Modeling criminal careers of different levels of offence

Abstract

We set up and analyse a mathematical model, the Serious Crime Model, which describes the interaction of mild and serious offenders and potential criminals. However we get more complete results for a simpler version of this model, the Mild Crime Model, with no serious offenders. For the full Serious Crime Model there are two key parameters R01 and R02 corresponding to the basic reproduction number in the mathematics of infectious diseases, which determine the behaviour of the system. For the Simpler Mild Crime Model there is just one such parameter R01. Both forward and backward bifurcation can occur for this second model with two subcritical non-trivial equilibria possible for R01<1 in the backwards case. For backwards bifurcation there is another threshold value R0* such that the upper non-trivial equilibrium is unstable for R01<R0* and stable for R01>R0*. For forwards bifurcation there is a second additional threshold value R0** such that the stability of the unique non-trivial equilibrium switches from unstable to stable as R01 passes through R0**. At the end we return to the full Serious Crime Model and discuss the behaviour of this model. The results are meaningful and interesting because in all of the other epidemiological and sociological models of which we are aware, analogous thresholds to R0* and R0** do not exist. For forwards bifurcation the unique non-trivial equilibrium, and for backwards bifurcation with two subcritical endemic equilibria the higher non-trivial equilibrium, are also usually always locally asymptotically stable. So our models exhibit unusual and interesting behaviour.