计算机病毒网络传播模型稳定性与控制研究
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摘要
随着信息技术应用越来越多的渗透到工程、商业和社交活动中，计算机病毒的威胁变成了日益重要的关注问题。了解和控制计算机病毒传播、发展规律，继而可以建立计算机病毒的动力学的数学模型。通过对模型的分析和仿真等操作，可以帮助揭示病毒流行的原因，得到计算机病毒的发展变化的规律，进而找到对病毒预防和控制的策略，对于抵御计算机病毒的侵害、维护良好的网络安全和信息安全对于人们来说是非常必要的。
本文的主要工作和研究成果如下：
1.考虑了网络中计算机病毒传播的状态转移过程，在此基础上建立了病毒传播的动力学模型，分析了模型及稳定性问题。为了方便描述病毒的动态特性，引入了“当量日”的概念。通过使用“当量日”的概念，建立起来一个描述计算机病毒的离散时间的数学模型。通过计算，得到建立的数学模型的有病平衡点和无病平衡点。然后，运用Lyapunov第一方法得到无病平衡点稳定性的充分条件，再由圆盘定理得出有病平衡点的充分条件。几个仿真证明了稳定性条件的有效性。
2.研究了带有免疫的计算机病毒传播模型及稳定性问题。首先，建立数学模型来描述计算机网络中带有免疫的病毒传播模型的动力学特性，由计算得出了模型的无病平衡点和有病平衡点。分别求得无病平衡点和有病平衡点稳定的条件。仿真结果表明了所求条件的有效性。
3.研究了网络中存在双病毒模型及稳定性问题。首先，建立离散的数学模型来描述计算机网络中双病毒传播模型的动力学特性，得出了模型的无病平衡点和有病平衡点。分别求得无病平衡点和有病平衡点稳定的条件。仿真结果表明了所求条件的有效性。
4.研究了网络中SEIQR模型及稳定性问题。首先，建立离散的数学模型来描述病毒在网络中动力学特性，得出了模型的无病平衡点和有病平衡点。分别求得无病平衡点和有病平衡点稳定的条件。仿真结果表明了所求条件的有效性。
5.借鉴了经典的HIV动力学模型，建立了带有时滞的离散病毒数学模型，首先给出了数学模型的无病平衡点和有病平衡点。然后求得了数学模型的无病平衡点的渐近稳定性条件。之后得出了有病平衡点渐近稳定的条件。最后仿真结果表明了得到的稳定性条件的有效性。
6.将控制项加入离散计算机病毒模型中，利用经典的求解最优控制律问题得到防控计算机病毒的最优控制律，仿真结果显示了提出的最优控制律的有效性。
With the increasing penetration of information technology (IT)applications to engineering, business and social activities, the threat ofcomputer viruses has become an increasingly important concern for people.Understand and control the spread and development of computer viruses, andestablish mathematical model of computer virus. Reveal the reasons for theprevalence of the virus through the analysis and simulation of the model, andthen find the method of prevention and control strategies for the virus. It isimportant for people to resist computer viruses, well-maintained networksecurity and information security.
The main research results of this paper are as follows:
1. The state transition of computer virus propagation in networks,stability analysis and establishing mathematical model of computer virus areconsidered. In order to describe the dynamic characteristic of the virus, aconcept of “equivalent day” is presented. By using “equivalent day”, amathematical model of discrete-time computer virus is established. Thedisease-free equilibrium and the disease equilibrium are first derived from themathematical model. Then the sufficient conditions of stability for thedisease-free equilibrium are obtained by the first Lyapunov method. And thesufficient conditions of stability for the disease equilibrium are given by disctheorem. Simulation results demonstrate the effectiveness of the stabilityconditions.
2. Stability analysis with vaccination of computer virus model innetworks is discussed. Firstly, establish mathematical model of computervirus. The disease-free equilibrium and the disease equilibrium are firstderived from the mathematical model. Then the sufficient conditions ofstability for the disease-free equilibrium and disease equilibrium are given.Simulation results demonstrate the effectiveness of the stability conditions.
3. Stability analysis of two-type computer viruses model in networks isdiscussed. Firstly, establish discrete-time mathematical model of computervirus. The disease-free equilibrium and the disease equilibrium are firstderived from the mathematical model. Then the sufficient conditions ofstability for the disease-free equilibrium and disease equilibrium are given.Simulation results demonstrate the effectiveness of the stability conditions.
4. Stability analysis of a discrete-time computer SEIQR model innetworks is discussed. Firstly, establish discrete-time mathematical model ofcomputer virus. The disease-free equilibrium and the disease equilibrium arefirst derived from the mathematical model. Then the sufficient conditions ofstability for the disease-free equilibrium and disease equilibrium are given.Simulation results demonstrate the effectiveness of the stability conditions
5. Discrete-time mathematical model of logic bomb computer virus withdelay is established by using HIV dynamic model for reference. Thedisease-free equilibrium and the disease equilibrium are first derived from themathematical model. Then the sufficient conditions of stability for thedisease-free equilibrium and disease equilibrium are given. Simulation resultsdemonstrate the effectiveness of the stability conditions.
