Dynamical analysis and effects of law enforcement in a social interaction model

Abstract

Crime is a rising problem all over the world. Over the years, several researches have been conducted to coin suitable model systems and intervention strategies that would decrease delinquent behavior and promote prosocial development. Being dynamic and complex process, the spread of crime requires a system level approach. In this paper, we formulate and analyze a series of dynamical model systems of the dilation of crime incorporating law enforcement. The population models of social interactions have been constructed based on predator–prey interaction models with Holling type II response function. Two different types of populations (criminal minded and non-criminal minded) have been assumed in the given community/society. According to law policy for crime control, we scrutinize the dynamic behavior of the model system concerning law enforcement on the criminal minded population. Along with analytical expressions for the existence of different equilibrium points and their stability, we have also provided their geometrical interpretations using isocline analysis. The expressions obtained for the existence and stability of equilibrium points have been used to investigate the effects of coefficient of law enforcement and the logistic growth term on the prevalence of crime. It is observed that for a threshold value of law enforcement, a stable limit cycle exists. In particular, a threshold of law enforcement is determined beyond which the associated place/community could be made crime free. A threshold value R0 (similar to basic reproduction number in epidemiology) has also been introduced. It is shown that when R0<1, crime free equilibrium is stable. The elimination and persistence of crime have been discussed via two parameters (law enforcement and half saturation constant) bifurcation diagram. Theoretical results have been supported via numerical simulations followed by discussion and societal impacts of the present study.