An improved two-layer model for rumor propagation considering time delay and event-triggered impulsive control strategy

The rapid development of the Internet has broadened the existing channels of rumor propagation, it has added many new features during the dynamic process of the rumor propagation that are different from those in the past. In this paper, we construct an improved XYZ−ISR two-layer model considering time delay to describe the dynamic process of rumor propagation in multiple channels. Firstly, we obtain the basic regeneration number R0 by the next generation matrix method. Secondly, we derive the stability condition of the equilibrium point by constructing Lyapunov functions. In order to reduce the negative impact caused by rumor propagation at minimum cost, we design an event-triggered impulsive control strategy and optimize the control parameters by using a particle swarm optimization algorithm. Then, by comparing the rumor propagation process in the XYZ−ISR two-layer model and the classical rumor DK model, we find that although rumors propagate faster in the XYZ−ISR model in the initial stage, the rumor spreads on a smaller scale than in the DK model in terms of the whole life cycle of the rumor due to the official supervision and timely release of corresponding rumor-debunking information. Besides, individuals need a certain amount of careful consideration time between receiving the information and deciding whether to forward it or not, and this time delay slows the behavior spreading and reduces its scale. Finally, we verify the reasonableness of the above theoretical results through numerical simulations. Moreover, by simulating a rumor case on the microblog with real data, we find that the model proposed in this paper can fit the real rumor propagation process well, and the event-triggered impulsive control strategy can control the rumor propagation more effectively in multiple channels.