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 epidemiologyand 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%DIFDELCMD < ] %%% 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 .
 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 %DIFDELCMD < ] %%% 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 .
D
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.
A
that we
R
17
15
R
that aims at unifying, generalizing, and enlarging the fields of application for epidemiologyand collective behavior models
, and that generalizes the SIR type models from epidemiology. Such systems are now also used to describe collective behaviors
R
on these apparently various
for these apparently diverse
R
,
:
R
, more general, activity
more generally activity,
R
: u is thought of as the actual
. We think of u as the actually
A
,
R
growth
dynamics
R
17
15
R
Here we place the emphasis on
We particularly emphasize here
D
defined in
D
17
D
the fact that the unrest has
D
positive feedback on the social tension (though no monotonicity condition is assumed). In
D
17
R
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
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
R
solution far from the propagating edge, that is, we give a description of the new regime of the system following the
system following an
D
in particular
D
: 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
R
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
provide a variety of numerical simulations to illustrate our results and to reveal further properties and open questions