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

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.