Time-Dependent Dynamical System Model for Prison Population Analysis

This study develops an applied mathematical modeling framework, implemented through system dynamics, to analyze the evolution of the Chilean prison population and support policy-oriented scenario analysis. The model represents the prison system through balance equations and distinguishes defendants in pretrial detention D(t), convicted prisoners C(t), and the total prison population T(t). Quarterly institutional transition rates were derived from administrative records and demographic data. The model was empirically parameterized and evaluated through in-sample reconstruction using quarterly administrative data for 2022–2024; no independent, out-of-sample validation was performed. It was then used to examine scenarios for 2025. The prison population increased from 40,098 in Q1 2022 to 58,669 in Q4 2024, mainly due to the accumulation of convicted prisoners and asymmetric release rates. The model reproduced the observed trajectory of T(t) with a mean absolute percentage error of 3.539%, consistent with in-sample reconstruction rather than external predictive validation. Scenario-based sensitivity analysis indicates that changes in the police identification rate PR(Q) generate amplified downstream responses, whereas reintegration scenarios produce moderate short-term reductions. The main contribution is an interpretable, stock-and-flow framework integrating quarterly, time-varying institutional rates, procedural differentiation, demographic normalization, and scenario simulation. The results should be interpreted as model-based scenarios rather than causal estimates.