Mathematical modeling of crime and corruption dynamics: a cross-national analysis of their interdependencies

Abstract This study presents a novel compartmental mathematical model for analyzing crime and corruption within socio-political systems, with a focus on Colombia, Guatemala, and Venezuela. Recognizing the mutually reinforcing nature of crime and corruption, the model incorporates key population compartments: vulnerable citizens, law enforcement, exposed individuals, corrupt agents, persistent criminals, prosecuted individuals, and those in correctional institutions. A system of nonlinear differential equations captures the dynamic transitions between these states, driven by socio-institutional and behavioral parameters. Qualitative analysis establishes the model’s positivity, boundedness, and equilibria analysis, while the basic reproduction number $$\mathcal {R}_0$$ is derived using the next-generation matrix method to assess the threshold conditions for persistence or eradication of corruption and crime. The model is calibrated using empirical data from the United Nations Office on Drugs and Crime (UNODC), with country specific simulations revealing divergent trajectories. Results indicate that high transition into corruption and low prosecution rates exacerbate systemic corruption, particularly in Guatemala. In contrast, Colombia’s strong enforcement and moderate transition rates yield better containment. Venezuela presents an intermediate case, with early gains diminishing over time. Policy simulations suggest that increasing prosecution and sentencing rates significantly reduce long-term corruption prevalence.