  State-space models have been widely used to model the dynamics of
communicable diseases in populations of interest by fitting to time-series
data. Particle filters have enabled these models to incorporate stochasticity
and so can better reflect the true nature of population behaviours. Relevant
parameters such as the spread of the disease, $R_t$, and recovery rates can be
inferred using Particle MCMC. The standard method uses a Metropolis-Hastings
random-walk proposal which can struggle to reach the stationary distribution in
a reasonable time when there are multiple parameters.
  In this paper we obtain full Bayesian parameter estimations using gradient
information and the No U-Turn Sampler (NUTS) when proposing new parameters of
stochastic non-linear Susceptible-Exposed-Infected-Recovered (SEIR) and SIR
models. Although NUTS makes more than one target evaluation per iteration, we
show that it can provide more accurate estimates in a shorter run time than
Metropolis-Hastings.
