中文摘要：

本文研究了一类具有饱和发生率的离散SIR传染病模型的动力学性态.我们利用再生矩阵的方法定义了模型的基本再生数;直接计算得到了平衡点的存在性;利用线性化矩阵和Jury判据讨论了平衡点的稳定性;利用中心流形定理讨论了平衡点处可能发生的分支,包括flip分支和Hopf分支.最后,通过数值模拟展示了所得到的理论结果和模型的复杂动力学性态.

英文摘要：

A discrete SIR epidemic model with the saturation incidence is formulated and studied.We define the basic reproductive number of the model by using the regeneration matrix,obtain the existence condition of equilibria by directly calculating model,and obtain the stability conditions of equilibria by the linearization matrix and Jury criterion.We also use the center manifold theorem to study the flip and Hopf bifurcations.Numerical simulations are conducted to demonstrate our theoretical results and complex dynamics of the model.