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Evacuations and disasters: modeling stressed crowds Alethea Barbaro - - PowerPoint PPT Presentation

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Evacuations and disasters: modeling stressed crowds Alethea Barbaro Joint work with: Daniel Balagu (CWRU), Jess Rosado (UCLM), Nicole Abaid (Virginia Tech), Sachit Butail (Northern Illinois University), Hye Rin Lindsay Lee (CWRU), and Jenny

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Evacuations and disasters: modeling stressed crowds

Alethea Barbaro Joint work with: Daniel Balagué (CWRU), Jesús Rosado (UCLM), Nicole Abaid (Virginia Tech), Sachit Butail (Northern Illinois University), Hye Rin Lindsay Lee (CWRU), and Jenny Brynjarsdottir (CWRU) Crowds: Models and Control (CIRM) 3 June 2019

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Motivation

  • A. Barbaro – Evacuations and disasters: modeling stressed crowds
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Cucker-Smale Model

  • A. Barbaro – Evacuations and disasters: modeling stressed crowds

Where

  • Well-studied in literature
  • Flocking ensured if 𝛿 ≤

# $

  • Captures the basic dynamics of a group of
  • rganisms deciding to move together
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SLIDE 4

Background

  • A. Barbaro – Evacuations and disasters: modeling stressed crowds

Tambe et al. (USC) considered the following 2D model: where

  • For particle 𝑗, 𝐻' is the set of agents within a given distance, 𝑟' is its level of emotion, 𝜁' is the

strength with which it transmits its emotion, 𝛾' is how receptive it is to influence, 𝛽', is the strength of the interactions between particles 𝑗 and 𝑘

  • Note that each particle moves in a straight line directly away from the origin
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Tambe et al. vs. Cucker-Smale

  • Choose 𝛾' = 1. Then

𝑒𝑟' 𝑒𝑢 = 2

,∈45

𝜁,𝛽', 𝑟, − 𝑟'

  • A. Barbaro – Evacuations and disasters: modeling stressed crowds
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Tambe et al. vs. Cucker-Smale

  • Let us now choose the neighborhoods 𝐻' so that

they all contain every particle, and let 𝜁

, = # 7 and 𝛽', = 𝑏',

  • A. Barbaro – Evacuations and disasters: modeling stressed crowds
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Tambe et al. vs. Cucker-Smale

  • The resulting equation for 𝑟' is exactly the

velocity evolution described by the Cucker-Smale model in 1D

  • A. Barbaro – Evacuations and disasters: modeling stressed crowds
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Our Particle Model

  • A. Barbaro – Evacuations and disasters: modeling stressed crowds
  • Flocking model with a contagious emotional component:

(1)

  • This model allows individuals to change direction over the course of the simulation
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Model Behaviors

  • A. Barbaro – Evacuations and disasters: modeling stressed crowds
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Why Kinetic Limits?

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Kinetic Model

  • Mean field limit of our microscopic model:

(2) where

  • A. Barbaro – Evacuations and disasters: modeling stressed crowds
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Notion of Solution

  • Hypotheses: Suppose that
  • 𝐺 𝑢, 𝑦, 𝑥, 𝑟 ≤ 𝐷# 1 + 𝑦 + 𝑟
  • 𝐻 𝑢, 𝑦, 𝑥, 𝑟 ≤ 𝐷$ 1 + 𝑦 + 𝑟
  • 𝐺 and 𝐻 are locally Lipschitz with respect to 𝑦 and 𝑟.
  • A. Barbaro – Evacuations and disasters: modeling stressed crowds

Definition 1. (Notion of solution) We say that a function 𝑔: 0, 𝑈 → 𝒬

# ℝF×ℝF×ℝ is a solution to Equation (2) with initial condition 𝑔 H

when

  • The fields 𝐺 and 𝐻 satisfy the Hypotheses above
  • 𝑔 = ℱJ,4

K #𝑔 H, where ℱJ,4 K

is the flow map associated to (1)

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Existence and Uniqueness of Solutions

  • Theorem. Let 𝑔

H ∈ 𝒬 # ℝF×ℝF×ℝ with compact support. Then, there exists a

unique solution 𝑔 ∈ 𝒟 0, +∞ , 𝒬

# ℝF×ℝF×ℝ

to equation (2) with initial condition 𝑔

H in the sense of Definition 1.

  • A. Barbaro – Evacuations and disasters: modeling stressed crowds
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Convergence in 𝜕 and 𝑟

  • Theorem. Let {(𝑦', 𝜕', 𝑟')(t)}'R#

7

be the solution to (1) with initial condition {(𝑦'

H, 𝜕' H, 𝑟' H)}'R# 7

and let 𝛿 ≤ #

$. Then

  • 𝑟' 𝑢 → S

𝑟 for all 1 ≤ 𝑗 ≤ 𝑂. (S 𝑟 is the average level of fear of the population).

  • If 𝑒 = 2,
  • define 𝜕V and 𝜕Wthe directions such that all other directions 𝜕' lie between

𝜕V and 𝜕W,

  • let 𝛽(𝑢) be the angle formed by these two directions at each time.
  • Then if 𝛽 0 ≤ 𝜌 and 𝛾 ≤ #

$, there exists 𝜕∗ such that 𝜕' → 𝜕∗ for all 1 ≤ 𝑗 ≤

𝑂.

