Rumor propagation on complex networks is rapidly affecting people’s life. As we all know, the regulatory control of rumors by regulators has a specific impact on the spread of rumors. Limited regulatory resources may saturate the regulatory level. Therefore, in this paper, we have introduced a saturation treatment function to model this phenomenon and further establish an SIS rumor propagation model with consideration of some comprehensive influence on rumor diffusion. First of all, we prove the boundedness of solutions, and the basic reproduction number $R_0$ is obtained by the method of the next generation matrix. Secondly, by constructing Lyapunov function and applying the linearization method of differential equations, the stability conditions of the equilibrium points are derived. Further, we determine the condition for the backward bifurcation of the rumor propagation model. Moreover, in order to control the spread of rumors, we propose targeted immunization control, acquaintance immunization control and optimal control strategies based on complex networks. Finally, the sensitivity analysis of the basic reproduction number is carried out, and the correctness of the theoretical results is verified by numerical simulations. Our results may provide us with useful insights into the dynamics of online rumor propagation. For example, the basic reproduction number $R_0$ gives the threshold of rumor propagation, which provides control conditions for suppressing the spread of rumors. Stability analysis indicates the likelihood of local or global outbreaks for rumors. In addition, the comparison of the effect for control strategies gives a variety of options for the control method.

谣言在复杂网络上的传播正在迅速影响人们的生活。众所周知，监管机构对谣言的监管控制对谣言的传播有特定的影响。有限的监管资源可能会使监管水平饱和。因此，在本文中，我们引入了饱和处理函数来对此现象进行建模，并考虑到对谣言扩散的一些综合影响，进一步建立了SIS谣言传播模型。首先，我们证明了解的有界性，以及基本的再现数${\mathrm{[R}}_0$通过下一代矩阵的方法获得。其次，通过构造李雅普诺夫函数并应用微分方程的线性化方法，得出平衡点的稳定性条件。此外，我们确定了谣言传播模型向后分叉的条件。此外，为了控制谣言的传播，我们提出了基于复杂网络的针对性免疫控制，熟人免疫控制和最佳控制策略。最后，对基本再生数进行了敏感性分析，并通过数值模拟验证了理论结果的正确性。我们的结果可能为我们提供有关在线谣言传播动态的有用见解。例如，基本复制编号${\mathrm{[R}}_0$给出了谣言传播的阈值，这提供了抑制谣言传播的控制条件。稳定性分析表明谣言在当地或全球爆发的可能性。此外，控制策略效果的比较为控制方法提供了多种选择。