  The recent coronavirus disease (COVID-19) outbreak has dramatically increased
the public awareness and appreciation of the utility of dynamic models. At the
same time, the dissemination of contradictory model predictions has highlighted
their limitations. If some parameters and/or state variables of a model cannot
be determined from output measurements, its ability to yield correct insights
-- as well as the possibility of controlling the system -- may be compromised.
Epidemic dynamics are commonly analysed using compartmental models, and many
variations of such models have been used for analysing and predicting the
evolution of the COVID-19 pandemic. In this paper we survey the different
models proposed in the literature, assembling a list of 36 model structures and
assessing their ability to provide reliable information. We address the problem
using the control theoretic concepts of structural identifiability and
observability. Since some parameters can vary during the course of an epidemic,
we consider both the constant and time-varying parameter assumptions. We
analyse the structural identifiability and observability of all of the models,
considering all plausible choices of outputs and time-varying parameters, which
leads us to analyse 255 different model versions. We classify the models
according to their structural identifiability and observability under the
different assumptions and discuss the implications of the results. We also
illustrate with an example several alternative ways of remedying the lack of
observability of a model. Our analyses provide guidelines for choosing the most
informative model for each purpose, taking into account the available knowledge
and measurements.
