The global dynamics for an age-structured tuberculosis transmission model with the exponential progression rate

Abstract

• We fixed the exponential function to describe the progression rate of TB. • The threshold value of disease persistent or not is given. • The existence and global stability of disease free equilibrium and endemic equilibrium both are proved. • The existence of global attractor, and persistence of disease are studied. This paper deals with the global dynamic analysis for a tuberculosis transmission model with age-structure and the exponential progress rate. The time delay in the progression from the latent individuals to becoming the infectious individuals is also considered in our model. The basic reproduction number R 0 of the model is defined. It is proved that R 0 = 1 is a threshold to determine the disease extinction or persistence. The disease free equilibrium is globally stable (unstable) if R 0 < 1 (if R 0 > 1). There exists an endemic equilibrium and the system is uniformly persistent if R 0 > 1. The global stability of the endemic equilibrium is given when R 0 > 1. At end, the model is applied to describe tuberculosis transmission in China. The number of total population, the number of the annual new TB cases, the annual PPD positive rate, and the prevalent rate all match the statistical data well.