Tuberculosis (TB) remains a major public health challenge, particularly in high-burden countries such as India, china, Indonesia, etc. In these regions, ongoing transmission and relapse hinder control efforts. This study develops and analyzes a compartmental \(SE_1E_2IRB\) model that incorporates environmental transmission and relapse pathways. The basic reproduction number \(R_0\) is derived through the next-generation matrix method. Lyapunov functions is used to establish the global asymptotic stability of the infection-free equilibrium. Bifurcation analysis examines system behavior at the critical threshold \(R_0 = 1\) . Parameter sensitivity analysis identifies the transmission rate ( \(\beta\) ), slow latent reactivation rate ( \(\rho_2\) ), and recovery rate ( \(\gamma\) ) as key drivers of TB dynamics. An increase in \(\beta\) or \(\rho_2\) by 10% raises \(R_0\) by 8.9% and 8.8%, respectively, while a 10% improvement in recovery reduces \(R_0\) by 8.6%. The impact of vaccination is further analyzed using an optimal control framework for \(R_0 = 1.43\) , representing sustained transmission in the absence of interventions. Using Pontryagin’s Maximum Principle, the optimal timing and intensity of vaccination are determined to minimize infections and maximize recoveries. Vaccination strategies, specifically Bacillus Calmette-Guérin (BCG), reduce the infected population (from 2,504 to 2,316 within 3 years) and substantially increase recoveries (from 2,748 to 199,783). Achieving vaccination coverage of 90% or higher yields near-optimal reductions in infection and substantial gains in population immunity. High-coverage BCG vaccination is a cost-effective strategy that can guide policy toward India’s TB elimination and 2030 Sustainable Development Goals and WHO elimination targets.
