本文選題：傳染病傳播模型 + 非線性發(fā)生率　； 參考：《中南大學(xué)》2012年碩士論文

【摘要】：傳染病歷來(lái)是人類的大敵,利用動(dòng)力學(xué)方法建立傳染病傳播的數(shù)學(xué)模型,并通過模型對(duì)傳染病進(jìn)行定性和定量的分析與研究已經(jīng)取得一些成果,主要集中在判定、預(yù)測(cè)疾病發(fā)展趨勢(shì)上。本文研究具有非線性發(fā)生率和非定常人口的傳染病傳播網(wǎng)絡(luò)模型。在本文考察的模型中,非定常人口因素使得傳染病傳播模型表現(xiàn)為網(wǎng)絡(luò)大系統(tǒng),非線性發(fā)生率又使得通常的李雅普諾夫分析顯得較為復(fù)雜�；谶@樣的觀察,本文提出一種將穩(wěn)定性判定和平衡點(diǎn)類型判定分開處理的方法,內(nèi)容如下： 1.將協(xié)調(diào)系統(tǒng)理論與狀態(tài)近似估計(jì)方法相結(jié)合,首先運(yùn)用協(xié)調(diào)系統(tǒng)理論整體地判定平衡點(diǎn)的全局漸近穩(wěn)定性,其次使用狀態(tài)近似估計(jì)的方法判定平衡點(diǎn)的類型,即是零平衡點(diǎn)或非零平衡點(diǎn),得到的判據(jù)都用系統(tǒng)參數(shù)明確表達(dá),易于驗(yàn)證。采用狀態(tài)估計(jì)的辦法還便于得到某些定量估計(jì),如定量估計(jì)康復(fù)率對(duì)疾病的影響； 2.將協(xié)調(diào)系統(tǒng)理論與Driessche閾值理論相結(jié)合,首先運(yùn)用協(xié)調(diào)系統(tǒng)理論整體判定平衡點(diǎn)的全局漸近穩(wěn)定性,其次使用Driessche的閾值理論判定平衡點(diǎn)的類型,得到模型的全局閾值條件。此方法避免了構(gòu)造李雅普諾夫函數(shù)帶來(lái)的困難。值得強(qiáng)調(diào)的是,Driessche閡值理論只能得到零平衡點(diǎn)的局部穩(wěn)定判據(jù),但是當(dāng)其與協(xié)調(diào)系統(tǒng)理論結(jié)合時(shí),本文得到的是平衡點(diǎn)的全局穩(wěn)定判據(jù)； 3.本文最后給出上述理論方法的一個(gè)實(shí)際應(yīng)用,即將其應(yīng)用在種群遷移中存在路途感染的兩斑塊模型中,證明模型的全局漸近穩(wěn)定性,給出判定平衡點(diǎn)類型的全局閾值條件。 本文的結(jié)論對(duì)于判定疾病的消失或流行及在流行情況下疾病的控制有理論指導(dǎo)意義。
[Abstract]:Infectious diseases have always been the great enemy of human beings. The mathematical model of infectious disease transmission has been established by using dynamic methods, and some achievements have been achieved in qualitative and quantitative analysis and research of infectious diseases through the model, which is mainly focused on the determination of infectious diseases. Predict the trend of disease development. In this paper, the transmission network model of infectious diseases with nonlinear incidence and unsteady population is studied. In the model investigated in this paper, the unsteady population factors make the infectious disease transmission model behave as the network large-scale system, and the nonlinear incidence makes the common Lyapunov analysis more complicated. Based on this observation, this paper proposes a method to deal with the stability judgment and the equilibrium type decision separately. The contents are as follows: 1. By combining the theory of coordinated systems with the method of approximate state estimation, the global asymptotic stability of equilibrium points is determined by the theory of coordinated systems, and the type of equilibrium points is determined by the method of state approximate estimation. That is, the zero equilibrium point or the non zero equilibrium point, the obtained criteria are clearly expressed by the system parameters, which is easy to verify. The use of state estimation also facilitates the availability of certain quantitative estimates, such as quantitative estimates of the impact of rehabilitation rates on disease; 2. By combining the theory of coordinated system with the theory of Driessche threshold, the global asymptotic stability of equilibrium point is determined by the theory of coordinated system theory, and then the type of equilibrium point is determined by Driessche's threshold theory, and the global threshold condition of the model is obtained. This method avoids the difficulty of constructing Lyapunov function. It is worth emphasizing that Driessche's boundary value theory can only obtain the local stability criterion of zero equilibrium point, but when it is combined with the coordinated system theory, the global stability criterion of equilibrium point is obtained in this paper. 3. In the end of this paper, a practical application of the above theoretical method is given, that is to say, it is applied to the two-patch model with path infection in population migration, the global asymptotic stability of the model is proved, and the global threshold condition for judging the type of equilibrium point is given. The conclusion of this paper is of theoretical significance in determining the disappearance or prevalence of diseases and the control of diseases under epidemic conditions.
【學(xué)位授予單位】：中南大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2012
【分類號(hào)】：O242.1;R311

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本文編號(hào)：1874411