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

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

Carlo Bianca, Marco Dalla Via, Christian Dogbe

To cite this version:

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. �hal-02151412�

PEDESTRIAN DYNAMICS

CARLO BIANCA^1,2,?, MARCO DALLA VIA^1,2, CHRISTIAN DOGBE³

1Laboratoire Quartz EA 7393, ´Ecole Sup´erieure d’Ing´enieurs en G´enie ´Electrique, Pro- ductique et Management Industriel, 95092 Cergy-Pontoise, France.

2Laboratoire de Recherche en Eco-innovation Industrielle et Energ´etique, ´Ecole Sup´erieure d’Ing´enieurs en G´enie ´Electrique, Productique et Management Industriel, 95092 Cergy- Pontoise, France.

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

?Corresponding Author. [email protected]

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 sideεand 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 lettingεgo 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.

2010Mathematics Subject Classification35C20, 76R50, 35Q84

Keywords: Crowd dynamics, Joint probability, Partial differential equations, Transition rate, Reaction-diffusion equation

1

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 op- timal theoretical framework should depict, at least at a qualitative level, the emerging phe- nomena; 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 parame- ters 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 sideε) containing a certain number of pedestrians. The vertical and horizontal di- rections 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 di- rections 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 timetthe crowd has a specific config- uration. 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 pedestrain evacuation from a domain, respectively. The first model is based on the main assumption that pedes- trians 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 par- ticular it is assumed that pedestrians move towards the higher density region (anisotropic model). Specifically it is shown that the macroscopic dynamics, obtained by lettingεgo

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 De ^⊂R². LetL∈R⁺be the largest length of the domainDe^,V∈R⁺the maximum velocity which a pedestrian can attain andT ∈R⁺the time necessary for a pedestrian to cover the maximum lengthLwith velocityV, i.e. L=V T. The following dimensionless variables are defined: The timet=t/T˜ ∈[0,+∞[, thex-position denoted byx=x/L˜ ∈[0,1]and they-position denoted byy=y/L˜ ∈[0,1], where ˜t is the dimensional time and(x,˜ y)˜ ∈De is the dimensional vector position. Accordingly the dimensionless domain writesD⁼ [0,1]×[0,1].

Letk∈N\ {0}. The domainDis divided intok²subdomains. Each subdomain is a square S_{i j}, fori,j∈ {1,2, . . . ,k}, with length sideε=1/k∈R⁺. The squareS_{i j}is thus defined as follows:

S_{i j}= [(i−1)ε,iε]×[(j−1)ε,jε], i,j∈ {1,2, . . . ,k}, (2.1) and it is centered at(x^c_i,y^c_j)∈D^{, where:}

x^c_i =ε

2(2i−1), y^c_j= ε

2(2j−1), i,j∈ {1,2, . . . ,k}. (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 criterionα, which influences both the vertical and horizontal movements, see Figure 1. Specifically:

• The vertical evolution of a pedestrian allocated inS_{i j} is modeled by introducing the transition rate per unit time T_{i j}^ω:=T_{i j}^ω[α](t), for ω∈ {U,D}, whereU and Ddenote the upper and down movements, respectively. SpecificallyT_{i j}^U denotes the transition rate of pedestrians from the square S_{i j} to the squareSi j+1 andT_{i j}^D denotes the transition rate from the squareS_{i j}to the squareSi j−1;

• The horizontal evolution of a pedestrian allocated inS_{i j}is described by the tran- sition rate per unit timeT_{i j}^ω:=T_{i j}^ω[α](t), forω∈ {R,L}, whereRandLdenote the right and left movements, respectively. In particularT_{i j}^Rdenotes the transition rate of pedestrians from the squareS_{i j}to the right squareS_i+1jandT_{i j}^Ldenotes the transition rate of pedestrians from the squareS_{i j}to the left squareSi−1j.

