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We consider state and parameter estimation for compartmental models havingboth time-varying and time-invariant parameters. Though the described Bayesiancomputational framework is general, we look at a specific application to thesusceptible-infectious-removed (SIR) model which describes a basic mechanismfor the spread of infectious diseases through a system of coupled nonlineardifferential equations. The SIR model consists of three states, namely, thethree compartments, and two parameters which control the coupling among thestates. The deterministic SIR model with time-invariant parameters has shown tobe overly simplistic for modelling the complex long-term dynamics of diseasestransmission. Recognizing that certain model parameters will naturally vary intime due to seasonal trends, non-pharmaceutical interventions, and other randomeffects, the estimation procedure must systematically permit these time-varyingeffects to be captured, without unduly introducing artificial dynamics into thesystem. To this end, we leverage the robustness of the Markov Chain Monte Carlo(MCMC) algorithm for the estimation of time-invariant parameters alongsidenonlinear filters for the joint estimation of the system state and time-varyingparameters. We demonstrate performance of the framework by first considering aseries of examples using synthetic data, followed by an exposition on publichealth data collected in the province of Ontario.

A Bayesian model calibration framework for stochastic compartmental models with both time-varying and time-invariant parameters