基于复杂网络的谣言传播模型稳定性和最优控制研究  

Exploration of Stability and Optimal Control for Rumor Propagation Models within Complex Networks

作　　者：庄洪萍 向会立 周敏 ZHUANG Hongping;XIANG Huili;ZHOU Min(School of Mathematics and Statistics,Hubei Minzu University,Enshi 445000,China)

机构地区：[1]湖北民族大学数学与统计学院,湖北恩施445000

出　　处：《湖北民族大学学报（自然科学版）》2024年第3期442-450,共9页Journal of Hubei Minzu University:Natural Science Edition

基　　金：国家自然科学基金项目(ZZG19001)。

摘　　要：为深入理解谣言传播规律和探究如何有效地控制谣言传播,基于复杂网络构建无知者-潜伏者-传播者-稳定者(uninformed-susceptible,latent-exposed,spreader-infected,stable-recovered,SEIR)谣言传播模型,研究其动力学行为和最优控制问题。首先,利用下一代矩阵法计算基本再生数,通过相应的雅可比矩阵和构造李雅普诺夫函数讨论无谣言平衡点的局部和全局渐近稳定性。其次,通过特征方程证明谣言盛行平衡点的全局渐近稳定性。最后,利用庞特里亚金最值原理,研究该模型的最优控制问题,证明最优控制解的存在性和具体形式;此外,通过正向和反向扫描方法得到最优控制问题的数值解。结果表明,谣言在系统中是否传播受网络异质性水平的影响;当控制■时,目标函数取得最小值,表明对无知者和潜伏者施加控制是有效的。该研究在谣言传播控制方面具有重要意义。To gain an in-depth understanding of the laws governing rumor propagation and to explore effective methods of controlling it,an uninformed-susceptible,latent-exposed,spreader-infected,stable-recovered(SEIR)rumor propagation model was constructed based on complex networks,and its dynamic behavior and optimal control problems were investigated.Initially,the next-generation matrix method was utilized to calculate the basic reproduction number,and the local and global asymptotic stability of the rumor-free equilibrium point was discussed using the corresponding Jacobian matrix and the construction of Lyapunov functions.Subsequently,the global asymptotic stability of the rumor-free equilibrium point was proven through the characteristic equation.Finally,the optimal control problem of the model was studied by using the Pontryagin optimal value principle,and the existence and concrete form of the optimal control solution were proved.Additionally,numerical solutions to the optimal control problem were obtained through forward and backward scanning methods.The results indicated that whether a rumor spreads in the system was influenced by the level of network heterogeneity.When the control u=u^was applied,the objective function reached its minimum,suggesting that controlling the uninformed and latent individuals was effective.This research holds significant importance in the control of rumor propagation.

关 键 词：复杂网络 谣言传播模型 网络异质性水平 稳定性 最优控制 数值模拟 

分 类 号：O175[理学—数学]