2D model equation of motion

In the 2D model, susceptible agents in area 1, or infected agents in area 3, only move according to social influences.  When outside of the preferred area, an individual moves by weighting both the personal location preference and social influences.  To represent motion in a 2D space and allow for group fission-fusion dynamics, we use a modified version of the zonal model of \cite{COUZIN_2002,Ioannou_2012,Couzin2005}.  An individual swims with a constant speed \(v\), and turning dynamics are given by an equation for angular velocity \cite{Gautrais_2008}: