The first case of Corona Virus Disease 2019 (COVID-19) was reported in Wuhan, China in December 2019. Since then, COVID-19 has quickly spread out to all provinces in China and over 150 countries or territories in the world. With the first level response to public health emergencies (FLRPHE) launched over the country, the outbreak of COVID-19 in China is achieving under control in China. We develop a mathematical model based on the epidemiology of COVID-19, incorporating the isolation of healthy people, confirmed cases and contact tracing measures. We calculate the basic reproduction numbers 2.5 in China (excluding Hubei province) and 2.9 in Hubei province with the initial time on January 30 which shows the severe infectivity of COVID-19, and verify that the current isolation method effectively contains the transmission of COVID-19. Under the isolation of healthy people, confirmed cases and contact tracing measures, we find a noteworthy phenomenon that is the second epidemic of COVID-19 and estimate the peak time and value and the cumulative number of cases. Simulations show that the contact tracing measures can efficiently contain the transmission of the second epidemic of COVID-19. With the isolation of all susceptible people or all infectious people or both, there is no second epidemic of COVID-19. Furthermore, resumption of work and study can increase the transmission risk of the second epidemic of COVID-19.

The first case of Corona Virus Disease 2019 (COVID-19) was reported in Wuhan, China in December 2019. Since then, COVID-19 has quickly spread out to all provinces in China and over 150 countries or territories in the world. With the first level response to public health emergencies (FLRPHE) launched over the country, the outbreak of COVID-19 in China is achieving under control in China. We develop a mathematical model based on epidemiology of COVID-19, incorporating the isolation of healthy people, confirmed cases and close contacts. We calculate the basic reproduction numbers 2.5 in China (excluding Hubei province) and 2.9 in Hubei province with the initial time on January 30 which show the severe infectivity of COVID-19, and verify that the current isolation method effectively contains the transmission of COVID-19. Under the isolation of healthy people, confirmed cases and close contacts, we find a noteworthy phenomenon that is the potential second epidemic of COVID-19, and estimate the peak time and value and the cumulative number of cases. Simulations show that the isolation of close contacts tracked measure can efficiently contain the transmission of the potential second epidemic of COVID-19. With isolation of all susceptible people or all infected people or both, there is no potential second epidemic of COVID-19. Furthermore, resumption of work and study can increase the transmission risk of the potential second epidemic of COVID-19.

In order to study the dissemination mechanism of knowledge, a SIRI dynamics model with the learning rate, the forgetting rate and the recalling rate is constructed in this paper. Stability of equilibria and global dynamics of the SIRI model are analyzed. Two thresholds that determine whether knowledge is disseminated are given. We describe the stability of the equilibria for the SIRI model in which there are an equilibrium and a line of equilibria. In particular, we find the dividing curve function which is used to partition invariant set in order to discuss the local stability, and obtain the equation of the wave peak value or wave trough value in the process of knowledge dissemination. Numerical simulations are provided to support the theoretical results. The complicated dynamics properties exhibit that the model is very sensitive to variation of parameters, which play an important role on controlling and administering the knowledge dissemination.

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.