Quantitative Methods for Hospital Planning and Control

The provision of hospital resources, such as beds, operating theatres and nurses, is a matter of considerable public and political concern and has been the subject of widespread debate. The political element of healthcare emphasises the need for objective methods and tools to inform the debate and provide a better foundation for decision-making. An appreciation of the dynamics governing a hospital system, and the flow of patients through it, point towards the need for sophisticated capacity models reflecting the complexity, uncertainty, variability and limited resources. A common current practice is to plan and manage hospital capacities through a simple deterministic models using average patient flows, average needs, average length-of-stay, average duration of surgical operations etc. Average analysis can be misleading since the underlying distribution is not symmetric. To overcome such limit it is described the probability distribution of some of the most important proxies for measuring the consumption of hospital resources such as discharge rate, admission rate, number of hospitalized patients, and Length of Stay (LoS). While model proposed to describe the LoS is an innovative generalization of models previously applied in this area, the model for the description of the discharge and admission rate is borrowed from the financial mathematics. It is assumed that exist an analogy between the default of a financial institution and discharge of a patient. This approach come up with a simple and closed-form formula for the distribution function of the discharge rate and the admission rate. Moreover some risk metrics, used in financial mathematics, are applied in order to analysis the tail of the distributions. In order to investigate the activities of the hospital departments, a deterministic analysis of numeric indicators is performed. Among the common measured clinical parameters, a robust metrics, characterizing the constituent entities and the best opportunity tools for the characterization of the results, have been identified. Using this approach is provided an application in the evolution of the department. Particularly, the attention is focused on the evolving of the medical team. Deterministic approached turns out to be not suitable for the development of decision support tools, mathematically speaking, a hospital corresponds to a complex stochastic system so that the common deterministic approach for planning and managing the system can be expected to be inadequate. Hence it is also provided a methodological approach to optimize the hospital resource allocation based on stochastic dynamic programming (SDP). SDP approach is well-positioned to model these types of problems because of the explicitly sequential nature of the decision policies they produce. The aim is to reduce the probability of having a number of patients different from a fixed level over a define interval of time. It is shown that the optimal policy proposed performs better than an empirical policy. In modern societies, the cost of healthcare are increasing year by year. The requirement is to cut costs without diminishing the quality of care. One solution is to increase efficiency; hospital need to plan their operations to use available resources in an optimal fashion. In order to analysed the relation between LoS, risk score, and costs, Real Option Approach (ROA) is applied. Physician has the right but not the obligation to discharge a patent if some efficiency conditions are not verified. In according financial yield curve models, a cost function is estimated and the results are compared whit value obtain from ROA. It is also proposed an application of the Lotka-Volterra model and an extension of the Heston model. The thesis has the following outline: Chapter 1 presents an introduction to healthcare systems and briefly discusses sources of fund and issues affecting healthcare quality and costs. It also highlights the importance of using quantitative models to analyse different healthcare delivery strategies and optimize costs. Chapter 2 focuses on healthcare financing systems in different countries and describes different methods of paying for healthcare providers. The strengths and weaknesses of the discussed methods are also pointed out. Chapter 3 introduces a generic framework for healthcare planning. This framework, encompassing 4 hierarchical levels of control and four managerial areas, is used to identify external and internal environmental characteristics affecting the organization of healthcare systems. Chapter 4 provides an overview of the main mathematical theories used in subsequent chapters. These include stochastic differential equations, real option analysis, option pricing and Poisson processes. Chapter 5 introduces a statistical model to describe the length of stay of hospital patients. The proposed model overcomes some of the limitations of previous models by using a Phase-Type Gamma distribution which is able to capture the data characteristics in a more accurate way. The model is tested on a case study based on the Campus Bio-Medico hospital database. Chapter 6 introduces some quality indicators as a tool to evaluate health systems performance and quality. A dynamic stochastic optimization model is then proposed to optimize hospital bed occupancy. Three different models to describe patient discharge probabilities are also proposed and then used to evaluate the optimal policies. Chapter 7 introduces three financial-like models to describe the variable costs associated with patient hospitalization: a Nelson-Siegel model; a Black-Scholes model and a Cox-Ingersoll-Ross model. The second part of the dissertation, Chapters 8 and 9, tackles different problems that the student has investigated during his doctoral studies and which are not related to healthcare system planning. Specifically, Chapter 8 describes a model to optimize the consumption of financial inspection resources for tax evasion by analysing the interaction between prevention/control activities and illegal behaviours. Chapter 9 proposes a new stochastic volatility model for the calibration of option prices.