Modeling propagation of epidemics, social unrest and other collective behaviors

Berestycki, Henri; Nordmann, Samuel; Rossi, Luca

A considerable literature is devoted to the introduction and analysis of variants of the SI epidemiology models. Similar models are also proposed to describe the spread of riots and, more generally, of collective behaviors in various social contexts. The use of epidemiology models to describe such social phenomena is based on the analogy between the mechanisms of contagion and social imitation. In turn, this analogy also points to the social nature of epidemics. This paper is concerned with a family of Reaction-Diffusion systems introduced in [17] that aims at unifying, generalizing, and enlarging the fields of application for epidemiology and collective behavior models. In this paper, we propose a modeling approach on these apparently various phenomena through the example of the dynamics of social unrest. The model involves two quantities, the level of social unrest, or, more general, activity u, and a field of social tension v, which play asymmetric roles: u is thought of as the actual observed or explicit quantity while v is an ambiant, sometimes implicit field of susceptibility that modulates the growth of u. In this article, we explore this class of model and prove several theoretical results based on the framework developed in [17], of which the present work is a companion paper. Here we place the emphasis on two subclasses of systems defined in [17]: tension inhibiting and tension enhancing. These are characterized by the fact that the unrest has respectively a negative or positive feedback on the social tension (though no monotonicity condition is assumed). In [17] we derive a threshold phenomenon in terms of the initial level of social tension: below a critical value, a small triggering event is quickly followed by a resumption of calm, while, above this value, it generates an eruption of social unrest spreading through space with an asymptotically constant speed. The new results we derive in the present paper concern the behavior of the solution far from the propagating edge, that is, we give a description of the new regime of the system following the initial surge of activity. We show in particular that the model can give rise to many diverse qualitative dynamics: ephemeral or limited-duration social movements-referred to as"riots"-in the tension inhibiting case, and persisting social movements-lasting upheavals-in the tension enhancing case, as well as other more complex behaviors in some mixed cases. We also investigate this model by numerical simulations that highlight the richness of this framework. We finally propose and study extensions of the model, such as spatially heterogeneous systems.