In this paper, we study the SIR epidemic model with vital dynamics , from the point of view of integrability. In the case of the death/birth rate , the SIR model is integrable, and we provide its general solutions by implicit functions, two Lax formulations and infinitely many Hamilton-Poisson realizations. In the case of , we prove that the SIR model has no polynomial or proper rational first integrals by studying the invariant algebraic surfaces. Moreover, although the SIR model with is not integrable and we cannot get its exact solution, based on the existence of an invariant algebraic surface, we give the global dynamics of the SIR model with .

中文翻译：

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具有生命动力学的SIR传染病模型的可积性。

在本文中，我们研究具有生命动力学的SIR流行病模型 ，从可集成性的角度来看。在死亡率/出生率的情况下， SIR模型是可积分的，我们通过隐函数，两个Lax公式和无数个Hamilton-Poisson实现提供通用解决方案。如果是，通过研究不变代数曲面，我们证明SIR模型没有多项式或适当的有理第一积分。而且，尽管SIR模型具有 是不可积的，我们不能得到它的精确解，基于不变代数曲面的存在，我们给出了SIR模型的全局动力学 。