Discrete-time compartmental models with partially observed data: a comparison among frequentist and Bayesian approaches for addressing likelihood intractability

Compartmentalmodelshaveemergedasusefultoolsinepidemiologyduetotheirmechanisticnature. Theyprovideinsights into complex dynamicsystemsandallowpredictionsunderdifferentscenarios. However, despite their widespread use, there is still a gap in the literature, concerning their statistical formalization and a systematic discussion of the statistical methods suitable for both tasks of inference and forecasting. In this work, starting from the fundamental distinction between deterministic and stochastic compartmental models, we focus on how the formulation of the likelihood function becomes a necessary and challenging step in the transition from a deterministic to a stochastic framework. We then analyse the various difficulties encountered in evaluating the likelihood function associated with discrete-time stochastic models. We distinguish two rea sons for the intractability of the likelihood function, the high dimension of missing data and the complexity of the model structure, and discuss suitable methods for addressing the problem both from a frequentist and Bayesian perspective. We overview likelihood-based methods andexploretheuseoflikelihood-free approaches in this framework, namely approximate Bayesian computation algorithms and a method that combines model calibrationwithaparametricbootstrapprocedure.Weemphasizetheirabilitytomakeinferencesfromdatathat are partially observed, or only observed in some aggregated form. To showcase their feasibility and reliability, we compare the likelihood-free and likelihood-based methods at work with a toy example of the Susceptible Infected-Removed. Finally, we explore the relevance of likelihood-free methods in a real-world framework through an example of a complex compartmental model developed to study smoking dynamics in Tuscany (Italy).