Si+1j

T_ij^R Sij

Si−1j

Sij+1

S_ij−1 T_ij^L

T_ij^U

T_ij^D

FIGURE 1. The pedestrian evolution from the square S_{i j} to the neigh- borhood squares.

It is assumed that total number of pedestrianN remains constant in time. Accordingly one has:

T_ik^U=0, ∀i∈ {1,2, . . . ,k}, (2.3) T_i1^D=0, ∀i∈ {1,2, . . . ,k}, (2.4) T_{k j}^R=0, ∀j∈ {1,2, . . . ,k}, (2.5) T_1j^L=0, ∀j∈ {1,2, . . . ,k}. (2.6) Letn= (n₁₁,n₁₂, . . . ,n_kk)∈N^k2 be a configuration of the crowd, namelyn_{i j}denotes the number of pedestrians into the squareS_{i j}. Accordingly one has:

k i=1

∑

k

∑

n_{i j}=N^{. (2.7)}

The probability that at timetthe configuration of the crowd isnwill be denoted byP(t,n) (joint probability) and it is such that:

∑

P(t,n):=

N

∑

n11=0 N

∑

n12=0

· · · N

∑

ni j=0

· · · N

∑

nkk=0

P(t,(n₁₁,n₁₂, . . . ,ni j, . . . ,n_kk)) =1. (2.8) The master equation (ME) fulfilled byP(t,n)is based on the definition of the following operators:

• The operatorU_{i j}:N^k

2 →N^k

2, for i∈ {1,2, . . . ,k}, j∈ {1,2, . . . ,k−1}, which allocates a pedestrian into the square S_{i j} and removes it from the above square

S_{i j+1}. Accordingly:

U_{i j}(n₁₁,n₁₂, . . . ,n_{i j}, . . . ,n_kk) = (n₁₁,n₁₂. . . ,n_{i j}+1,n_{i j+1}−1, . . . ,n_kk). (2.9)

• The operator D_{i j}:N^k

2 →N^k

2, fori∈ {1,2, . . . ,k}, j∈ {2,3. . . ,k}, which allo- cates a pedestrian into the squareS_{i j}and removes it from the below squareSi j−1. Accordingly:

D_{i j}(n₁₁,n₁₂, . . . ,n_{i j}, . . . ,n_kk) = (n₁₁,n₁₂, . . . ,ni j−1−1,n_{i j}+1, . . . ,n_kk). (2.10)

• The operator R_{i j}:N^k

2 →N^k

2, fori∈ {1,2, . . . ,k−1},j∈ {1,2, . . . ,k}, which allocates a pedestrian into the square S_{i j} and removes it from the right square Si+1j. Accordingly:

R_{i j}(n₁₁,n₁₂, . . . ,n_{i j}, . . . ,n_kk) = (n₁₁,n₁₂, . . . ,ni j−1,n_{i j}+1,n_{i j+1}, . . . ,n_i+1j−1, . . . ,n_kk).

(2.11)

• The operator Li j:N^k

2 →N^k

2, fori∈ {2,3, . . . ,k},j∈ {1,2, . . . ,k}, which allo- cates a pedestrian into the square Si j and removes it from the left square Si−1j. Accordingly:

L_{i j}(n₁₁,n₁₂, . . . ,n_{i j}, . . . ,n_kk) = (n₁₁,n₁₂, . . . ,ni−1j−1, . . . ,ni j−1,n_{i j}+1,ni j+1. . . ,n_kk).

