# A dynamic nonlinear flow algorithm to model patient flow

## TL;DR

This paper introduces a new algorithm to model patient flow in hospitals, identifying bottlenecks and analyzing hospital performance dynamically.

## Contribution

A modified dynamic flow algorithm is proposed to capture patient flow variability in hospitals, incorporating ward capacities, staff, and service time.

## Key findings

- The algorithm creates a dynamic residual graph to measure bottleneck persistency and severity.
- It identifies root causes of bottlenecks and wards’ nonlinear behavior over time.
- The method provides a quick holistic view of hospital performance.

## Abstract

Hospitals are complex systems, and the flow of patients is dynamic and nonlinear in such systems. Network representation allows flow algorithms to observe bottlenecks as candidates for optimisation. To model the dynamic behaviour of the patient flow, we need to consider the variability in arrival rates and service times (length of stay). Previously proposed dynamic flow algorithms mainly focused on arrival and departure rates, inflow and outflow, edges’ and vertices’ capacity, and routing, with applications mainly in transportation and telecommunication. In hospitals, bottlenecks that emerge from the patients’ flow are a result of the vertices (wards) behaviour defined by capacity (beds), number of servers (staff), service time variability, and edges (care pathways) distribution probability. We offer a modified flow algorithm that takes a hospital network, iterates over the patients’ arrival rates, and measures the flow with respect to vertices’ capacities, servers, service time variability, edge capacity, and distribution probability. The result is a dynamic residual graph to measure the bottlenecks’ persistency and severity, identify the root causes of bottlenecks, and wards’ dynamic nonlinear behaviour. The algorithm provides a quick holistic view of hospital performance and the analysis of the edges and vertices’ behaviour over time.