Modeling School Crowd Flow With Math

ISEF Category: Mathematics

Subcategory: Other  ·  Difficulty: Advanced  ·  Setup: School Lab  ·  Time: Full Year

The Hook

A hallway can turn into a traffic jam in seconds. One late bell, one narrow stairwell, and movement changes from walking to crowding. You can model that shift with math, then test whether your model predicts what students actually do.

What Is It?

This project studies how people move when a school lets out. Instead of treating each student as a random dot, you can model the crowd in two ways at once. One part is discrete choice, which means each person picks between paths, exits, or speeds. The other part is a Hughes-type PDE, a type of equation that treats the crowd like a flowing material whose density changes over space and time.

Think of it like this. A few students near a door are like cars in a parking lot. A packed hallway is more like water in a pipe. Your model tries to blend both views. The discrete-choice part captures decisions, while the PDE part captures congestion, bottlenecks, and pressure from nearby crowding.

You can track real movement with smartphone video and turn that footage into data points. Then you can compare your predictions with a simulation tool such as SUMO, which can model how agents move through a space. The result is a math project about human behavior, flow, and prediction.

Why This Is a Good Topic

This is a strong science fair topic because you can change real, measurable features, like hallway width, exit choice, or crowd density, and see how the flow changes. You also get to build a model, test it, and judge how close it comes to real movement. That gives you both theory and data. The topic connects to school safety, building design, and evacuation planning, so your results have clear real-world value.

Research Questions

• How does hallway width affect predicted and observed crowd density during dismissal?
• What is the effect of exit choice rules on bottleneck formation near doors?
• Does adding a discrete-choice component improve prediction accuracy over a pure density-based model?
• To what extent does the calibrated PDE model match smartphone-tracked walking speeds in different hallway zones?
• Which model parameters most strongly change evacuation time in SUMO simulations?
• How does crowd density at a stairwell change the error between observed and simulated flow?

Basic Materials

• Smartphone with video recording, for tracking pedestrian movement in school hallways.
• Tripod or stable phone mount, for fixed overhead or side-view filming.
• Printed floor plan or hallway sketch, for mapping zones and paths.
• Tape measure, for estimating hallway and doorway dimensions.
• Spreadsheet software, for organizing counts, distances, and timings.
• Graph paper or digital annotation tool, for marking paths and zones on frames.
• Consent and school approval forms, for any recording of students.
• Stopwatch or timing app, for checking event timing against video.

Advanced Materials

• Multiple synchronized smartphones, for wider coverage of the same crowd event.
• Overhead camera setup, for cleaner tracking in dense areas.
• Laptop with Python, for tracking, fitting, and simulation analysis.
• ImageJ, for frame-by-frame motion measurement.
• SUMO, for agent-based evacuation and route simulation.
• MATLAB or R, for fitting model parameters and comparing residuals.
• Network or floor-plan digitization software, for turning the hallway layout into a model.
• Access to school safety or building layout data, for realistic boundary conditions.

Software & Tools

• Python: Processes tracking data, fits model parameters, and compares predicted and observed crowd flow.
• ImageJ: Helps you mark student positions frame by frame from video.
• SUMO: Simulates pedestrian or evacuation movement in a built environment.
• R: Runs statistical tests, error comparisons, and sensitivity checks.
• GeoGebra: Lets you sketch and test simplified flow equations and boundaries.

Experiment Steps

1. Define the space you will model, then choose the hallway, stairwell, or exit zone that matters most.
2. Decide which movement features you will measure, such as density, speed, route choice, or queue length.
3. Build a simple data pipeline that turns video into position and time records.
4. Fit a baseline model first, then add the discrete-choice and PDE pieces one by one.
5. Plan controls that separate normal walking from congestion effects and avoid mixing different dismissal patterns.
6. Compare model output with SUMO and real footage, then test which assumptions reduce the error most.

Common Pitfalls

• Recording from a bad angle, which hides people behind others and breaks position tracking.
• Mixing different dismissal days, which changes crowd behavior and makes the data inconsistent.
• Measuring only total exit time, which misses where congestion starts and why it grows.
• Fitting the model to one hallway section, then claiming it predicts the whole campus.
• Ignoring school schedule changes, which can shift crowd patterns more than the model itself.

What Makes This Competitive

A strong version of this project does more than fit one curve. You can compare at least two model families, test them on separate dismissal days, and report which assumptions hold up. You can also do sensitivity analysis to show which variables matter most, like exit width, turning angle, or initial density. If you connect the math back to a real safety question, your work feels useful, not just theoretical.

Project Variations

• Focus on one stairwell instead of the whole dismissal route, then test whether vertical bottlenecks change the best-fit model.
• Compare morning arrival flow with afternoon dismissal flow to see whether the same equations work in both directions.
• Replace video tracking with manually counted density zones, then see how much accuracy you lose with a simpler measurement method.

Learn More

• MIT OpenCourseWare, 18.01 Single Variable Calculus: Use it to review rate of change and accumulation ideas that appear in flow models, found by searching MIT OpenCourseWare.
• NOAA, Office of Response and Restoration resources: Use crowd-evacuation and hazard-planning examples to connect math models to real safety problems, found on NOAA pages.
• USGS, modeling and mapping resources: Learn how spatial boundaries and geometry feed into environmental and movement models, found on USGS pages.
• PubMed: Search for review articles on pedestrian dynamics, evacuation modeling, or crowd flow to see how researchers validate these models.
• Transportation Research Part C: Read peer-reviewed papers on traffic and pedestrian modeling through your school library or journal search tools.
• NASA Open Science data and analysis guides: Practice handling real-world datasets and comparing model predictions with observations, found through NASA resources.