are dynamically consistent since they maintain essential properties of the corresponding continuous IBVP model. Finally, numerical simulations are performed from which it is demonstrated that the proposed NSFD method is advantageous over the standard finite difference (SFD) method.
Author Foko, Severin
Tadmon, Calvin
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SubjectTerms Asymptotic properties
Boundary value problems
Computer simulation
diffusion
Discretization
Finite difference method
global stability
HBV infection
Hepatitis
Hepatitis B
Liapunov functions
Lyapunov function
Mathematical analysis
Mathematical models
Matrix methods
non-standard finite difference scheme
numerical simulations
positivity
Viruses
Title Non-standard finite difference method applied to an initial boundary value problem describing hepatitis B virus infection
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