State estimation for non-linear transmission models of Tuberculosis

Abstract

Given the high prevalence of Tuberculosis (TB) and the mortality rate associated with the disease, numerous models, such as the Gammaitoni and Nucci (GN) model, were developed to model the risk of transmission. These models typically rely on a quanta generation rate as a measurement of infectivity. However this state cannot be measured directly.

Since the quanta generation rate cannot be measured directly, the unique contribution of this work is the development of state estimators to estimate the quanta generation rate from available measurements. Towards this end, the GN model is adapted into an augmented single-room GN model, and a simplified two-room GN model. A sensitivity analysis is performed on both models to determine the effects of deviation of parameters and the effect thereof on the uncertainty of the quanta state. An algebraic identifiability analysis is performed on the models to determine whether the parameters are identifiable and distinguishable from one another.

An observability analysis shows that both models are observable, i.e. it is theoretically possible to estimate the number of quanta (the quanta state) and the quanta generation rate given available measurements. An additional measurement (rate of change of the measurable variable) is added to increase the observability of the models. Kalman filters are used to estimate the quanta state.

First, a continuous-time extended Kalman filter (CEKF) is used for both adapted models using a simulation and measurement time of 60s. Reasonable quanta state estimates are achieved in both cases. A more realistic scenario, with a measurement rate of 1 day, is used next. For these estimates, a hybrid extended Kalman filter (HEKF) is used. Performance of the filter degrades for the quanta state estimates of the HEKFs. The effects of filter tuning and a greater deviation in initial estimates are also investigated and compared.

The CEKFs, the adapted models, and real-time measurements could potentially be used in a control system feedback loop to reduce the transmission of TB in confined spaces such as hospitals.