A Comparative Study of Nonstandard Finite Difference (NSFD) and Classical Methods for a Tuberculosis Susceptible–Infected–Recovered (SIR) Model
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Abstract
Background: Tuberculosis (TB), caused by Mycobacterium tuberculosis infection, remains one of the most severe airborne infectious diseases worldwide, imposing a substantial public health burden due to its high morbidity and mortality rates. Understanding its transmission dynamics through reliable mathematical modeling is essential for effective disease control and prevention strategies. We aimed to develops and analyzes a nonlinear deterministic Susceptible–Infected–Recovered (SIR) model for tuberculosis transmission and evaluate the dynamical consistency of a Nonstandard Finite Difference (NSFD) scheme in comparison with classical numerical methods.
Methods: A nonlinear SIR model incorporating recruitment, standard incidence transmission, natural and disease-induced mortality, and recovery mechanisms was formulated. The basic reproduction number was derived using the nextgeneration matrix approach. Qualitative analysis was performed to examine positivity, boundedness, and stability of equilibria. A dynamically consistent NSFD scheme was constructed and compared with the forward Euler and fourth-order Runge-Kutta (RK4) methods through numerical simulations.
Results: The disease-free equilibrium is stable when , while an endemic equilibrium exists when . Numerical simulations demonstrate that Euler and RK4 methods might produce instability and non-physical negative solutions for larger step sizes. In contrast, the NSFD scheme preserves positivity, boundedness, and stability independently of the time step size and consistently reproduces the qualitative behavior of the continuous system.
Conclusion: The NSFD approach provides a robust and reliable computational framework for modeling tuberculosis transmission. It outperforms classical numerical methods in preserving the dynamical properties of the system and is particularly suitable for long-term epidemiological simulations.
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References
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