In this paper, we analyze the global dynamics of an endemic mathematical model that incorporates direct immunity by vaccination, as well as the shift from the asymptomatic to the symptomatic group in complex networks. By analyzing the Jacobian matrix and constructing suitable Lyapunov functionals, the stability of the disease-free equilibrium and the endemic equilibrium is determined with respect to the basic reproduction number $R_0$. Numerical simulations in scale-free and Poisson network environments are presented. The results validate the correctness of our theoretical analyses.