Threshold Dynamics of a SIRI Model with Reinfection: Averaged and Periodic Systems and Application to Tuberculosis Data

Tuberculosis (TB) remains a major public health challenge in high-burden regions, where reinfection and seasonal variation play important roles in disease transmission. In this paper, we study a tuberculosis transmission model with reinfection based on the SIRI framework, with particular emphasis on the intrinsic relationship between the averaged system and the periodic system. The averaged system is shown to characterize the long-term epidemiological behavior, whereas the periodic system captures short-term seasonal fluctuations. From a theoretical perspective, we prove that the periodic system and its corresponding averaged system share the same basic reproduction number. We analyze the threshold dynamics of the seasonal model and investigate the dynamical properties of the averaged system, including the existence and stability of equilibria and the occurrence of backward bifurcation. In particular, we show that disease persistence may occur even when the basic reproduction number ($R_0$) is less than one, and we examine the stability of equilibrium points at the critical threshold ($R_0=1$). These results reveal how transmission and reinfection jointly determine the disease burden and equilibrium structure. To validate the theoretical findings, numerical simulations are performed using tuberculosis incidence data from Yunnan Province, China, covering the period from 2005 to 2020. The numerical simulations suggest that the seasonal model provides a better fit to the data, while the averaged model may overestimate the transmission potential of the disease. Under the condition that the two models share the same basic reproduction number, a constrained numerical simulation is performed. The results show that, under certain parameter settings, the endemic equilibrium of the averaged system can approximate the mean prevalence of the periodic solution. However, such an approximation cannot be guaranteed in general.