Abstract: This thesis researches SIRS models with delayed and saturation incidence by using the bifurcation method of differential equations. By means of the Hopf bifurcation theorem and considering the delay [τ] as a bifurcation parameter, the endemic equilibrium is locally stable when [τ] is small enough and Hopf bifurcation occurs when [τ] passes through some critical values. Matlab is employed to carry out numerical simulation to verify the results.

Key words: time delayed, Hopf bifurcation, saturation incidence, SIRS models

摘要： 应用微分方程分支理论，研究了具有时滞和饱和接触率的SIRS模型，以时滞[τ]为分支参数，运用Hopf分支理论，得到当时滞[τ]充分小时正平衡点是局部渐近稳定的，当[τ]经过一系列临界值时模型出现Hopf分支。用Matlab软件进行数值仿真验证了结论的正确性。

关键词: 时滞, Hopf分支, 饱和接触率, SIRS模型