(2.12) Bearing all above in mind, the time evolution of the joint probabilityP(t,n)is described by the following ME:

∂tP(t,n) =T^U[P](t,n) +T^D[P](t,n) +T^R[P](t,n) +T^L[P](t,n), (2.13) where:

T^U[P](t,n) =G^U[P](t,n)−L^U[P](t,n)

=

k i=1

∑

k−1

∑

T_{i j}^U (n_{i j}+1)P(t,U_{i j}(n))

−

k

∑

∑

T_{i j}^U n_{i j}P(t,n) ,(2.14) T^D[P](t,n) =G^D[P](t,n)−L^D[P](t,n)

=

k i=1

∑

k

∑

T_{i j}^D (n_{i j}+1)P(t,Di j(n))

−

k i=1

∑

k

∑

T_{i j}^D ni jP(t,n) ,(2.15) T^R[P](t,n) =G^R[P](t,n)−L^R[P](t,n)

=

k−1

∑

i=1 k

∑

j=1

T_{i j}^R (n_{i j}+1)P(t,R_{i j}(n))

−

k−1

∑

i=1 k

∑

j=1

T_{i j}^R n_{i j}P(t,n) ,(2.16) T^L[P](t,n) =G^L[P](t,n)−L^L[P](t,n)

=

k

∑

i=2 k

∑

j=1

T_{i j}^L (n_{i j}+1)P(t,L_{i j}(n))

−

k

∑

i=2 k

∑

j=1

T_{i j}^L n_{i j}P(t,n)

. (2.17) Specifically:

• The operatorT^U[P](t,n)models the all possible up-hand movements. In particular G^U[P](t,n)is the gain term whileL^U[P](t,n)is the loss term;

• The operatorT^D[P](t,n)models the all possible down-hand movements. In par- ticularG^D[P](t,n)is the gain term whileL^D[P](t,n)is the loss term;

• The operatorT^R[P](t,n)models the all possible right-hand movements. In partic- ularG^R[P](t,n)is the gain term whileL^R[P](t,n)is the loss term;

• The operatorT^L[P](t,n)models the all possible left-hand movements. In particu- larG^L[P](t,n)is the gain term whileL^L[P](t,n)is the loss term.

The average number of pedestriansN_{i j}(t):=N(t,x^c_i,y^c_j), fori,j∈ {1,2, . . . ,k}, at timet and into the squareS_{i j}is defined as follows:

N_{i j}(t) =

∑

n

n_{i j}P(t,n). (2.18)

The matrix of the average number of pedestrians (density) is denoted byN(t) = [N_{i j}(t)].

Theorem 2.1. Let T_{i j}^ω, forω∈ {U,D,L,R}, i,j∈ {1,2, . . . ,k}, be the transition rate.

The matrixN(t)is solution of the following system of partial differential equations:



































∂tN11(t) =T₁₂^DN12(t) +T₂₁^LN21(t)− T₁₁^U+T₁₁^R N11(t)

∂_tN₁₂(t) =T₁₁^UN₁₁(t) +T₁₃^DN₁₃(t) +T₂₂^LN₂₂(t)− T₁₂^U+T₁₂^D+T₁₂^R N₁₂(t) ...

∂tN_{i j}(t) =T_{i j−1}^U Ni j−1(t) +T_{i j+1}^D N_{i j+1}(t) +T_i−1j^R Ni−1j(t) +T_i+1j^L N_i+1j(t)

−

T_{i j}^U+T_{i j}^D+T_{i j}^R+T_{i j}^L N_{i j}(t) ...

∂tN_kk(t) =T_kk−1^U Nkk−1(t) +T_k−1k^R Nk−1k(t)− T_kk^D+T_kk^L N_kk(t).

(2.19)

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

∂_tN_hl(t) =

∑

n

n_hl∂_tP(t,n), h,l∈ {1,2, . . . ,k}. (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:

∂tN_hl(t) =

∑

n

n_hl k

i=1

∑

k−1

∑

T_{i j}^U (n_{i j}+1)P(t,U_{i j}(n))−n_{i j}P(t,n)

+

k i=1

∑

k

∑

T_{i j}^D (n_{i j}+1)P(t,Di j(n))−ni jP(t,n) +

k−1

∑

i=1 k

∑

j=1

T_{i j}^R (n_{i j}+1)P(t,R_{i j}(n))−n_{i j}P(t,n)

