A new crossover dynamics mathematical model of monkeypox disease based on fractional differential equations and the Ψ-Caputo derivative: Numerical treatments

A novel crossover model for monkeypox disease that incorporates $\Psi$-Caputo fractional derivatives is presented here, where we use a simple nonstandard kernel function $\Psi \left(t\right)$. We can be obtained the Caputo and Caputo–Katugampola derivatives as special cases from the proposed derivative. The crossover dynamics model defines four alternative models: fractal fractional order, fractional order, variable order, and fractional stochastic derivatives driven by fractional Brownian motion over four time intervals. The $\Psi$-nonstandard finite difference method is designed to solve fractal fractional order, fractional order, and variable order mathematical models. Also, the nonstandard modified Euler Maruyama method is used to study the fractional stochastic model. A comparison between $\Psi$-nonstandard finite difference method and $\Psi$-standard finite difference method is presented. Moreover, numerous numerical tests and comparisons with real data were conducted to validate the methods’ efficacy and support the theoretical conclusions.