Circulating the truth and quarantining a subset of rumor spreaders are two major rumor-quelling strategies. In practice, a mixture of the two strategies may be more effective than any one of the two strategies. This paper focuses on effectiveness analysis of the mixed strategy. For this purpose, we are going to establish a rumor-truth competing model on two-layer network. First, we introduce a Markov model characterizing the stochastic dynamics of the rumor-truth competing process, and write the corresponding Kolmogorov model capturing the expected dynamics of the rumor-truth competing process. Second, we give a bilinear model as the first approximation to the Kolmogorov model, and suggest a generic model as a more accurate approximation to the Kolmogorov model. The two models are the focus of concern in this work. For ease in treatment, we describe a limit system of the generic model. By studying the limit model, we present a criterion for the rumor to subside, a criterion for the rumor not necessarily to subside, and a criterion for the rumor to persist, respectively. These findings are instructive to the quelling of false rumors. Finally, through computer experiments we find that when a rumor subsides, the bilinear model is a good approximation to the Kolmogorov model. (C) 2018 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.