  We consider state and parameter estimation for compartmental models having
both time-varying and time-invariant parameters. Though the described Bayesian
computational framework is general, we look at a specific application to the
susceptible-infectious-removed (SIR) model which describes a basic mechanism
for the spread of infectious diseases through a system of coupled nonlinear
differential equations. The SIR model consists of three states, namely, the
three compartments, and two parameters which control the coupling among the
states. The deterministic SIR model with time-invariant parameters has shown to
be overly simplistic for modelling the complex long-term dynamics of diseases
transmission. Recognizing that certain model parameters will naturally vary in
time due to seasonal trends, non-pharmaceutical interventions, and other random
effects, the estimation procedure must systematically permit these time-varying
effects to be captured, without unduly introducing artificial dynamics into the
system. To this end, we leverage the robustness of the Markov Chain Monte Carlo
(MCMC) algorithm for the estimation of time-invariant parameters alongside
nonlinear filters for the joint estimation of the system state and time-varying
parameters. We demonstrate performance of the framework by first considering a
series of examples using synthetic data, followed by an exposition on public
health data collected in the province of Ontario.
