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

Abstract

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.