Where to treat the older patient? Can Markov models help us better understand the relationship between hospital and community care?

We have previously used Markov models to describe movements of patients between hospital states; these may be actual or virtual and described by a phase-type distribution. Here we extend this approach to a Markov reward model for a healthcare system with Poisson admissions and an absorbing state, typically death. The distribution of costs is evaluated for any time and expressions derived for the mean and variances of costs. The average cost at any time is then determined for two scenarios: the Therapeutic and Prosthetic models, respectively. This example is used to illustrate the idea that keeping acute patients longer in hospital to ensure fitness for discharge, may reduce costs by decreasing the number of patients that become long-stay. In addition we develop a Markov Reward Model for a healthcare system including states, where the patient is in hospital, and states, where the patient is in the community. In each case, the length of stay is described by a phase-type distribution, thus enabling the representation of durations and costs in each phase within a Markov framework. The model can be used to determine costs for the entire system thus facilitating a systems approach to the planning of healthcare and a holistic approach to costing. Such models help us to assess the complex relationship between hospital and community care.     

Time-dependent analysis for refused admissions in clinical wards

For capacity planning issues in health care, such as the allocation of hospital beds, the admissions rate of patients is commonly assumed to be constant over time. In addition to the purely random fluctuations, there is also typically a predictable pattern in the number of arriving patients. For example, roughly 2/3 of the admitted patients at an Intensive Care Unit arrives during office hours. Also, most of the scheduled admissions occur during weekdays instead of during the weekend.     

Modelling the requirement for supplementary nurses in an intensive care unit

The intensive care unit (ICU) of a hospital is an essential yet costly resource. Consequently, intensive care modelling has become increasingly prevalent in recent years in attempts to increase efficiency and reduce costs. Previous models have usually assumed that the numbers of beds available are restricted; when all beds are occupied, any additional patients are referred elsewhere or elective surgeries are cancelled. In this study, activities at the ICU at a large teaching hospital were modelled using data relating to all admissions to the ICU during the year 2000—a total of 1084 admissions. The unit is unusual in that the majority of patients referred for intensive care therapy are admitted. Bed numbers are increased when necessary to cope with demand. However, nurses are a restricted resource. In order to maintain the required nurse:patient ratio of at least one:one, supplementary nurses are employed during busy periods. Supplementary nurse costs are substantial and so nurse utilization must be closely monitored. The development of a model that calculates the required number of supplementary nurses per shift, and also encapsulates the time-dependent nature of elective surgery admissions and complex duration-of-stay profiles, is presented in this paper. In particular, the model is used to determine the number of rostered nurses that are required to minimize overall nursing staff costs.     

Using a continuous-time Markov model with Poisson arrivals to describe the movements of geriatric patients

The population of geriatrics in a given hospital district is relatively stable and therefore we may model the movement of geriatric patients by considering both their stays in hospital and subsequent releases back into the community. The care of the elderly in departments of geriatric medicine may be generally classified into two forms of clinical care, acute/rehabilitative and long stay. Our paper describes the movement of pateints through departments of geriatric medicine and subsequent stays in the community by a four-stage continuous-time Markov model, where the stages represent acute/rehabilitative patients, long-stay patients, ex-patients in the community and former patients who are now dead, respectively. Admissions are modelled as a Poisson stream and expressions are calculated for the distribution, mean and variance of numbers of patients in each compartment at any time. Using these expressions the model is then fitted to a large data set of hospital spells containing over 10 000 admissions. © 1998 John Wiley & Sons, Ltd.     

A multi-objective model to determine efficient resource levels in a medical assessment unit

This paper details models that determine the efficient allocation of resources on a medical assessment unit (MAU) of a general hospital belonging to the National Health Service (NHS) UK. The MAU was established to improve the quality of care given to acute medical patients on admission, and also provide the organizational means of rapid assessment and investigation in order to avoid unnecessary admissions. To analyse the performance of the MAU, doctors, nurses and beds are considered as the three main resources. Then a model is developed using the goal programming approach in multiobjective decision making and solved to deal with MAU performance. The developed model is solved under three different sets of patient admissions with the same resource levels using past data from the MAU. The results of the model are used to analyse the needed resource levels. Conclusions as to the appropriate staffing levels and functions of the MAU are drawn.     

