Any epidemiological compartmental model with constant population is shown to be a Hamiltonian dynamical system in which the total population plays the role of the Hamiltonian function. Moreover, some particular cases within this large class of models are shown to be bi-Hamiltonian. New interacting compartmental models among different populations, which are endowed with a Hamiltonian structure, are introduced. The Poisson structures underlying the Hamiltonian description of all these dynamical systems are explicitly presented, and their associated Casimir functions are shown to provide an efficient tool in order to find exact analytical solutions for epidemiological models, such as the ones describing the dynamics of the COVID-19 pandemic.

任何具有恒定人口的流行病学区室模型都被证明是一个哈密顿动力系统，其中总人口扮演哈密顿函数的角色。此外，这一大类模型中的一些特殊情况被证明是双汉密尔顿的。引入了具有哈密顿结构的不同种群之间新的相互作用的隔室模型。所有这些动力系统的哈密顿描述背后的泊松结构都被明确地呈现出来，并且它们相关的 Casimir 函数被证明可以提供一种有效的工具，以便为流行病学模型找到精确的分析解决方案，例如描述 COVID- 19 大流行。