Epidemic models are normally used to describe the spread of infectious diseases. In this paper, we will discuss an epidemic model with time delay. Firstly, the existence of the positive fixed point is proven; and then, the stability and Hopf bifurcation are investigated by analyzing the distribution of the roots of the associated characteristic equations. Thirdly, the theory of normal form and manifold is used to drive an explicit algorithm for determining the direction of Hopf bifurcation and the stability of the bifurcation periodic solutions. Finally, some simulation results are carried out to validate our theoretic analysis.

中文翻译：

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时滞流行病模型的稳定性及Hopf分岔分析

流行病模型通常用于描述传染病的传播。在本文中，我们将讨论具有时间延迟的流行病模型。首先证明正不动点的存在性；然后，通过分析相关特征方程根的分布来研究稳定性和Hopf分岔。第三，使用范式和流形理论来驱动确定Hopf分岔方向和分岔周期解稳定性的显式算法。最后，进行了一些仿真结果来验证我们的理论分析。