  Prediction of the progression of an infectious disease outbreak in a
population is an important task. Differential equations are often used to model
an epidemic outbreak's behaviour but are challenging to parametrise.
Furthermore, these models can suffer from misspecification, which biases the
estimates. Stochastic models can help with misspecification but are even more
expensive to simulate and perform inference with. Here, we develop an
explicitly likelihood-based variation of the generalised profiling method as a
tool for prediction and inference under model misspecification. Our approach
allows us to carry out identifiability analysis and uncertainty quantification
using profile likelihood-based methods without the need for marginalisation. We
provide additional justification for this approach by introducing a new
interpretation of the model approximation component as a stochastic constraint.
This interpretation preserves the rationale for using profiling rather than
integration to remove nuisance parameters while still providing a link back to
stochastic models. We applied an initial version of this method during an
outbreak of measles in Samoa in 2019-2020 and found that we achieved relatively
fast, accurate predictions. Here present the most recent version of our method
and its application to this measles outbreak, along with additional validation.
