A strategic optimal control SVITR Model for tuberculosis (TB) disease in Pakistan by using numerical and fractional schemes.

Tuberculosis (TB) continues to be a major global public health concern, posing huge difficulties to healthcare systems and causing significant morbidity and mortality globally. The continued spread of tuberculosis necessitates immediate attention and efficient intervention techniques despite continuous control attempts. Pakistan is one of the high-TB-burden nations where the disease still poses a concern to public health. Therefore, understanding transmission dynamics and creating efficient control policies can be greatly aided by mathematical modeling in conjunction with optimal control procedures. This study focuses on a deterministic mathematical SVITR-TB model in Pakistan. We establish the mathematical soundness of the proposed model via existence and uniqueness analysis, then derive the basic reproduction number (R0) to project long-term epidemiological trends. Stability in disease-free and endemic conditions is assessed using Routh-Hurwitz and Lyapunov function criteria. The sensitivity analysis is performed to identify the most sensitive parameters. The NSFD numerical scheme is implemented for numerical simulation. We also develop the fractional model, ensure its well-posedness, use the fractional-order RK4 method for the fractional SVITR-TB model, and analyze a more accurate and realistic scenario with a smaller fractional order α. The optimal control model is constructed to analyze disease dynamics further, and the RK4 backward scheme is used to simulate the optimal-control SVITR-TB model, which shows that the disease is more rapidly controllable than in the SVITR-TB model. Pontryagin's method assesses optimal control strategies, including an awareness campaign and increased vaccination coverage. The efficiency analysis is performed to determine the most suitable strategy. The findings demonstrate that strategy 1 is relatively effective than the other control strategies for controlling TB.