Structure-Preserving Schemes for a Fractional SVIR Epidemic Model with a Hybrid Mittag-Leffler-Caputo-Fabrizio Operator

Abstract

This paper proposes and analyzes a fractional-order SVIR epidemic model based on a hybrid Mittag-Leffler-Caputo-Fabrizio (MLCF) fractional operator with a nonsingular kernel. This model captures short- and long-term memory effects in epidemic transmission dynamics. The positivity and boundedness of the solutions are proven through an integrated formulation of the MLCF operator and a fractional Gronwall inequality. The basic reproduction number $\mathcal{R}_0$, equilibrium points, and their local and global stability properties are rigorously investigated through Jacobian analysis, logarithmic Lyapunov functionals, and a fractional LaSalle invariance principle. To approximate the model, a $θ$-weighted nonstandard finite difference (NSFD) method is developed. This method preserves the continuous system's key qualitative properties, including positivity and boundedness, and is unconditionally stable in the fully implicit case. Consistency and first-order convergence are also proven. Numerical experiments, together with sensitivity and bifurcation analyses, illustrate the impact of fractional memory parameters on epidemic evolution and demonstrate the effectiveness of the proposed approach.