Interpretability of Epidemiological Models : The Curse of Non-Identifiability

Abstract

Interpretability of epidemiological models is a key consideration, especially when these models are used in a public health setting. Interpretability is strongly linked to the identifiability of the underlying model parameters, i.e., the ability to estimate parameter values with high confidence given observations. In this paper, we define three separate notions of identifiability that explore the different roles played by the model definition, the loss function, the fitting methodology, and the quality and quantity of data. We define an epidemiological compartmental model framework in which we highlight these non-identifiability issues and their mitigation. INTRODUCTIONThe global COVID-19 pandemic has spurred intense interest in epidemiological compartmental models (Thompson, 2020;Brauer, 2008). The use of epidemiological models is driven by four main criteria: expressivity to faithfully capture the disease dynamics; learnability of parameters conditioned on the available data; interpretability to understand the evolution of the pandemic; and generalizability to future scenarios by incorporating additional information.Compartmental models are popular because they are relatively simple and known to be highly expressive and generalizable. However, their interpretability and learnability depend strongly on the alignment between data observations and model complexity. Different choices of model parameters can often lead to (approximately) the same forecast case counts, leading to what is commonly referred to as non-identifiability (Raue et al., 2009;Jacquez & Perry, 1990). The problem of nonidentifiability is only exacerbated with increased model complexity.This lack of identifiability is detrimental because (a) parameter distributions estimated from the observed data tend to be biased with large variances, thus precluding easy interpretation, and (b) nonidentifiable models typically have reduced accuracy on long-time forecasts due to the high parameter variance. This phenomenon is illustrated in Figure 1, where the forecasting errors between a nonidentifiable model and its reparametrized version (later shown to be identifiable) are compared.Non-identifiability in epidemiological models is rooted in the model dynamics, in the fitting loss function and methodology, and in the quality and quantity of data available. Identifiability is typically broadly classified into structural (i.e., purely model-dependent) (Reid, 1977;Massonis et al., 2020) and practical (i.e., data-, loss-and fitting methodology-dependent), with the latter often defined vaguely, and in the context of specific loss functions (Raue et al., 2009; Wieland et al., 2021).Contributions. This paper delineates various general notions of model identifiability that are contextualized in compartmental epidemiological models, including a novel notion of statistical identifiability that depends on the loss function optimized in estimation, and an empirical framework to assess practical identifiability in terms of the highest posterior density intervals. We study...

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