In this paper, we improve an Ignorant-Lurker-Spreader-Removal (ILSR) rumor propagation model as in [Yang et al., 2019] in social networks with consideration to Logistic growth and two discrete delays. First, we prove the existence of equilibrium points by calculating the basic reproduction number according to the next generation matrix. Regarding the two discrete delays as bifurcating parameters, the local asymptotical stability and Hopf bifurcation of the positive equilibrium point are discussed for six different scenarios by analyzing the characteristic equations of linearized systems. Applying the normal form theory and the center manifold theorem, some important conclusions about the stability and direction of bifurcating periodic solution are given when the two time delays are equal. Subsequently we study the global stability of the equilibrium points by constructing Lyapunov functions when the two delays disappear. Finally, we verify the conclusions through numerical simulations and perform sensitivity analysis on the basic reproduction numbers.

中文翻译：

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社交网络中具有两个离散时滞的谣言传播模型的全局稳定性和分岔分析

在本文中，我们改进了社交网络中的 [Yang et al., 2019] 中的 Ignorant-Lurker-Spreader-Removal (ILSR) 谣言传播模型，同时考虑了 Logistic 增长和两个离散延迟。首先，我们通过根据下一代矩阵计算基本再生数来证明平衡点的存在。以两个离散时滞为分岔参数，通过分析线性化系统的特征方程，讨论了六种不同场景下正平衡点的局部渐近稳定性和Hopf分岔。应用范式理论和中心流形定理，给出了两个时滞相等时分岔周期解的稳定性和方向性的一些重要结论。随后，我们通过构造 Lyapunov 函数在两个延迟消失时研究平衡点的全局稳定性。最后，我们通过数值模拟验证了结论，并对基本再生数进行了敏感性分析。