# Particle Methods Simulations by Kinetic Theory Models of Human Crowds Accounting for Stress Conditions

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## Abstract

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## 1. Plan and Aims of the Paper

## 2. Selection of a Crowd Model and Simulations

#### 2.1. Requirements of the Computational Approach

#### 2.2. Selection of a Computational Model

- $\alpha $ models the quality of the venue, where $\alpha =0$ corresponds to worst quality which, in practice, prevents motion, while $\alpha =1$ models high quality which can allows high speeds, however depending on the local density, as increasing density reduces locally the speed.
- $\beta $ models the level of stress, where $\beta =0$ corresponds to absence of concentration, $\beta =0.5$ to a normal level of stress which is needed by a proper self-organization of the motion, and $\beta =1$ models the highest level of stress. The connection between level of stress and local density can be modeled by simple functions which depict how $\beta $ is not a fixed parameter, but it grows with the local density when the said density pass a critical threshold, e.g., ${\rho}_{c}\ge 0.5$.

**Interaction rate:**A simple model of the rate of interaction between walkers is obtained by assuming that this rate is constant.**Walking strategy:**Walkers first select the walking direction, which ends up to correspond to the velocity direction, and subsequently adapt the speed to the local density pattern in their visibility zone.**Selection of the velocity direction:**According to [7], the strategy which leads to the velocity direction $\phi $ is a weighted selection out of three directions: (i) towards the trajectory going to the exit, (ii) attraction by the low gradient path, (iii) attraction by the main stream. The selection is driven by the local density $\rho $ and the parameter $\beta $ which models the level of stress. In more detail, attraction (ii) is enhanced by $\rho $, while attraction (iii) is enhanced by $\beta $.**Perception of the density along the velocity direction:**Once a velocity direction has been selected walkers perceive a density different from the real one as follows:$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\partial}_{\phi}\left[\rho \right]\ge 0\phantom{\rule{14.22636pt}{0ex}}\Rightarrow \phantom{\rule{14.22636pt}{0ex}}{\rho}_{\phi}\left[\rho \right]=\rho +\frac{{\partial}_{\phi}\left[\rho \right]}{\sqrt{1+{\left({\partial}_{\phi}\left[\rho \right]\right)}^{2}}}\phantom{\rule{0.166667em}{0ex}}(1-\rho )\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\partial}_{\phi}\left[\rho \right]<0\phantom{\rule{14.22636pt}{0ex}}\Rightarrow \phantom{\rule{14.22636pt}{0ex}}{\rho}_{\phi}\left[\rho \right]=\rho +\frac{{\partial}_{\phi}\left[\rho \right]}{\sqrt{1+{\left({\partial}_{\phi}\left[\rho \right]\right)}^{2}}}\phantom{\rule{0.166667em}{0ex}}\rho \phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$$${\partial}_{\phi}\left[\rho \right]\to \infty \Rightarrow {\rho}_{\phi}\to 1\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}{\partial}_{\phi}\left[\rho \right]=0\Rightarrow {\rho}_{\phi}=\rho \phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}{\partial}_{\phi}\left[\rho \right]\to -\infty \Rightarrow {\rho}_{\phi}\to 0.$$**Adjustment of the speed:**Walkers, after perception of the density, adjust their speed to the perceived density conditions. The new speed ${v}_{\phi}$ can be modeled as follows:$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\partial}_{\phi}\left[\rho \right]\ge 0\phantom{\rule{14.22636pt}{0ex}}\Rightarrow \phantom{\rule{14.22636pt}{0ex}}{v}_{\phi}\left[\rho \right]=v-\alpha \phantom{\rule{0.166667em}{0ex}}{\rho}_{\phi}\left[\rho \right]\phantom{\rule{0.166667em}{0ex}}v\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\partial}_{\phi}\left[\rho \right]<0\phantom{\rule{14.22636pt}{0ex}}\Rightarrow \phantom{\rule{14.22636pt}{0ex}}{v}_{\phi}\left[\rho \right]=v-\alpha \phantom{\rule{0.166667em}{0ex}}{\rho}_{\phi}\left[\rho \right]\phantom{\rule{0.166667em}{0ex}}(1-v)\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$**Stress-density reciprocal influence:**A key feature of the dynamics is the interaction between local density and level of stress as when the local density tends to the maximal value also the local stress tends to limit value $\alpha =1$.**Boundary conditions:**The statement of the mathematical problem requires the implementation of boundary conditions which depend on the distance from the wall measured along the direction of motion. When this distance tends to zero the attraction by the trajectory towards the exit is enhanced with respect to the other directions.

#### 2.3. Simulations of Crowds over a Bridge with Internal Obstacles

- The number of pedestrians in the simulations is of a higher order, namely of the order f ${10}^{3}$ instead of ${10}^{2}$.
- The velocity is continuous rather than discrete in a fixed number of directions, while the speed is not related to the velocity diagram, but it accounts for local gradients.
- The individual decision processes which leads to the selection of the velocity direction account for the presence of obstacles.
- The dynamics accounts for the different level of stress and of their relation to local density.
- A stochastic computational method has been developed to take into account all specific features indicated in the above items.

#### 2.4. Flows of Crowds Moving to Opposite Directions

## 3. Critical Analysis towards Safety Problems

**broad variety of the possible**actions that can effectively reduce the crisis situation. In addition, the database can be used also to train leaders who in the crowd can contribute, by addressing the walking dynamics, to reduce the risk of safety problems. More in general, simulations can provide a detailed description of the role of the geometry and quality of the venue over the dynamics in space of the aforementioned density patterns.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**Flow patterns, at different times, in the interactions of two streams moving towards opposite directions.

**Figure 4.**Flow patterns, for large times, in the interactions of two streams moving towards opposite directions.

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Elaiw, A.; Al-Turki, Y.
Particle Methods Simulations by Kinetic Theory Models of Human Crowds Accounting for Stress Conditions. *Symmetry* **2020**, *12*, 14.
https://doi.org/10.3390/sym12010014

**AMA Style**

Elaiw A, Al-Turki Y.
Particle Methods Simulations by Kinetic Theory Models of Human Crowds Accounting for Stress Conditions. *Symmetry*. 2020; 12(1):14.
https://doi.org/10.3390/sym12010014

**Chicago/Turabian Style**

Elaiw, Ahmed, and Yusuf Al-Turki.
2020. "Particle Methods Simulations by Kinetic Theory Models of Human Crowds Accounting for Stress Conditions" *Symmetry* 12, no. 1: 14.
https://doi.org/10.3390/sym12010014