Modeling dynamics of the spread of crime in a society

Abstract

Crime is a social epidemic. The spread of crime in a population is very much dependent on the social structure of the society. Although there are several factors which influence the dynamics of the spread of crime in a society, it is an established fact that crime spreads in a society like an infectious disease. Here, a nonlinear mathematical model is formulated and analyzed to study the dynamics of crime by considering simple mass-action type incidence and constant recruitment and death type demography. The basic reproduction number R₀ of the proposed model is computed and all possible equilibria of the model are obtained. Stability analysis of the model is discussed in detail. The nontrivial equilibrium exists only when the basic reproduction number $R_0>1$ and it is locally asymptotically stable under some restriction on parameters. Further, this model is extended to delay differential equation model by incorporating the delay in catching the criminals. In the presence of delay the local stability of nontrivial equilibrium point is preserved only upto certain value of delay and beyond this Hopf-bifurcation occurs and the model system exhibits oscillatory behavior. Further, the ODE model is converted to stochastic model and results of stochastic and deterministic models are compared using numerical simulation.

Keywords:

• Delay model
• reproduction number
• stability analysis
• stochastic model

Disclosure Statement

The authors declare that they have no conflicts of interest.