Using a Markov reward model to estimate spend-down costs for a geriatric department

A Markov model is used to describe movements of geriatric patients within a hospital system where the states of the Markov chain are acute/rehabilitative, long-stay care, discharge or death. By assigning costs to the states of this model, we can estimate the spend-down costs of running down services given that there are no more admissions and different costs are assigned to acute/rehabilitative and long-stay care. The model is used to estimate the spend-down costs using data previously validated for three Departments of Geriatric Medicine in the South West Thames Region of England. Our approach allows hospital planners to identify cost-effective strategies which take into account the fact that some geriatric patients remain in long-stay care for very long periods of time.     

Nursing care demand prediction based on a decomposed semi-markov population model

The semi-markovian population model introduced by Kao for the planning of progressive care hospitals is adapted to the prediction of nursing care demand at the level of a care unit in a general hospital. Assuming a feedback admission policy which refills the unit as soon as discharges occur, it is shown that the care unit can be decomposed into B independent subsystems corresponding to each of the B beds in the unit.For each bed the semi-Markov model permits the computation of the expected care demand and its variance for each of the seven forthcoming days. The model permits also the prediction of admissions of new patients. A prediction formula can thus be obtained where the expected care demand is expressed as a linear function of the expected number of admissions in the forthcoming days.Finally this methodology is illustrated on real data obtained in the gynaecology department of the Montreal Jewish General Hospital.     

Surgical case scheduling as a generalized job shop scheduling problem

Surgical case scheduling allocates hospital resources to individual surgical cases and decides on the time to perform the surgeries. This task plays a decisive role in utilizing hospital resources efficiently while ensuring quality of care for patients. This paper proposes a new surgical case scheduling approach which uses a novel extension of the Job Shop scheduling problem called multi-mode blocking job shop (MMBJS). It formulates the MMBJS as a mixed integer linear programming (MILP) problem and discusses the use of the MMBJS model for scheduling elective and add-on cases. The model is illustrated by a detailed example, and preliminary computational experiments with the CPLEX solver on practical-sized instances are reported.     

Looking in the wrong place for healthcare improvements: A system dynamics study of an accident and emergency department

Accident and Emergency (A&E) units provide a route for patients requiring urgent admission to acute hospitals. Public concern over long waiting times for admissions motivated this study, whose aim is to explore the factors which contribute to such delays. The paper discusses the formulation and calibration of a system dynamics model of the interaction of demand pattern, A&E resource deployment, other hospital processes and bed numbers; and the outputs of policy analysis runs of the model which vary a number of the key parameters. Two significant findings have policy implications. One is that while some delays to patients are unavoidable, reductions can be achieved by selective augmentation of resources within, and relating to, the A&E unit. The second is that reductions in bed numbers do not increase waiting times for emergency admissions, their effect instead being to increase sharply the number of cancellations of admissions for elective surgery. This suggests that basing A&E policy solely on any single criterion will merely succeed in transferring the effects of a resource deficit to a different patient group.     

Combining simulation and goal programming for healthcare planning in a medical assessment unit

This paper describes a detailed simulation model for healthcare planning in a medical assessment unit (MAU) of a general hospital belonging to the national health service (NHS), UK. The MAU is established to improve the quality of care given to acute medical patients on admission, and to provide the organisational means of rapid assessment and investigation in order to avoid unnecessary admissions. The simulation model enables different scenarios to be tested to eliminate bottlenecks in order to achieve optimal clinical workflow. The link between goal programming (GP) and simulation for efficient resource planning is explored. A GP model is developed for trade-off analysis of the results obtained from the simulation. The implications of MAU management preferences to various objectives are presented.     

