Backward bifurcation and periodic dynamics in a tuberculosis model with integrated control strategies

In this study, we present a unified mathematical model for tuberculosis (TB) that integrates key interventions: Mask use and media campaigns to raise community awareness and promote vaccine booster uptake. The model also incorporates slow–fast disease progression and limited treatment capacity. A mathematical analysis was conducted to determine the existence and stability of equilibrium points. From the mathematical analysis on the stability criteria of the TB-free equilibrium point, we show that TB can be eradicated if the basic reproduction number is below one. However, due to insufficient treatment capacity, a backward bifurcation may occur when the reproduction number equals one, enabling the coexistence of endemic and disease-free equilibria even when the reproduction number is below one. The parameter estimation is based on TB incidence data per 100,000 individuals in Indonesia. Sensitivity analysis reveald that although both interventions are effective, media campaigns combined with vaccine boosters are more impactful in reducing TB transmission than the use of masks. Numerical simulations further suggest the possibility of periodic outbreaks, indicating potential seasonal TB patterns. To explore adaptive intervention strategies, we extended the model using an optimal control framework. Our findings suggested that combined implementation of face masks and media campaigns is more effective than using either alone, particularly when the likelihood of rapid disease progression increases.