Abstract By drawing an analogy to the spreading dynamics of an infectious disease, the authors derive a fractional-order susceptible-infected-removed (SIR) model to examine the user adoption and abandonment of online social networks, where adoption is analogous to infection, and abandonment is analogous to recovery. They modify the traditional SIR model with demography, so that both infectious and noninfectious abandonment dynamics are incorporated into the model. More precisely, they consider two types of abandonment: (i) infectious abandonment resulting from interactions between an abandoned and an adopted member, and (ii) noninfectious abandonment which is not influenced by an abandoned member. In addition, they study the existence and uniqueness of nonnegative solutions of the model, as well as the existence and stability of its equilibria. They establish a nonnegative threshold quantity R 0 α for the model and show that if R 0 α 1 , the user-free equilibrium E 0 is locally asymptotically stable. In addition, they find a region of attraction for E 0 . If R 0 α > 1 , they prove that the model has a unique user-prevailing equilibrium E ∗ that is globally asymptotically stable. Their stability results also show that the infectious abandonment dynamics do not contribute to the stability of the user-free and user-prevailing equilibria, and that it only affects the location of the user-prevailing equilibrium. The Jacobian matrix technique and the Lyapunov function method are used to show the stability of the equilibria. They perform numerical simulations to verify these theoretical results. Finally, they conduct a case study of fitting their model to some historical Instagram user data to show the effectiveness of the model.

中文翻译：

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分数阶在线社交网络模型的稳定性分析

摘要 通过对传染病的传播动态进行类比，作者推导出了一个分数阶易感感染移除 (SIR) 模型来检查用户对在线社交网络的采用和放弃，其中采用类似于感染，并且放弃类似于恢复。他们用人口统计学修改了传统的 SIR 模型，以便将传染性和非传染性遗弃动态纳入模型。更准确地说，他们考虑了两种类型的遗弃：（i）由被遗弃成员和被收养成员之间的相互作用引起的感染性遗弃，以及（ii）不受遗弃成员影响的非感染性遗弃。此外，他们还研究了模型非负解的存在性和唯一性，以及其均衡点的存在性和稳定性。他们为模型建立了一个非负的阈值量 R 0 α 并表明如果 R 0 α 1 ，则无用户均衡 E 0 是局部渐近稳定的。此外，他们找到了 E 0 的吸引区域。如果 R 0 α > 1 ，则证明该模型具有唯一的用户主导均衡 E*，该均衡是全局渐近稳定的。他们的稳定性结果还表明，传染性放弃动态对无用户和用户主导均衡的稳定性没有贡献，它只影响用户主导均衡的位置。雅可比矩阵技术和李雅普诺夫函数方法用于显示平衡的稳定性。他们进行数值模拟以验证这些理论结果。最后，