本文選題：倉室模型　切入點：基本再生數(shù)　出處：《東南大學(xué)》2016年博士論文

【摘要】：盡管經(jīng)典的疾病傳播動力學(xué)模型在預(yù)測某些具體疾病方面取得了一定的成功,但是它們往往過于簡單且忽視了一些重要的方面,如多階段／多群體、接觸人數(shù)和其它的疾病狀態(tài)等。本文考慮了在復(fù)雜網(wǎng)絡(luò)框架下的病毒和流行病傳播模型,討論了多階段／多群體模型的全局穩(wěn)定性,復(fù)雜網(wǎng)絡(luò)上幾類模型的全局動力學(xué)以及度相關(guān)網(wǎng)絡(luò)上SIR疾病傳播的建模問題。全文共五章。第二章討論耦合網(wǎng)絡(luò)上的多階段／多群體傳染病模型。第三章討論了復(fù)雜網(wǎng)絡(luò)上幾類傳播模型。第四章為基于網(wǎng)絡(luò)連邊的SIR疾病傳播建模問題。在第二章,首先,研究了一個多階段水傳播疾病模型平衡點的存在唯一性及全局穩(wěn)定性,并在此基礎(chǔ)上,進一步提出了一類具有普適性的多階段霍亂傳播模型,在合理的生物學(xué)假設(shè)下,推導(dǎo)了基本再生數(shù),利用全局Lyapunov函數(shù)、Kirchhoff矩陣樹定理和LaSalle不變性原理研究了平衡點的全局穩(wěn)定性。其次,研究了具有間接傳播途徑多群體SEI動物疾病模型的全局動力學(xué)。在合理的生物學(xué)假設(shè)下,推導(dǎo)了模型的基本再生數(shù)并證明了無病平衡點的全局穩(wěn)定性；另一方面,由于加權(quán)有向圖的權(quán)重矩陣是可約的,故結(jié)合全局Lyapunov函數(shù)和Kirchhoff矩陣樹定理,利用了一個新的組合等式來討論地方病平衡點的全局穩(wěn)定性。在第三章,利用比較原理和有向圖中的Kirchhoff矩陣樹定理研究了復(fù)雜網(wǎng)絡(luò)上幾類傳播模型的全局動力學(xué)。首先,討論了一個帶有出生與死亡網(wǎng)絡(luò)水傳播疾病模型的全局動力學(xué)及各種免疫策略對傳播的影響。其次,研究了一個考慮平衡出生與死亡事件的異質(zhì)網(wǎng)絡(luò)中染病期和攜帶期都具有傳染力疾病模型的全局穩(wěn)定性,且當(dāng)不考慮個體出生與死亡時,得到了疾病的最終規(guī)模表達式,并利用數(shù)值模擬比較了不同免疫策略對疾病傳播的影響。最后,基于Lyapunov函數(shù)和Kirchhc off黽陣樹定理,討論了一個基于網(wǎng)絡(luò)的計算機病毒模型有毒平衡點的全局穩(wěn)定性問題,并利用比較原理,給出了無毒平衡點全局漸近穩(wěn)定的一個更簡潔證明。在第四章,首先回顧了配置網(wǎng)絡(luò)上基于連邊的SIR疾病傳播模型和度相關(guān)網(wǎng)絡(luò)上兩個基于節(jié)點的SIR疾病傳播模型,隨后介紹了兩個會導(dǎo)致度相關(guān)性出現(xiàn)的增長網(wǎng)絡(luò)模型。利用連續(xù)時間隨機模擬算法,對比了度相關(guān)網(wǎng)絡(luò)上基于連邊及基于節(jié)點SIR模型的預(yù)測結(jié)果和100次隨機模擬SIR結(jié)果的均值。仿真結(jié)果表明,在度相關(guān)網(wǎng)絡(luò)上,僅利用度分布信息的基于連邊的SIR模型預(yù)測結(jié)果在許多方面如疾病初始指數(shù)增長率、峰值和峰值到達的時間,要優(yōu)于利用度相關(guān)信息的基于節(jié)點的SIR模型預(yù)測結(jié)果,這說明配置網(wǎng)絡(luò)上基于連邊的SIR模型可能具有更廣闊的適用范圍。
[Abstract]:Although the epidemic model of classic and has achieved some success in predicting some specific diseases, but they are often too simple and ignore some important aspects, such as multi stage / multi group, contact number and other disease states. The virus and epidemic spreading model in complex networks under the framework of the discussion the global stability of multi stage / multi population model, the global dynamics of several kinds of models on complex networks and SIR network modeling problem of the spread of the disease. This paper consists of five chapters. The second chapter discusses the multi stage / multi group coupling network epidemic model. The third chapter discusses the propagation models of several complex the fourth chapter is the SIR network. The spread of the disease based on the network modeling problem of edges. In the second chapter, firstly, study the existence and uniqueness of a multi stage model of waterborne disease equilibrium And global stability, and on this basis, put forward a kind of multi stage model of cholera spread universality, in biology reasonable assumption, derived the basic reproduction number, using the global Lyapunov function, global stability theorem and LaSalle invariance principle of equilibrium matrix Kirchhoff tree. Secondly, the global dynamics the study has indirect transmission SEI group of animal disease model in biology. Reasonable assumption, the basic reproduction number derived model and prove the global stability of the disease-free equilibrium; on the other hand, the weighted directed graph weight matrix is reducible, so with the global Lyapunov function and Kirchhoff matrix the tree theorem, using a new combined equation to discuss the global stability of the endemic equilibrium. In the third chapter, using the comparison principle and theorem to the Kirchhoff matrix in the tree graph Several classes of complex network propagation model of global dynamics is studied. Firstly, the influence of birth and death with a network of waterborne disease model of global dynamics and various immunization strategies on propagation are discussed. Secondly, the study has global stability of infectious disease model with a consideration of the balance of birth and death events in different diseases quality of network and carry, and when not considering the individual birth and death, the ultimate expression of the disease scale, and the effect of different immunization strategies for the spread of the disease were compared by numerical simulation. Finally, based on the Lyapunov function and Kirchhc off array Strider tree theorem, we discuss a global stability problem of computer virus model of network based on toxic balance, and using the comparison principle, given no global asymptotic stability of the equilibrium point of a more concise proof. In the fourth chapter, first review The configuration of the network based on the SIR model and the spread of disease related to network even on the edge of the two SIR based models of disease transmission node, then introduces two growth network model leads to correlation appears. Using the continuous time stochastic simulation algorithm, compares the mean edges and prediction results based on SIR model and 100 nodes random simulation of SIR based on the results of correlation network. Simulation results show that the degree of correlation in the network, using only the degree distribution information based on SIR model predictions of edges in many aspects such as the initial disease index growth rate, peak value and peak arrival time is superior to the utilization of information related to the prediction results of SIR model based on node, the network configuration of SIR model on the edges may have a wider applicable range based on.

【學(xué)位授予單位】：東南大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O157.5
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本文編號：1654649