自适应网络中病毒传播的稳定性和分岔行为研究

自适应复杂网络是以节点状态与拓扑结构之间存在反馈回路为特征的网络. 针对自适应网络病毒传播模型, 利用非线性微分动力学系统研究病毒传播行为; 通过分析非线性系统对应雅可比矩阵的特征方程, 研究其平衡点的局部稳定性和分岔行为, 并推导出各种分岔点的计算公式. 研究表明, 当病毒传播阈值小于病毒存在阈值, 即R₀0^c时, 网络中病毒逐渐消除, 系统的无病毒平衡点是局部渐近稳定的; R₀^c0<1时, 网络出现滞后分岔, 产生双稳态现象, 系统存在稳定的无病毒平衡点、较大稳定的地方病平衡点和较小不稳定的地方病平衡点; R₀>1时, 网络中病毒持续存在, 系统唯一的地方病平衡点是局部渐近稳定的. 研究发现, 系统先后出现了鞍结分岔、跨临界分岔、霍普夫分岔等分岔行为. 最后通过数值仿真验证所得结论的正确性. 关键词： 自适应网络 稳定性 分岔 基本再生数

Stability and bifurcation of epidemic spreading on adaptive network

Adaptive network is characterized by feedback loop between states of nodes and topology of the network. In this paper, for adaptive epidemic spreading model, epidemic spreading dynamics is studied by using a nonlinear differential dynamic system. The local stability and bifurcation behavior of the equilibrium in this network model are investigated and all kinds of bifurcation point formula are obtained by analyzing its corresponding characteristic equation of Jacobian matrix of the nonlinear system. It is shown that, when the epidemic threshold is less than epidemic persistence threshold R₀0^c, the disease always dies out and the disease-free equilibrium is asymptotically locally stable. If R₀^c0<1, a backward bifurcation leading to bistability possibly occurs, and there are possibly three equilibria: a stable disease-free equilibrium, a larger stable endemic equilibrium, and a smaller unstable endemic equilibrium. If R₀>1, the disease is uniformly persistent and only one endemic equilibrium is asymptotically locally stable. It is also found that the system has saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation. Numerical simulations are given to verify the results of theoretical analysis.
Keywords： adaptive network stability bifurcation basic reproduction number