We consider an agent-based model of emotional contagion coupled with motion in one dimension that has recently been studied in the computer science community. The model involves movement with a speed proportional to a “fear” variable that undergoes a temporal consensus averaging based on distance to other agents. We study the effect of Riemann initial data for this problem, leading to shock dynamics that are studied both within the agent-based model as well as in a continuum limit. We examine the behavior of the model under distinguished limits as the characteristic contagion interaction distance and the interaction timescale both approach zero. The limiting behavior is related to a classical model for pressureless gas dynamics with “sticky” particles. In comparison, we observe a threshold for the interaction distance vs. interaction timescale that produce qualitatively different behavior for the system - in one case particle paths do not cross and there is a natural Eulerian limit involving nonlocal interactions and in the other case particle paths can cross and one may consider only a kinetic model in the continuum limit.
Bibliographical noteFunding Information:
The authors thank Milind Tambe and Jason Tsai for suggesting this problem. We also thank Maria D’Orsogna for useful discussions early on. This work has been supported by ARO MURI Grant W911NF-11-1-0332, NSF Grant DMS-0968309, NSF Grant DMS-0907931, and ARO Grant W911NF1010472.
© 2014, Springer Science+Business Media New York.
- Conservation laws
- Contagion model
- Riemann problem
- Traffic flow