Kalman Filtering and Bayesian Inference in Enhancing Pandemic State Estimation

Abstract

The COVID-19 pandemic is a contemporary challenge that requires long-term prediction and sustainable management strategies to effectively deal with its consequences. Developing mathematical models for the COVID-19 pandemic is subject to uncertainties and limitations due to inherently inaccurate phenomena. In this context, this thesis has the following aims: Firstly, an epidemiological model is proposed as a nonlinear ordinary differential equation (ODE), named the SEIQRD model (Susceptible-Exposed-Infected-Quarantined-Recovered-Deceased). This model extends the simple SIR (Susceptible-Infected-Recovered) model to predict the transmission dynamics of COVID-19. The recursive estimator Kalman filters help to accurately extract information for the states or quantities of interest from noisy measurements. Subsequently, the extended Kalman filter (EKF), designed to handle nonlinear dynamical systems, is integrated with the proposed SEIQRD model. This integration enhances estimation accuracy, minimising uncertainties for underlying dynamic systems and providing estimates for unmeasurable hidden states. Additionally, improvement is made to the proposed model by incorporating further parameters, resulting in the improved-SEIQRD model. Moving forward, the aim is to enhance the accuracy of the EKF algorithm by introducing the extended skew Kalman filter (ESKF) algorithm based on a skewness distribution within the improved-SEIQRD model. This is crucial as COVID-19 data may include outliers that could result in inaccurate estimations using traditional Gaussian Kalman filtering approximation. The necessity arises from the asymmetry in the posterior distribution, where the effectiveness of the ESKF algorithm has been proven in capturing skewness in both states and noise distributions. Secondly, the generalised Bayesian inference method, known as the nested sampling algorithm, is employed to conduct realistic parameter estimation, especially in complex and high-dimensional parameter spaces or when the posterior distribution exhibits irregular shapes. This enhancement in estimation is achieved compared to standard Markov Chain Monte Carlo (MCMC) methods by utilising the mean posterior distribution of the quantity of interest and estimating uncertainties as well. Then, these probabilities are used to fit the epidemiological models presented in this thesis. This thesis evaluates the proposed method using numerical simulation results for active cases and death cases in Saudi Arabia. Additionally, within the context of the EKF algorithm, an open question in the Kalman filter framework is explored by tuning its coefficients. A method is proposed for estimating the measurement noise covariance matrix in the EKF algorithm. This is achieved by fitting the error between the reported data and the mean SEIQRD model, demonstrating improvement over arbitrarily chosen values. Thirdly, an effective Bayesian model comparison approach is employed, utilising Bayesian evidence approximated by nested sampling, to compare the SEIQRD models proposed in this thesis with the traditional SIRD model. This comparison is supported by evaluating the EKF performance for each chosen model. It is demonstrated that the proposed technique can mitigate variations between the models' predictions and assess the required complexity levels. Overall, the main contribution is developing a generalised approach encompassing deterministic/stochastic model alterations, parameter estimation, noise distribution, and model comparison in a single pipeline. Finally, relevant conclusions and future development trends are provided for addressing unknown pandemics, along with the potential utilisation of non-Gaussian Kalman filters.