一类计算机病毒SIR模型的传播动力学

发布时间：2018-03-14 05:17

本文选题：计算机病毒　切入点：SIR模型　出处：《安徽师范大学》2017年硕士论文　论文类型：学位论文

【摘要】：计算机病毒总是伴随着计算机的使用而不断出现,它对人类生活的影响也日益显著.虽然我们可以通过各种方法将出现的病毒杀死,但是对于病毒的扩散却难以预防.为了控制计算机病毒的传播,许多专家和学者对它的传播机制进行了研究.因为计算机病毒的传播规律与生物病毒的传染性比较相似,所以我们可以采用传染病模型的原理来描述计算机病毒传播的方式,并对计算机病毒传播中的某些现象给出合理的解释,这有助于我们更好地理解计算机病毒的传播机制,以便于高效地预防和控制计算机病毒的传播.我们主要研究了一类均质空间下具有不同传播模式的计算机病毒SIR模型,其中S表示计算机中的易感个体的密度,I表示已感染个体的密度,R表示修好恢复个体的密度.我们将在模型建立过程中介绍传染病动力学中的相关概念,并对模型所涉及的问题进行了系统的研究,给出了 S、I、R这三个变量的长时间变化规律,也即是相应的传播动力学结论.本文主要由下面五个部分组成:第一章具体介绍了与本文研究有关的背景知识、文献来源和现有文献中已取得的成果,我们将结合计算机病毒的特点,考虑不同因素,构建出具有不同传播方式的SIR模型.在这些因素中,我们不仅考虑了病毒的空间扩散性,而且注意到被感染区域随时间变化这一特征,使得我们构建的模型更符合病毒在现实中的传播规律.在第二章中,我们用与空间变量无关的常微分方程系统来描述SIR模型,不仅计算出所构建模型的阈值R0—传染病动力学中的一个重要指标,而且根据R0的取值范围探讨了无病平衡点和染病平衡点的局部稳定性.在第三章中,我们考虑病毒的传播不仅与时间有关,而且与空间也有关,于是我们将空间扩散性这一因素加入到模型中,利用具齐次Neumann边界条件的反应—扩散方程组来描述病毒的传播规律.在该模型中,我们依然围绕阈值R0的大小来探讨无病平衡点和染病平衡点的稳定性情况,既得到了平衡点局部稳定性的结论,也结合其它条件得到了平衡点全局稳定性的结论.在第四章中,我们引入了自由边界条件,该因素的考虑源于被病毒感染的区域随时间变化这一事实.与前两章的阈值R0为常数这一结论不同的是,具自由边界条件的SIR模型的阈值是与时间有关的函数ROF(t),我们同样运用阈值并结合相关的条件,给出了病毒能逐渐消退的充分条件.为了使我们所获得的理论结果更具体形象,我们将在第五章中对文中的部分结论进行数值模拟,可以进一步证实我们的理论结果.同时,我们也将对全文不同传播模式下的SIR模型所获得的传播动力学行为给出传染病学解释,分析出相关传染病学参数的不同取值对病毒传播所起到的决定性作用,也意味着人们对这些参数的重视,将有利于控制计算机病毒.
[Abstract]:Computer viruses have always appeared with the use of computers, and their impact on human life has become increasingly significant, although we can kill them in various ways. But it is difficult to prevent the spread of virus. In order to control the spread of computer virus, many experts and scholars have studied its transmission mechanism, because the law of transmission of computer virus is similar to that of biological virus. So we can use the principle of infectious disease model to describe the transmission of computer virus, and give a reasonable explanation of some phenomena in the transmission of computer virus, which is helpful for us to better understand the transmission mechanism of computer virus. In order to prevent and control the spread of computer viruses efficiently and efficiently, we mainly study a class of computer virus SIR models with different transmission modes in homogeneous space. Where S denotes the density of susceptible individuals in a computer, I represents the density of infected individuals and R means the density of individuals repaired to recover. We will introduce the related concepts of infectious disease dynamics in the process of modeling. The problems involved in the model are systematically studied, and the long-time variation law of these three variables is given. This paper mainly consists of the following five parts: the first chapter introduces the background knowledge, the source of the literature and the achievements in the existing literature. We will combine the characteristics of computer viruses and consider different factors to construct SIR models with different modes of transmission. In these factors, we will not only consider the spatial diffusion of viruses. In chapter 2, we use a system of ordinary differential equations independent of spatial variables to describe the SIR model. Not only an important index of threshold R0-infectious disease dynamics of the model is calculated, but also the local stability of disease-free equilibrium point and disease-free equilibrium point is discussed according to the range of R0. In the third chapter, We considered that the spread of the virus was not only related to time, but also to space, so we added the element of spatial diffusion to the model. In this model, the stability of disease-free equilibrium and disease-free equilibrium is discussed by using the reaction-diffusion equations with homogeneous Neumann boundary conditions. In chapter 4th, we introduce the free boundary condition. The consideration of this factor stems from the fact that the region infected by the virus changes over time. In contrast to the conclusion that the threshold R0 is constant in the previous two chapters, The threshold of SIR model with free boundary conditions is a time-related function, ROFFT. We also use the threshold value and combine the relevant conditions to give a sufficient condition that the virus can recede gradually. In order to make our theoretical results more specific, We will simulate some of the conclusions in Chapter 5th, which can further confirm our theoretical results. At the same time, We will also explain the dynamics of transmission obtained by the SIR model under different transmission modes, and analyze the decisive effect of the different parameters of infectious diseases on the transmission of the virus. Also means that people attach importance to these parameters, will be conducive to the control of computer viruses.
【学位授予单位】：安徽师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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