基于分数阶微分方程的手机病毒传播模型研究

An Epidemic Model Research of the Mobile Phone Virus Based on Fractional Differential Equations

摘要: 随着手机普及率的提高，手机病毒的传播也愈发严重。因此，对于病毒传播规律的研究必不可少。本文建立并研究了一类基于分数阶微分方程的手机病毒传播模型，利用分数阶微分方程的相关理论，详细分析了平衡点的存在性及其局部稳定性，并通过数值试验验证了理论结果的正确性。通过研究，我们得到在基本再生数小于1的情况下，未感染平衡点是局部渐进稳定的，病毒会消亡；在基本再生数大于1时，感染平衡点局部渐近稳定，病毒将扩散。根据所得到的理论结果给出了控制手机病毒传播的有效措施，为手机病毒的预测、控制和防治提供了重要的参考依据。

Abstract: With the increasing popularity of mobile phones, the spread of mobile virus has become increas-ingly serious. Therefore, the study of the spread of the virus is necessary. In this paper, we first establish an epidemic model of mobile phone virus based on the fractional differential equation. Then by means of the theory of fractional differential equations, we analyse the existence and sta-bility condition of the equilibrium of the model. It is showed that if the basic reproduction number is less than 1, the infection free equilibrium is locally asymptotically stable and virus will die out, and if the basic reproductive number is greater than 1, the infection equilibrium is stable and the virus will spread. Some numerical experiments are carried out to confirm the obtained results. In addition, some effective measures are given to control the spread of the mobile virus, which pro-vides referential support for virus prediction, control and prevention.