+

k

∑

i=2 k

∑

j=1

T_{i j}^L (n_{i j}+1)P(t,L_{i j}(n))−n_{i j}P(t,n)

. (2.21)

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

∑

n_hl





k i=1

∑

k−1 j=1

∑

T_{i j}^U (n_{i j}+1)P(t,U_{i j}(n))−n_{i j}P(t,n)



. (2.22)

By considering the terms corresponding toi=h,j=lin the equation (2.22), one has:

T_hl^U

∑

n

n_hl (n_hl+1)P(t,U_hl(n))−n_hlP(t,n)

. (2.23)

By changing the indices in the first sumU_hl(n)→n, the equation (2.23) writes:

T_hl^U

∑

n

(n_hl−1)n_hlP(t,n)−n²_hlP(t,n)

=−T_hl^UN_hl(t). (2.24) By considering the terms corresponding toi=h,j=l−1 in the equation (2.22), one has:

T_hl−1^U

∑

n

n_hl (nhl−1+1)P(t,Uhl−1(n))−nhl−1P(t,n)

. (2.25)

By changing the indices in the first sumUhl−1(n)→n, the equation (2.25) writes:

T_hl−1^U

∑

n

(n_hl+1)nhl−1P(t,n)−n_hlnhl−1P(t,n)

=T_hl^UNhl−1(t). (2.26) The other terms of the equation (2.22), which correspond toi6=h,j6={l,l−1}, are equal to zero. Therefore the equation (2.22) writes:

∑

n_hl





k i=1

∑

k−1

∑

T_{i j}^U (n_{i j}+1)P(t,U_{i j}(n))−n_{i j}P(t,n)



=T_hl−1^U Nhl−1(t)−T_hl^UN_hl(t).

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

∑

n_hl





k

∑

i=1 k

∑

j=2

T_{i j}^D (n_{i j}+1)P(t,D_{i j}(n))−n_{i j}P(t,n)



=T_hl+1^D N_hl+1(t)−T_hl^DN(2.28)_hl(t),

∑

n_hl





k−1

∑

i=1 k

∑

j=1

T_{i j}^R (n_{i j}+1)P(t,R_{i j}(n))−n_{i j}P(t,n)



=T_h−1l^R Nh−1l(t)−T_hl^RN_hl(2.29)(t),

∑

n_hl





k i=2

∑

k

∑

T_{i j}^L (n_{i j}+1)P(t,L_{i j}(n))−ni jP(t,n)



=T_h+1l^L Nh+1l(t)−T_hl^LN_hl(2.30)(t).

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).

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_{i j}^ω[α](t),forω∈ {U,D,R,L},i,j∈ {1,2, . . . ,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 inS_{i j}to the initial density. Specifically one has:

T_{i j}^ω=D₁σ_{i j}

ε² , ω∈ {U,D,R,L}, i,j∈ {1,2, . . . ,k}, (3.1)

where

σ_{i j}:=σ(x^c_i,y^c_j) =N_{i j}(0), i,j∈ {1,2, . . . ,k}, (3.2) andD1 is the diffusion coefficient. According to this isotropic dynamics, a pedestrian moves from the squareSi jwhich is highly crowded, to the neighborhood squares that are less crowded (see Figure 2), and the possibility for a pedestrian to leave the squareS_{i j}is in inverse proportion to the areaε²ofS_{i j}. By replacing the equations (3.1) into the system

Si+1j

Sij

Si−1j

Sij+1

Sij−1

Si+1j

Sij

Si−1j

Sij+1

Sij−1

FIGURE2. The pedestrians move from the squareS_{i j}, which is initially highly crowded (left panel), to the neighborhood squares which are less crowded. The new configuration is shown in the right panel.

(2.19), one has:

∂tN_{i j}(t) =D₁

ε² σi j−1Ni j−1(t) +σi j+1N_{i j+1}(t) +σi−1jNi−1j(t) +σi+1jN_i+1j(t)−4σi jN_{i j}(t) , (3.3) fori,j∈ {1,2, . . . ,k}.

