Hybrid Multi-Stage Analysis of Fractional p-Laplacian System: An Application to the SEIR Epidemic Model

Abstract

This manuscript is dedicated to the qualitative analysis of a novel class of nonlinear fractional differential equations designed to model multi-stage phenomena. The main focus is on a system that is controlled by an advanced piecewise hybrid fractional derivative and a nested p-Laplacian operator. This operator captures dynamic regime shifts by successively using the modified ABC derivatives, Atangana-Baleanu Caputo (ABC), and traditional integer-order derivatives over different time intervals. In order to establish strict requirements for the existence and uniqueness of the solution, we use the Banach Fixed-Point Theorem to reformulate the issue into an analogous system of Volterra integral equations. Additionally, the system's resilience is ensured by a detailed investigation of its Ulam-Hyers (U-H) stability. An application of this theoretical framework to a multi-stage SEIR epidemic model demonstrates its usefulness, as the piecewise operator successfully replicates the long-term effects of public health measures.