Online social networks (OSNs), as multilingual communication platforms, provide great convenience for rumor spreading. However, a great deal of negative impacts on individuals, society and economy will be caused during the spread of rumor. In order to explore the rules of rumor propagation on online social networks and formulate the corresponding control strategies, in this paper, we propose new 2I2SR rumor propagation models with and without time-delay based on multilingual environment. Firstly, we calculate the basic reproduction number which completely determines the local and global stability of the rumor free equilibrium. Secondly, our theoretical analysis reveals the existence and stability of the rumor equilibria, and some critical conditions are obtained for the existence of Hopf bifurcation by selecting critical parameters and delay as bifurcation parameters. Furthermore, by using Pontryagin’s Maximum principle, a real-time optimization method that minimizes the cost of restraining rumor is proposed to eliminate the rumor within an expected time period. Finally, some numerical simulations and a practical application are given to prove the correctness of the theoretical results.

在线社交网络（OSN）作为多语言交流平台，为谣言传播提供了极大的便利。但是，谣言在传播过程中会对个人、社会和经济造成很大的负面影响。为了探索在线社交网络上的谣言传播规律并制定相应的控制策略，本文提出了基于多语言环境的新的有时滞和无时滞的2I2SR谣言传播模型。首先，我们计算完全决定无谣言均衡的局部和全局稳定性的基本再生数。其次，我们的理论分析揭示了谣言均衡的存在性和稳定性，得到了谣言均衡存在的一些临界条件。Hopf 分岔通过选择关键参数和延迟作为分岔参数。此外，利用庞特里亚金的极大值原理，提出了一种抑制谣言的成本最小的实时优化方法，以在预期的时间段内消除谣言。最后通过数值模拟和实际应用证明了理论结果的正确性。