Geographical simulation modelling for the regional planning of oral and maxillofacial surgery across London

Oral and maxillofacial surgery (OMFS) is a recognized surgical specialty, with its foundations in dentistry. The current configuration of OMFS services across London has evolved over time and reflects historical rather than contemporary patterns of care. The creation of a London Health Region in 1998 provided the opportunity for rational planning of hospital services to serve the resident population of London (7.2 million) and beyond, with recent change focusing on London's five sectors that are represented within this planning model. A detailed geographical simulation model has been developed and has enabled planners to consider a number of OMFS service configurations and evaluate their impact on providers, variations in caseload, travelling distances and times for patients, and thus inform consultation over change. The research confirms that any in-patient service rationalization which concentrates care in one designated hub (main centre) per sector, involves a significant increase in caseload for the designated hub. Average travelling distances and times for in-patient admissions also increase significantly. However, it does suggest that current commissioned provision of day surgery patterns may not be well aligned to the geographical distribution of need for services, resulting in many patients travelling further than necessary for day surgery treatment. These may be overcome by sending patients to their local centre, which may be out with their sector of residence.     

Efficiency and welfare implications of managed public sector hospital waiting lists

A queuing model for public health service waiting lists is developed, and the implications for patient welfare of different systems for managing the waiting list are analysed. If patients are admitted to hospital on a first-come-first-served basis, a welfare gain is achieved by moving from a system of implicit to one of explicit rationing of access to the waiting list. If individual waiting times and hospital admissions are dependent on clinical priority, a further welfare gain is achievable without the use of explicit rationing, by reallocating the total waiting time from the more towards the less seriously ill. On efficiency and welfare criteria, a maximum waiting time guarantee does not appear to be a desirable development.     

Improving the Use of Hospital Maternity Beds

Two studies are described dealing with the problem of hospital admission and duration of stay for maternity care. Maternity admissions and deliveries are shown to be strongly influenced by the general availability of hospital beds; duration of stay is not affected by bed scarcity. A statistical method of assessing the perinatal mortality risk of individual cases and selecting women for hospital care is outlined.     

A queueing model for bed-occupancy management and planning of hospitals

The aim of this paper is, on the one hand, to describe the movement of patients through a hospital department by using classical queueing theory and, on the other hand, to present a way of optimising the use of hospital resources in order to improve hospital care. A queueing model is used to determine the main characteristics of the access of patients to hospital, such as mean bed occupancy and the probability that a demand for hospital care is lost because all beds are occupied. Moreover, we present a technique for optimising the number of beds in order to maintain an acceptable delay probability at a sufficiently low level and, finally, a way of optimising the average cost per day by balancing costs of empty beds against costs of delayed patients.     

Improving operational effectiveness of tactical master plans for emergency and elective patients under stochastic demand and capacitated resources

This paper develops a two-stage planning procedure for master planning of elective and emergency patients while allocating at best the available hospital resources. Four types of resources are considered: operating theatre, beds in the medium and in the intensive care units, and nursing hours in the intensive care unit. A tactical plan is obtained by minimizing the deviations of the resources consumption to the target levels of resources utilization, following a goal programming approach. The MIP formulation to get this tactical plan is specifically designed to account for emergency care since it allows for the reservation of some capacity for emergency patients and possible capacity excess. To deal with the deviation between actually arriving elective patients and the average number of patients on which the tactical plan is based, we consider the possibility of planning a higher number of patients than the average to create operating slots in the tactical plan (slack planning). These operating slots are then filled in the operational plan following several flexibility rules. We consider three options for slack planning that lead to three different tactical plans on which we apply three flexibility rules to get finally nine alternative weekly schedules of elective patients. We then develop an algorithm to modify this schedule on a daily basis so as to account for emergency patients’ arrivals. Scheduled elective patients may be cancelled and emergency patients may be sent to other hospitals. Cancellation rules for both types of patients rely on the possibility to exceed the available capacities. Several performance indicators are defined to assess patient service and hospital efficiency. Simulation results show a trade-off between hospital efficiency and patient service.     

