Analysis of a Fractional-Order Mathematical Model on Crime Dynamics Incorporating Police Force and Rehabilitation

This study addresses the limitation of traditional integer-order crime models that fail to capture memory-dependent dynamics in criminal behavior. Our objective is to develop and analyze a novel fractional-order model incorporating media influence, police force, and rehabilitation strategies using the Liouville−Caputo derivative. The novelty lies in simultaneously integrating multiple societal interventions within a memory-dependent framework and developing a new generalized telephone polynomial collocation method (GTPCM) for numerical solution. We employ fractional calculus theory for stability and bifurcation analysis alongside our spectral numerical method. The main contribution is a complete mathematical framework bridging theoretical criminology and computational mathematics. Key findings reveal that fractional memory effects alter crime persistence through damped oscillations, identify critical parameters via sensitivity analysis, and demonstrate through optimal control that combined preventive and rehabilitative strategies to reduce criminal populations by approximately 78.1% compared to no intervention.