Numerical Approximation for a Dynamical System of the SIR System

Document Type : Research Paper

Authors
Jafar Ahmadi Shali 1
Shahbaz Mustafa 2
1 Faculty of Mathematics, Statistics and Computer Science, University of Tabriz, Tabriz, Iran
2 Department of Mathematics, Institute of Numerical Sciences, Gomal University, Dera Ismail Khan 29050, KPK, Pakistan
10.22055/jacm.2025.48071.4963
Abstract
In this paper, we are going to approximate the dynamical system of the SIR mathematical model, numerically. The expression for basic reproduction number R0 is obtained which plays main role in the local and global stability. On the other hand, if R0 > 1 then the equilibrium point is unstable. Our numerical simulations show that our proposed NSFD method provides reliable and consistent, positive and converging results at all step sizes as compared to the traditional RK-4 method and Euler method which diverge at large step sizes and produce negative and unstable results with large oscillations. The local stability of the equilibrium point is proved by Jacobean method; however, Schur-Cohn conditions are used to discuss the local stability for the discrete NSFD scheme. To prove the global stability of equilibria, the Enatsu criterion and Lyapunov function are used. This paper presents theoretical findings and numerical simulations that can be used as a powerful instrument for forecasting the spread of added infectious illnesses. 
Keywords
Stability analysis
Dynamical system
Global warming
Basic reproduction number
Subjects

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Journal of Applied and Computational Mechanics
Volume 11, Issue 4
Autumn 2025
Pages 1103-1111