Qualitative And Stability Analysis For Several Types Of Differential Equation Models

In this paper, by using stability theory of differential equation, we studied a SIRmodel and a SIRS model with nonlinear epidemic rate, as well as SEIRS modelrespectively. The existence of equilibrium was ifrstly discussed, then the localstability and global stability of the equilibrium were researched for these models.First of all, through analysis of the law of infection and transmission of epidemicdiseases, we established the SIR model with nonlinear infection rate. The existence ofequilibrium was then discussed, and the threshold of the system was deifnedaccording to the condition of the disease-free equilibrium and epidemic equilibrium.Secondly，by obtaining the linear system corresponding to the dynamic system ofthe SIRS model and using dynamics theory of the Lyapunov method and LaSalleinvariance principle, we constructed the corresponding Lyapunov function of thesystem and obtained the local and global asymptotic stability of disease-freeequilibrium and endemic equilibrium under the condition of different threshold.Thirdly，relative to the SIR epidemic model，we set up a SIRS model withnonlinear epidemic rate in susceptible people on the condition of re-infection. Theconditions of existence of equilibrium and the stability of the equilibrium point underdifferent threshold were studied. In order to validate stability of system underdifferent threshold, we used the computer software MATLAB to make numericalsimulation of the SIRS model with speciifc numerical example. At the same time, byassigning different values to the parameters of the system, the change of equilibriumpoint was compared and the law of development of epidemic diseases was obtained.Next, When the above SIRS model has only one equilibrium, we analyzed thebifurcation and determined that the equilibrium of the system was a Bogdanov-Takens singularity. Finally, a SEIRS epidemic model was introduced and the localand global asymptotic stability of endemic equilibrium were analyzed by the stabilitytheory of differential equation.