Using phase-type models to cost stroke patient care across health,social and community services

Stroke disease places a heavy burden on society, incurring long periods of time in hospital and community care, and associated costs. Also stroke is a highly complex disease with diverse outcomes and multiple strategies for therapy and care. Previously a modeling framework has been developed which clusters patients into classes with respect to their length of stay (LOS) in hospital. Phase-type models were then used to describe patient flows for each cluster. Also multiple outcomes, such as discharge to normal residence, nursing home, or death can be permitted. We here add costs to this model and obtain the Moment Generating Function for the total cost of a system consisting of multiple transient phase-type classes with multiple absorbing states. This system represents different classes of patients in different hospital and community services states. Based on stroke patients’ data from the Belfast City Hospital, various scenarios are explored with a focus on comparing the cost of thrombolysis treatment under different regimes. The overall modeling framework characterizes the behavior of stroke patient populations, with a focus on integrated system-wide costing and planning, encompassing hospital and community services. Within this general framework we have developed models which take account of patient heterogeneity and multiple care options. Such complex strategies depend crucially on developing a deep engagement with the health care professionals and underpinning the models with detailed patient-specific data.     

An integrated approach for scheduling health care activities in a hospital

To effectively utilise hospital beds, operating rooms (OR) and other treatment spaces, it is necessary to precisely plan patient admissions and treatments in advance. As patient treatment and recovery times are unequal and uncertain, this is not easy. In response, a sophisticated flexible job-shop scheduling (FJSS) model is introduced, whereby patients, beds, hospital wards and health care activities are respectively treated as jobs, single machines, parallel machines and operations. Our approach is novel because an entire hospital is describable and schedulable in one integrated approach. The scheduling model can be used to recompute timings after deviations, delays, postponements and cancellations. It also includes advanced conditions such as activity and machine setup times, transfer times between activities, blocking limitations and no wait conditions, timing and occupancy restrictions, buffering for robustness, fixed activities and sequences, release times and strict deadlines. To solve the FJSS problem, constructive algorithms and hybrid meta-heuristics have been developed. Our numerical testing shows that the proposed solution techniques are capable of solving problems of real world size. This outcome further highlights the value of the scheduling model and its potential for integration into actual hospital information systems.     

Dimensioning hospital wards using the Erlang loss model

How many beds must be allocated to a specific clinical ward to meet production targets? When budgets get tight, what are the effects of downsizing a nursing unit? These questions are often discussed by medical professionals, hospital consultants, and managers. In these discussions the occupancy rate is of great importance and often used as an input parameter. Most hospitals use the same target occupancy rate for all wards, often 85%. Sometimes an exception is made for critical care and intensive care units. In this paper we demonstrate that this equity assumption is unrealistic and that it might result in an excessive number of refused admissions, particularly for smaller units. Queuing theory is used to quantify this impact. We developed a decision support system, based on the Erlang loss model, which can be used to evaluate the current size of nursing units. We validated this model with hospital data over the years 2004–2006. Finally, we demonstrate the efficiency of merging departments.     

Patient mix optimization in tactical cardiothoracic surgery planning: a case study

Cardiothoracic surgery planning involves different resourcessuch as operating theatre (OT) time, medium care beds, intensivecare beds and nursing staff. Within cardiothoracic surgery differentcategories of patients can be distinguished with respect totheir requirements of resources. The mix of patients is, therefore,an important aspect of decision making for the hospital to managethe use of these resources. A master OT schedule is used atthe tactical level of planning for deriving the weekly OT plan.It defines for each day of a week the number of OT hours availableand the number of patients operated from each patient category.We develop a model for this tactical level planning problem,the core of which is a mixed integer linear program. The modelis used to evaluate scenarios for surgery planning at tacticalas well as strategic levels, demonstrating the potential ofinteger programming for providing recommendations for change.     

A simple graphical decision aid for the placement of elderly people in long-term care

This paper describes the construction of a graphical decision tool to aid placement decisions of a multidisciplinary review panel for admissions to long-term care in a London borough in the UK. First we construct a prediction model of placement decisions based on an applicant's attributes. Using data from the London borough, a composite model comprising syndromic decision rules followed by a two-stage hierarchical logistic regression model is proposed. The model proved to be robust in differentiating cases needing residential home care and nursing home care. Placement outcomes generated by the model are then represented graphically on a triangle plot. This approach could potentially be used as a decision support tool by managers of long-term care for continuous monitoring and assessment of the appropriateness of placements with respect to residents’ needs.