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RESEARCH ARTICLE

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

Moch. Fandi Ansori1*
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1 Department of Mathematics, Faculty of Science and Mathematics, Universitas Diponegoro, Semarang , Indonesia
IJOCTA, 026310189 https://doi.org/10.36922/IJOCTA026310189
Received: 27 July 2026 | Revised: 22 October 2026 | Accepted: 23 October 2026 | Published online: 6 November 2026
© 2026 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
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Abstract

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.

Keywords
Tuberculosis
Nonstandard finite difference scheme
Saturated treatment
Backward bifurcation
Real data
Parameter estimation
Funding
This research is funded by the Indonesian Endowment Fund for Education (LPDP) on behalf of the Indonesian Ministry of Higher Education, Science and Technology and managed under the EQUITY Program (Contract No. 4306/B3/DT.03.08/2025).
Conflict of interest
The author declares there are no competing interests.
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An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing
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