两类网络蠕虫的模型建立及动力学性态分析

Modeling and Analysis of Dynamical Behaviors for a Two-Worm Model

网络蠕虫之间存在着复杂的关系,它们对蠕虫的传播和演化等动力学行为有着重要的影响,刻画这些关系有助于找到更好的控制和预防策略.本文建立了两类蠕虫(蠕虫I、蠕虫II)传播的数学模型,通过分析得到两个阈值条件R_1和R_2,当R_1<1和R_2<1,无病平衡点全局渐近稳定,意味着两类蠕虫最终均被清除;当R_2<1<R1边界平衡点Q_1全局渐近稳定,也即蠕虫II灭绝,蠕虫I将持续存在;当R_1<1<R2边界平衡点Q2全局渐近稳定,也即蠕虫I灭绝,蠕虫II将持续存在;当R_1>1和R2>1时,存在惟一正平衡点且全局渐近稳定,即两类蠕虫(蠕虫I与蠕虫II)同时持续存在.通过理论分析可以得到要控制蠕虫病毒可以通过控制参数来实现,进一步给出控制蠕虫病毒相对应的措施.最后通过数值模拟验证了理论分析结果.

The complex relationship among interuet worms have great impact on the dynamics of worms. To describe the propagation of worm, it is necessary to characterize these interactions. In the paper, A two-worm model is presented, and we got the thresholds R1 and R2, if R1 〈 1 and R2 〈 1, the disease-free equilibrium is globally asymptotically stable, the worms will be died out. If R2 〈 1 〈 R1 the boundary equilibrium Q1 is globally asymptotically stable which means that the worm I will exist and the worm II will die out. If R1 〈 1 〈 R2 the boundary equilibrium Q2 is globally asymptotically stable, i.e., the worm II will exist and the worm I will die out. If R1 〉 1 and R2 〉 1 ,there is an unique equilibrium, which is globally asymptotically stable implying persistent immune responses. By theoretical analysis, we get that the worm can be controlled by adjusting the parameters. Furthermore, we can give measures of controlling worms. In the end, we verify the theoretical analysis results by numerical simulations.