# A MASTER EQUATION-BASED FRAMEWORK FOR THE MODELING OF PEDESTRIAN DYNAMICS

Carlo Bianca, Marco Dalla Via, Christian Dogbe

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Carlo Bianca, Marco Dalla Via, Christian Dogbe. A MASTER EQUATION-BASED FRAMEWORK FOR THE MODELING OF PEDESTRIAN DYNAMICS. Mathematics in Engineering, Science and Aerospace (MESA), 2019, Vol. 10, (No. 1), pp.129-142. [https://hal-02151412](https://hal-02151412)

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Distributed under a Creative Commons Attribution 4.0 International License
A MASTER EQUATION-BASED FRAMEWORK FOR THE MODELING OF
PEDESTRIAN DYNAMICS

CARLO BIANCA¹,²,*, MARCO DALLA VIA¹,², CHRISTIAN DOGBE³

¹ Laboratoire Quartz EA 7393, École Supérieure d'Ingénieurs en Génie Électrique, Productique et Management Industriel, 95092 Cergy-Pontoise, France.

² Laboratoire de Recherche en Eco-innovation Industrielle et Energétique, École Supérieure d'Ingénieurs en Génie Électrique, Productique et Management Industriel, 95092 Cergy-Pontoise, France.

³ Department of Mathematics, Université de Caen-Normandie, LMNO, CNRS, UMR 6139, 14032 Caen cedex, France.

* Corresponding Author. c.bianca@ecam-epmi.com

ABSTRACT. This paper is devoted to the derivation of a master equation-based frame-
work for the modeling of the pedestrian dynamics into a bounded domain of the plane.
The framework is based on the domain decomposition into squares with length side ε and
on the definition of the transition rates in the admissible directions that define the mathe-
matical operators of the master equation fulfilled by the joint probability. In particular two
specific mathematical models are derived within the new framework: An isotropic model
for the pedestrian dynamics in the checkout area of a supermarket where the pedestrians
move towards low-density regions of the domain, and an anisotropic model describing the
movement of pedestrians during an evacuation where high-density regions of the domain
are reached. The macroscopic dynamics, obtained by letting ε go to zero, is described by
reaction-diffusion equations.

# 1 Introduction

In the last two decades, the design of theoretical frameworks for the modeling of crowd
dynamics has been an interesting and challenging research field which has shown a fun-
damental interaction between mathematicians and physicists. The crowd dynamics is the
result of the interactions among the different individuals, usually pedestrians, composing
the system. Accordingly, a detailed analysis of the crowd dynamics has shown the typical
properties of a complex system, i.e. large ensemble of interacting elements, ability to per-
form a strategy, impossibility to describe the collective behaviors by the knowledge of the
dynamics and interaction of a few entities. The interested reader is referred to [1, 2] for a
detailed definition and the analysis of the main characteristics of a complex system. More
precisely the crowd dynamics is the emerging phenomenon of the complex interactions
between the pedestrians that are able to perform a specific strategy.

<footnote n="2010">²⁰¹⁰ Mathematics Subject Classification 35C20, 76R50, 35Q84

Keywords: Crowd dynamics, Joint probability, Partial differential equations, Transition rate, Reaction-diffusion equation</footnote>
The manuscript by Henderson [3] can be considered, to the best of our knowledge, as the seminal paper in the theoretical modeling of crowds. However, considering the interest that nowadays the control of crowd has attracted, the research activity on the crowd dynamics has recently increased. As a complex system, the dynamics of a crowd can be modeled at different levels (scales). Specifically three main modeling approaches have been developed: A microscopic approach, which is based on the derivation of a differential equation, usually an ODE, for each pedestrian composing the crowd (see [4, 5, 6, 7]); a macroscopic approach, which is based on the derivation of evolution equations (PDE) for the macroscopic observable quantities (such as the mass density, the linear momentum and the kinetic energy, see [8, 9, 10, 11, 12, 13, 14]); a mesoscopic approach based on the derivation of kinetic equations (or more in general nonlinear integro-differential equations) fulfilled by the statistical distribution function defined over the microscopic quantities, i.e. the position and velocity variables (see the recent contributions [15, 16] and the references cited therein). The reader interested in more details and a further literature is referred to the review papers [17, 18].

The definition of a suitable mathematical framework is an open problem considering that each of the above modeling approaches shows advantages and disadvantages. An optimal theoretical framework should depict, at least at a qualitative level, the emerging phenomena; the empirical data should not be plugged artificially into the models, but should be reproduced after a suitable choice of the parameters (usually a limited number of parameters related to well defined physical phenomena). It is worth stressing that data are obtained at the macroscopic scale, while the individual behaviors are not generally observed.

Recently the cellular automata approach has been proposed for the modeling of crowd dynamics, see, among others, papers [19, 20, 21, 22, 23]. A cellular automata model can be considered as a discrete model at the microscopic scale where specific rules guide the pedestrian transition from one subdomain (cell) to another subdomain. The model simulates pedestrians as entities (automata) into cells. The walkway is modeled as grid cells and a pedestrian is represented as a circle.

It is worth stressing that further approaches can be found in the pertinent literature. The reader is referred to papers [24, 25, 26, 27] and therein references.

