Modeling Crowd Dynamics Brent Morgan1, 2, Krista Parry1, 3, Andy Platta1, 3, Mitch Wilson1, 4 1Mathematics, 2Engineering Physics, 3Chemistry, 4Mechanical Engineering

1
Modeling Crowd DynamicsBrent Morgan1, 2, Krista
Parry1, 3, Andy Platta1, 3, Mitch Wilson1,
41Mathematics, 2Engineering Physics, 3Chemistry,
4Mechanical Engineering

• Project Description
• Goal To accurately model the movement of
  individuals in a crowd out of a room through a
  single or multiple exits.
• Take into account different levels of urgency
  with regard to the context of the exit
  circumstances.
• Expand the model to include different parameters
  such as individuals pushing or shoving, as well
  as the possibility for injuries.
• Take into account human behavior in crowd
  situations such as maintaining a certain amount
  of personal space and increased urgency as
  proximity to the exit increases.
• Scientific Challenges Potential Applications
• To accurately model an individuals movement
  while taking into account a certain degree of
  intelligence and free will.
• Planning escape routes for buildings.
• Determining room capacities for building coding.

Results
Based on our implications of the Swarm and
Kirchner models, we found that even with our
normalized parameters Bij and Rij, we were able
to recreate graphs and movement patterns as in
the two comparable models. We were able to
observe similar trends regarding parameter
values, such as comparing ks and kd. We were
also able to verify patterns relating to the
injury threshold and the percentage of people
that escape given an increasing number of initial
individuals. See the accompanying figures for
illustrations.
Figure 2 Series of screen shots at increasing
time steps during one trial with Ks3, Kd2.
Active agents denoted in blue, injured agents
denoted in red.

• Published Models for Crowd Dynamics
• Simulation of Evacuation Processes Using a
  Bionics-Inspired Cellular Automaton Model for
  Pedestrian Dynamics, by Ansgar Kirchner
• Introduces the Kirchner Field Model. This model
  simulates the evacuation of people from a room.
  Peoples movements are determined
  probabilistically with the probability
  distribution depending on proximity to the door
  and the movement of other agents. We will refer
  to this model as the Kirchner model.
• Macroscopic Effects of Microscopic Forces Between
  Agents in Crowd Models, by Colin M. Henein
• Implements many of the ideas used in the
  Kirchner Field Model but adds to it with the
  addition of a force field that simulates agents
  pushing. This allows the addition of injuries to
  the model which render agents immobile. We will
  refer to this model as the Swarm Force model.

• Glossary of Technical Terms
• Cellular Automata A regular array of identical
  finite state automata whose next state is
  determined solely by their current state and the
  state of their neighbors.
• Bosons Elementary particles that give the agents
  information about their surroundings. These
  particles are dropped by the agents when they
  occupy a cell, and other individuals are
  attracted to these particles. This can be likened
  to ants leaving a pheromone trail to signal a
  pathway for others to follow.

Figure 4 Plot of number injured as a function of
the injury threshold. Ks3, Kd3
Figure 3 Graphs of people remaining in room as a
function of time. Kd, see key, Ks3.

• Methodology
• Used method of direct computer simulation and
  cellular automata to construct a grid
  representation of a room with a single exit.
• Constructed a probability distribution of an
  individuals 8 neighboring cells, based on
  factors such as the proximity to the door, and
  previous paths of individuals. (Equations shown
  on right).
• Determine which individual moves first at each
  time step based on a Force Determining equation
  which is inversely related to the distance to the
  door.
• Use concepts from Kirchner Field Model 4 and
  Swarm Force Model 1 use a dynamic and static
  field to determine the individuals most probable
  move and include injuries of individuals due to
  pushing or crowding.
• The static field will be inversely related to the
  distance to the door, thus the cell that is
  closer to the door will have a higher probability
  that the individual will move to it.
• The dynamic field keeps track of the bosons that
  individuals have left when previously occupying
  the cell. This raises the probability of an
  individual moving to the cell to simulate
  individuals following others if a path is
  successful.
• The dynamic and static fields are weighted
  differently based on proximity to the door.
• Pushing is allowed, and if an individual is
  pushed past the threshold, they will become
  permanently injured, indicated by a red dot in
  the following graphs.

Equations
Parameters Pij Probability of individual moving
to cell (i, j) N Normalization coefficient Bij
Normalized dynamic preference factor ?ij
Occupancy of the cell (i,j) kD Dynamic field
coefficient kS Static field coefficient Rij
Normalized static preference factor Dij
Literature dynamic preference factor Sij
Literature static preference factor ?ij Swarm
Force parameter, based on occupancy ?ij Swarm
Force parameter, based on walls Fij Force in
determining movement order of individuals (X, Y)
Coordinates of the exit door

• Force Determining equation

Figure 5 Plot of percentage of individuals
leaving the room as a function of density of
individuals in the room. Ks3, Kd2

• Modified Swarm Force equation

• References
• Henin, C. M. White, T. Macroscopic Effects of
  Microscopic Forces Between Agents in Crowd
  Models. Physica. A 373, 694-712 (2007).
• Weng, W.G., Pan, L.L., Shen, S.F., Yuan, H.Y.
  Small-grid Analysis of Discrete Model for
  Evacuation from a Hall. Physica. A 374, 821-826
  (2007).
• Seyfried, A., Steffen, B., Lippert, T. Basics of
  Modelling the Pedestrian Flow. Physica A 368,
  232-238 (2006).
• Kirchner, Ansgar Schadschneider, Andreas.
  Simulation of Evacuation Processes Using a
  Bionics-Inspired Cellular Automaton Model for
  Pedestrian Dynamics. Physica. A 312, 260-276
  (2002).

Figure 1 Method determining order of individual
movement per time step, depending on the Force
Determining equation.

• Literature Swarm Force equation

• Acknowledgments
• This project was mentored by Jorge Ramirez,
  whose help is acknowledged with great
  appreciation.
• Support from a University of Arizona TRIF
  (Technology Research Initiative Fund) grant to J.
  Lega is also gratefully acknowledged.