Dynamics of tuberculosis via NSFD scheme with nonlinear incidence, relapse, reinfection, and saturated treatment

Tuberculosis (TB) remains one of the most prevalent infectious diseases worldwide and continues to pose a major public health challenge. This study presents a TB model using a nonstandard finite difference (NSFD) scheme that incorporates nonlinear incidence, relapse, reinfection, and saturated treatment effects. The discrete model is designed to preserve the key qualitative properties of the continuous system, including positivity, boundedness, and invariance of the feasible region. The disease-free and endemic equilibria are analyzed, and their local stability conditions are established using the Jury criterion. The basic reproduction number is derived and is consistent with that of the corresponding continuous model. Furthermore, the bifurcation analysis reveals that the model can exhibit either forward or backward bifurcation, indicating the possibility of bistability and disease persistence even when the basic reproduction number is less than one. To better reflect real-world dynamics, the model parameters are estimated using TB incidence data from Indonesia for the period 2000–2024. Numerical simulations, including the dynamical behavior around equilibria, global sensitivity analysis, epidemiological outcome sensitivity, and bifurcation diagrams, further support the analytical results. The robustness of the proposed NSFD scheme and its performance relative to classical numerical methods are also examined. The results highlight the importance of incorporating treatment saturation and nonlinear transmission in understanding TB dynamics while providing a reliable discrete-time modeling framework.