Global dynamics of a tuberculosis model with age-dependent latency and time delays in treatment

Since there exists heterogeneity in incubation periods of tuberculosis and a time lag between treatment and recovery. In this study, we develop a tuberculosis model that takes into account age-dependent latency and time delays in treatment to describe the transmission of tuberculosis. We first show that the solution semi-flow of the model is well-posed and has a global attractor D within an infinite dimensional space Γ . Then we define the basic reproduction number R and prove that it determines the global dynamics of the model. If R< 1 , the global attractor D reduces to the disease-free equilibrium state, indicating that the disease-free equilibrium state is globally asymptotically stable. When R> 1 , the semi-flow generated by the model is uniformly persistent, and there exists an interior global attractor D for this uniformly persistent model. By constructing a suitable Lyapunov function and applying LaSalle’s Invariance Principle, we show that the global attractor D is reduced to the endemic equilibrium state, which means that the endemic equilibrium state is globally asymptotically stable. Based on the tuberculosis data in China from 2007 to 2020, we simulate the parameters and initial values of the proposed model. Furthermore, we calculate the sensitivity of R to the parameters and find the most sensitive parameters to R . Finally, we present an improved strategy to achieve the WHO’s goal of reducing the incidence of tuberculosis by 90% by 2035 compared to 2015.