Measuring differences between phenomenological growth models applied to epidemiology

Abstract

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.