The effective reproduction number ($R_t$) is widely used to track epidemic dynamics in real time. The standard estimation framework uses "instantaneous $R_t$," defined via the renewal equation, which relates new infections to past infections through a generation interval distribution. Compartmental models like SEIR yield a seemingly distinct quantity, "mechanistic $R_t$," based on the effective contact rate and duration of infectiousness. We prove these two definitions are equivalent under homogeneous mixing, the standard assumption in compartmental modeling. We also derive the generation interval distribution implied by SEIR dynamics. A practical consequence is that generation intervals, often treated as assumption-light inputs to renewal equation estimators, in fact encode specific compartmental structure.

翻译：有效再生数($R_t$)被广泛用于实时追踪流行病传播动态。标准估计框架采用"瞬时$R_t$"，其通过更新方程定义，该方程利用代际间隔分布将新感染与既往感染相关联。SEIR等仓室模型基于有效接触率和传染期时长，产生一个看似不同的量——"机制$R_t$"。我们证明在仓室建模的标准假设——均匀混合条件下，这两种定义是等价的。我们还推导了SEIR动力学隐含的代际间隔分布。这一结论的实际意义在于：代际间隔常被视为更新方程估计器中需要较少假设的输入参数，而实际上其编码了特定的仓室结构特征。