This paper focuses on the dynamics of pedestrians within a bounded domain of the plane. Specifically the given domain is divided into a number of subdomains (squares with length side $\epsilon$) containing a certain number of pedestrians. The vertical and horizontal directions are the admissible movements, and a certain criterion (e.g. the leader-following dynamics) is established for the movement of a pedestrian among the squares. Bearing all above in mind, the mathematical operators describing the vertical and horizontal directions are obtained by defining the transition rates per unit of time. The mathematical operators are employed for the derivation of the evolution equation fulfilled by the joint probability, which stands for the probability that at time $t$ the crowd has a specific configuration. The mathematical framework, which refers to the matrix of the average number of pedestrians (density), consists into a system of partial differential equations. In order to further show the applicability of the proposed framework, two specific models are derived with application to the checkout area of a supermarket and to the pedestrian evacuation from a domain, respectively. The first model is based on the main assumption that pedestrians move towards low-density regions; consequently a privileged direction is missing (isotropic model); the second model is based on the leader-following dynamics [28], in particular it is assumed that pedestrians move towards the higher density region (anisotropic model). Specifically it is shown that the macroscopic dynamics, obtained by letting $\epsilon$ go
to zero, is described by reaction-diffusion equations. It is worth to precise that the new framework proposed in the present paper takes advantage from the contents of the papers [29, 30, 31, 32].

The present paper is organized into four more sections which follow this introduction. Specifically Section 2 is devoted to the derivation of the mathematical framework. The domain decomposition, the transition rates and the related mathematical operators, and the joint probability are defined within the section. Section 3 and Section 4 deal with the derivation of an isotropic and an anisotropic models, respectively. Finally Section 5 concludes the paper with further discussions and research perspectives.

## 2 The mathematical framework

This section is concerned with the derivation of a master equation-based framework that can be proposed for the modeling of pedestrian dynamics within a two-dimensional domain $\tilde{D} \subset \mathbb{R}^2$. Let $L \in \mathbb{R}^+$ be the largest length of the domain $\tilde{D}$, $V \in \mathbb{R}^+$ the maximum velocity which a pedestrian can attain and $T \in \mathbb{R}^+$ the time necessary for a pedestrian to cover the maximum length $L$ with velocity $V$, i.e. $L = VT$. The following dimensionless variables are defined: The time $t = \tilde{t}/T \in [0, +\infty[$, the $x$-position denoted by $x = \tilde{x}/L \in [0, 1]$ and the $y$-position denoted by $y = \tilde{y}/L \in [0, 1]$, where $\tilde{t}$ is the dimensional time and $(\tilde{x}, \tilde{y}) \in \tilde{D}$ is the dimensional vector position. Accordingly the dimensionless domain writes $\mathcal{D} = [0, 1] \times [0, 1]$.

Let $k \in \mathbb{N} \setminus \{0\}$. The domain $\mathcal{D}$ is divided into $k^2$ subdomains. Each subdomain is a square $S_{ij}$, for $i, j \in \{1, 2, \dots, k\}$, with length side $\varepsilon = 1/k \in \mathbb{R}^+$. The square $S_{ij}$ is thus defined as follows:

$$S_{ij} = [(i-1)\varepsilon, i\varepsilon] \times [(j-1)\varepsilon, j\varepsilon], \quad i, j \in \{1, 2, \dots, k\}, \qquad (2.1)$$

and it is centered at $(x_i^c, y_j^c) \in \mathcal{D}$, where:

$$x_i^c = \frac{\varepsilon}{2}(2i-1), \quad y_j^c = \frac{\varepsilon}{2}(2j-1), \quad i, j \in \{1, 2, \dots, k\}. \qquad (2.2)$$

It is assumed that a pedestrian can move among the squares only by vertical and horizontal directions. The movement of pedestrians follows a certain criterion $\alpha$, which influences both the vertical and horizontal movements, see Figure 1. Specifically:

* The vertical evolution of a pedestrian allocated in $S_{ij}$ is modeled by introducing the transition rate per unit time $T_{ij}^{\omega} := T_{ij}^{\omega}[\alpha](t)$, for $\omega \in \{U, D\}$, where $U$ and $D$ denote the upper and down movements, respectively. Specifically $T_{ij}^U$ denotes the transition rate of pedestrians from the square $S_{ij}$ to the square $S_{ij+1}$ and $T_{ij}^D$ denotes the transition rate from the square $S_{ij}$ to the square $S_{ij-1}$;

* The horizontal evolution of a pedestrian allocated in $S_{ij}$ is described by the transition rate per unit time $T_{ij}^{\omega} := T_{ij}^{\omega}[\alpha](t)$, for $\omega \in \{R, L\}$, where $R$ and $L$ denote the right and left movements, respectively. In particular $T_{ij}^R$ denotes the transition rate of pedestrians from the square $S_{ij}$ to the right square $S_{i+1j}$ and $T_{ij}^L$ denotes the transition rate of pedestrians from the square $S_{ij}$ to the left square $S_{i-1j}$.
<figure><img src="image_2.png" /><figcaption>FIGURE 1. The pedestrian evolution from the square $S_{ij}$ to the neighborhood squares.</figcaption></figure>

It is assumed that total number of pedestrian $\mathcal{N}$ remains constant in time. Accordingly one has:

$$T_{ik}^{U} = 0, \quad \forall i \in \{1, 2, \dots, k\}, \qquad (2.3)$$

$$T_{i1}^{D} = 0, \quad \forall i \in \{1, 2, \dots, k\}, \qquad (2.4)$$

$$T_{kj}^{R} = 0, \quad \forall j \in \{1, 2, \dots, k\}, \qquad (2.5)$$

$$T_{lj}^{L} = 0, \quad \forall j \in \{1, 2, \dots, k\}. \qquad (2.6)$$

Let $\mathbf{n} = (n_{11}, n_{12}, \dots, n_{kk}) \in \mathbb{N}^{k^2}$ be a configuration of the crowd, namely $n_{ij}$ denotes the number of pedestrians into the square $S_{ij}$. Accordingly one has:

$$\sum_{i=1}^{k} \sum_{j=1}^{k} n_{ij} = \mathcal{N}. \qquad (2.7)$$

The probability that at time $t$ the configuration of the crowd is $\mathbf{n}$ will be denoted by $P(t, \mathbf{n})$ (joint probability) and it is such that:

$$\sum_{\mathbf{n}} P(t, \mathbf{n}) := \sum_{n_{11}=0}^{\mathcal{N}} \sum_{n_{12}=0}^{\mathcal{N}} \cdots \sum_{n_{ij}=0}^{\mathcal{N}} \cdots \sum_{n_{kk}=0}^{\mathcal{N}} P(t, (n_{11}, n_{12}, \dots, n_{ij}, \dots, n_{kk})) = 1. \qquad (2.8)$$

