本文選題：結(jié)核病 + 全局漸近穩(wěn)定性��； 參考：《華中師范大學(xué)》2013年碩士論文

【摘要】：結(jié)核病是一種由結(jié)核桿菌引起的傳染病.結(jié)核病傳染性極強(qiáng),全球每年都會(huì)有數(shù)百萬人死于結(jié)核病.一旦感染結(jié)核病,自身將長期忍受病痛的折磨,甚至?xí)＜吧?因此研究結(jié)核病的發(fā)病機(jī)理,傳播規(guī)律,治療措施等都具有很重要的實(shí)際意義. 數(shù)學(xué)模型為研究結(jié)核病的傳播以及評估醫(yī)療干預(yù)帶來的結(jié)果提供了一種有效的工具.本文討論的是一類雙線性發(fā)生率的結(jié)核病動(dòng)力系統(tǒng)模型.我們利用再生矩陣的方法得到了模型的基本再生數(shù)R0,并且利用構(gòu)造李雅普諾夫函數(shù)的方法,證明了當(dāng)基本再生數(shù)R0≤1時(shí),系統(tǒng)的無病平衡點(diǎn)是全局漸近穩(wěn)定的；利用復(fù)合系統(tǒng)理論,證明了當(dāng)基本再生數(shù)R0＞1時(shí),系統(tǒng)的唯一的流行病平衡點(diǎn)是全局漸近穩(wěn)定的. 本文的組織結(jié)構(gòu)如下：第一章引言,簡單介紹了結(jié)核病相關(guān)的醫(yī)學(xué)背景,以及結(jié)核病動(dòng)力系統(tǒng)模型的主要研究成果.第二章數(shù)學(xué)工具,列舉了本文中相關(guān)的微分方程動(dòng)力系統(tǒng)理論知識.第三章,建立了一個(gè)典型的結(jié)核病傳播SEI模型,通過計(jì)算求得這個(gè)動(dòng)力系統(tǒng)模型的無病平衡點(diǎn),以及這個(gè)系統(tǒng)的的基本再生數(shù)；并且證明了系統(tǒng)的無病平衡點(diǎn)是全局穩(wěn)定的；然后給出了該模型流行病平衡點(diǎn)的存在和穩(wěn)定性條件.第四章對上述模型進(jìn)行了改進(jìn),建立了新的模型,運(yùn)用類似的方法對該模型進(jìn)行了定性分析.第五章,數(shù)值模擬.最后,第六章是文章的結(jié)論.
[Abstract]:Tuberculosis is an infectious disease caused by TB bacilli. Tuberculosis is very infectious. Millions of people die from tuberculosis every year in the world. Once it is infected, it will endure the pain and even endanger life for a long time. Therefore, it is very important to study the pathogenesis of tuberculosis, the law of transmission and treatment, and so on. Significance.
The mathematical model provides an effective tool to study the spread of tuberculosis and to evaluate the results of medical intervention. This paper discusses a class of bilinear incidence of the model of the dynamic system of tuberculosis. We use the method of regeneration matrix to obtain the basic regeneration number R0 of the model, and use the method of constructing Lyapunov function. It is proved that the disease-free equilibrium point of the system is globally asymptotically stable when the basic regeneration number is R0 less than 1. By using the composite system theory, it is proved that the only epidemic equilibrium point of the system is globally asymptotically stable when the basic regeneration number is R0 > 1.
The organizational structure of this paper is as follows: the first chapter, introduction, briefly introduces the medical background of tuberculosis and the main research results of the model of the dynamic system of tuberculosis. The second chapter is a mathematical tool, which enumerates the theoretical knowledge of the dynamic system of differential equations in this paper. The third chapter builds a typical SEI model for tuberculosis transmission. The disease-free equilibrium point of the dynamic system model and the basic regeneration number of the system are calculated, and the disease free equilibrium point is proved to be global stable. Then the existence and stability conditions of the epidemic equilibrium point of the model are given. In the fourth chapter, the above model is improved, a new model is established and the application class is used. The qualitative analysis of the model is carried out in a similar way. The fifth chapter is numerical simulation. Finally, the sixth chapter is the conclusion of the article.
【學(xué)位授予單位】：華中師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2013
【分類號】：R52;O242.1

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本文編號：1967276