Implications of mixed exponential occupancy distributions and patient flow models for health care planning.

There is considerable evidence that the distribution of the length of time that a patient occupies a bed in a hospital department is best described by a sum of two or three exponential terms, because of the presence of acute care, rehabilitation, and possibly long term care patients in the department. The patient flow models implied by these mixed exponential distributions are presented and fitting them to observed data when the admission rate fluctuates is discussed. Unlike single exponential distributions, mixed exponential distributions imply that the average length of stay of patients currently resident in the department is much longer than the average length of stay of a group of patients discharged over a period of time, so that the latter way of measuring will not correctly indicate what portion of the resources are being used by rehabilitation and long term care patients. Also, the expected additional length of stay increases dramatically with the time already spent in the department. Applications to predicting the effects of policy changes and to long term monitoring of hospital departments are presented. Two American hospitals are analyzed. The occupancy times in the government supported hospital follow a mixed exponential distribution similar to those found in the United Kingdom, but in the private hospital they fit a single exponential distribution, indicating markedly different management practices.

Modelling hospital and social care bed occupancy and use by elderly people in an English health district.

The paper confirms that exponential equations can be used to model the total system and sub-systems of institutional health and social care for elderly people using bed occupancy census data for 6,068 elderly aged 65 and over. Two streams of flow were present in NHS acute hospitals, Local Authority residential homes and independent sector nursing homes. Three streams of flow were present in the overall data set and in the NHS geriatric hospital beds, NHS psychiatry beds and independent sector residential care homes. In total 22% of patients/residents stayed an average of 24 days (short stay), 69% for 825 days (medium stay) and 9% for 3,384 days (long stay). In both sexes, the older a patient/resident, the longer the time they occupied short stay beds and the shorter the time they occupied long stay beds.

Balancing acute and long-term care: the mathematics of throughput in departments of geriatric medicine.

The empirical distribution of length of stay of patients in departments of geriatric medicine is fit extremely well by a sum of two exponentials. Most of the patients in a geriatric department are rehabilitated and discharged or they die within a few weeks of admission, but the few who become long-stay patients remain for months or even years. A model is presented for the flow of patients through a geriatric department, which has analogies to models of drug flow in pharmacokinetics. The theoretical model explains why the empirical pattern of length of stay in the occupied beds fits a sum of two exponentials; conversely, the empirical distribution, obtained from the midnight bed state report, can be used to study the effect of various policy decisions on both immediate and future admission rates for the department, and shows the benefits of policies which reduce long-stay patient numbers by improving long-stay rehabilitation.