Dynamical behavior and optimal impulse control analysis of a stochastic rumor spreading model

The Internet era has brought great convenience to our life and communication. Meanwhile, it also makes a bunch of rumors propagate faster and causes even more harm to human life. Therefore, it is necessary to perform effective control mechanisms to minimize the negative social impact from rumors. Thereout, firstly, we formulate a rumor spreading model considering psychological factors and thinking time, then, we add white noise (i.e., stochastic interference) and two pulse control strategies which denote education mechanism and refutation mechanism into the model. Secondly, we obtain the global positive solutions and demonstrate the global exponential stability of the unique positive periodic rumor-free solution. Thirdly, we discuss the extinction and persistence of rumor. Moreover, we use Pontriagin's minimum principle to explore the optimal impulse control. Finally, several numerical simulations are carried out to verify the effectiveness and availability of the theoretical analysis. We conclude that the pulse control strategies have a great influence on controlling rumor spreading, and different control strategies should be adopted under different transmission scenarios.