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False rumors can lead to huge economic losses or/and social instability. Hence, mitigating the impact of bogus rumors is of primary importance. This paper focuses on the problem of how to suppress a false rumor by use of the truth. Based on a set of rational hypotheses and a novel rumor-truth mixed spreading model, the effectiveness and cost of a rumor-containing strategy are quantified, respectively. On this basis, the original problem is modeled as a constrained optimization problem (the RC model), in which the independent variable and the objective function represent a rumor-containing strategy and the effectiveness of a rumor-containing strategy, respectively. The goal of the optimization problem is to find the most effective rumor-containing strategy subject to a limited rumor-containing budget. Some optimal rumor-containing strategies are given by solving their respective RC models. The influence of different factors on the highest cost effectiveness of a RC model is illuminated through computer experiments. The results obtained are instructive to develop effective rumor-containing strategies. equation. Due to the exact modeling and the perfect accommodation of the network topology, this model characterizes the SIS epidemics more accurately as compared with all previous SIS models. Moreover, this model is analytically tractable, resulting in profound conclusions. In recent years, this individual-based modeling approach has been widely applied to areas such as epidemic spreading [42][43][44], malware propagation [45][46][47][48][49][50][51][52], cyber security [53][54][55][56][57], and message transmission [58]. To our knowledge, to date the rumor-containing problem has not yet been studied under individual-level rumor spreading models.Clarifying rumors by releasing truths is a common way to inhibit rumors [59,60]. In this context, every individual may choose to believe the rumor or believe the truth or be uncertain. This paper focuses on the problem of how to inhibit a false rumor by use of the truth. Based on a novel individual-level rumor-truth mixed spreading model, the effectiveness and cost of a rumor-containing strategy are quantified. Thereby, the original problem is modeled as a constrained optimization problem (the rumor-containing (RC) model), in which the independent variable and the objective function represent a rumor-containing strategy and the effectiveness of a rumor-containing strategy, respectively. The goal of the optimization problem is to find the most effective rumor-containing strategy subject to a limited rumor-containing budget. Some optimal rumor-containing strategies are given by solving their respective RC models. The influence of different factors on the highest cost effectiveness of a RC model is uncovered through computer simulations. These results are condusive to the design of practical rumor-containing strategies. To our knowledge, this is the first time the rumor-containing problem is treated in this way.The subsequent materials of this work are organized in this fashion: ...

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This paper examines the dual-channel supply chain in which the differentiation of the environmental sustainability of channels is considered. We analyze the influences of the level of environmental sustainability of channels on the pricing policies for the supply chain members in both centralized and decentralized models using the Stackelberg game model under inconsistent price policy. We obtain the optimal level of environmental sustainability of channels and pricing decisions for the players in the centralized and decentralized dual-channel supply chains. Results show that the influence mechanisms of the level of environmental sustainability of channels on the pricing decisions are different in the centralized and decentralized models. Furthermore, numerical analysis has been conducted to investigate the effects of the cross-environmental-sustainability sensitivity factor and the initial proportion of consumers who prefer the retail channel on the level of environmental sustainability of channels, pricing policies and players' profits.