Qualitative Analysis For Epidemic Models With Pulse Vaccination And Distributed Delay

In biological mathematics, dynamic behaviors of epidemic models are the most interesting topics by establishing the mathematical models to study the spread of infectious diseases and the trend of change. In order to make the model more practical, it is necessary to consider some factors such as the pulse vaccination and incubation period in the model. In recent years, some epidemic models with impulse and delays attracted more and more people’s attention.This paper mainly studies dynamical behaviors for two classes of epidemic models with pulse vaccination and distributed delays by using theory of delayed differential equations and impulsive differential equations.In the first part, the stability of a class of infectious disease model with distributed delay in SIR is studied. The existence and global stability of the disease-free periodic solution is obtained by using the pulse type comparison principle and the analysis technique. Furthermore, the persistence is obtained for the delayed models.In the second part, a class of SEIR epidemic model with impulsive vaccination and distributed delay is proposed. The existence of the periodic solution of the system is obtained by the analysis technique. The global stability of the model is obtained by using the pulse comparison principle. In the end, the oscillation of the impulsive model are discussed, and the permanence of the model is obtained.