Accelerated inference for stochastic compartmental models with over-dispersed partial observations

An assumed density approximate likelihood is derived for a class of partially observed stochastic compartmental models which permit observational over-dispersion. This is achieved by treating time-varying reporting probabilities as latent variables and integrating them out using Laplace approximations within Poisson Approximate Likelihoods (LawPAL), resulting in a fast deterministic approximation to the marginal likelihood and filtering distributions. We derive an asymptotically exact filtering result in the large population regime, demonstrating the approximation's ability to recover latent disease states and reporting probabilities. Through simulations we: 1) demonstrate favorable behavior of the maximum approximate likelihood estimator in the large population and time horizon regime in terms of ground truth recovery; 2) demonstrate order of magnitude computational speed gains over a sequential Monte Carlo likelihood based approach and explore the statistical compromises our approximation implicitly makes. We conclude by embedding our methodology within the probabilistic programming language Stan for automated Bayesian inference to develop a model of practical interest using data from the Covid-19 outbreak in Switzerland.