Let∂D⁼A^∪Bbe the boundary of the domainD, where the setsA^andBare defined as follows:

A^{={(0,y),(1,y):y∈[0,1]}, (3.4)} B^{={(x,0),(x,1):x∈[0,1]}. (3.5)} SpecificallyA denotes the vertical boundaries of the domainD ^andB denotes the hori- zontal boundaries of the domainD^.

Proposition 3.1. Let T_{i j}^ω, forω∈ {U,D,R,L}, i,j∈ {1,2, . . . ,k}, be the transition rate.

Let N_{i j}(t), for i,j∈ {1,2, . . . ,k}, be the solution of the equation(3.3). Assume that

ε→0limN_{i j}(t) =ρ(t,x,y), (3.6)

ε→0limσi j=ρ0(x,y). (3.7)

Thenρ(t,x,y)is the solution of the following initial value problem with Neumann condi- tions:















∂_tρ(t,x,y) =D1∆ ρ0(x,y)ρ(t,x,y)

(t,x,y)∈[0,+∞[×D^, ρ(0,x,y) =ρ0(x,y) (x,y)∈D^,

∂_x ρ0(x,y)ρ(t,x,y)

=0 (t,x,y)∈[0,+∞[×A^,

∂y ρ0(x,y)ρ(t,x,y)

=0 (t,x,y)∈[0,+∞[×B^.

(3.8)

Proof. By expanding the functionsσ(x,y)andN(t,x,y)at(x^c_i,y^c_j)and by evaluating them in the points(x^c_i±1,y^c_j)and(x_i^c,y^c_j±1), respectively, one recovers the termsσi±1jNi±1j(t) andσi j±1Ni j±1(t)of the equation (3.3) that now write as follows:

σi±1jNi±1j(t) =σ(x^c_i±1,y^c_j)N(t,x^c_i±1,y^c_j)

=σ(x^c_i,y^c_j)N(t,x^c_i,y^c_j)±ε∂(σN)

∂x (t,x^c_i,y^c_j) +1

2ε²∂²(σN)

∂x² (t,x^c_i,y^c_j) +o(ε(3.9)²), σi j±1Ni j±1(t) =σ(x^c_i,y^c_j±1)N(t,x^c_i,y^c_j±1)

=σ(x^c_i,y^c_j)N(t,x^c_i,y^c_j)±ε∂(σN)

∂y (t,x^c_j,y^c_i) +1

2ε²∂²(σN)

∂y² (t,x^c_i,y^c_j) +o(ε(3.10)²), and replacing the equations (3.9) and (3.10) into the equation (3.3), one has:

∂tNi j(t) =D₁σi j

ε² ε²∂²(σN)

∂x² (t,x^c_i,y^c_j) +D₁σi j

ε² ε²∂²(σN)

∂y² (t,x^c_i,y^c_j) +o(ε²), (3.11) By assumptions (3.6) and (3.7), the equation (3.11) rewrites as follows:

∂tρ(t,x,y) = D₁∂x ρ(t,x,y)∂_xρ0(x,y) +ρ0(x,y)∂_xρ(t,x,y) +D₁∂y ρ(t,x,y)∂_yρ0(x,y) +ρ0(x,y)∂_yρ(t,x,y)

= D₁∆ ρ0(x,y)ρ(t,x,y)

, (3.12)

with the initial condition:

ρ(0,x,y) =ρ0(x,y), (x,y)∈D^{. (3.13)} Taking into account that the diffusion outside of the domainD is not possible, see the equations (2.3), (2.4), (2.5) and (2.6), the following boundary condition has to be imposed:

∂ ρ0(x,y)ρ(t,x,y)

∂nˆ =0, ∀(t,x,y)∈[0,+∞[×∂D^{, (3.14)} where ˆn:=n(x,ˆ y)∈R², for(x,y)∈∂D, denotes the normal vector to the boundary∂D^. The condition (4.14) rewrites as follows:

∂x ρ0(x,y)ρ(t,x,y)

=0, (t,x,y)∈[0,+∞[×A^{, (3.15)}

∂y ρ0(x,y)ρ(t,x,y)

=0, (t,x,y)∈[0,+∞[×B^{, (3.16)} where the setsA ^andBare defined in the equation (3.4) and (3.5).

