Efficient relaxation scheme for the SIR and related compartmental models

In this paper, we introduce a novel numerical approach for approximating the Susceptible–Infectious–Recovered (SIR) model in epidemiology. Our method enhances the existing linearization procedure by incorporating a suitable relaxation term to tackle the transcendental equation of nonlinear type. Developed within the continuous framework, our relaxation method is explicit and easy to implement, relying on a sequence of linear differential equations. This approach yields accurate approximations in both discrete and analytical forms. Through rigorous analysis, we prove that, with an appropriate choice of the relaxation parameter, our numerical scheme is non-negativity-preserving; moreover, it is strongly convergent to the true solution. We also extend the applicability of our relaxation method to handle some variations of the traditional SIR model. Finally, we present numerical examples using simulated data to demonstrate the effectiveness of our proposed method.