Modeling tuberculosis transmission with kernel functions: effects of vaccination and time delay

This study investigates the dynamics of tuberculosis (TB) transmission in high-density populations with suboptimal hygiene conditions, focusing on India as a representative setting. We develop a transmission model incorporating both weak and strong kernel functions to represent distributed time delays in TB latency, along with vaccination effects. By analyzing stability around the TB-free equilibrium with respect to the time delay arising from the latent phase, we identify critical bifurcation thresholds for local stability. The model exhibits a Hopf bifurcation at the tuberculosis-existence equilibrium when the latency delay exceeds this critical value, indicating oscillatory disease dynamics. Numerical simulations validate our analytical findings and provide insights into the interplay between vaccination parameters, latency periods, and TB control. The results offer valuable guidance for optimizing vaccination strategies and timing interventions in TB-endemic regions.