Optimal control analysis of a tuberculosis model with drug-resistant population

Tuberculosis (TB), caused by Mycobacterium tuberculosis, stands as one of the most infectious diseases globally, predominantly affecting the lungs (known as pulmonary tuberculosis). It manifests in two primary forms based on bacterial drug sensitivity: drug-sensitive TB (DS-TB) and drug-resistant TB (DR-TB). DS-TB remains susceptible to medication, whereas DR-TB has developed resistance. This study explores a mathematical model explaining the spread of tuberculosis within a drug-resistant population, proposing optimal control strategies to curb its dissemination through educational initiatives and enhancements in healthcare facilities. The stability analysis reveals that disease-free equilibrium points are locally asymptotically stable when R₀ < 1, while endemic equilibrium points prevail and are locally asymptotically stable if R₀ > 1. Additionally, sensitivity analysis identifies important parameters within the model. By using the Pontryagin Maximum Principle, control variables are integrated and numerically solved. Through simulations and cost assessments, we illustrate the efficacy of employing both control strategies concurrently, effectively reducing the populations susceptible to exposure, DS-TB, and DR-TB infections.