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Considering the ability of network users to distinguish and refute the rumor, we establish a Susceptible–Believed–Denied (SBD) rumor spreading model with nonlinear incidence rate and time delay in complex networks. Specifically, two dynamical systems of rumor propagation are given based on the mean field theory in homogeneous and heterogeneous networks respectively. Then, we obtain the equilibrium points of the systems and calculate the basic reproduction number R 0 according to the next generation matrix in homogeneous networks and the existence of a positive equilibrium point in heterogeneous networks. The local and global asymptotic stabilities of the rumor-free equilibrium points are demonstrated by constructing Lyapunov functions under the given condition R 1 ≤ 1. Further, we perform representative numerical simulations to illustrate the theoretical results. Through simulation experiments, the dynamics around the rumor-spreading equilibrium points when R 0 > 1 are also discussed to complement some dynamical properties of our models. • This paper presents a novel SBD rumor spreading model combined with classifying thought. • Comparative analysis of homogeneous and heterogeneous networks is numerically given. • Stability is proved in a strict mathematical way. [ABSTRACT FROM AUTHOR]