Dynamical study and event-triggered impulsive control of rumor propagation model on heterogeneous social network incorporating delay

Considering the multilingual environment of rumors propagation, we propose a novel I2SR rumor propagation model with time delay on heterogeneous networks. Firstly, the basic reproduction number, depending on the network structure, is given to ensure the existence of the equilibrium points. Meanwhile, the dynamic behaviors of the rumor free equilibrium point (RFEP) are analyzed via the local linearization method and comparison principle. We also prove that the rumor equilibrium point (REP) is globally asymptotically stable (G-stable) by employing the Lyapunov stability theory and irreducible matrix properties and demonstrate the effect of time delay on the outbreak point and stability. Secondly, in order to reduce the cost of limited public resources, a new event-triggered impulsive control (ETIC) is applied to restrain the propagation of rumor. The event-triggering function is carefully designed and some criteria are obtained to guarantee that the rumor disappeared. Finally, we recognize the most sensitive parameter to the model and numerically simulate the model with some useful parameter sets to justify the theoretical results. Furthermore, using a rumor example of Sina Weibo in 2017, we find that our model can well fit the real rumor spreading process, and the ETIC can effectively inhibit the rumor propagation.