Population Dynamics And Nonlinear Infection Rate Of The Infectious Disease Model

The paper deals with three kinds of epidemic models with population dynamics and Nonlinear incidence rate.We have established an SI epidemic model with constant inputand nonlinear incidence rate (βI~pS)/(1+αI~p), an SI epidemic model with constant input andnonlinear incidence rate βI~pS~q and an SI epidemic model with a nonlinear incidencerate and Smith growth in susceptible class.Compared with the former epidemic models with nonlinear incidence rate ,the three models all introduce population dynamics .That means the total amount of populations is no longer a constant ,so they can describe the communicable diseases' transmitting law more accurately.We discussed the positive invariant set,the exsistence and the stability of the equilibrium by the stability theory of ordinary differential equation and the conditions for the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium are obtained.Farther,for some range the parameters,they can undergo Hopf bifurcation.At last, a comparison is also made between the three models.