本文選題：計(jì)算機(jī)病毒　切入點(diǎn)：SIR模型　出處：《安徽師范大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：計(jì)算機(jī)病毒總是伴隨著計(jì)算機(jī)的使用而不斷出現(xiàn),它對(duì)人類生活的影響也日益顯著.雖然我們可以通過(guò)各種方法將出現(xiàn)的病毒殺死,但是對(duì)于病毒的擴(kuò)散卻難以預(yù)防.為了控制計(jì)算機(jī)病毒的傳播,許多專家和學(xué)者對(duì)它的傳播機(jī)制進(jìn)行了研究.因?yàn)橛?jì)算機(jī)病毒的傳播規(guī)律與生物病毒的傳染性比較相似,所以我們可以采用傳染病模型的原理來(lái)描述計(jì)算機(jī)病毒傳播的方式,并對(duì)計(jì)算機(jī)病毒傳播中的某些現(xiàn)象給出合理的解釋,這有助于我們更好地理解計(jì)算機(jī)病毒的傳播機(jī)制,以便于高效地預(yù)防和控制計(jì)算機(jī)病毒的傳播.我們主要研究了一類均質(zhì)空間下具有不同傳播模式的計(jì)算機(jī)病毒SIR模型,其中S表示計(jì)算機(jī)中的易感個(gè)體的密度,I表示已感染個(gè)體的密度,R表示修好恢復(fù)個(gè)體的密度.我們將在模型建立過(guò)程中介紹傳染病動(dòng)力學(xué)中的相關(guān)概念,并對(duì)模型所涉及的問(wèn)題進(jìn)行了系統(tǒng)的研究,給出了 S、I、R這三個(gè)變量的長(zhǎng)時(shí)間變化規(guī)律,也即是相應(yīng)的傳播動(dòng)力學(xué)結(jié)論.本文主要由下面五個(gè)部分組成:第一章具體介紹了與本文研究有關(guān)的背景知識(shí)、文獻(xiàn)來(lái)源和現(xiàn)有文獻(xiàn)中已取得的成果,我們將結(jié)合計(jì)算機(jī)病毒的特點(diǎn),考慮不同因素,構(gòu)建出具有不同傳播方式的SIR模型.在這些因素中,我們不僅考慮了病毒的空間擴(kuò)散性,而且注意到被感染區(qū)域隨時(shí)間變化這一特征,使得我們構(gòu)建的模型更符合病毒在現(xiàn)實(shí)中的傳播規(guī)律.在第二章中,我們用與空間變量無(wú)關(guān)的常微分方程系統(tǒng)來(lái)描述SIR模型,不僅計(jì)算出所構(gòu)建模型的閾值R0—傳染病動(dòng)力學(xué)中的一個(gè)重要指標(biāo),而且根據(jù)R0的取值范圍探討了無(wú)病平衡點(diǎn)和染病平衡點(diǎn)的局部穩(wěn)定性.在第三章中,我們考慮病毒的傳播不僅與時(shí)間有關(guān),而且與空間也有關(guān),于是我們將空間擴(kuò)散性這一因素加入到模型中,利用具齊次Neumann邊界條件的反應(yīng)—擴(kuò)散方程組來(lái)描述病毒的傳播規(guī)律.在該模型中,我們依然圍繞閾值R0的大小來(lái)探討無(wú)病平衡點(diǎn)和染病平衡點(diǎn)的穩(wěn)定性情況,既得到了平衡點(diǎn)局部穩(wěn)定性的結(jié)論,也結(jié)合其它條件得到了平衡點(diǎn)全局穩(wěn)定性的結(jié)論.在第四章中,我們引入了自由邊界條件,該因素的考慮源于被病毒感染的區(qū)域隨時(shí)間變化這一事實(shí).與前兩章的閾值R0為常數(shù)這一結(jié)論不同的是,具自由邊界條件的SIR模型的閾值是與時(shí)間有關(guān)的函數(shù)ROF(t),我們同樣運(yùn)用閾值并結(jié)合相關(guān)的條件,給出了病毒能逐漸消退的充分條件.為了使我們所獲得的理論結(jié)果更具體形象,我們將在第五章中對(duì)文中的部分結(jié)論進(jìn)行數(shù)值模擬,可以進(jìn)一步證實(shí)我們的理論結(jié)果.同時(shí),我們也將對(duì)全文不同傳播模式下的SIR模型所獲得的傳播動(dòng)力學(xué)行為給出傳染病學(xué)解釋,分析出相關(guān)傳染病學(xué)參數(shù)的不同取值對(duì)病毒傳播所起到的決定性作用,也意味著人們對(duì)這些參數(shù)的重視,將有利于控制計(jì)算機(jī)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.
【學(xué)位授予單位】：安徽師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175

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