Abstract
<abstract> This paper is concerned with a family of Reaction-Diffusion systems that we introduced in <sup>[<xref ref-type="bibr" rid="b15">15</xref>]</sup>, and that generalizes the <italic>SIR</italic> type models from epidemiology. Such systems are now also used to describe collective behaviors. In this paper, we propose a modeling approach for these apparently diverse phenomena through the example of the dynamics of social unrest. The model involves two quantities: the level of social unrest, or more generally activity, $ u $, and a field of social tension $ v $, which play asymmetric roles. We think of $ u $ as the actually observed or explicit quantity while $ v $ is an ambient, sometimes implicit, field of susceptibility that modulates the dynamics of $ u $. In this article, we explore this class of model and prove several theoretical results based on the framework developed in <sup>[<xref ref-type="bibr" rid="b15">15</xref>]</sup>, of which the present work is a companion paper. We particularly emphasize here two subclasses of systems: <italic>tension inhibiting</italic> and <italic>tension enhancing</italic>. These are characterized by respectively a negative or a positive feedback of the unrest on social tension. We establish several properties for these classes and also study some extensions. In particular, we describe the behavior of the system following an initial surge of activity. We show that the model can give rise to many diverse qualitative dynamics. We also provide a variety of numerical simulations to illustrate our results and to reveal further properties and open questions. </abstract>
Highlights
We propose a mathematical model inspired from [13, 16, 18] to account for the dynamics of SU and ST
Let us indicate that the threshold phenomenon on the initial level of social tension may not involve the sign of vb − v, but rather the sign of λb defined as the lowest eigenvalue of the operator, ∀φ ∈ C2(Ω), and given by the expression
An increasing number of papers consider systems of Reaction-Diffusion equations to model the dynamics of epidemics or collective behaviors such as riots
Summary
Introduced by Kermack and McKendrick [48] as one particular instance of a family of models, the SIR compartmental type model and its host of variants are basic tools of epidemiology. “There is some similarity between the present models and models used in epidemiology, the diffusion of information and innovations, and the evolution of behavior in groups over time.” This approach is developed by Burbeck et al [23] in their pioneering paper about the dynamics of riots:. We focus on two important classes of models: the tension inhibiting systems (Section 4), which generate ephemeral movements of social unrest, and the tension enhancing systems (Section 5), which give rise to time-persisting movements of social unrest In both cases, we present some new results about the behavior of solutions far from the leading edge of the propagating front, as well as for traveling wave solutions. In the caption of each figure, we give a clickable URL link and the reference to a video of the simulation available online
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