Opinion formation Models: from individual interactions to the asymptotic behaviour of the whole population

Abstract: Agent based models are of special interest for interdisciplinary collaboration with specialists on the field, since they allow to consider specifically which are the features of the interactions behind the processes, when drawing the (discrete) master equations.  

Mean field equations could be derived (e.g. at cellular level), taking into account diverse heterogeneities of the agents (cells). However, mean field simplification is not appropriate for certain interactions among individuals (e.g. disease spreading), neuronal processes, etc. Instead of working on purely complex networks, we can add to each agent a degree of connectivity deducing systems of coupled PDEs.  

For the study of populations trough long terms, we work with trait and age structured systems of particles. The inference of the deterministic model lies on taking hydrodynamic limits in the corresponding Markovian process, giving rise to the underlying PDE.   

In all of the cases the resulting PDE contains nonlinear and nonlocal terms which are non standard in the more classical literature. Therefore, numerical simulations are often crucial to understand the asymptotic behaviour of the solutions and give us hints to find appropriate arguments to show our results.  

We have mainly applied these techniques to opinion formation models. However, they could be applied to a broad range of topics in Biology and Medicine. Indeed, some members of the mentioned group have implemented these ideas to a SIR model on random graphs, with successful results