Stability and optimal control for delayed rumor-spreading model with nonlinear incidence over heterogeneous networks

On the multilingual online social networks of global information sharing, the wanton spread of rumors has an enormous negative impact on people's lives. Thus, it is essential to explore the rumor-spreading rules in multilingual environment and formulate corresponding control strategies to reduce the harm caused by rumor propagation. In this paper, considering the multilingual environment and intervention mechanism in the rumor-spreading process, an improved ignorants-spreaders-1-spreaders-2-removers (I2SR) rumor-spreading model with time delay and the nonlinear incidence is established in heterogeneous networks. Firstly, based on the mean-field equations corresponding to the model, the basic reproduction number is derived to ensure the existence of rumor-spreading equilibrium. Secondly, by applying Lyapunov stability theory and graph theory, the global stability of rumor-spreading equilibrium is analyzed in detail. In particular, aiming at the lowest control cost, the optimal control scheme is designed to optimize the intervention mechanism, and the optimal control conditions are derived using the Pontryagin's minimum principle. Finally, some illustrative examples are provided to verify the effectiveness of the theoretical results. The results show that optimizing the intervention mechanism can effectively reduce the densities of spreaders-1 and spreaders-2 within the expected time, which provides guiding insights for public opinion managers to control rumors.