An SI Epidemic Model with Smith Increasing and Nonlinear Incidence Rate

The paper deals with a kind of epidemic models with Smith increasing and Nonlinear incidence rate.Compared with the former epidemic models with nonlinear incidence rate,the model induces population dynamics.That means the total amount of populations is no longer a constant and increases by Simith function,so they can describe the communicable diseases' transmitting law more accurately.Discussed the positive invariant set,the existence 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.Further,for some range the parameters,they can undergo Hopf bifurcation.