Exact solution structure and dynamic preservation in a nonstandard finite-difference formulation of the SIR model

Authors

• Ramsha Shafqat, Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathum Thani 12120, ThailandFollow
• Wutiphol Sintunavarat, Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathum Thani 12120, ThailandFollow

Abstract

A new nonstandard finite difference method is developed for discretizing the classical Susceptible--Infected--Removed (SIR) epidemic model. In the proposed approach, the first derivatives in the continuous SIR system are replaced by two carefully constructed nonstandard denominator functions, yielding a dynamically consistent discrete model with improved numerical stability while preserving consistency with the original continuous system. It is proved that the proposed scheme preserves the fundamental dynamical properties and qualitative behavior of the continuous SIR model, including positivity and the long-term dynamics of the population classes. Moreover, the resulting discrete system admits explicit analytical solutions, providing further insight into the model and enabling a direct verification of the proposed numerical method. Finally, a series of numerical experiments is presented to validate the theoretical results and to illustrate the long-term evolution of the susceptible, infected, and removed populations, thereby demonstrating the effectiveness of the proposed scheme.

Recommended Citation

Shafqat, Ramsha and Sintunavarat, Wutiphol (2026) "Exact solution structure and dynamic preservation in a nonstandard finite-difference formulation of the SIR model," Mathematical Modelling and Numerical Simulation with Applications: Vol. 6: Iss. 3, Article 3.
DOI: https://doi.org/10.53391/2791-8564.1035
Available at: https://mmnsa.researchcommons.org/journal/vol6/iss3/3