Fractional-Order Delayed Epidemic Models in Memory-Based Probabilistic Controlled Metric Spaces via Fixed Point Theory

Abstract

We introduce memory-based probabilistic controlled metric spaces and establish a non-Markovian contraction principle that extends the Banach fixed point theorem by depending on a finite number of previous iterates. This framework is suited for systems with delay, hereditary effects, and fractional dynamics. We prove unique existence and convergence of Picard iterations under memory-based probabilistic contractions. As an application, we analyze a fractional-order delayed epidemic model with memory and incubation effects, showing convergence to a unique equilibrium. These results provide a unified tool for fractional epidemic dynamics with delay and memory.