Efficient relaxation scheme for the SIR and related compartmental models

TL;DR

This paper presents a new explicit relaxation scheme for the SIR epidemiological model that improves numerical approximation accuracy, preserves non-negativity, and converges globally, with extensions to model variations and demonstrated effectiveness.

Contribution

The paper introduces a novel relaxation-based numerical method for the SIR model that is explicit, easy to implement, and guarantees non-negativity and convergence, addressing gaps in existing solvers.

Findings

The method accurately approximates SIR models in discrete and analytical forms.

It guarantees non-negativity and global convergence with proper relaxation parameters.

Numerical examples confirm the method's effectiveness on simulated data.

Abstract

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.