With the emergence of new media, the spread of rumors has become increasingly easy. Quickly controlling the spread of rumors has become an urgent problem to be solved. In this paper, we consider a rumor model and establish sufficient conditions for the extinction of the rumor based on discrete-time observation. By constructing an auxiliary system and estimating the error, we obtain the upper bound of the time delay. Then, the extinction of the rumor is established by using the perturbation method. Both deterministic and stochastic cases are considered. Finally, numerical examples verify our theoretic results.