4 An anisotropic dynamics model

This section deals with a pedestrian dynamics model where the criterion is the move- ment towards high-density regions of the domainD. This dynamics occurs during evac- uation 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 inSi j to the density of the neighborhood squares. Specifically one assumes that:

T_{i j}^U =D2

ε² σi j+1, T_{i j}^D=D2

ε² σi j−1, T_{i j}^R=D2

ε² σi+1j, T_{i j}^L=D2

ε²σi−1j, (4.1) whereD₂∈R⁺ is the diffusion coefficient and the densityσi j, for i,j∈ {1,2, . . . ,k}, is given by the equation (3.2). According to this anisotropic dynamics, a pedestrian moves from the squareS_{i j}, which is less crowded, to the neighborhood squares that are highly crowded, see Figure 3.

Si+1j

Sij

Si−1j

Sij+1

Sij−1

Si+1j

Sij

Si−1j

Sij+1

Sij−1

FIGURE 3. A pedestrian moves from the square S_{i j}, which is initially less crowded (left panel) to the squareSi+1jthat is highly crowded. The new configuration is shown in the right panel.

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

∂tN_{i j}(t) = D2σ_{i j}

ε² Ni j−1(t) +N_{i j+1}(t) +Ni−1j(t) +N_i+1j(t)

−D₂

ε² σi j−1+σi j+1+σi−1j+σi+1j

N_{i j}(t), (4.2) fori,j∈ {1,2, . . . ,k}.

Proposition 4.1. Let T_{i j}^ω, forω∈ {U,D,R,L}, i,j∈ {1,2, . . . ,k}, be the transition rate.

Let N_{i j}(t), for i,j∈ {1,2, . . . ,k}, be the solution of the equation(4.2). Assume that

ε→0limN_{i j}(t) =ρ(t,x,y), (4.3)

ε→0limσi j=ρ0(x,y). (4.4)

Thenρ(t,x,y)is the solution of the following initial value problem with Neumann condi- tions:















∂tρ(t,x,y) =D2ρ0(x,y) ∆ρ(t,x,y)

−D2ρ(t,x,y) ∆ρ0(x,y)

(t,x,y)∈[0,+∞[×D^,

ρ(0,x,y) =ρ0(x,y) (x,y)∈D^,

∂_x ρ0(x,y)ρ(t,x,y)

=0 (t,x,y)∈[0,+∞[×A^,

∂y ρ0(x,y)ρ(t,x,y)

=0 (t,x,y)∈[0,+∞[×B^.

(4.5) Proof. By expanding the functionsσ(x,y)andN(t,x,y)at(x^c_i,y^c_j)and by evaluating them in the points (x^c_i±1,y^c_j) and(x^c_i,y^c_j±1), respectively, one recovers the terms σi±1j,σi j±1, Ni±j(t),Ni j±1(t)of the equation (4.2) that now write as follows:

σi±1j=σ(x^c_i±1,y^c_j) =σ(x^c_i,y^c_j)±ε∂σ

∂x(x^c_i,y^c_j) +1 2ε²∂²σ

∂x²(x^c_i,y^c_j) +o(ε²), (4.6) σi j±1=σ(x^c_i,y^c_j±1) =σ(x^c_i,y^c_j)±ε∂σ

∂y(x^c_j,y^c_i) +1 2ε²∂²σ

∂y²(x^c_i,y^c_j) +o(ε²), (4.7) Ni±1j(t) =N(t,x^c_i±1,y^c_j) =N(t,x^c_i,y^c_j)±ε∂N

