  Addressing the challenge of scaling-up epidemiological inference to complex
and heterogeneous models, we introduce Poisson Approximate Likelihood (PAL)
methods. In contrast to the popular ODE approach to compartmental modelling, in
which a large population limit is used to motivate a deterministic model, PALs
are derived from approximate filtering equations for finite-population,
stochastic compartmental models, and the large population limit drives
consistency of maximum PAL estimators. Our theoretical results appear to be the
first likelihood-based parameter estimation consistency results which apply to
a broad class of partially observed stochastic compartmental models and address
the large population limit. PALs are simple to implement, involving only
elementary arithmetic operations and no tuning parameters, and fast to
evaluate, requiring no simulation from the model and having computational cost
independent of population size. Through examples we demonstrate how PALs can be
used to: fit an age-structured model of influenza, taking advantage of
automatic differentiation in Stan; compare over-dispersion mechanisms in a
model of rotavirus by embedding PALs within sequential Monte Carlo; and
evaluate the role of unit-specific parameters in a meta-population model of
measles.
