Stability Analysis Of Two Epidemic Model With Nonlinear Incidence Rate

In the subject of biomathematics, the epidemic dynamics has been the focus of thestudy. The epidemic model with nonlinear incidence rate has been recently attractedwidely attention in the research of epidemic dynamics.In this paper, we study the stability and Hopf bifurcation of the equilibrium for theepidemic model with nonlinear incidence rate and time delays. We extend the modelwith nonlinear incidence rate in previous publications, and discuss the stability andHopf bifurcation for two epidemic models with nonlinear incidence rate and delay. Themain contents are follows.In the first part, we study an SIRI epidemic model with nonlinear incidence rateand latent period, which describes the psychological effect of certain serious diseaseson the community when the sizes of infective is getting larger. By utilizing the basicreproduction number, we first obtain the threshold dynamics on the global stability ofthe equilibrium for the model without latent period, then analyze the stability and Hopfbifurcation for the model with the latent period. The results show the influence ofnonlinear incidence rate and latent period on the dynamical behaviors of the SIRImodel.In the second part, an SIRS epidemic model with modified saturated incidence rateand delay situation is investigated. By utilizing the basic reproduction number,Lyapunov functional method and the iteration technique, we obtain the thresholddynamics on the global stability of the equilibrium for this model.