In this work, we reconsider a susceptible-infected-removed-type (SIR-type) epidemiological model with time-varying transmission and recovery rates proposed by Wacker and Schlüter in (Mathematical Biosciences and Engineering 20(7):12923–12954 (2023)). We shortly summarize analytical properties of this dynamical system and present some results on a first-order non-standard finite-difference-method for its numerical solution suggested in Wacker and Schlüter (Mathematical Biosciences and Engineering 20(7):12923–12954 (2023)). As our main contribution, we propose some ideas to construct high-order non-standard finite-difference-methods for simulation of this non-autonomous, non-linear system of differential equations as a model for the spread of epidemic diseases which preserve important properties of the time-continuous case, for example, non-negativity and preservation of total population size. The presented methods are based on Richardson extrapolation as our main construction tool. Conclusively, we compare our suggested non-standard finite-difference-methods with classical explicit time integration methods such as the explicit Eulerian time-stepping schme or different explicit Runge-Kutta time-stepping methods.