[Paper Review] Modeling the propagation of riots, collective behaviors, and epidemics

Henri Berestycki, Samuel Nordmann+1 more|arXiv (Cornell University)|May 20, 2020

Opinion Dynamics and Social Influence12 references4 citations

TL;DR

This paper introduces a reaction-diffusion model that unifies the dynamics of riots, collective behaviors, and epidemics by coupling an observable activity level $u$ with an underlying social tension field $v$. It identifies two key regimes—tension-inhibiting (leading to short-lived riots) and tension-enhancing (leading to lasting upheavals)—and demonstrates how initial conditions and tension thresholds determine long-term outcomes, including wave propagation speed and spatial spread.

ABSTRACT

This paper is concerned with a family of Reaction-Diffusion systems that we introduced in [15], and that generalizes the SIR 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 ambiant, 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 [15], of which the present work is a companion paper. We particularly emphasize here two subclasses of systems: tension inhibiting and tension enhancing. These are characterized by respectively a negative or a positivefeedback 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.

Motivation & Objective

• To develop a unified mathematical framework for modeling social unrest, collective behaviors, and epidemics using reaction-diffusion systems.
• To analyze the role of social tension $v$ as a latent, modulating field influencing the dynamics of observable activity $u$.
• To classify and compare two distinct dynamical regimes: tension-inhibiting (short-lived riots) and tension-enhancing (sustained movements).
• To investigate the impact of initial conditions, including the magnitude of triggering events and spatial heterogeneity, on propagation and long-term behavior.
• To extend the model to include non-local diffusion and compartmental structures for greater realism in social dynamics modeling.

Proposed method

• Formulates a reaction-diffusion system coupling $u$ (activity) and $v$ (social tension), with asymmetric roles: $u$ is observed, $v$ is ambient and modulating.
• Introduces two subclasses: tension-inhibiting (negative feedback: $v$ decreases with $u$) and tension-enhancing (positive feedback: $v$ increases with $u$).
• Uses traveling wave solutions to analyze propagation speed and spatial spread, with analytical results derived for large-time behavior.
• Employs numerical simulations to explore threshold phenomena, wave speed, and the effect of initial conditions on system outcomes.
• Extends the model to non-local diffusion (e.g., fractional Laplacian or convolution kernels) to capture long-range social influence and media effects.
• Considers spatial heterogeneity by introducing non-uniform initial tension $v_0(x)$ and obstacles (e.g., gaps in propagation) via variable coefficients.

Experimental results

Research questions

• RQ1 How does the feedback mechanism between social tension $v$ and activity $u$ determine whether a social movement remains short-lived or becomes sustained?
• RQ2 What is the critical threshold of initial social tension $v_b$ that determines whether a triggering event leads to a riot or a lasting upheaval?
• RQ3 How do the speed and shape of traveling waves depend on the parameters of the $v$-equation and the initial magnitude of the unrest?
• RQ4 What happens to the propagation of social unrest when spatial heterogeneity—such as regional differences in baseline tension or physical obstacles—is introduced?
• RQ5 Can non-local diffusion operators better capture long-range social contagion, such as media-driven spread, compared to classical diffusion?

Key findings

• In the tension-inhibiting case, if the baseline tension $v_b > v_*$, the system returns to calm; if $v_b < v_*$, a riot emerges and propagates as a traveling wave with speed $c_1 = 2\sqrt{r(1)f(1) - \omega}$.
• In the tension-enhancing case, a sufficiently large initial surge can trigger a lasting upheaval even when $v_b < v_*$, leading to a sustainable excited state.
• Numerical simulations show that for $v_b < v_*$, the system may exhibit two opposite traveling waves with intermediate speeds between $c_b$ and $c_1$, depending on parameters.
• The model exhibits a double threshold effect in mixed cases: depending on initial conditions, the system can transition between calm, riot, or lasting upheaval.
• When $v_0(x)$ is non-uniform, spatial heterogeneity significantly affects propagation, with obstacles blocking movement only if the gap is large and $v_b < v_*$.
• Non-local diffusion, such as fractional Laplacian, can be incorporated to model long-range social influence and media effects, potentially leading to anomalous propagation behaviors.