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A PDE-Based Model of Runner Flow in Mass Participation Events
Alice Molinari, Brian Block, Stephanie Stockar, Simona Sacone
TL;DR
Marathon organizers need estimates of runner movement despite sparse checkpoint observations, while large events can pose safety and logistical challenges. This paper models marathon flow with a convection-diffusion PDE and a time-varying mean speed; the improved model reduces average checkpoint-arrival-time error by up to 2%.
Problem
Large pedestrian events can create safety and logistical challenges, and marathon runner locations are observed at course checkpoints rather than continuously.
Source: §1 · p. 1Method
Results
The improved mean-speed calculation reduced average time to reach checkpoints along the course by up to 2%.
Source: §6 · p. 6Takeaways & Limitations
Checkpoint arrival-time estimates are relevant to volunteer allocation and decisions about when to reopen roads.
Source: §4 · p. 5The model assumes piecewise-constant parameters because data are available only at checkpoints and initializes the race with an empty course.
Source: §2 · p. 2
Abstract
from arXiv · showhide
This paper presents a macroscopic approach for modeling pedestrian movement. A convection-diffusion partial differential equation (PDE) is described and used to model the flow of runners along a marathon course. A marathon is a good modeling case study for human movement because the location of runners is only known in sparse locations, so an accurate model is needed to know where people are at all times. As well, marathons offer many safety and logistical problems that can be solved by modeling human movement. Data is obtained from the 2023 Columbus Marathon, and model improvements are made to include a temporally varying convection term. Results show an increase in model accuracy which can allow for better resource allocation and safety measures.
1. INTRODUCTION
Large pedestrian events can bring safety and logistical challenges, motivating models of crowd movement. This paper develops a macroscopic PDE approach for marathon flow, improving an earlier convection-diffusion model to represent runner speed and dispersion.
1. INTRODUCTION
- Large pedestrian events can produce congestion, safety problems, and logistical delays, making crowd management an important modeling motivation.Examples include large events and trains unloading many tourists at small stations.Source: §1 · p. 1
- Microscopic models represent individual movement but can be computationally costly, while macroscopic PDE models represent pedestrian movement as flow along routes.Prior macroscopic approaches have included fluid-based models and models for specific high-density or panic scenarios.Source: §1 · p. 1
- The paper develops a marathon-course macroscopic model that improves a convection-diffusion PDE by using space- and time-dependent mean speed and a method for determining runner dispersion.The model is motivated by the need to estimate flow where course data are unavailable, with possible uses in volunteer assignment, road reopening, and identifying choke points.Source: §1 · pp. 1–2
- The paper presents the convection-diffusion model, validates it with 2023 Columbus Marathon data, and then updates the mean-speed calculation to represent runner slowdown and different runner classes.The base model uses a temporally constant mean speed before the later time-varying extension.Source: §1 · p. 2
2. CONVECTION DIFFUSION PEDESTRIAN MODEL
The model treats a marathon as one-dimensional runner flow governed by density conservation, with convection representing forward movement and diffusion representing spreading. It simplifies coefficients using checkpoint data, specifies race conditions, and solves the discretized equation numerically.
2. CONVECTION DIFFUSION PEDESTRIAN MODEL
- The course is treated as one-dimensional and the model is formulated as a conservation equation for runner density.Distance and time define the model domain; density and flow are its state quantities.Source: §2 · p. 2
- Runner movement is modeled macroscopically through density and flow, combining forward convective transport with diffusion that spreads runners as their speeds vary.The model tracks the whole runner pack rather than individual runner trajectories.Source: §2 · p. 2
- Diffusion represents runners spreading from high- to low-density regions, while mean speed drives organized movement toward the finish line.The model initially treats the diffusion coefficient and mean speed as space-dependent parameters.Source: §2 · p. 2
- The model assumes D(x) and v(x) are piecewise constant because measurements are available only at course checkpoints.Source: §2 · p. 2
- The race begins with an empty-course initial condition, runners enter through a time-dependent feed rate, and an open finish boundary lets them exit.The feed rate describes runner entry after the starting gun.Source: §2 · p. 2
- The equation is nondimensionalized using race time, course length, and runner count, then discretized with the second-order Crank–Nicolson method.The continuous model is discretized on temporal and spatial grids.Source: §2 · pp. 2–3
- Boundary nodes use Taylor-series expansions rather than the interior central-difference formula when constructing the numerical updates.The start uses a forward expansion and the finish uses a backward expansion.Source: §2 · p. 3
- After discretization and boundary-condition application, the equations form a tridiagonal matrix system solved by direct Gauss–Jordan elimination.Source: §2 · p. 3
3. DATA ANALYSIS
The 2023 Columbus Marathon data records runner times at checkpoints and the race start, supporting analysis of runner flow and density. Arrival times show increasing separation between the fastest and slowest runners over the course.
