Mathematical analysis of dynamical systems involving Atangana–Baleanu piecewise derivative

Most mathematical models of epidemiology often assume initial conditions to be either zero or constant. However, this paper focuses on analyzing a mathematical model that addresses real-world problems encompassing diverse domains and varying initial data. Firstly, we discuss scenarios where the initial population can take on different values. In this study, we establish the existence of solutions for the proposed fractional differential dynamical system with integral-type initial conditions involving the Atangana–Baleanu piecewise derivative. The existence of at least one solution is proven using the Schauder fixed point theorem, while the uniqueness of the solution is discussed employing the Banach contraction principle. Additionally, we examine the criteria for Hyers–Ulam stability of the Atangana–Baleanu-piecewise fractional differential system. To approximate the solution, we propose a numerical scheme and provide an example to validate both our results and the numerical scheme employed.