∂x(t,x^c_i,y^c_j) +1 2ε²∂²N

∂x²(t,x^c_i,y^c_j) +o(ε(4.8)²), Ni j±1(t) =N(t,x^c_i,y^c_j±1) =N(t,x^c_i,y^c_j)±ε∂N

∂y(t,x^c_j,y^c_i) +1 2ε²∂²N

∂y²(t,x^c_i,y^c_j) +o(ε(4.9)²), and replacing the equations (4.6), (4.7), (4.8) and (4.9) into the equation (4.2), one has:

∂tN_{i j}(t) =D₂

ε² ε²σ(x^c_i,y^c_j) ∂²N

∂x²(t,x^c_i,y^c_j) +∂²N

∂y²(t,x^c_i,y^c_j)

!

−D₂

ε² ε²N(t,x^c_i,y^c_j) ∂²σ

∂x²(x^c_i,y^c_j) +∂²σ

∂y²(x^c_i,y^c_j)

!

+o(ε²). (4.10) By assumptions (4.3) and (4.4), the equation (4.10) rewrites as follows:

∂_tρ(t,x,y) = D₂∂_x

ρ₀(x,y)∂_xρ(t,x,y)−ρ(t,x,y)∂_xρ₀(x,y) +D₂∂_y

ρ₀(x,y)∂_yρ(t,x,y)−ρ(t,x,y)∂_yρ₀(x,y)

, (4.11)

= D₂ρ₀(x,y) ∆ρ(t,x,y)

−D₂ρ(t,x,y) ∆ρ₀(x,y)

, (4.12) with the initial condition:

ρ(0,x,y) =ρ0(x,y), (x,y)∈D^{. (4.13)} Taking into account that the diffusion outside the domainDis not possible, see the equa- tions (2.3), (2.4), (2.5) and (2.6), the following boundary condition has to be imposed:

∂ ρ₀(x,y)ρ(t,x,y)

∂nˆ =0, ∀(t,x,y)∈[0,+∞[×∂D^{, (4.14)} where ˆn:=n(x,ˆ y)∈R², for(x,y)∈∂D, denotes the normal vector to the boundary∂D^. The condition (4.14) rewrites as follows:

∂_x ρ₀(x,y)ρ(t,x,y)

=0, (t,x,y)∈[0,+∞[×A^{, (4.15)}

∂_y ρ₀(x,y)ρ(t,x,y)

=0, (t,x,y)∈[0,+∞[×B^{, (4.16)}

where the setsA ^andBare defined in the equation (3.4) and (3.5).

5 Discussions and research directions

The main contribution of the present paper has been the definition of a general mathe- matical framework for the modeling of crowd dynamics, which is based on the behavior of pedestrians. Specifically by defining the admissible directions of a pedestrian (micro- scopic 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 forεthat 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 coefficientsD₁andD₂. Indeed the assumption that the coefficientsD₁andD₂are constants can be relaxed by assuming a pointwise dependence. For instance, for the isotropic model defined in Section 3, one can assume that:

T_{i j}^ω=D_{i j}

ε² σi j, ω∈ {U,D,R,L},i,j∈ {1,2, . . . ,k}. (5.1) By performing the same calculations made in the Proposition 3.1, the reaction-diffusion equation fulfilled by the pedestrian densityρ(t,x,y)now reads:

∂tρ(t,x,y) =∆ D(x,y)ρ₀(x,y)ρ(t,x,y)

, (t,x,y)∈[0,+∞[×D^{, (5.2)} where

limε→0D_{i j}=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 quantityσ_{i j}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 mod- els 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 numer- ical analysis point of view by performing numerical simulations. However an important analytical investigation refers to the possibility to derive the macroscopic dynamics de- scribed 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 sys- tem 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 vehi- cles 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 pedestri- ans 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 functionσ_{i j}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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