Phenomenological growth models (PGMs) provide a framework for characterizing epidemic trajectories, estimating key transmission parameters, gaining insight into the contribution of various transmission pathways, and providing long-term and short-term forecasts. Such models only require a small number of parameters to describe epidemic growth patterns. They can be expressed by an ordinary differential equation (ODE) of the type $C^{\prime }\left(t\right)=f(t,C;\Theta )$ for $t>0$, $C\left(0\right)=C_0$, where $t$ is time, $C\left(t\right)$ is the total size of the epidemic (the cumulative number of cases) at time $t$, $C_0$ is the initial number of cases, $f$ is a model-specific incidence function, and $\Theta$ is a vector of parameters. The current COVID-19 pandemic is a scenario for which such models are of obvious importance. In Bürger et al. (2019) it is demonstrated that some PGMs are better at fitting data of specific epidemic outbreaks than others even when the models have the same number of parameters. This situation motivates the need to measure differences in the dynamics that two different models are capable of generating. The present work contributes to a systematic study of differences between PGMs and how these may explain the ability of certain models to provide a better fit to data than others. To this end a so-called empirical directed distance (EDD) is defined to describe the differences in the dynamics between different dynamic models. The EDD of one PGM from another one quantifies how well the former fits data generated by the latter. The concept of EDD is, however, not symmetric in the usual sense of metric spaces. The procedure of calculating EDDs is applied to synthetic data and real data from influenza, Ebola, and COVID-19 outbreaks.

现象学增长模型 (PGM) 为表征流行病轨迹、估计关键传播参数、深入了解各种传播途径的贡献以及提供长期和短期预测提供了一个框架。这样的模型只需要少量的参数来描述流行病的增长模式。它们可以用以下类型的常微分方程 (ODE) 表示$C^{‘}\left(吨\right)=F(吨,C;\Theta )$为了$吨>0$,$C\left(0\right)=C_0$， 在哪里 $吨$是时候了，$C\left(吨\right)$ 是当时流行病的总规模（累计病例数） $吨$,$C_0$ 是初始案例数，$F$ 是特定于模型的关联函数，并且$\Theta$ 是一个参数向量。在当前的 COVID-19 大流行中，此类模型显然具有重要意义。在 Bürger 等人。(2019) 结果表明，即使模型具有相同数量的参数，一些 PGM 比其他模型更擅长拟合特定流行病爆发的数据。这种情况激发了测量两种不同模型能够产生的动力学差异的需要。目前的工作有助于系统研究 PGM 之间的差异，以及这些差异如何解释某些模型比其他模型更适合数据的能力。为此，定义了所谓的经验定向距离 (EDD) 来描述不同动力学模型之间的动力学差异。一种 PGM 与另一种 PGM 的 EDD 量化了前者与后者生成的数据的拟合程度。然而，EDD 的概念在通常的度量空间意义上并不是对称的。计算 EDD 的过程适用于流感、埃博拉和 COVID-19 爆发的合成数据和真实数据。