Impact of vaccination and reinfection on tuberculosis dynamics under fractional order modeling

Tuberculosis (TB) continues to pose a significant problem to global health, characterized by its intricate transmission dynamics, latency, and treatment resistance. This study introduces an innovative fractal–fractional compartmental model designed to effectively represent the delayed effects of transmission and recovery in TB dynamics. The model integrates vaccination, reinfection, relapse, and seasonality, while also incorporating memory effects via fractal–fractional derivatives. The mathematical analyses confirm the positivity, boundedness, and Hyers–Ulam stability of the TB model, while the reproduction number and equilibrium states offer valuable insights into transmission thresholds. The analysis of sensitivity underscores the critical parameters that affect the persistence of TB. Theoretical results are validated through numerical simulations employing the Newton polynomial approximation method, highlighting the significant influence of fractional orders on the dynamics of epidemic trajectories. Effective control strategies that combine vaccination and treatment interventions show significant decreases in infection rates and enhancements in recovery outcomes. The results highlight the effectiveness of fractional-order models in formulating evidence-driven strategies for the mitigation and eradication of TB.