The impact of age and environment on tuberculosis transmission

This paper presents a tuberculosis transmission model incorporating age structure and dual time delays to examine the joint effects of environmental pathways and infection-stage heterogeneity. Theoretical analysis establishes threshold dynamics dominated by the basic reproduction number $\mathcal{R}_{0}$, demonstrating that when $\mathcal{R}_{0}<1$, the system is globally asymptotically stable at the disease-free equilibrium $E_{0}$ regardless of variations in $\tau _{1}$ and $\tau _{2}$. When $\mathcal{R}_{0}>1$, the system is globally asymptotically stable at the endemic equilibrium $E_{*}$ under conditions where $\tau _{1}=\tau _{2}=0$ or $\tau _{1}=\tau _{2}=\tau >0$. Numerical simulations not only validate these global stability conclusions but also reveal the differential regulatory effects of dual time delays. This study clarifies the regulatory mechanism of dual age-dependent infection delays on tuberculosis transmission, providing a dynamical basis for formulating relevant prevention and control strategies.