6. Control item is added to discrete-time mathematical model ofcomputer virus. The optimal control law of controlling computer virus usingclassical optimal control law is presented. Simulation results demonstrate theeffectiveness of optimal control law.
本文的主要工作和研究成果如下：
1.考虑了网络中计算机病毒传播的状态转移过程，在此基础上建立了病毒传播的动力学模型，分析了模型及稳定性问题。为了方便描述病毒的动态特性，引入了“当量日”的概念。通过使用“当量日”的概念，建立起来一个描述计算机病毒的离散时间的数学模型。通过计算，得到建立的数学模型的有病平衡点和无病平衡点。然后，运用Lyapunov第一方法得到无病平衡点稳定性的充分条件，再由圆盘定理得出有病平衡点的充分条件。几个仿真证明了稳定性条件的有效性。
2.研究了带有免疫的计算机病毒传播模型及稳定性问题。首先，建立数学模型来描述计算机网络中带有免疫的病毒传播模型的动力学特性，由计算得出了模型的无病平衡点和有病平衡点。分别求得无病平衡点和有病平衡点稳定的条件。仿真结果表明了所求条件的有效性。
3.研究了网络中存在双病毒模型及稳定性问题。首先，建立离散的数学模型来描述计算机网络中双病毒传播模型的动力学特性，得出了模型的无病平衡点和有病平衡点。分别求得无病平衡点和有病平衡点稳定的条件。仿真结果表明了所求条件的有效性。
4.研究了网络中SEIQR模型及稳定性问题。首先，建立离散的数学模型来描述病毒在网络中动力学特性，得出了模型的无病平衡点和有病平衡点。分别求得无病平衡点和有病平衡点稳定的条件。仿真结果表明了所求条件的有效性。
5.借鉴了经典的HIV动力学模型，建立了带有时滞的离散病毒数学模型，首先给出了数学模型的无病平衡点和有病平衡点。然后求得了数学模型的无病平衡点的渐近稳定性条件。之后得出了有病平衡点渐近稳定的条件。最后仿真结果表明了得到的稳定性条件的有效性。
6.将控制项加入离散计算机病毒模型中，利用经典的求解最优控制律问题得到防控计算机病毒的最优控制律，仿真结果显示了提出的最优控制律的有效性。
With the increasing penetration of information technology (IT)applications to engineering, business and social activities, the threat ofcomputer viruses has become an increasingly important concern for people.Understand and control the spread and development of computer viruses, andestablish mathematical model of computer virus. Reveal the reasons for theprevalence of the virus through the analysis and simulation of the model, andthen find the method of prevention and control strategies for the virus. It isimportant for people to resist computer viruses, well-maintained networksecurity and information security.
The main research results of this paper are as follows:
1. The state transition of computer virus propagation in networks,stability analysis and establishing mathematical model of computer virus areconsidered. In order to describe the dynamic characteristic of the virus, aconcept of “equivalent day” is presented. By using “equivalent day”, amathematical model of discrete-time computer virus is established. Thedisease-free equilibrium and the disease equilibrium are first derived from themathematical model. Then the sufficient conditions of stability for thedisease-free equilibrium are obtained by the first Lyapunov method. And thesufficient conditions of stability for the disease equilibrium are given by disctheorem. Simulation results demonstrate the effectiveness of the stabilityconditions.
2. Stability analysis with vaccination of computer virus model innetworks is discussed. Firstly, establish mathematical model of computervirus. The disease-free equilibrium and the disease equilibrium are firstderived from the mathematical model. Then the sufficient conditions ofstability for the disease-free equilibrium and disease equilibrium are given.Simulation results demonstrate the effectiveness of the stability conditions.
3. Stability analysis of two-type computer viruses model in networks isdiscussed. Firstly, establish discrete-time mathematical model of computervirus. The disease-free equilibrium and the disease equilibrium are firstderived from the mathematical model. Then the sufficient conditions ofstability for the disease-free equilibrium and disease equilibrium are given.Simulation results demonstrate the effectiveness of the stability conditions.
4. Stability analysis of a discrete-time computer SEIQR model innetworks is discussed. Firstly, establish discrete-time mathematical model ofcomputer virus. The disease-free equilibrium and the disease equilibrium arefirst derived from the mathematical model. Then the sufficient conditions ofstability for the disease-free equilibrium and disease equilibrium are given.Simulation results demonstrate the effectiveness of the stability conditions
5. Discrete-time mathematical model of logic bomb computer virus withdelay is established by using HIV dynamic model for reference. Thedisease-free equilibrium and the disease equilibrium are first derived from themathematical model. Then the sufficient conditions of stability for thedisease-free equilibrium and disease equilibrium are given. Simulation resultsdemonstrate the effectiveness of the stability conditions.
6. Control item is added to discrete-time mathematical model ofcomputer virus. The optimal control law of controlling computer virus usingclassical optimal control law is presented. Simulation results demonstrate theeffectiveness of optimal control law.
引文
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