3. DATA ANALYSIS
- The dataset covers 2,967 runners and includes chip start times, checkpoint crossing times, and finish times.Checkpoints are at 7 km, 15 km, 21.1 km, and 32.2 km.Source: §3 · p. 3
- Runner flow and density are calculated from the data, with the start feed rate serving as the boundary condition.The start plot shows cumulative entrants and feed rate; organizers use an S-shaped release with two waves separated by 3 minutes, and all runners enter within 20 minutes.Source: §3 · pp. 3–4, Figure 3 · p. 4
- At the finish, the first-to-last runner gap exceeds four hours, up from about forty minutes at 7 km.Fig. 2 presents arrival times at each checkpoint and the finish line; the slower runners begin to fatigue between halfway and 20 miles.Source: §3 · p. 3, Figure 2 · p. 4
4. BASE MODEL RESULTS
The base model uses spatially varying mean speed and diffusivity to describe runner flow. It captures observed flow trends, while its estimated mean checkpoint times show increasing error toward the finish.
4. BASE MODEL RESULTS
- Mean runner speed mostly decreases with course distance, except between 15 km and 13.1 miles, where finishing half-marathon runners influence the speeds.Fig. 4 shows mean speed as a function of course distance.Source: §4 · p. 4, Figure 4 · p. 4
- Diffusivity increases with course distance, consistent with runners dispersing along the course.The model estimates diffusivity with particle swarm optimization to minimize the error between measured and modeled flow.Source: §4 · p. 4, Figure 5 · p. 5
- The constant-speed model captures the main measured flow trends, including temporal spreading at each checkpoint as runners disperse.Fig. 6 compares measured and modeled flow rates at each checkpoint.Source: §4 · p. 4, Figure 6 · p. 5
- The model overestimates the first checkpoint time and underestimates the finishing time by almost 4.4%.Table 1 compares mean times at each checkpoint.Source: §4 · p. 5, Table 1 · p. 5
5. IMPROVED MODEL RESULTS
The improved model uses a mean speed that varies over both time and distance to represent differences between earlier and later runners. Applied to the 2023 marathon data, it estimates average crossing times more accurately than the earlier model.
5. IMPROVED MODEL RESULTS
- The model uses time- and space-varying mean speed because averaging across all runners underestimates leaders’ speeds and overestimates those at the back.The paper links this difference to runners’ previous training and nutrition and to hitting the wall.Source: §5 · p. 5
- With a one-hour interval, runners are sorted by checkpoint arrival time, grouped by hour, and each group’s mean speed is computed from its runners’ velocities.A different mean-speed vector is used at each hour.Source: §5 · pp. 5–6
- Fig. 7 plots mean speed by distance for seven race hours; the second-hour curve spans the widest range, from runners near 10 km to others near 35 km.The plotted speeds fall to zero where no runners are present; no runner goes beyond about 20 km in the first hour.Source: §5 · p. 5
- The variable-speed model represents the measured flow-rate data well at each checkpoint, as shown in Fig. 8.The comparison is between measured and modeled flow rates.Source: Figure 8 · p. 6, §5 · p. 6
- Roughly 2%: the improved model decreases first-checkpoint and average finishing-time errors, while estimating average crossing times almost exactly at other checkpoints.Table 2 reports the comparison for the variable-mean-speed model.Source: Table 2 · p. 6, §5 · p. 6
6. CONCLUSION
The paper applies a pedestrian-flow PDE to Columbus Marathon data and improves its mean-speed calculation to vary across time and distance. The revised calculation better models runner movement and reduces average checkpoint-time errors by up to 2%.
6. CONCLUSION
- The spatially and temporally varying mean-speed calculation better captures runners slowing toward the end of the race and reduces average checkpoint-time errors by up to 2%.The study uses data from the Columbus Marathon to validate the base model before making the improvement.Source: §6 · p. 6
- Future work will examine reducing high-density areas, changing race starts, planning aid stations and resources, and extending the model to other large events.The proposed additional settings include concerts, fairs, parades, and festivals, with crowd control and safe evacuation as planning aims.Source: §6 · p. 6