The Impact of Vaccination and Environmental Interventions on Tuberculosis Control: Theory and Applications

Tuberculosis (TB), caused by Mycobacterium tuberculosis, remains a leading cause of death from a single infectious agent, particularly in low-resource settings. Classical models often fail to capture the memory-dependent and complex dynamics of TB transmission. This study introduces a novel fractional-order mathematical model using the Atangana-Baleanu-Caputo (ABC) derivative to incorporate vaccination, reinfection, and environmental interventions. The existence and uniqueness of solutions are established using Schauder and Banach fixed-point theorems. Numerical simulations showed that decreasing the fractional order significantly reduces the rate of disease spread, lowers the peak incidence, and extends the duration of the epidemic curve, reflecting the hereditary and temporal properties captured by fractional calculus, while higher fractional orders increase the basic reproduction number (R0) due to enhanced memory effects. Sensitivity analysis revealed that vaccination, timely treatment, and environmental hygiene are the most effective measures for reducing both (R0) and cumulative incidence. The study showed that fractional-order models provide more realistic epidemic trajectories than classical integer-order models and offer a robust framework for guiding TB control strategies and policy planning.