The master equation (ME) fulfilled by $P(t, \mathbf{n})$ is based on the definition of the following operators:

* The operator $U_{ij}: \mathbb{N}^{k^2} \to \mathbb{N}^{k^2}$, for $i \in \{1, 2, \dots, k\}, j \in \{1, 2, \dots, k-1\}$, which allocates a pedestrian into the square $S_{ij}$ and removes it from the above square
$S_{ij+1}$. Accordingly:

$$U_{ij}(n_{11}, n_{12}, \dots, n_{ij}, \dots, n_{kk}) = (n_{11}, n_{12}, \dots, n_{ij} + 1, n_{ij+1} - 1, \dots, n_{kk}). \quad (2.9)$$

* The operator $D_{ij}: \mathbb{N}^{k^2} \to \mathbb{N}^{k^2}$, for $i \in \{1, 2, \dots, k\}$, $j \in \{2, 3, \dots, k\}$, which allocates a pedestrian into the square $S_{ij}$ and removes it from the below square $S_{ij-1}$. Accordingly:

$$D_{ij}(n_{11}, n_{12}, \dots, n_{ij}, \dots, n_{kk}) = (n_{11}, n_{12}, \dots, n_{ij-1} - 1, n_{ij} + 1, \dots, n_{kk}). \quad (2.10)$$

* The operator $R_{ij}: \mathbb{N}^{k^2} \to \mathbb{N}^{k^2}$, for $i \in \{1, 2, \dots, k-1\}$, $j \in \{1, 2, \dots, k\}$, which allocates a pedestrian into the square $S_{ij}$ and removes it from the right square $S_{i+1,j}$. Accordingly:

$$R_{ij}(n_{11}, n_{12}, \dots, n_{ij}, \dots, n_{kk}) = (n_{11}, n_{12}, \dots, n_{ij-1}, n_{ij} + 1, n_{ij+1}, \dots, n_{i+1j} - 1, \dots, n_{kk}). \quad (2.11)$$

* The operator $L_{ij}: \mathbb{N}^{k^2} \to \mathbb{N}^{k^2}$, for $i \in \{2, 3, \dots, k\}$, $j \in \{1, 2, \dots, k\}$, which allocates a pedestrian into the square $S_{ij}$ and removes it from the left square $S_{i-1,j}$. Accordingly:

$$L_{ij}(n_{11}, n_{12}, \dots, n_{ij}, \dots, n_{kk}) = (n_{11}, n_{12}, \dots, n_{i-1j} - 1, \dots, n_{ij-1}, n_{ij} + 1, n_{ij+1}, \dots, n_{kk}). \quad (2.12)$$

Bearing all above in mind, the time evolution of the joint probability $P(t, \mathbf{n})$ is described by the following ME:

$$\partial_t P(t, \mathbf{n}) = \mathbf{T}^U[P](t, \mathbf{n}) + \mathbf{T}^D[P](t, \mathbf{n}) + \mathbf{T}^R[P](t, \mathbf{n}) + \mathbf{T}^L[P](t, \mathbf{n}), \quad (2.13)$$

where:

$$\begin{aligned} \mathbf{T}^U[P](t, \mathbf{n}) &= \mathbf{G}^U[P](t, \mathbf{n}) - \mathbf{L}^U[P](t, \mathbf{n}) \\ &= \sum_{i=1}^{k} \sum_{j=1}^{k-1} T_{ij}^U((n_{ij} + 1)P(t, U_{ij}(\mathbf{n}))) - \sum_{i=1}^{k} \sum_{j=1}^{k-1} T_{ij}^U(n_{ij}P(t, \mathbf{n})), \quad (2.14) \end{aligned}$$

$$\begin{aligned} \mathbf{T}^D[P](t, \mathbf{n}) &= \mathbf{G}^D[P](t, \mathbf{n}) - \mathbf{L}^D[P](t, \mathbf{n}) \\ &= \sum_{i=1}^{k} \sum_{j=2}^{k} T_{ij}^D((n_{ij} + 1)P(t, D_{ij}(\mathbf{n}))) - \sum_{i=1}^{k} \sum_{j=2}^{k} T_{ij}^D(n_{ij}P(t, \mathbf{n})), \quad (2.15) \end{aligned}$$

$$\begin{aligned} \mathbf{T}^R[P](t, \mathbf{n}) &= \mathbf{G}^R[P](t, \mathbf{n}) - \mathbf{L}^R[P](t, \mathbf{n}) \\ &= \sum_{i=1}^{k-1} \sum_{j=1}^{k} T_{ij}^R((n_{ij} + 1)P(t, R_{ij}(\mathbf{n}))) - \sum_{i=1}^{k-1} \sum_{j=1}^{k} T_{ij}^R(n_{ij}P(t, \mathbf{n})), \quad (2.16) \end{aligned}$$

$$\begin{aligned} \mathbf{T}^L[P](t, \mathbf{n}) &= \mathbf{G}^L[P](t, \mathbf{n}) - \mathbf{L}^L[P](t, \mathbf{n}) \\ &= \sum_{i=2}^{k} \sum_{j=1}^{k} T_{ij}^L((n_{ij} + 1)P(t, L_{ij}(\mathbf{n}))) - \sum_{i=2}^{k} \sum_{j=1}^{k} T_{ij}^L(n_{ij}P(t, \mathbf{n})). \quad (2.17) \end{aligned}$$

Specifically:

