Global asymptotic stability of one time-continuous model and its time-discrete variant for basic virus dynamics

In this article, we investigate global asymptotic stability of both equilibria, namely the virus-free and the virus-endemic equilibria, of a classical within-host HIV dynamical model. At first, we show global asymptotic stability of these equilibria for the time-continuous setting by introducing two suitable Lyapunov-functions. Afterwards, we introduce a time-discrete variant of this model by applying the implicit Eulerian time-stepping scheme and demonstrate that it can be recasted in an explicit manner. As our main contribution, we show that the same Lyapunov-functions of the time-continuous model transfer to the time-discrete case to establish global asymptotic stability of both equilibria where even non-equidistant time grids can be used. Finally, we compare our suggested non-standard finite-diffence-method with the classical explicit Eulerian time-stepping method, one second-order Runge-Kutta time-stepping scheme and an explicit-implicit non-standard finite-difference-method. We see that only both non-standard finite-difference-methods conserve non-negativity for arbitrary time-step sizes and that they behave similarly for small time-step sizes.