Numerical Treatments for a Crossover Cholera Mathematical Model Combining Different Fractional Derivatives Based on Nonsingular and Singular Kernels

This study introduces a novel mathematical model to describe the progression of cholera by integrating fractional derivatives with both singular and non-singular kernels alongside stochastic differential equations over four distinct time intervals. The model incorporates three key fractional derivatives: the Caputo-Fabrizio fractional derivative with a non-singular kernel, the Caputo proportional constant fractional derivative with a singular kernel, and the Atangana-Baleanu fractional derivative with a non-singular kernel. We analyze the stability of the core model and apply various numerical methods to approximate the proposed crossover model. To achieve this, the approximation of Caputo proportional constant fractional derivative with Grünwald-Letnikov nonstandard finite difference method is used for the deterministic model with a singular kernel, while the Toufik-Atangana method is employed for models involving a non-singular Mittag-Leffler kernel. Additionally, the integral Caputo-Fabrizio approximation and a two-step Lagrange polynomial are utilized to approximate the model with a non-singular exponential decay kernel. For the stochastic component, the Milstein method is implemented to approximate the stochastic differential equations. The stability and effectiveness of the proposed model and methodologies are validated through numerical simulations and comparisons with real-world cholera data from Yemen. The results confirm the reliability and practical applicability of the model, providing strong theoretical and empirical support for the approach.