Stabilitiy Analysis Epidemic Models With A General Contact Rate

The prevention of epidemic diseases is an important issue which is critical to the human health and national economy. The quantification reseach on the principle of epidemic diseases is the critical basis of the prevention mechanism. Models are presented to analyze the spread mechanism of epidemic diseases. The influence of infectious force in the latent period to the disease and the spread mechanism of epidemic diseases are investigated, employing the stability methods of the differential equation and the theory of compound matrices.In the first section, the SEI epidemic model with a general contact rate and infectious force in the latent period is investigated. Through the application of existence of theorem of zero root, the existence unique conditions of the equilibrium points of the model are presented. Using Routh-Hurwitz criterion, the local stability of the endemic equilibrium point is obtained. Through the construction of Liapunov function and the theory of compound matrices, the global stability of the disease-free equilibrium point and the endemic equilibrium point are given. Through ayalysis of the stability of the model, the threshold which determines whether the disease is extinct or not is obtained. Using the Matlab software, the numerical simulation is presented.In the second section, considering the permanent immunity factor, the SEIR epidemic model with infectious force in the latent period which makes the model more universal is investigated. Through the application of existence of theorem of zero root, the existence unique conditions of the equilibrium points of the model are presented. Through the construction of Liapunov function, the global stability of the disease-free equilibrium point is given. Introducing the change of variable, the four-dimensional system is reduced the three-dimensional system. Using Routh-Hurwitz criterion, the local stability of the positive equilibrium point of the limit system of the three-dimensional system is obtained. Through the theory of compound matrices, the global stability of the positive equilibrium point is obtained, the global stability of the endemic equilibrium point of the four-dimensional system is given by the limit equation theory. Through analysis of the stability of the model, the threshold which determines whether the disease is extinct or not is obtained. Using the Matlab software, the numerical simulation of the limit system is presented.