In this paper, we consider a mathematical model of a coronavirus disease involving the Caputo–Fabrizio fractional derivative by dividing the total population into the susceptible population $\mathcal{S}(t)$ S ( t ) , the vaccinated population $\mathcal{V}(t)$ V ( t ) , the infected population $\mathcal{I}(t)$ I ( t ) , the recovered population $\mathcal{R}(t)$ R ( t ) , and the death class $\mathcal{D}(t)$ D ( t ) . A core goal of this study is the analysis of the solution of a proposed mathematical model involving nonlinear systems of Caputo–Fabrizio fractional differential equations. With the help of Lipschitz hypotheses, we have built sufficient conditions and inequalities to analyze the solutions to the model. Eventually, we analyze the solution for the formed mathematical model by employing Krasnoselskii’s fixed point theorem, Schauder’s fixed point theorem, the Banach contraction principle, and Ulam–Hyers stability theorem.