Recently, piecewise differential operators have been introduced to capture crossover dynamics in physical systems. In the evolution of corruption, the underlying dynamics can shift across different regimes. These crossovers occur due to policy changes, economic shocks, or shifts in social behavior. To demonstrate the crossover dynamics of a corruption mathematical system, we use a piecewise operator. The piecewise operator is divided into three pieces: a classic or integer order operator, a fractional operator, and a stochastic operator. For the fractional order case, we use the constant proportional Caputo (CPC) operator, which is a straightforward linear combination of the Riemann–Liouville (RL) integral and the Caputo derivative. Theoretical analysis such as existence and uniqueness of solutions for the fractional case under CPC derivative, is elucidated via notions of fixed point theory, specifically the implication of Perov’s fixed point result and for the stochastic model using Ito calculus. Numerical results are presented for the proposed model. Graphical analysis of the corruption model is performed using PW operators across three distinct intervals to portray the crossover dynamics of the considered system. Also, the influence of various parameters on the crossover dynamics of the corruption model is illustrated via numerical simulations. Sensitivity of parameters is demonstrated via some statistical experiments, such as scatter plots and Pearson correlation coefficients, quantifying the relationship between key parameters of the system. The validity of the result is verified by comparing the system dynamics with real data dynamics via 2D graphs.
Laila A. Al‐Essa (Sat,) studied this question.