لخّصلي

Modeling the dynamics of stress propagation in disaster situations is crucial for effective emergency response planning and risk management.In this paper, we present a macroscopic first-order compartmental advection-diffusion-reaction model that describes how populations in two separate interconnected zones ?1 and ?2, via their human behaviors (compartments) that represent the stressed and non-stressed populations respectively, interact via stress contagion within each zone and migration between these two zones, spatial propagation is also considered.

Modeling the dynamics of stress propagation in disaster situations is crucial for effective emergency response planning and risk management. Recent works have extensively studied crowds
under stress using mathematical modeling and differential equations. These studies range from
microscopic modeling using systems of ODEs for low-density crowds to macroscopic modeling
using PDEs for high-density crowds. For more details on these models, refer to recent works
[24, 25, 27, 4, 5, 18, 21, 19] and references therein. The macroscopic approach we adopt in
this paper considers a crowd as a locally entire quantity, without recognizing individual differences locally, and is therefore more suitable for studying the movement of an extremely large
number of pedestrians. In particular, first-order macroscopic models, introduced by Hughes [6]
(see also [33]), are based on a continuity-type equation and a density-velocity closure equation
with suitable boundary conditions. Furthermore, several models are devoted to studying the
dynamics of multiple pedestrian species in the context of macroscopic first-order systems, see
[31, 32, 7, 5, 8, 19, 6, 9, 17] and references therein. For instance, in [19], the authors introduced a macroscopic first-order model describing stress (panic) propagation (in one spatial zone)
by considering interactions between different human behaviors (alert, panic, and control) with
advection, diffusion, intrinsic transitions, and imitation parameters and their impact to the propagation of stress in one spatial zone. These models primarily focus on stress propagation and
influence within a single spatial zone during emergency situations.
In this paper, we present a macroscopic first-order compartmental advection-diffusion-reaction
model that describes how populations in two separate interconnected zones Ω1 and Ω2, via
their human behaviors (compartments) that represent the stressed and non-stressed populations
respectively, interact via stress contagion within each zone and migration between these two
zones, spatial propagation is also considered. More precisely, we consider two bounded zones Ω1
and Ω2 subsets of R
2
such that ∂Ω1 ∩ ∂Ω2 = ∅, with two compartments in each zone, one for
the stressed population and the other for the non-stressed population. The mathematical model
includes, in each zone, diffusion, which represents the way in which people move randomly in
all directions, and non-linear advection, which models the way in which people move closer to
departure regions in order to migrate towards arrival regions. It also includes reaction terms
(linear and non-linear) that describe intrinsic transitions in human behaviors that reflect natural
reductions or increases in stress levels in pedestrians without taking into account interactions
between the two human behaviors (stress and non-stress), while imitation refers to the way stress
is propagated by eye contact (in the presence of interactions). This modeling approach has been
described in [19] to describe the spatial propagation of stress by interactions with other human
behaviors in one spatial zone, see also references therein for similar reaction-diffusion models.
However, in this article, we also consider migration between the two zones, meaning that the
two zones are linked by migration, allowing part of the population to move from one zone to the
other one in both directions, taking into account the capacity of each zone. Each zone includes
one (or more) departure and reception region(s) to facilitate migration between zones. These
may be two countries, two cities or two districts linked by road, rail or air. Departure and arrival
areas, such as railway stations or airports, this model is illustrated well in Figure 2. This type
of interconnected coupling between two zones follows the approaches described in [16, 13] in the
case of reaction-diffusion systems. A key feature of our model is that migration is treated as an
internal connection, not by a boundary condition. We also use nonlinear advection along with