A Hybrid Predictive Framework for Zika Dynamics Using the Normalized Caputo–Fabrizio Operator

Source abstract

We create and test a Zika transmission model that includes vector‐borne, sexual, and vertical channels under seasonal forcing and a fixed pregnancy‐to‐birth delay. All human and vector compartments are modeled using the normalized Caputo–Fabrizio (NCF) fractional derivative, allowing for nonsingular memory effects while maintaining classical initial conditions. Casting the system into an equivalent Volterra form, we establish existence by Schauder’s fixed‐point theorem and uniqueness on finite horizons under a Lipschitz–time contraction ( L T < 1); moreover, we prove Ulam–Hyers stability with an explicit linear‐in‐time bound specific to the normalized kernel. For computation, we develop a first‐order Adams–Moulton predictor–corrector scheme specifically adapted to the NCF–Volterra formulation; owing to the exponential nonsingular NCF kernel, the memory weights are obtained in closed form, which simplifies the history integral and permits an efficient treatment of the delayed terms. Numerical experiments show how the fractional order modulates epidemic timing and peak size across female, male, child, and vector classes under seasonal variation. This bridges classical ODE behavior ( α ⟶1) and strong‐memory regimes in a transparent manner.