Semianalytical Solutions of a Caputo–Fabrizio Fractional Epidemic Model via the q-Kushare Homotopy Transform Method

In this study, an enhanced semianalytical framework is developed for solving nonlinear fractional differential equations involving the Caputo–Fabrizio fractional operator. The proposed Caputo–Fabrizio q-Kushare homotopy analysis transform method (CFq-KHATM) combines the Caputo–Fabrizio derivative, Kushare transform, and q-homotopy analysis within a unified operator-consistent framework. Unlike conventional hybrid approaches, the method preserves the nonsingular exponential memory structure of the Caputo–Fabrizio derivative and provides additional convergence-control flexibility through the auxiliary parameter ℠and q-deformation parameter n. To demonstrate its applicability, a Caputo–Fabrizio fractional epidemic model with memory-dependent transmission dynamics is investigated. Numerical simulations for different fractional-order values show that memory effects significantly influence epidemic evolution, infection persistence, and environmental viral dynamics. Furthermore, comparisons with existing methods together with absolute and relative error analyses confirm the accuracy, stability, and convergence efficiency of the proposed framework. The obtained results indicate that CFq-KHATM is a flexible and computationally reliable tool for nonlinear fractional epidemic systems and other memory-dependent dynamical models.