Two Types Of Delay The Dynamics Of Infectious Disease Model

Epidemic dynamical model is an important part of mathematical models in biology. Exploring epidemic spreading and predicting its development trend are main objectives for epidemical study, and they are the theoretical basis of control policy adopted by the government and medical department. The aim of this work is to construct two epidemic mathematic models by the method of compartment model and study their dynamical behaviors.The full text is divided into three chapters. In the first chapter, we introduce the purpose and significance of the epidemic models, and review the status and progress of the study of infectious disease. In the second chapter, a delayed SEIRS epidemic model with pulse vaccination and nonlinear incidence is formulated. We obtain the exact formulation of the infection free period solution of the impulsive epidemic system. We obtain two new threshold values R^* and R_* . According to the comparison theorem of impulsive differential equations, we obtain the explicit formulation of R^* and R_* . Under the condition R_* < 1, the infection free period solution is globally attractive, and R_* > 1 implies that the model is permanence. Theoretical results show that a large vaccination rate or a short pulse of vaccination or a long latent period or a long immune period will benefit the extinction of the disease. Otherwise, a small vaccination rate or a short immune period will benefit the persistence of the disease. In the third chapter, a delayed SIA epidemic model with nonlinear incidence is established. We obtain the existence of the disease free equilibrium and endemic equilibrium. The basic reproductive number ( R₀ ) of the model is determined. By analyzing the local asymptotically stability and global attractively of the disease-free equilibrium, we obtain that the disease-free equilibrium is globally asymptotically stable if R₀≤1, which is the condition of the extinction of the disease. Moreover, we show that under p > 1, the disease is permanent if R₀ > 1. By using Liapunov functional approach, the sufficient conditions for the endemic equilibrium which is globally asymptotically stable are obtained when p = 1.