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

A new tuberculosis model consisting of ordinary differential equations and partial differential equations is established in this paper. The model includes latent age (i.e., the time elapsed since the individual became infected but not infectious) and relapse age (i.e., the time between cure and reappearance of symptoms of tuberculosis). We identify the basic reproduction number [Formula: see text] for this model, and show that the [Formula: see text] determines the global dynamics of the model. If [Formula: see text], the disease-free equilibrium is globally asymptotically stable, which means that tuberculosis will disappear, and if [Formula: see text], there exists a unique endemic equilibrium that attracts all solutions that can cause the spread of tuberculosis. Based on the tuberculosis data in China from 2007 to 2018, we use Grey Wolf Optimizer algorithm to find the optimal parameter values and initial values of the model. Furthermore, we perform uncertainty and sensitivity analysis to identify the parameters that have significant impact on the basic reproduction number. Finally, we give an effective measure to reach the goal of WHO of reducing the incidence of tuberculosis by 80% by 2030 compared to 2015.