* The operator $\mathbf{T}^U[P](t, \mathbf{n})$ models the all possible up-hand movements. In particular $\mathbf{G}^U[P](t, \mathbf{n})$ is the gain term while $\mathbf{L}^U[P](t, \mathbf{n})$ is the loss term;

* The operator $\mathbf{T}^D[P](t, \mathbf{n})$ models the all possible down-hand movements. In particular $\mathbf{G}^D[P](t, \mathbf{n})$ is the gain term while $\mathbf{L}^D[P](t, \mathbf{n})$ is the loss term;

* The operator $\mathbf{T}^R[P](t, \mathbf{n})$ models the all possible right-hand movements. In particular $\mathbf{G}^R[P](t, \mathbf{n})$ is the gain term while $\mathbf{L}^R[P](t, \mathbf{n})$ is the loss term;
• The operator $\mathbf{T}^L[P](t, \mathbf{n})$ models the all possible left-hand movements. In particular $\mathbf{G}^L[P](t, \mathbf{n})$ is the gain term while $\mathbf{L}^L[P](t, \mathbf{n})$ is the loss term.

The average number of pedestrians $N_{ij}(t) := N(t, x_i^c, y_j^c)$, for $i, j \in \{1, 2, \dots, k\}$, at time $t$ and into the square $S_{ij}$ is defined as follows:

$$N_{ij}(t) = \sum_{\mathbf{n}} n_{ij} P(t, \mathbf{n}). \qquad (2.18)$$

The matrix of the average number of pedestrians (density) is denoted by $\mathbf{N}(t) = [N_{ij}(t)]$.

**Theorem 2.1.** Let $T_{ij}^\omega$, for $\omega \in \{U, D, L, R\}$, $i, j \in \{1, 2, \dots, k\}$, be the transition rate. The matrix $\mathbf{N}(t)$ is solution of the following system of partial differential equations:

$$\left\{
\begin{aligned}
\partial_t N_{11}(t) &= T_{12}^D N_{12}(t) + T_{21}^L N_{21}(t) - (T_{11}^U + T_{11}^R) N_{11}(t) \\
\partial_t N_{12}(t) &= T_{11}^U N_{11}(t) + T_{13}^D N_{13}(t) + T_{22}^L N_{22}(t) - (T_{12}^U + T_{12}^D + T_{12}^R) N_{12}(t) \\
\vdots \\
\partial_t N_{ij}(t) &= T_{ij-1}^U N_{ij-1}(t) + T_{i,j+1}^D N_{i,j+1}(t) + T_{i-1,j}^R N_{i-1,j}(t) + T_{i+1,j}^L N_{i+1,j}(t) \\
&\quad - (T_{ij}^U + T_{ij}^D + T_{ij}^R + T_{ij}^L) N_{ij}(t) \\
\vdots \\
\partial_t N_{kk}(t) &= T_{kk-1}^U N_{kk-1}(t) + T_{k-1,k}^R N_{k-1,k}(t) - (T_{kk}^D + T_{kk}^L) N_{kk}(t).
\end{aligned}
\right.
\qquad (2.19)$$

*Proof.* From the equation (2.18), one has:

$$\partial_t N_{hl}(t) = \sum_{\mathbf{n}} n_{hl} \partial_t P(t, \mathbf{n}), \quad h, l \in \{1, 2, \dots, k\}. \qquad (2.20)$$

By replacing the equation (2.13) into the equation (2.20) and by considering the equations (2.14), (2.15), (2.16) and (2.17) one has:

$$\partial_t N_{hl}(t) = \sum_{\mathbf{n}} n_{hl} \left[ \sum_{i=1}^{k} \sum_{j=1}^{k-1} T_{ij}^U \left( (n_{ij} + 1)P(t, U_{ij}(\mathbf{n})) - n_{ij}P(t, \mathbf{n}) \right) \right. \\
+ \sum_{i=1}^{k} \sum_{j=2}^{k} T_{ij}^D \left( (n_{ij} + 1)P(t, D_{ij}(\mathbf{n})) - n_{ij}P(t, \mathbf{n}) \right) \\
+ \sum_{i=1}^{k-1} \sum_{j=1}^{k} T_{ij}^R \left( (n_{ij} + 1)P(t, R_{ij}(\mathbf{n})) - n_{ij}P(t, \mathbf{n}) \right) \\
\left. + \sum_{i=2}^{k} \sum_{j=1}^{k} T_{ij}^L \left( (n_{ij} + 1)P(t, L_{ij}(\mathbf{n})) - n_{ij}P(t, \mathbf{n}) \right) \right]. \qquad (2.21)$$

The first term of the right hand side of the equation (2.21) reads:

$$\sum_{\mathbf{n}} n_{hl} \left( \sum_{i=1}^{k} \sum_{j=1}^{k-1} T_{ij}^U \left( (n_{ij} + 1)P(t, U_{ij}(\mathbf{n})) - n_{ij}P(t, \mathbf{n}) \right) \right). \qquad (2.22)$$

By considering the terms corresponding to $i = h, j = l$ in the equation (2.22), one has:

$$T_{hl}^U \sum_{\mathbf{n}} n_{hl} \left( (n_{hl} + 1)P(t, U_{hl}(\mathbf{n})) - n_{hl}P(t, \mathbf{n}) \right). \qquad (2.23)$$
By changing the indices in the first sum $U_{hl}(\mathbf{n}) \to \mathbf{n}$, the equation (2.23) writes:

$$T_{hl}^{U} \sum_{\mathbf{n}} ((n_{hl} - 1)n_{hl}P(t, \mathbf{n}) - n_{hl}^{2}P(t, \mathbf{n})) = -T_{hl}^{U}N_{hl}(t). \quad (2.24)$$

By considering the terms corresponding to $i = h, j = l - 1$ in the equation (2.22), one has:

$$T_{hl-1}^{U} \sum_{\mathbf{n}} n_{hl} ((n_{hl-1} + 1)P(t, U_{hl-1}(\mathbf{n})) - n_{hl-1}P(t, \mathbf{n})). \quad (2.25)$$

