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.

翻译：本文提出一种用于流行病学SIR模型逼近的新型数值方法。该方法通过引入合适的松弛项来处理非线性超越方程，从而改进现有线性化流程。我们在连续框架下发展的松弛方法具有显式形式且易于实现，其核心基于一系列线性微分方程。该方法可获得离散与解析形式下的精确逼近。通过严谨分析，我们证明了在恰当选择松弛参数时，该数值方案具有非负保持性且全局强收敛至真实解。现有多种SIR求解器对此类理论性质的研究尚不充分。我们还拓展了松弛方法的适用范围，使其能处理传统SIR模型的若干变体。最后，通过模拟数据的数值算例验证了所提方法的有效性。