Pulse Under The Action Of Several Types Of Delay Epidemic Model

This paper mainly studies several types of epidemic models with nonlinear incidence rate, where the first type is the impulsive epidemic model with two time delays, and the other two types are the impulsive models with a time delay and two parallel infectious stages. By using the theories of discrete dynamics, continuous dynamics and impulsive differential equations, and by incorporating with methods of nonlinear analysis, we obtain the sufficient conditions for for the elimination of the disease (the global attractivity of the infection free periodic solution) and permanence of the models.The paper is divided into 3 parts: the first chapter mostly introduces the researching background and current situation of epidemic dynamics. Meanwhile, it also illustrates the background of the models and shows some pre-knowledge needed for the study.In the second chapter, we focuses on an impulsive model with nonlinear incidence rate (βIS / (1 +αIh) ( h > 0))and two time delays. By using stroboscopic map and the impulsive differential comparison theorem, some results are obtained: when Rφ< 1, the infection free periodic solution of the model is global attractive; when R* > 1, the endemic disease is permanent.In the third chapter, we mainly study the impulsive delayed epidemic models with nonlinear incidence (βSI /φ( I)(φ(0) = 1,φ′( I) > 0)and two parallel infectious stages. According to the methods of nonlinear analysis, we demonstrate that if Rφ1 < 1and Rφ2 < 1, the infection free periodic solution of the model is global attractive; if R1φ> 1 and R2φ> 1, the endemic disease is permanent. Finally, we also investigate an impulsive delayed epidemic model with vertical transmission and two parallel infectious stage, and show some corresponding results.