Several newly nonlinear models for describing dynamics of COVID-19 pandemic have been proposed and investigated in literature recently. In light of these models, we attempt to reveal the role of fractional calculus in describing the growth of COVID-19 dynamics implemented on Saudi Arabia's society over 107 days; from 17 Dec 2020 to 31 March 2021. Above is achieved by operating two fractional-order differential operators, Caputo and the Caputo-Fabrizio operators, instead of the classical one. One of expanded SEIR models is utilized for achieving our purpose. With the help of using the Generalized Euler Method (GEM) and Adams-Bashforth Method (ABM), the numerical simulations are performed respectively in view of the Caputo and Caputo-Fabrizio operators. Accordance with said, the stability analysis of the two proposed fractional-order models is discussed and explored in view of obtaining the equilibrium points, determining the reproductive number ( $R_0$) and computing the elasticity indices of $R_0$. Several numerical comparisons reveal that the fractional-order COVID-19 models proposed in this work are better than that of classical one when such comparisons are performed between them and some real data collected from Saudi Arabia's society. This inference together with the cases predictions that could easily deduced from the proposed fractional-order models can allow primary decision makers and influencers to set the right plans and logic strategies that should be followed to face this pandemic.