Content area
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.
Details
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, 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