本文選題：計算機(jī)病毒　切入點：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】：計算機(jī)病毒總是伴隨著計算機(jī)的使用而不斷出現(xiàn),它對人類生活的影響也日益顯著.雖然我們可以通過各種方法將出現(xiàn)的病毒殺死,但是對于病毒的擴(kuò)散卻難以預(yù)防.為了控制計算機(jī)病毒的傳播,許多專家和學(xué)者對它的傳播機(jī)制進(jìn)行了研究.因為計算機(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描述計算機(jī)病毒傳播的方式,并對計算機(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算機(jī)病毒的傳播機(jī)制,以便于高效地預(yù)防和控制計算機(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算機(jī)病毒SIR模型,其中S表示計算機(jī)中的易感個體的密度,I表示已感染個體的密度,R表示修好恢復(fù)個體的密度.我們將在模型建立過程中介紹傳染病動力學(xué)中的相關(guān)概念,并對模型所涉及的問題進(jìn)行了系統(tǒng)的研究,給出了 S、I、R這三個變量的長時間變化規(guī)律,也即是相應(yīng)的傳播動力學(xué)結(jié)論.本文主要由下面五個部分組成:第一章具體介紹了與本文研究有關(guān)的背景知識、文獻(xiàn)來源和現(xiàn)有文獻(xiàn)中已取得的成果,我們將結(jié)合計算機(jī)病毒的特點,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)實中的傳播規(guī)律.在第二章中,我們用與空間變量無關(guān)的常微分方程系統(tǒng)來描述SIR模型,不僅計算出所構(gòu)建模型的閾值R0—傳染病動力學(xué)中的一個重要指標(biāo),而且根據(jù)R0的取值范圍探討了無病平衡點和染病平衡點的局部穩(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的大小來探討無病平衡點和染病平衡點的穩(wěn)定性情況,既得到了平衡點局部穩(wěn)定性的結(jié)論,也結(jié)合其它條件得到了平衡點全局穩(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)域隨時間變化這一事實.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模型的閾值是與時間有關(guān)的函數(shù)ROF(t),我們同樣運用閾值并結(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文中的部分結(jié)論進(jìn)行數(shù)值模擬,可以進(jìn)一步證實我們的理論結(jié)果.同時,我們也將對全文不同傳播模式下的SIR模型所獲得的傳播動力學(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)的不同取值對病毒傳播所起到的決定性作用,也意味著人們對這些參數(shù)的重視,將有利于控制計算機(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)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175

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