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

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.