  Prediction of the progression of an infectious disease outbreak is important
for planning and coordinating a response. Differential equations are often used
to model an epidemic outbreak's behaviour but are challenging to parameterise.
Furthermore, these models can suffer from misspecification, which biases
predictions and parameter 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 justification for this approach by
introducing a new interpretation of the model approximation component as a
stochastic constraint. This preserves the rationale for using profiling rather
than integration to remove nuisance parameters while also 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 it achieved relatively
fast, accurate predictions. Here we present the most recent version of our
method and its application to this measles outbreak, along with additional
validation.
