Recearch On The Stability And Bifurcation Problems Of A Class Of Discrete Epidemic Model

In realistic life, infection diseases always seriously endanger human health, which has been concerned by each country of the world. In this paper, the differential equation theory is used for studing the stability and bifurcation problem of a class of discrete epidemic model. The main work is as following:Firstly, a discrete SIRS model with distributed time delay is established by considering the effect of the general framework of nonlinear incidence. Find the threshold of disease extinction or not by calculating the solution of model and get the sufficient conditions for existence of disease-free equilibrium and endemic equilibrium. By constructing appropriate lyapunov function and combining the boundedness of the solution, we prove the global stability of the disease-free equilibrium and the endemic equilibrium, respectively. Uniform persistence of the endemic equilibrium was studied with the inequality zooming and reduction to absurdity. Whereafter, the conclusion was testified by using the digital simulation tool of matlab software.Lastly, what the incidence of functional form of the model plays a key role in model behavior. However, the effect of Saturation incidence of infections disease model has a practical significance. A discrete SIR epidemic model with saturation incidence is established. Using Jury criteria and analysis of the eigenvalues of the linearized system, get the local stability of the equilibrium point and bifurcation point. The branch of the model is discussed by the ably transformation of model and using the Neimark-Sacker bifurcation existence theory. Lastly, the conclusion was testified by using the digital simulation tool of matlab software.