Coxian phase-type modelling and time series forecasting of patient length of stay and hospital overcrowding

Hospital overcrowding is defined as the condition where the num ber of patients exceeds the healthcare system capacity and resources (Hoot and Aronsky, 2008). Long durations of stay are the main cause of overcrowding, especially in the emergency department (Mentzoni et al., 2019). The levels of overcrowding can be measured by using the number of patients who receive their treatments on trolleys, rather than in hospital beds. The concern in this thesis is the general problem of queueing caused by hospital overcrowding. The main focus is on enhancing exist ing statistical techniques to help reduce the overcrowding problem arising in the emergency department. The problem is approached from two different perspectives: (i) by modelling patient length of stay using a series of conditional Coxian phase-type distributions, and (ii) by forecasting the levels of overcrowding using times series models. Modelling emergency department length of stay and forecasting the overcrowding will help measure the forthcoming workload and improve the management of material and human resources. Coxian phase-type distributions are becoming increasingly popular in healthcare applications, especially the analysis of time-to-event data. In this work, we develop a model based on a series of Coxian phase-type distributions to model compartmental healthcare systems. The model is applied to emergency department length of stay data from University Hospital Limerick in Ireland. The drawback in Coxian models is that they have non-unique representations and parameter estimation is computationally expensive, which then makes it difficult to incorporate covariates into the model parameters. We re-visit the non-uniquness problem and develop new theorems and properties that are specific to Coxian distributions. We reformulate the Coxian models through a finite mixture of density functions, which facilitates the inclusion of covariates and reduces the computational time. Time series analysis provides a useful tool for forecasting over crowding in hospitals. The trolley count data for all Irish hospitals show a double seasonality with a long seasonal period, as well as moving trends. We use models which combine both autoregressive integrated moving average and trigonometric components to handle short-term autocorrelation and regular seasonal trends. The observed moving trends are handled using dummy variables.