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Analysis of a spatio-temporal advection-diffusion model for human behaviors during a catastrophic event

K. Khalil, V. Lanza, D. Manceau, M. A. Aziz-Alaoui, D. Provitolo

Abstract

In this work, using the theory of first-order macroscopic crowd models, we introduce a compartmental advection-diffusion model, describing the spatio-temporal dynamics of a population in different human behaviors (alert, panic and control) during a catastrophic event. For this model, we prove the local existence, uniqueness and regularity of a solution, as well as the positivity and $L^1$--boundedness of this solution. Then, in order to study the spatio-temporal propagation of these behavioral reactions within a population during a catastrophic event, we present several numerical simulations for different evacuation scenarios.

Analysis of a spatio-temporal advection-diffusion model for human behaviors during a catastrophic event

Abstract

In this work, using the theory of first-order macroscopic crowd models, we introduce a compartmental advection-diffusion model, describing the spatio-temporal dynamics of a population in different human behaviors (alert, panic and control) during a catastrophic event. For this model, we prove the local existence, uniqueness and regularity of a solution, as well as the positivity and --boundedness of this solution. Then, in order to study the spatio-temporal propagation of these behavioral reactions within a population during a catastrophic event, we present several numerical simulations for different evacuation scenarios.
Paper Structure (15 sections, 7 theorems, 161 equations, 7 figures, 3 tables)

This paper contains 15 sections, 7 theorems, 161 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. A spatio-temporal advection-diffusion model for human behaviors during a catastrophic event
  3. The temporal model of the human behaviors of a population during a catastrophic event
  4. First-order macroscopic pedestrians models
  5. The spatio-temporal model corresponding to \ref{['eq:main1System APC2 ODEs']}
  6. Local existence, positivity and $L^1$--boundedness of the spatio-temporal model \ref{['eq:main1System APC21']}-\ref{['initial condition']}
  7. The abstract formulation and the associated boundary value Cauchy problem
  8. Preliminary results
  9. Local existence and regularity
  10. Positivity
  11. $L^1$-boundedness
  12. Numerical Simulations
  13. Conclusion
  14. Proofs of the preliminary results of Section \ref{['sec:3']}
  15. Tables of the functions and the parameters of the APC model

Key Result

Proposition 3.2

The following assertions hold: (i) The closed operator generates a contraction holomorphic $C_0$-semigroup $(\mathcal{T}(t))_{t\geq 0}$ on $X$. (ii) The semigroup $(\mathcal{T}(t))_{t\geq 0}$ generated by $\mathcal{A}_0$ is compact and positive. Moreover, the semigroup $(\mathcal{T}(t))_{t\geq 0}$ i where, for each $i=1,\cdots,5$, $(\mathcal{T}_{i}(t) )_{t\geq 0}$ is the semigroup generated by the

Figures (7)

  • Figure 1: The transfer diagram of the APC model. The arrows indicate the transitions among the compartments.
  • Figure 2: The function $\xi$ involved in the imitation terms: the imitation starts very slowly, then it accelerates before slowing down and saturating.
  • Figure 3: Example of the functions $\gamma$ and $\phi$, which describe the transition from the daily to the alert behaviors, and from the control to everyday life behaviors: $\gamma(t)=\zeta(t,1,3)$ and $\phi(t)=\zeta(t,20,70)$ respectively.
  • Figure 4: The direction vector $\vec{\nu}(x_1,x_2)$ given in \ref{['Direction of the mouvment nu']} describing the desired direction of pedestrians to reach the point $(x_1^p,x_2^p)$ which is located outside the domain $\overline{\Omega}$ since the population looks to escape from the exit $\Gamma_{2}$ towards this point.
  • Figure 5: Initial conditions: initial location of the population for each scenario: (a) the population is concentrated in a single group in the center of the domain; (b) the population is subdivided into three groups; (c) an obstacle is located between the exit and the population, which is concentrated in a single group within the domain. We recall that the exit is on the right of the domain, see Figure \ref{['Direction']}.
  • ...and 2 more figures

Theorems & Definitions (17)

  • Remark 2.1
  • Definition 3.1
  • Proposition 3.2
  • Remark 3.3
  • Proposition 3.4
  • Definition 3.5
  • Proposition 3.6
  • Remark 3.7
  • Lemma 3.8
  • Definition 3.9
  • ...and 7 more