Fractional-order crime dynamics: Linear vs. nonlinear transmission models using physics-informed neural networks

Crime dynamics are inherently nonlinear and coupled with the social, economic, and behavioral interactions between the non-criminals, criminals, and rehabilitated people. The conventional integer-order differential equation models cannot capture the memory effects and hereditary behavior inherent in human society. This work is devoted to the development and analysis of four fractional-order crime-transmission models that consider different nonlinear interaction terms, namely: (i) bilinear interaction, (ii) saturation in non-criminals, (iii) saturation in criminals, and (iv) combined saturation. The Caputo fractional operator has been utilized to model the memory-dependent transitions among the non-criminals S(t), criminals C(t), and prisoners/rehabilitated individuals P(t). Moreover, bifurcation and sensitivity analyses are conducted to investigate the influence of critical parameters on stability switching and crime persistence. Additionally, the convergence behavior of the proposed numerical scheme is examined to validate its accuracy and reliability. A framework of physics-informed neural networks (PINNs) is implemented for estimating the parameters and fractional orders and predicting future levels of crime. The PINNs is trained by a simultaneous minimization of the data-fitting loss and residual loss of the differential equation, hence learning hidden dynamics with sparse or noisy crime-rate data. Results have shown that models incorporating nonlinear saturation provide more stable and realistic predictions, while the fractional order (η) significantly affects long-term memory and persistence of crime. A comparison study shows that Model 4 has the least residual error and best prediction accuracy due to its most general form. The result implies that the fractional-order PINNs frameworks are a strong tool in the prediction of crime trends and testing the impact of the intervention parameters such as rehabilitation rate, transmission rate of crime, and recruitment rate into society.