Numerical Solution to SIR Model using SBP-SAT operators

Abstract

Accurate and stable numerical simulation of epidemic dynamics is essential for translating mathematical models into reliable computational tools for public health. The Susceptible-Infectious-Recovered (SIR) model remains a cornerstone of mathematical epidemiology, yet the robustness of its numerical treatment strongly influences predictive reliability in applications ranging from outbreak forecasting to intervention assessment. Here, we introduce a high-order Summation-By-Parts (SBP) framework with Simultaneous Approximation Terms (SAT) for the numerical solution of the SIR system. Unlike standard integrators, the SBP-SAT method enforces stability at the discrete level while retaining high accuracy, offering a principled approach for handling nonlinear epidemic dynamics. We validate the scheme against the analytical solution of the classical SIR model, demonstrating both accuracy and robustness. By situating SBP-SAT methods within the growing landscape of numerical approaches for SIR-type models--including Runge-Kutta, finite difference, and fractional-order solvers--this work establishes SBP-SAT as a powerful and generalizable alternative for computational epidemiology. Beyond the classical SIR model, these results pave the way for applying SBP-SAT schemes to more complex epidemic models where stability and accuracy are critical, thereby advancing the methodological foundations of computational biology.