By changing the indices in the first sum $U_{hl-1}(\mathbf{n}) \to \mathbf{n}$, the equation (2.25) writes:

$$T_{hl-1}^{U} \sum_{\mathbf{n}} ((n_{hl} + 1)n_{hl-1}P(t, \mathbf{n}) - n_{hl}n_{hl-1}P(t, \mathbf{n})) = T_{hl}^{U}N_{hl-1}(t). \quad (2.26)$$

The other terms of the equation (2.22), which correspond to $i \neq h, j \neq \{l, l-1\}$, are equal to zero. Therefore the equation (2.22) writes:

$$\sum_{\mathbf{n}} n_{hl} \left( \sum_{i=1}^{k} \sum_{j=1}^{k-1} T_{ij}^{U} ((n_{ij} + 1)P(t, U_{ij}(\mathbf{n})) - n_{ij}P(t, \mathbf{n})) \right) = T_{hl-1}^{U} N_{hl-1}(t) - T_{hl}^{U} N_{hl}(t). \quad (2.27)$$

The other terms of the equation (2.21) can be manipulated analogously to the equation (2.22). In particular one has:

$$\sum_{\mathbf{n}} n_{hl} \left( \sum_{i=1}^{k} \sum_{j=2}^{k} T_{ij}^{D} ((n_{ij} + 1)P(t, D_{ij}(\mathbf{n})) - n_{ij}P(t, \mathbf{n})) \right) = T_{hl+1}^{D} N_{hl+1}(t) - T_{hl}^{D} N_{hl}(t). \quad (2.28)$$

$$\sum_{\mathbf{n}} n_{hl} \left( \sum_{i=1}^{k-1} \sum_{j=1}^{k} T_{ij}^{R} ((n_{ij} + 1)P(t, R_{ij}(\mathbf{n})) - n_{ij}P(t, \mathbf{n})) \right) = T_{hl-1}^{R} N_{hl-1}(t) - T_{hl}^{R} N_{hl}(t). \quad (2.29)$$

$$\sum_{\mathbf{n}} n_{hl} \left( \sum_{i=2}^{k} \sum_{j=1}^{k} T_{ij}^{L} ((n_{ij} + 1)P(t, L_{ij}(\mathbf{n})) - n_{ij}P(t, \mathbf{n})) \right) = T_{hl+1}^{L} N_{hl+1}(t) - T_{hl}^{L} N_{hl}(t). \quad (2.30)$$

By replacing the equations (2.27), (2.28), (2.29) and (2.30) into the equation (2.21), one has the partial differential equations system (2.19).

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

The system of partial differential equations (2.19) constitutes the mathematical framework that can be proposed as a general paradigm for the derivation of a specific model for the pedestrian dynamics. A specific model is based on the definition of the transition rate $T_{ij}^{\omega}[\alpha](t)$, for $\omega \in \{U, D, R, L\}$, $i, j \in \{1, 2, \dots, k\}$.

### 3 An isotropic dynamics model

This section is devoted to the derivation of a specific model for the pedestrian dynamics within the mathematical framework (2.19). Specifically the model refers to the movement of pedestrians that want to reach a target. The criterion is to move towards low-density regions of the domain. This dynamics is typical at the exit of a metro station or in the checkout area of a supermarket.

The dynamics can be modeled by linking the movement of pedestrians in $S_{ij}$ to the initial density. Specifically one has:

$$T_{ij}^{\omega} = \frac{D_{1}\sigma_{ij}}{\varepsilon^{2}}, \quad \omega \in \{U, D, R, L\}, \quad i, j \in \{1, 2, \dots, k\}, \quad (3.1)$$
where

$$ \sigma_{ij} := \sigma(x_i^c, y_j^c) = N_{ij}(0), \quad i, j \in \{1, 2, \dots, k\}, \qquad (3.2) $$

and $D_1$ is the diffusion coefficient. According to this isotropic dynamics, a pedestrian moves from the square $S_{ij}$ which is highly crowded, to the neighborhood squares that are less crowded (see Figure 2), and the possibility for a pedestrian to leave the square $S_{ij}$ is in inverse proportion to the area $\varepsilon^2$ of $S_{ij}$. By replacing the equations (3.1) into the system

<figure><img src="image_5.png" /><figcaption>FIGURE 2. The pedestrians move from the square $S_{ij}$, which is initially highly crowded (left panel), to the neighborhood squares which are less crowded. The new configuration is shown in the right panel.</figcaption></figure>

(2.19), one has:

$$ \partial_t N_{ij}(t) = \frac{D_1}{\varepsilon^2} (\sigma_{ij-1}N_{ij-1}(t) + \sigma_{ij+1}N_{ij+1}(t) + \sigma_{i-1j}N_{i-1j}(t) + \sigma_{i+1j}N_{i+1j}(t) - 4\sigma_{ij}N_{ij}(t)), \qquad (3.3) $$

for $i, j \in \{1, 2, \dots, k\}$.

Let $\partial D = \mathcal{A} \cup \mathcal{B}$ be the boundary of the domain $D$, where the sets $\mathcal{A}$ and $\mathcal{B}$ are defined as follows:

$$ \mathcal{A} = \{(0, y), (1, y) : y \in [0, 1]\}, \qquad (3.4) $$

$$ \mathcal{B} = \{(x, 0), (x, 1) : x \in [0, 1]\}. \qquad (3.5) $$

Specifically $\mathcal{A}$ denotes the vertical boundaries of the domain $D$ and $\mathcal{B}$ denotes the horizontal boundaries of the domain $D$.

