Global asymptotic stability of an age-structured tuberculosis model: An analytical method to determine kernel coefficients in Lyapunov functional

Tuberculosis (TB) is a slowly progressive and chronic long-lasting communicable disease and several significant age-dependent heterogeneous factors are involved in TB transmission. To be specific, the individual susceptibility to TB varies with the ages during their life time, the current BCG vaccine effectiveness wanes since vaccination. And the rate of progression from latent TB infection to active TB, and the infectiousness of TB patients, the rate of treatment success for the patients and the risk of relapse for recovered individuals are also dependent on the sojourn time elapsed since the individuals enter into related compartmental classes. Based on the model framework developed by Li et al. (2022) a novel PDE TB model structured by ages of susceptibility, vaccination, latency, infection, treatment and relapse is formulated to evaluate the influences of the class-age structure on TB transmission. Global dynamical properties, including local stability, uniform persistence and global asymptotic stability, are systematically investigated and worked out by application of some new analytical techniques. Meanwhile, we develop an analytical method to determine kernel coefficients in Lyapunov functional. Our theoretical findings manifest that the age-structured PDE TB model remains global threshold stability characterized by the basic reproduction number. Some numerical simulations underscore the need for substantially improving vaccination rate for susceptible adolescents and adults population to effectively contain even end TB spread if new effective vaccines are available.