Mathematical Modeling of the Role of Imitation in Crime Dynamics

TL;DR

This paper develops a nonlinear differential equation model to understand how imitation influences crime spread, revealing conditions for crime eradication or persistence and emphasizing the importance of reducing relapse rates.

Contribution

It introduces a novel mathematical model incorporating imitation effects in crime dynamics and analyzes stability and bifurcation phenomena.

Findings

Crime-free equilibrium is stable when the basic reproduction number is less than one.

The model exhibits backward bifurcation with coexistence of endemic and crime-free states.

Reducing relapse rates significantly impacts the potential to eliminate crime.

Abstract

Crime remains one of the significant problems that countries are grappling with globally. With shrinking economies and increasing poverty, crime has been on the rise in many countries. In this paper, we propose a system of non-linear ordinary differential equations to model crime dynamics in the presence of imitation. The model consists of four independent compartments: individuals who are not at risk of committing a crime, individuals at risk of committing a crime, individuals committing a crime, and individuals convicted and jailed for a crime. The model is analyzed using the basic reproduction number. The analysis shows the system has a locally asymptotically stable crime-free equilibrium when the basic reproduction number is less than unity. The model exhibits a backward bifurcation in which two endemic equilibria coexist with the crime-free equilibrium. When the basic reproduction number exceeds unity, the system has a locally asymptotically stable endemic equilibrium, and the crime-free becomes unstable. Numerical simulations are carried out to verify the analytical results. The sensitivity analysis shows that the relapse rate highly influences the basic reproduction number of our model. This indicates that the proportion of individuals leaving prisons and becoming criminals should be minimized to minimize crime.