## Preprint 2020-12

## Raimund Bürger, Gerardo Chowell, Leidy Y. Lara-Diaz:

### Measuring the distance between epidemic growth models

### Abstract:

Phenomenological growth models 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 $C'(t) = f (t; \Theta )$ for $t >0$, $C(0) = C_0$, where $t$ is time, $C(t)$ 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 [R. Bürger, G. Chowell, L.Y. Lara-Díaz, Math. Biosci. Eng. 16 (2019) 4250-4273] it is demonstrated that some models 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 quantify the distance between two models as a measure of differences in the dynamics that each model is capable of generating. The present work contributes to a systematic study of differences between models and how such differences may explain the ability of certain models to provide a better fit to data than others. To this end metrics are defined that describe the differences in the dynamics between different dynamic models. These metrics are based on a concept of distance of one growth model from another one that quantifies how well the formerfits data generated by the latter. This concept of distance is, however, not symmetric. The procedure of calculating distances is applied to synthetic data and real data from influenza, Ebola, and COVID-19 outbreaks.