  Stochastic epidemic models (SEMs) fit to incidence data are critical to
elucidating outbreak dynamics, shaping response strategies, and preparing for
future epidemics. SEMs typically represent counts of individuals in discrete
infection states using Markov jump processes (MJPs), but are computationally
challenging as imperfect surveillance, lack of subject-level information, and
temporal coarseness of the data obscure the true epidemic. Analytic integration
over the latent epidemic process is impossible, and integration via Markov
chain Monte Carlo (MCMC) is cumbersome due to the dimensionality and
discreteness of the latent state space. Simulation-based computational
approaches can address the intractability of the MJP likelihood, but are
numerically fragile and prohibitively expensive for complex models. A linear
noise approximation (LNA) that approximates the MJP transition density with a
Gaussian density has been explored for analyzing prevalence data in
large-population settings, but requires modification for analyzing incidence
counts without assuming that the data are normally distributed. We demonstrate
how to reparameterize SEMs to appropriately analyze incidence data, and fold
the LNA into a data augmentation MCMC framework that outperforms deterministic
methods, statistically, and simulation-based methods, computationally. Our
framework is computationally robust when the model dynamics are complex and
applies to a broad class of SEMs. We evaluate our method in simulations that
reflect Ebola, influenza, and SARS-CoV-2 dynamics, and apply our method to
national surveillance counts from the 2013--2015 West Africa Ebola outbreak.
