Differential equations modeling of patients and physicians dynamics in emergency rooms: Optimal control policies and heuristic implementation methods
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Hospital emergency rooms are difficult to manage because of the complexity of allocating costly clinical resources, mainly physicians, in light of uncertainty in the arrivals of patients, the increasingly crowded emergency rooms, and the costs of delaying medical treatment. Without effective clinical resource management, emergency rooms are less prepared to deal with unanticipated surge in service demand and are also less able to reduce crowding of patients. Crowding of patients is a particular concern at the national level because when emergency rooms are crowded, medical care is delayed and the likelihood of poor quality care is high. We propose a model using ordinary differential equations to evaluate the natural dynamics of patients and physicians in emergency rooms. From understanding the natural dynamics, we use optimal control theory to determine the optimal dynamics of physicians to increase the benefits of treating more patients while controlling the costs of medical care. Numerical simulations of our model, using actual hospital data, suggest a continuous physicians’ optimal control policy. To implement such continuous policy in a practical way, we use discretization methods based on the concepts of the least square methods and the mean value theorem for integration applied to the previously determined optimal control policy.