Bifurcation Analysis Of Two Kinds Of Epidemic Models

Kermark and Mckendrick proposed the classical SIR model in 1927,the infectious disease models have attracted the attention of many researchers,because the study has played an important theoretical role in disease prevention and control.Based on the qualitative theory of differential equation and difference equation,the dynamic properties of two kinds of epidemic models were studied in this paper.Firstly,an SIS epidemic model with a generalized non-monotone and saturated incidence rate was studied.Generalized non-monotone saturated incidence rate describes that when the number of infected individual is large enough and there is a group effect,the infection rate of susceptible people is reduced due to the inhibitory effect caused by psychological factors.The system has rich dynamic properties.When psychological factors have a great influence,there is no endemic equilibrium except for one disease-free equilibrium,and the disease-free equilibrium is globally asymptotically stable;when the influence of psychological factors is small,there are two endemic equilibria in the system.In addition,the system has saddle-node bifurcation and the Hopf bifurcation.When the parameter is the critical value,it has unique endemic equilibrium,which is a codimension two cusp point.When two bifurcation parameters are perturbated,the system undergoes Bogdanov-Takens bifurcation.Numerical simulations are presented to illustrate the conclusion.Secondly,a discrete SIR model with saturated treatment function was studied.The saturated treatment function is a description of the effect of delayed treatment on infected patients in the case of limited medical resources and large number of infected patients.We prove that the model undergoes Flip bifurcation and Neimark-Sacker bifurcation in the case of parameter perturbation,and demonstrate the correctness of the conclusion directly through numerical simulation.