New Crossover Monkeypox Disease Mathematical Model Based on Fractal Fractional Caputo-Katugampola Derivative: Numerical Solutions

Abstract

This paper introduces a novel crossover epidemiological model for the transmission dynamics of monkeypox, integrating fractal-fractional Caputo-Katugampola derivatives with stochastic perturbations driven by fractional Brownian motion. The proposed framework captures complex memory, hereditary, and environmental effects within two temporally distinct regimes: a deterministic fractal-fractional phase and a stochastic fractional-order phase. We establish the existence and uniqueness of solutions using the Perov fixed-point theorem within generalized Banach spaces. To numerically solve the system, we construct a $\kappa$-nonstandard finite difference scheme and adapted modified Euler-Maruyama method adapted to the fractional stochastic setting.Sensitivity and bifurcation analyses reveal that transmission-related parameters are the primary drivers of disease dynamics, leading to the emergence of a forward bifurcation that marks the transition from disease-free to endemic states. Numerical simulations, calibrated with real outbreak data from the United States, demonstrate excellent agreement and highlight the critical role of memory and stochasticity in shaping infection trajectories. The results underscore the utility of fractal-fractional stochastic modeling for improving predictive power and informing targeted intervention strategies in emerging epidemics.