Analysis of a non-standard finite-difference-method for the classical target cell limited dynamical within-host HIV-model - Numerics and applications

Benjamin Wacker; Jan Christian Schlüter; Benjamin Wacker; Jan Christian Schlüter

doi:10.3934/mbe.2025086

Mathematical Biosciences and Engineering

2025, Volume 22, Issue 9: 2360-2390. doi: 10.3934/mbe.2025086

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Research article

Analysis of a non-standard finite-difference-method for the classical target cell limited dynamical within-host HIV-model - Numerics and applications

Benjamin Wacker ^{1
,
,},
Jan Christian Schlüter ^{2,3
,
,}

1.
Department of Engineering and Natural Sciences, University of Applied Sciences Merseburg, Eberhard-Leibnitz-Str. 2, D-06217 Merseburg, Germany
2.
Chair of Data Science, Faculty of Management, Social Work and Construction, HAWK, Haarmannplatz 3, D-37603 Holzminden, Germany
3.
Computational Epidemiology and Public Health Research Group, Institute for Medical Epidemiology, Biometrics and Informatics, Interdisciplinary Center for Health Sciences, Martin Luther University Halle-Wittenberg, Magdeburger Str. 8, D-06112 Halle, Germany

Received: 12 April 2025 Revised: 24 June 2025 Accepted: 02 July 2025 Published: 22 July 2025

Mathematical modeling and numerical simulation are valuable tools for getting theoretical insights into dynamic processes such as, for example, within-host virus dynamics or disease transmission between individuals. In this work, we propose a new time discretization, a so-called non-standard finite-difference-method, for numerical simulation of the classical target cell limited dynamical within-host HIV-model. In our case, we use a non-local approximation of our right-hand-side function of our dynamical system. This means that this right-hand-side function is approximated by current and previous time steps of our non-equidistant time grid. In contrast to classical explicit time stepping schemes such as Runge-Kutta methods which are often applied in these simulations, the main advantages of our novel time discretization method are preservation of non-negativity, often occurring in biological or physical processes, and convergence towards the correct equilibrium point, independently of the time step size. Additionally, we prove boundedness of our time-discrete solution components which underline biological plausibility of the time-continuous model, and linear convergence towards the time-continuous problem solution. We also construct higher-order non-standard finite-difference-methods from our first-order suggested model by modifying ideas from Richardson's extrapolation. This extrapolation idea improves accuracy of our time-discrete solutions. We finally underline our theoretical findings by numerical experiments.
- Boundedness,
- convergence,
- non-negativity,
- non-standard finite-difference-method,
- numerical simulation,
- within-host HIV-model
Citation: Benjamin Wacker, Jan Christian Schlüter. Analysis of a non-standard finite-difference-method for the classical target cell limited dynamical within-host HIV-model - Numerics and applications[J]. Mathematical Biosciences and Engineering, 2025, 22(9): 2360-2390. doi: 10.3934/mbe.2025086

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Abstract

Mathematical modeling and numerical simulation are valuable tools for getting theoretical insights into dynamic processes such as, for example, within-host virus dynamics or disease transmission between individuals. In this work, we propose a new time discretization, a so-called non-standard finite-difference-method, for numerical simulation of the classical target cell limited dynamical within-host HIV-model. In our case, we use a non-local approximation of our right-hand-side function of our dynamical system. This means that this right-hand-side function is approximated by current and previous time steps of our non-equidistant time grid. In contrast to classical explicit time stepping schemes such as Runge-Kutta methods which are often applied in these simulations, the main advantages of our novel time discretization method are preservation of non-negativity, often occurring in biological or physical processes, and convergence towards the correct equilibrium point, independently of the time step size. Additionally, we prove boundedness of our time-discrete solution components which underline biological plausibility of the time-continuous model, and linear convergence towards the time-continuous problem solution. We also construct higher-order non-standard finite-difference-methods from our first-order suggested model by modifying ideas from Richardson's extrapolation. This extrapolation idea improves accuracy of our time-discrete solutions. We finally underline our theoretical findings by numerical experiments.

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