An advection-diffusion-reaction model for stress propagation between two interconnected zones

TL;DR

This paper develops a mathematical model using advection-diffusion equations to describe stress spread between two interconnected zones during disasters, incorporating interactions, migrations, and control strategies.

Contribution

It introduces a novel compartmental advection-diffusion network model with rigorous mathematical analysis and simulations for stress propagation in interconnected populations.

Findings

Proved existence, uniqueness, and regularity of solutions.

Established positivity and boundedness of stress levels.

Demonstrated effectiveness of a local control strategy.

Abstract

In this work, we introduce a compartmental advection-diffusion network model to describe the propagation of stress in a population situated in two interconnected spatial zones during a disaster situation. The model accounts for interactions enabled by intrinsic transitions and imitations within each zone, as well as migrations between these zones. Using semigroup theory and abstract evolution equations, we prove the local existence, uniqueness, and regularity of the solutions. Additionally, we establish the positivity and $L^1$--boundedness of the solutions. Various numerical simulations are provided to illustrate different stress propagation scenarios, including the implementation of a local control strategy designed to minimize stress levels in a population facing a low-risk culture during a dangerous situation.