A rumor propagation model in multilingual environment with time and state dependent impulsive control

In this paper we are interested in the problem of modeling, analysis and control of rumor spreading on online social networks. As the diversity of languages and the difference of user nodes in the network, a heterogeneous rumor spreading model with two language spreaders is proposed, and the impulsive isolation control strategy is considered. Firstly, the model with periodic impulsive isolation control and time-delay recovery rate is discussed. Based on the stability theory of impulsive differential equations, some conditions regarding the impulsive period and control gains are given for the rumor asymptotically disappearance and persistent existence. Secondly, a new type of event-triggered impulsive isolation control method is developed to address the tradeoff between restraining rumor and minimizing the cost of control, and some conditions are given to make the rumor disappear asymptotically. Furthermore, an improved event-triggered mechanism is established based on the fixed-time stability theory, which provides the conditions for the rumor to disappear within a fixed-time. Finally, some numerical simulations and an actual example are given to verify the rationality of the model and the effectiveness of the control methods.