**Proposition 3.1.** Let $T_{ij}^\omega$, for $\omega \in \{U, D, R, L\}$, $i, j \in \{1, 2, \dots, k\}$, be the transition rate. Let $N_{ij}(t)$, for $i, j \in \{1, 2, \dots, k\}$, be the solution of the equation (3.3). Assume that

$$ \lim_{\varepsilon \to 0} N_{ij}(t) = \rho(t, x, y), \qquad (3.6) $$

$$ \lim_{\varepsilon \to 0} \sigma_{ij} = \rho_0(x, y). \qquad (3.7) $$
Then $\rho(t,x,y)$ is the solution of the following initial value problem with Neumann conditions:

$$ egin{cases} 
\partial_t \rho(t,x,y) = D_1 \Delta (\rho_0(x,y)\rho(t,x,y)) & (t,x,y) \in [0, +\infty[ \times \mathcal{D}, \\ 
\rho(0,x,y) = \rho_0(x,y) & (x,y) \in \mathcal{D}, \\ 
\partial_x (\rho_0(x,y)\rho(t,x,y)) = 0 & (t,x,y) \in [0, +\infty[ \times \mathcal{A}, \\ 
\partial_y (\rho_0(x,y)\rho(t,x,y)) = 0 & (t,x,y) \in [0, +\infty[ \times \mathcal{B}. 
\end{cases} 
\quad (3.8) $$

*Proof.* By expanding the functions $\sigma(x,y)$ and $N(t,x,y)$ at $(x_i^c, y_j^c)$ and by evaluating them in the points $(x_{i\pm1}^c, y_j^c)$ and $(x_i^c, y_{j\pm1}^c)$, respectively, one recovers the terms $\sigma_{i\pm1j}N_{i\pm1j}(t)$ and $\sigma_{ij\pm1}N_{ij\pm1}(t)$ of the equation (3.3) that now write as follows:

$$ \begin{aligned} \sigma_{i\pm1j} N_{i\pm1j}(t) &= \sigma(x_{i\pm1}^c, y_j^c) N(t, x_{i\pm1}^c, y_j^c) \\ &= \sigma(x_i^c, y_j^c) N(t, x_i^c, y_j^c) \pm \varepsilon \frac{\partial(\sigma N)}{\partial x}(t, x_i^c, y_j^c) + \frac{1}{2} \varepsilon^2 \frac{\partial^2(\sigma N)}{\partial x^2}(t, x_i^c, y_j^c) + o(\varepsilon^2) \end{aligned} \quad (3.9) $$

$$ \begin{aligned} \sigma_{ij\pm1} N_{ij\pm1}(t) &= \sigma(x_i^c, y_{j\pm1}^c) N(t, x_i^c, y_{j\pm1}^c) \\ &= \sigma(x_i^c, y_j^c) N(t, x_i^c, y_j^c) \pm \varepsilon \frac{\partial(\sigma N)}{\partial y}(t, x_i^c, y_j^c) + \frac{1}{2} \varepsilon^2 \frac{\partial^2(\sigma N)}{\partial y^2}(t, x_i^c, y_j^c) + o(\varepsilon^2) \end{aligned} \quad (3.10) $$

and replacing the equations (3.9) and (3.10) into the equation (3.3), one has:

$$ \partial_t N_{ij}(t) = \frac{D_1 \sigma_{ij}}{\varepsilon^2} \varepsilon^2 \frac{\partial^2(\sigma N)}{\partial x^2}(t, x_i^c, y_j^c) + \frac{D_1 \sigma_{ij}}{\varepsilon^2} \varepsilon^2 \frac{\partial^2(\sigma N)}{\partial y^2}(t, x_i^c, y_j^c) + o(\varepsilon^2), \quad (3.11) $$

By assumptions (3.6) and (3.7), the equation (3.11) rewrites as follows:

$$ \begin{aligned} \partial_t \rho(t,x,y) &= D_1 \partial_x (\rho(t,x,y)\partial_x \rho_0(x,y) + \rho_0(x,y)\partial_x \rho(t,x,y)) \\ &\quad + D_1 \partial_y (\rho(t,x,y)\partial_y \rho_0(x,y) + \rho_0(x,y)\partial_y \rho(t,x,y)) \\ &= D_1 \Delta (\rho_0(x,y)\rho(t,x,y)), \end{aligned} \quad (3.12) $$

with the initial condition:

$$ \rho(0, x, y) = \rho_0(x, y), \quad (x, y) \in \mathcal{D}. \quad (3.13) $$

Taking into account that the diffusion outside of the domain $\mathcal{D}$ is not possible, see the equations (2.3), (2.4), (2.5) and (2.6), the following boundary condition has to be imposed:

$$ \frac{\partial (\rho_0(x,y)\rho(t,x,y))}{\partial \hat{n}} = 0, \quad \forall (t,x,y) \in [0, +\infty[ \times \partial \mathcal{D}, \quad (3.14) $$

where $\hat{n} := \hat{n}(x,y) \in \mathbb{R}^2$, for $(x,y) \in \partial \mathcal{D}$, denotes the normal vector to the boundary $\partial \mathcal{D}$. The condition (4.14) rewrites as follows:

$$ \partial_x (\rho_0(x,y)\rho(t,x,y)) = 0, \quad (t,x,y) \in [0, +\infty[ \times \mathcal{A}, \quad (3.15) $$

$$ \partial_y (\rho_0(x,y)\rho(t,x,y)) = 0, \quad (t,x,y) \in [0, +\infty[ \times \mathcal{B}, \quad (3.16) $$

where the sets $\mathcal{A}$ and $\mathcal{B}$ are defined in the equation (3.4) and (3.5).

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>
## 4 An anisotropic dynamics model

This section deals with a pedestrian dynamics model where the criterion is the movement towards high-density regions of the domain $\mathcal{D}$. This dynamics occurs during evacuation from a domain (bridge, station). Usually the base of the dynamics is the leader-following model.

