Tuberculosis and COVID-19 are highly contagious and potentially life-threatening diseases that present major challenges to the world and it gets worse when co-infection occurs. In this study, we have devised an original TB–COVID-19 co-infection epidemic model ( \(SVI_{C} I_{T} I_{TC} QMR_{C} R_{T} R_{TC} )\) using a non-linear dynamical system, subdividing the human population into ten distinct compartments. The aim of the study is to understand the intricate dynamics of these two diseases and their interactions. Since the dynamics of the whole model is influenced by the two sub models (viz. TB only and COVID only), so a complete analysis of the two sub models have been carried out. Our analysis reveals the existence and stability conditions of disease-free and endemic equilibrium points, offering crucial information about disease persistence and mitigation strategies. Local stability is analyzed using the Jacobian matrix, and global stability is assessed with Lyapunov and Dulac functions. Our study reveals that quarantine and vaccination help in suppressing COVID-19 spread and TB–COVID-19 co infection; similarly adequate treatment for tuberculosis helps to reduce TB infection as well TB–COVID-19 co infection. We have performed sensitivity analysis, which highlights the most influential factors affecting basic reproduction number and eventually co-infection. Optimal control theory, using Hamiltonian function is applied. Using optimal control theory, we found that early and aggressive measures-such as quarantine, mask use, social distancing and adequate treatment-during initial outbreak stages are critical for minimizing co-infection and reducing their overall impact on public health. Numerical simulations performed using MATLAB validate our theoretical findings, offering insights for effectively managing the co-infection spread.