  • A. Barbaro – Evacuations and disasters: modeling stressed crowds
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Corollary (Rate of Convergence of 𝛽 𝑢 )

From the proof of the previous theorem we obtain

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with (Exponential convergence of the square of the difference)

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Numerical Validation in 2D

−4 −3 −2 −1 1 2 3 4 5 10 15 20 Convergence of α(t)

α0 = 8

8 π

α0 = 11

8 π

α0 = 5

8 π

α0 = 14

8 π

α0 = 9

8 π

α0 = 1

8 π

α0 = 1

4 π

α0 = 1

3 π

α0 = 7

8 π

α0 = 16

8 π

α0 = 10

8 π

α0 = 13

8 π

α0 = 3

8 π

α0 = 3

4 π

α0 = 12

8 π

α0 = 2

4 π

α0 = 2

3 π

  • A. Barbaro – Evacuations and disasters: modeling stressed crowds

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5 10 15 20 Convergence of q(t)

α0 = 8

8 π

α0 = 11

8 π

α0 = 5

8 π

α0 = 14

8 π

α0 = 9

8 π

α0 = 1

8 π

α0 = 1

4 π

α0 = 1

3 π

α0 = 7

8 π

α0 = 16

8 π

α0 = 10

8 π

α0 = 13

8 π

α0 = 3

8 π

α0 = 3

4 π

α0 = 12

8 π

α0 = 2

4 π

α0 = 2

3 π

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Examining Convergence of Alpha

−1.5 −1 −0.5 0.5 1 1.5 5 10 15 20 Convergence of α(t) with γ = 0.25 and α0 = 7

8π

β = 0.125 β = 0.25 β = 0.5 β = 1

  • A. Barbaro – Evacuations and disasters: modeling stressed crowds

−1.5 −1 −0.5 0.5 1 1.5 5 10 15 20 Convergence of α(t) with β = 0.25 and α0 = 7

8π

γ = 0.125 γ = 0.25 γ = 0.5 γ = 1

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Examining Convergence of Emotion

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5 10 15 20 Convergence of q(t) with γ = 0.25 and α0 = 7

8π

β = 0.125 β = 0.25 β = 0.5 β = 1

  • A. Barbaro – Evacuations and disasters: modeling stressed crowds

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5 10 15 20 Convergence of q(t) with β = 0.25 and α0 = 7

8π

γ = 0.125 γ = 0.25 γ = 0.5 γ = 1

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Observation: Rates of Convergence

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SWITCHING GEARS Experimental Evaluation of the Social Force Model

  • The ultimate goal of modeling: simulate real-world situations
  • Social force models are commonly used/referenced in the literature
  • Worth seeing how well they work before testing a new model
  • We use Helbing et al.’s social force model to simulate an experiment
  • A. Barbaro – Evacuations and disasters: modeling stressed crowds
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The Experiment (Sachit Butail et al.)

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The Experiment (Sachit Butail et al.)

  • Evacuation experiment
  • Participants asked to leave the room
  • Some instructed to leave quickly (Rush), others were given no instruction (No Rush)
  • Five conditions: 0% Rush, 25% Rush, 50% Rush, 75% Rush, 100% Rush
  • Videos were taken
  • Trajectories were found by post-processing
  • A. Barbaro – Evacuations and disasters: modeling stressed crowds
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Experimental Results

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The Question

How well does the Helbing Model perform in this context?

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Helbing’s Social Force Model

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Goal Force

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Where

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Interaction Force

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Wall Force

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A Sample Simulation

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Results of Helbing Model Using Helbing’s Parameters

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Parameter Choices

  • Parameters from Helbing et al.
  • 𝐵 = 2000 N
  • 𝐶 = 0.08 m
  • 𝑤H varied: 0.6 m/s (relaxed), 1 m/s (normal), 1.5 m/s (nervous)
  • Our Parameter Range
  • Create 𝐵^ and 𝐵7^, 𝐶^ and 𝐶7^, 𝑤^

H and 𝑤7^ H

  • Test 𝐵' between 0 and 4000 N
  • Test 𝐶' between 0 and 0.16 m
  • Test 𝑤'

H between 0.1 and 2 m/s

  • A. Barbaro – Evacuations and disasters: modeling stressed crowds
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Our Numerical Experiments

  • Latin Hypercube to choose sets of parameters for testing
  • We chose 600 representative points in parameter space
  • Ensembles of 100 for each choice of parameters
  • Compare distributions of average velocity between experiment and ensemble
  • Note: Needs to be done for both Rush and No Rush agents
  • A. Barbaro – Evacuations and disasters: modeling stressed crowds
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Comparing Experiments to Simulations: Wasserstein Distance

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Color-Coded Plots of Parameter Space

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34

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Color-Coded Plots of Parameter Space

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Best Fit

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The Best-Fitting Parameters

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Continuing Work

  • Exploring variations on contagion model
  • Cone of vision
  • Fixed radius of interaction
  • Confinement-dependent fear levels
  • Experiments!
  • A. Barbaro – Evacuations and disasters: modeling stressed crowds
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Thank You!