Bearing all above in mind, the dynamics can be modeled by linking the movement of pedestrians in $S_{ij}$ to the density of the neighborhood squares. Specifically one assumes that:

$$T_{ij}^{U} = \frac{D_2}{\varepsilon^2} \sigma_{ij+1}, \quad T_{ij}^{D} = \frac{D_2}{\varepsilon^2} \sigma_{ij-1}, \quad T_{ij}^{R} = \frac{D_2}{\varepsilon^2} \sigma_{i+1j}, \quad T_{ij}^{L} = \frac{D_2}{\varepsilon^2} \sigma_{i-1j}, \quad (4.1)$$

where $D_2 \in \mathbb{R}^+$ is the diffusion coefficient and the density $\sigma_{ij}$, for $i, j \in \{1, 2, \dots, k\}$, is given by the equation (3.2). According to this anisotropic dynamics, a pedestrian moves from the square $S_{ij}$, which is less crowded, to the neighborhood squares that are highly crowded, see Figure 3.

<figure><img src="image_8.png" /><figcaption>FIGURE 3. A pedestrian moves from the square $S_{ij}$, which is initially less crowded (left panel) to the square $S_{i+1j}$ that is highly crowded. The new configuration is shown in the right panel.</figcaption></figure>

By replacing the equations (4.1) in the system (2.19), one has:

$$\partial_t N_{ij}(t) = \frac{D_2 \sigma_{ij}}{\varepsilon^2} (N_{i,j-1}(t) + N_{i,j+1}(t) + N_{i-1,j}(t) + N_{i+1,j}(t)) - \frac{D_2}{\varepsilon^2} (\sigma_{ij-1} + \sigma_{ij+1} + \sigma_{i-1,j} + \sigma_{i+1,j}) N_{ij}(t), \quad (4.2)$$

for $i, j \in \{1, 2, \dots, k\}$.

**Proposition 4.1.** Let $T_{ij}^{\omega}$, for $\omega \in \{U, D, R, L\}$, $i, j \in \{1, 2, \dots, k\}$, be the transition rate. Let $N_{ij}(t)$, for $i, j \in \{1, 2, \dots, k\}$, be the solution of the equation (4.2). Assume that

$$\lim_{\varepsilon \to 0} N_{ij}(t) = \rho(t, x, y), \quad (4.3)$$

$$\lim_{\varepsilon \to 0} \sigma_{ij} = \rho_0(x, y). \quad (4.4)$$
Then $\rho(t,x,y)$ is the solution of the following initial value problem with Neumann conditions:

$$ egin{cases} 
\partial_t \rho(t,x,y) = D_2 \rho_0(x,y) (\Delta \rho(t,x,y)) - D_2 \rho(t,x,y) (\Delta \rho_0(x,y)) & (t,x,y) \in [0, +\infty[ \times \mathcal{D}, \\ 
\rho(0,x,y) = \rho_0(x,y) & (x,y) \in \mathcal{D}, \\ 
\partial_x (\rho_0(x,y)\rho(t,x,y)) = 0 & (t,x,y) \in [0, +\infty[ \times \mathcal{A}, \\ 
\partial_y (\rho_0(x,y)\rho(t,x,y)) = 0 & (t,x,y) \in [0, +\infty[ \times \mathcal{B}. 
\end{cases} 
\quad (4.5) $$

*Proof.* By expanding the functions $\sigma(x,y)$ and $N(t,x,y)$ at $(x_i^c, y_j^c)$ and by evaluating them in the points $(x_{i\pm 1}^c, y_j^c)$ and $(x_i^c, y_{j\pm 1}^c)$, respectively, one recovers the terms $\sigma_{i\pm 1j}$, $\sigma_{ij\pm 1}$, $N_{i\pm j}(t)$, $N_{ij\pm 1}(t)$ of the equation (4.2) that now write as follows:

$$ \sigma_{i\pm 1j} = \sigma(x_i^c, y_j^c) = \sigma(x_i^c, y_j^c) \pm \epsilon \frac{\partial \sigma}{\partial x}(x_i^c, y_j^c) + \frac{1}{2} \epsilon^2 \frac{\partial^2 \sigma}{\partial x^2}(x_i^c, y_j^c) + o(\epsilon^2), \quad (4.6) $$

$$ \sigma_{ij\pm 1} = \sigma(x_i^c, y_{j\pm 1}^c) = \sigma(x_i^c, y_j^c) \pm \epsilon \frac{\partial \sigma}{\partial y}(x_j^c, y_i^c) + \frac{1}{2} \epsilon^2 \frac{\partial^2 \sigma}{\partial y^2}(x_i^c, y_j^c) + o(\epsilon^2), \quad (4.7) $$

$$ N_{i\pm 1j}(t) = N(t, x_i^c, y_j^c) = N(t, x_i^c, y_j^c) \pm \epsilon \frac{\partial N}{\partial x}(t, x_i^c, y_j^c) + \frac{1}{2} \epsilon^2 \frac{\partial^2 N}{\partial x^2}(t, x_i^c, y_j^c) + o(\epsilon^2) \quad (4.8) $$

$$ N_{ij\pm 1}(t) = N(t, x_i^c, y_{j\pm 1}^c) = N(t, x_i^c, y_j^c) \pm \epsilon \frac{\partial N}{\partial y}(t, x_i^c, y_j^c) + \frac{1}{2} \epsilon^2 \frac{\partial^2 N}{\partial y^2}(t, x_i^c, y_j^c) + o(\epsilon^2) \quad (4.9) $$

and replacing the equations (4.6), (4.7), (4.8) and (4.9) into the equation (4.2), one has:

$$ \partial_t N_{ij}(t) = \frac{D_2}{\epsilon^2} \epsilon^2 \sigma(x_i^c, y_j^c) \left( \frac{\partial^2 N}{\partial x^2}(t, x_i^c, y_j^c) + \frac{\partial^2 N}{\partial y^2}(t, x_i^c, y_j^c) \right) - \frac{D_2}{\epsilon^2} \epsilon^2 N(t, x_i^c, y_j^c) \left( \frac{\partial^2 \sigma}{\partial x^2}(x_i^c, y_j^c) + \frac{\partial^2 \sigma}{\partial y^2}(x_i^c, y_j^c) \right) + o(\epsilon^2). \quad (4.10) $$

