  In this paper, we introduce a novel numerical approach for approximating the
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 and globally
strongly convergent towards the true solution. These theoretical findings have
not received sufficient attention in various existing SIR solvers. 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.
