Mathematical model analysis of drug resistance TB dynamics with immigrants and impact of media efficiency

Tuberculosis (TB) is a deadly and contagious disease that can be found all over the world. The purpose of this research is to analyze the dynamics of drug-resistant tuberculosis (DR-TB) involving infected individuals and media efficacy. Moreover, the study investigates the impact of media efficiency and immigration on TB disease transmission, as well as the role of early treatment and immunity enhancement in controlling the spread of the disease. In this paper, well-posedness (the boundedness and positivity) of the solutions of the mathematical model is verified to make the mathematical model biologically meaningful. To characterize the stability of the disease locally and globally, both TB-free and endemic equilibrium points were computed. The basic reproduction number of the model is constructed to reveal the dynamics of TB distribution and to help public health officials design effective strategies to control and prevent both the DS-TB and DR-TB epidemics. The local stability (at both disease-free and endemic equilibrium points) of the mathematical model was also verified by using the Routh-Hurwitz criterion. The local stability of the TB-free and endemic equilibrium points indicates that a bifurcation has occurred, and we have verified its existence using the Castillo-Chavez and Song theorem. The optimal control problem was analyzed using Pontryagin's maximum principle to characterize the protection control, first-line treatment control, and second-line treatment control. We compared our model with and without optimal control and found that the model with control strategies significantly reduced the disease. Sensitivity analysis is also performed to identify the most sensitive parameters, such as the transmission rate [Formula: see text] and media efficiency (m). Numerical simulations are conducted to identify effective combinations of control strategies, demonstrating that the disease can be controlled through these interventions. The study also examined the impacts of media efficiency and migration on TB transmission. Finally, the research concludes with a discussion and summary of the findings.