Analysis Of Epidemic Models With Vaccin-ation And Varying Total Population Size

The prevention and controlling of epidemic diseases is a very important issue. Vaccination is an effective way to control the epidemic disease. Theories of differential calculus are presented to study the epidemic model with vaccination. The threshold which determines the spread of the disease is found. Main results are as follows:Considering the effect of varying total population size, an epidemic model with vertical infection and vaccination is investigated. Using analysis method ,The existences of the disease-free equilibrium point and the endemic equilibrium point are discussed, The threshold which determines the outcome of the disease is found. Using Eigenvalue method and Hurwitz criterion respectively, the local stability of the disease-free equilibrium point and the endemic equilibrium point are obtained. Through the construction of Lyapunov function, the sufficient condition for the global asymptotical stability of disease-free equilibrium point is given. Using LaSalle Invariant theory , The sufficient condition for the global asymptotical stability of the endemic equilibrium point is obtained. Using the Matlab software, the numberical simulation is presented.Considering the effect of vaccination's validity on the model, we study an epidemic model with vaccination and varying total population size. Using analysis method ,The existence of the disease-free equilibrium point and the endemic equilibrium point are discussed. The threshold which determines the spread of the disease is found. Using Hurwitz criterion, the sufficient condition for the local stability of the disease-free equilibrium point is obtained. Through the construction of Lyapunov function, the global asymptotical stability of the disease-free equilibrium point and the endemic equilibrium point are given. The existence of Hopf bifurcation on the endemic equilibrium point is discussed by using Hopf bifurcation theorems. Using the center manifold theory and norms-based method, the sufficient conditions for the direction and stability of Hopf bifurcation are obtained. Using the Matlab software, the numberical simulation is presented.The research about the epidemic model with vaccination is investigated. A better way to control the spread of the epidemic can be found. It will have a very important theoretical and practical significance to the human health and the controlling of epidemic diseases .