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Considering the media coverage and age‐dependent education, a deterministic and a stochastic class‐age‐structured rumor propagation models are studied, respectively. First, the deterministic rumor propagation model is characterized by a coupled system of ordinary and partial differential equations. The positivity and boundness of solutions are proved, and the basic reproduction number is derived. Second, the stochastic rumor propagation model is formulated by stochastic differential equations. The existence of global positive solutions in model is discussed with the Itô's formula and stochastic Lyapunov function. Additionally, by utilizing comparison principle of stochastic differential equations and the strong law of large numbers, several sufficient conditions for extinction and persistence of the rumor are derived. Finally, numerical simulations are carried out for illustrating the results. [ABSTRACT FROM AUTHOR]