Educational programs and crime: a compartmental model approach
Alessandro Ramponi, M. Elisabetta Tessitore
TL;DR
This study formulates a compartmental, epidemic-style model for crime by partitioning the population into X (susceptible), I (incarcerated not in education), and E (incarcerated in education). It derives a basic reproduction number $R_0 = \frac{\alpha \Lambda}{\mu(\gamma_I+\mu)}$ and identifies three equilibria: DF (delinquency-free), EF (education-free), and CE (coexistence); their stability is governed by $R_0$ and the threshold $\overline{C} = 1 + \frac{\alpha(\gamma_E+\mu)}{\rho(\gamma_I+\mu)}$, revealing distinct regimes depending on parameter values. Numerical simulations corroborate the theoretical thresholds, showing transitions among DF, EF, and CE as $\gamma_I$ and other parameters vary, including damped oscillations near CE. An empirical application to Italian prison data (1992–2024) involves discretization and constrained regression to estimate $\alpha$ and $\rho$, yielding $\hat{R}_0 \approx 1.025$ and $\hat{\overline{C}} \approx 1.096$, suggesting proximity to the education-free regime and highlighting data limitations. Overall, the work provides a quantitative framework for evaluating rehabilitation strategies in correctional settings and motivates extensions to include control interventions and richer education structures.
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.