Open Access

ARTICLE

A Time-Continuous Model for an Untreated HIV-Infection and a Novel Non-Standard Finite-Difference-Method for Its Discretization

Benjamin Wacker¹, Jan Christian Schlüter^2,3,*

1 Department of Engineering and Natural Sciences, University of Applied Sciences Merseburg, Eberhard-Leibnitz-Str. 2, Merseburg, 06217, Germany
2 Faculty of Management, Social Work and Construction, HAWK, Haarmannplatz 3, Holzminden, 37603, 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, Halle, 06112, Germany

* Corresponding Author: Jan Christian Schlüter. Email:

(This article belongs to the Special Issue: Advances in Mathematical Modeling: Numerical Approaches and Simulation for Computational Biology)

Computer Modeling in Engineering & Sciences 2025, 144(2), 2191-2229. https://doi.org/10.32604/cmes.2025.067397

Received 02 May 2025; Accepted 08 July 2025; Issue published 31 August 2025

Abstract

In this work, we re-investigate a classical mathematical model of untreated HIV infection suggested by Kirschner and introduce a novel non-standard finite-difference method for its numerical solution. As our first contribution, we establish non-negativity, boundedness of some solution components, existence globally in time, and uniqueness on a time interval for an arbitrary for the time-continuous problem which extends known results of Kirschner’s model in the literature. As our second analytical result, we establish different equilibrium states and examine their stability properties in the time-continuous setting or discuss some numerical tools to evaluate this question. Our third contribution is the formulation of a non-standard finite-difference method which preserves non-negativity, boundedness of some time-discrete solution components, equilibria, and their stabilities. As our final theoretical result, we prove linear convergence of our non-standard finite-difference-formulation towards the time-continuous solution. Conclusively, we present numerical examples to illustrate our theoretical findings.

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Keywords

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