Educational programs and crime: a compartmental model approach

TL;DR

This paper introduces a compartmental mathematical model to analyze how peer interactions influence delinquent behavior and evaluate the impact of educational programs on reducing crime rates.

Contribution

It develops a novel epidemic-inspired model with three population groups to study the effects of educational programs on criminal behavior dynamics.

Findings

Identified three equilibrium states including crime-free and coexistence scenarios.

Derived the basic reproduction number $R_0$ to assess system stability.

Numerical simulations demonstrate how parameters affect delinquency levels.

Abstract

In this paper, we present a mathematical model to describe the temporal evolution of delinquent behavior, treating it as a socially transmitted phenomenon influenced by peer interactions, thus similar to an epidemic. We consider a compartmental framework involving three ordinary differential equations to describe the dynamics among the three population groups: individuals not incarcerated (susceptible), incarcerated offenders, and incarcerated offenders participating in an educational program. Transitions between the groups are governed by interaction-based mechanisms that capture the influence of peer effects in the spread of criminal behavior. The model revealed three equilibrium states: a delinquence free equilibrium, an equilibrium where no criminals attend an educational program, and a coexistence equilibrium. The basic reproduction number, $R_0$, was derived, and a sensitivity analysis revealed the key parameters that influence the system's stability. The model thus provides a quantitative basis for evaluating the effectiveness of rehabilitation strategies in correctional settings. Numerical simulations and an empirical application illustrate the qualitative properties of the model and show how parameter variations influence system behavior.