By assumptions (4.3) and (4.4), the equation (4.10) rewrites as follows:

$$ \partial_t \rho(t,x,y) = D_2 \partial_x [\rho_0(x,y) \partial_x \rho(t,x,y) - \rho(t,x,y) \partial_x \rho_0(x,y)] + D_2 \partial_y [\rho_0(x,y) \partial_y \rho(t,x,y) - \rho(t,x,y) \partial_y \rho_0(x,y)], \quad (4.11) $$

$$ = D_2 \rho_0(x,y) (\Delta \rho(t,x,y)) - D_2 \rho(t,x,y) (\Delta \rho_0(x,y)), \quad (4.12) $$

with the initial condition:

$$ \rho(0, x, y) = \rho_0(x, y), \quad (x, y) \in \mathcal{D}. \quad (4.13) $$

Taking into account that the diffusion outside the domain $\mathcal{D}$ is not possible, see the equations (2.3), (2.4), (2.5) and (2.6), the following boundary condition has to be imposed:

$$ \frac{\partial (\rho_0(x,y)\rho(t,x,y))}{\partial \hat{n}} = 0, \quad \forall (t,x,y) \in [0, +\infty[ \times \partial \mathcal{D}, \quad (4.14) $$

where $\hat{n} := \hat{n}(x,y) \in \mathbb{R}^2$, for $(x,y) \in \partial \mathcal{D}$, denotes the normal vector to the boundary $\partial \mathcal{D}$. The condition (4.14) rewrites as follows:

$$ \partial_x (\rho_0(x,y)\rho(t,x,y)) = 0, \quad (t,x,y) \in [0, +\infty[ \times \mathcal{A}, \quad (4.15) $$

$$ \partial_y (\rho_0(x,y)\rho(t,x,y)) = 0, \quad (t,x,y) \in [0, +\infty[ \times \mathcal{B}, \quad (4.16) $$
where the sets $\mathcal{A}$ and $\mathcal{B}$ are defined in the equation (3.4) and (3.5).

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

# 5 Discussions and research directions

The main contribution of the present paper has been the definition of a general mathematical framework for the modeling of crowd dynamics, which is based on the behavior of pedestrians. Specifically by defining the admissible directions of a pedestrian (microscopic scale), a system of partial differential equation has been derived for the matrix of the average number of pedestrians. As already mentioned, the mathematical framework is based on the definition of the different transition rates. By expansion methods and the limit for $\epsilon$ that goes to zero, the macroscopic dynamics has been derived for the specific models discussed in Section 3 and 4, see the equations (3.8) and (4.5).

Further discussions can be addressed to the models derived in this paper, in particular for what concerns the choice of the transition rates and the coefficients $D_1$ and $D_2$. Indeed the assumption that the coefficients $D_1$ and $D_2$ are constants can be relaxed by assuming a pointwise dependence. For instance, for the isotropic model defined in Section 3, one can assume that:

$$T_{ij}^{\omega} = \frac{D_{ij}}{\epsilon^2} \sigma_{ij}, \quad \omega \in \{U, D, R, L\}, i, j \in \{1, 2, \dots, k\}. \quad (5.1)$$

By performing the same calculations made in the Proposition 3.1, the reaction-diffusion equation fulfilled by the pedestrian density $\rho(t, x, y)$ now reads:

$$\partial_t \rho(t, x, y) = \Delta (D(x, y)\rho_0(x, y)\rho(t, x, y)), \quad (t, x, y) \in [0, +\infty[ \times \mathcal{D}, \quad (5.2)$$

where

$$\lim_{\epsilon \to 0} D_{ij} = D(x, y).$$

The main assumptions on the transition rates (see equations (2.3)-(2.6)) can be also relaxed in the case of a domain with exits. Accordingly the main target of the pedestrian is the exit and then the model needs to be further generalized by taking into account that the condition on the quantity $\sigma_{ij}$ cannot be the only to be defined. Moreover the boundary conditions need to be further detailed.

From the research perspectives viewpoint, many further investigations can be pursued from the theoretical and applications points of view. Indeed the reaction-diffusion models derived in the Sections 3 and 4 can be explored from a mathematical analysis point of view (existence and uniqueness of the solutions, asymptotic analysis) and from a numerical analysis point of view by performing numerical simulations. However an important analytical investigation refers to the possibility to derive the macroscopic dynamics described by the general mathematical framework (2.19). Finally an important improvement of the mathematical framework is the possibility to introduce further admissible directions to pedestrians for instance by considering a triangular or a hexagonal lattice.

From the application viewpoint, the pedestrian dynamics is not the only complex system that can be modeled by employing the general framework (2.19). Indeed the swarm dynamics shares many properties with the crowd dynamics, see [33, 34]. The flow of vehicles along a road with different lanes can be also modeled within the framework (2.19); the literature in the vehicular traffic modeling is vast and covers the three representation scales discussed in the introduction section, see, among others, papers [35, 36, 37, 38, 39, 40].

Finally, an important research perspective is the possibility to couple the mathematical framework introduced in the present paper with the discrete thermostatted kinetic theory
for the active particles proposed and analyzed in [41] where the evolution of the pedestrians is linked to the interaction rates among the pedestrians. Indeed in the mathematical models proposed in Sections 3 and 4, the transition rates of the mathematical framework (2.19) depends only on the function $\sigma_{ij}$ and not on the microscopic interactions among the pedestrians. This is a work in progress and the results will be presented in due course.

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