  Epidemics are inherently stochastic, and stochastic models provide an
appropriate way to describe and analyse such phenomena. Given temporal
incidence data consisting of, for example, the number of new infections or
removals in a given time window, a continuous-time discrete-valued Markov
process provides a natural description of the dynamics of each model component,
typically taken to be the number of susceptible, exposed, infected or removed
individuals. Fitting the SEIR model to time-course data is a challenging
problem due incomplete observations and, consequently, the intractability of
the observed data likelihood. Whilst sampling based inference schemes such as
Markov chain Monte Carlo are routinely applied, their computational cost
typically restricts analysis to data sets of no more than a few thousand
infective cases. Instead, we develop a sequential inference scheme that makes
use of a computationally cheap approximation of the most natural Markov process
model. Crucially, the resulting model allows a tractable conditional parameter
posterior which can be summarised in terms of a set of low dimensional
statistics. This is used to rejuvenate parameter samples in conjunction with a
novel bridge construct for propagating state trajectories conditional on the
next observation of cumulative incidence. The resulting inference framework
also allows for stochastic infection and reporting rates. We illustrate our
approach using synthetic and real data applications.
