# Analyzing Client Behavior in a Syringe Exchange Program

## TL;DR

This paper models and analyzes client visit behaviors in a syringe exchange program over ten years, using a phase-type distribution linked to client demographics to improve personalized intervention timing.

## Contribution

It introduces a novel phase-type distribution model that incorporates client features to predict inter-visit times in syringe exchange programs.

## Key findings

- Model accurately predicts client return times.
- Personalized predictions aid in intervention planning.
- System simulation enhances understanding of client behavior.

## Abstract

Multiple syringe exchange programs serve the Chicago metropolitan area, providing support for drug users to help prevent infectious diseases. Using data from one program over a ten-year period, we study the behavior of its clients, focusing on the temporal process governing their visits to service locations and on their demographics. We construct a phase-type distribution with an affine relationship between model parameters and features of an individual client. The phase-type distribution governs inter-arrival times between reoccurring visits of each client and is informed by characteristics of a client including age, gender, ethnicity, and more. The inter-arrival time model is a sub-model in a simulation that we construct for the larger system, which allows us to provide a personalized prediction regarding the client's time-to-return to a service location so that better intervention decisions can be made with the help of simulation.