This study explores the intricacies of the COVID-19 pandemic by employing a four-compartment model with a fractal-fractional derivative based on Caputo concept. The analysis hinges on Schauder fixed point theorem, used to qualitatively examine the solutions and ascertain their existence and uniqueness within the model. The fundamental reproduction number is determined through the next-generation matrix approach. This study delves into the stability of equilibrium points and conducts a sensitivity analysis of model parameters. The equilibrium without infections is locally and globally stable when the basic reproduction number is less than 1. Also, this equilibrium becomes unstable when the basic reproduction number exceeds 1. Applying Lyapunov principles and the Routh–Hurwitz criteria, it is established that the endemic equilibrium point is globally stable for the basic reproduction number values greater than 1. The proposed model incorporates Ulam-Hyers stability through nonlinear functional analysis. Lagrange interpolation method estimates solutions for the fractal-fractional order COVID-19 model. Numerical simulations are performed using MATLAB software to exemplify the model behavior in the context of the Italian case study. Furthermore, fractal-fractional calculus techniques hold significant promise for comprehending and predicting the pandemic’s global dynamics in other countries.
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