Analysis of an age-space structured tuberculosis model with treatment and relapse

This paper concerns with the global threshold dynamics of a Tuberculosis (TB) model incorporating age-space structure, treatment, and relapse. The original model is converted into a hybrid system comprising two Volterra integral equations and two partial differential equations by integrating along the characteristic line. The well-posedness for the model is demonstrated by using the fixed point theory in conjunction with the induction method. In order to discuss whether a disease is persistent or extinct, we provide the explicit formulation of the basic reproduction number. By analyzing the distribution of the characteristic roots of the characteristic equations and constructing the proper Lyapunov functionals, the local and global stability for the steady states are addressed. Numerical simulations are conducted to confirm the conclusions of our analytical results and reveal that reduction of the TB transmission coefficient, reduction of infectiousness of treated individuals infected with TB, and increasing the treatment rate of infectious class are three feasible measures to control the transmission of TB.