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

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

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

Cucker-Smale Model Well-studied in literature • # • Flocking ensured if 𝛿 ≤ $ Where Captures the basic dynamics of a group of • organisms deciding to move together A. Barbaro – Evacuations and disasters: modeling stressed crowds

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

Tambe et al. vs. Cucker-Smale • Choose 𝛾 ' = 1 . Then 𝑒𝑟 ' 𝑒𝑢 = 2 𝜁 , 𝛽 ', 𝑟 , − 𝑟 ' ,∈4 5 A. Barbaro – Evacuations and disasters: modeling stressed crowds

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

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

Our Particle Model Flocking model with a contagious emotional component : • (1) This model allows individuals to change direction over the course of the simulation • A. Barbaro – Evacuations and disasters: modeling stressed crowds

Model Behaviors A. Barbaro – Evacuations and disasters: modeling stressed crowds

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

Kinetic Model • Mean field limit of our microscopic model: (2) where A. Barbaro – Evacuations and disasters: modeling stressed crowds

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 . Definition 1. (Notion of solution) We say that a function 𝑔: 0, 𝑈 → # ℝ F ×ℝ F ×ℝ is a solution to Equation (2) with initial condition 𝑔 𝒬 H when The fields 𝐺 and 𝐻 satisfy the Hypotheses above • K #𝑔 K 𝑔 = ℱ J,4 H , where ℱ J,4 • is the flow map associated to (1) A. Barbaro – Evacuations and disasters: modeling stressed crowds

Existence and Uniqueness of Solutions # ℝ F ×ℝ F ×ℝ with compact support. Then, there exists a Theorem. Let 𝑔 H ∈ 𝒬 # ℝ F ×ℝ F ×ℝ unique solution 𝑔 ∈ 𝒟 0, +∞ , 𝒬 to equation (2) with initial condition 𝑔 H in the sense of Definition 1. A. Barbaro – Evacuations and disasters: modeling stressed crowds

Convergence in 𝜕 and 𝑟 7 Theorem. Let {(𝑦 ' , 𝜕 ' , 𝑟 ' )(t)} 'R# be the solution to (1) with initial condition and let 𝛿 ≤ # H , 𝜕 ' H , 𝑟 ' H )} 'R# 7 {(𝑦 ' $ . 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 𝜕 W the 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

Corollary (Rate of Convergence of 𝛽 𝑢 ) From the proof of the previous theorem we obtain with (Exponential convergence of the square of the difference) A. Barbaro – Evacuations and disasters: modeling stressed crowds

Numerical Validation in 2D Convergence of q ( t ) Convergence of α ( t ) 5 α 0 = 8 4 8 π α 0 = 8 α 0 = 11 8 π 4 . 5 8 π α 0 = 11 α 0 = 5 8 π 8 π 3 α 0 = 5 α 0 = 14 8 π 4 8 π α 0 = 14 α 0 = 9 8 π 8 π α 0 = 9 α 0 = 1 2 8 π 3 . 5 8 π α 0 = 1 α 0 = 1 8 π 4 π α 0 = 1 α 0 = 1 4 π 3 3 π 1 α 0 = 1 α 0 = 7 3 π 8 π α 0 = 7 α 0 = 16 8 π 2 . 5 8 π α 0 = 16 α 0 = 10 0 8 π 8 π α 0 = 10 α 0 = 13 8 π 2 8 π α 0 = 13 α 0 = 3 8 π 8 π − 1 α 0 = 3 α 0 = 3 8 π 4 π 1 . 5 α 0 = 3 α 0 = 12 4 π 8 π α 0 = 12 α 0 = 2 − 2 8 π 4 π 1 α 0 = 2 α 0 = 2 4 π 3 π α 0 = 2 3 π 0 . 5 − 3 0 − 4 0 5 10 15 20 0 5 10 15 20 A. Barbaro – Evacuations and disasters: modeling stressed crowds

Examining Convergence of Alpha Convergence of α ( t ) with β = 0 . 25 and α 0 = 7 Convergence of α ( t ) with γ = 0 . 25 and α 0 = 7 8 π 8 π 1 . 5 1 . 5 γ = 0 . 125 β = 0 . 125 γ = 0 . 25 β = 0 . 25 1 1 γ = 0 . 5 β = 0 . 5 β = 1 γ = 1 0 . 5 0 . 5 0 0 − 0 . 5 − 0 . 5 − 1 − 1 − 1 . 5 − 1 . 5 0 5 10 15 20 0 5 10 15 20 A. Barbaro – Evacuations and disasters: modeling stressed crowds

Examining Convergence of Emotion Convergence of q ( t ) with β = 0 . 25 and α 0 = 7 8 π Convergence of q ( t ) with γ = 0 . 25 and α 0 = 7 8 π 5 5 γ = 0 . 125 β = 0 . 125 4 . 5 4 . 5 γ = 0 . 25 β = 0 . 25 4 γ = 0 . 5 4 β = 0 . 5 γ = 1 3 . 5 β = 1 3 . 5 3 3 2 . 5 2 . 5 2 2 1 . 5 1 . 5 1 1 0 . 5 0 . 5 0 0 0 5 10 15 20 0 5 10 15 20 A. Barbaro – Evacuations and disasters: modeling stressed crowds

Observation: Rates of Convergence A. Barbaro – Evacuations and disasters: modeling stressed crowds

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

The Experiment (Sachit Butail et al.) A. Barbaro – Evacuations and disasters: modeling stressed crowds

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

Experimental Results A. Barbaro – Evacuations and disasters: modeling stressed crowds

The Question How well does the Helbing Model perform in this context? A. Barbaro – Evacuations and disasters: modeling stressed crowds

Helbing’s Social Force Model A. Barbaro – Evacuations and disasters: modeling stressed crowds

Goal Force Where A. Barbaro – Evacuations and disasters: modeling stressed crowds

Interaction Force A. Barbaro – Evacuations and disasters: modeling stressed crowds

Wall Force A. Barbaro – Evacuations and disasters: modeling stressed crowds

A Sample Simulation A. Barbaro – Evacuations and disasters: modeling stressed crowds

Results of Helbing Model Using Helbing’s Parameters A. Barbaro – Evacuations and disasters: modeling stressed crowds

Parameter Choices • Parameters from Helbing et al. • 𝐵 = 2000 N • 𝐶 = 0.08 m • 𝑤 H varied: 0.6 m/s (relaxed), 1 m/s (normal), 1.5 m/s (nervous) • Our Parameter Range H and 𝑤 7^ H • Create 𝐵 ^ and 𝐵 7^ , 𝐶 ^ and 𝐶 7^ , 𝑤 ^ • Test 𝐵 ' between 0 and 4000 N • Test 𝐶 ' between 0 and 0.16 m H between 0.1 and 2 m/s • Test 𝑤 ' A. Barbaro – Evacuations and disasters: modeling stressed crowds

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

Comparing Experiments to Simulations: Wasserstein Distance A. Barbaro – Evacuations and disasters: modeling stressed crowds

Color-Coded Plots of Parameter Space A. Barbaro – Evacuations and disasters: modeling stressed crowds 34

Color-Coded Plots of Parameter Space A. Barbaro – Evacuations and disasters: modeling stressed crowds

Best Fit A. Barbaro – Evacuations and disasters: modeling stressed crowds

The Best-Fitting Parameters A. Barbaro – Evacuations and disasters: modeling stressed crowds

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

Thank You!