【摘要】：傳染病模型是生物數(shù)學(xué)研究的主要內(nèi)容,運(yùn)用傳染病動(dòng)力學(xué)知識(shí),建立傳染病數(shù)學(xué)模型,并進(jìn)行數(shù)值模擬,得到傳染病的傳播規(guī)律,分析傳染病爆發(fā)和流行的主要原因,從而找到預(yù)防傳染病的最好方法。本文的主要研究?jī)?nèi)容如下:首先,建立了一類具有CTL免疫的乙肝病毒模型,研究分析了該模型的平衡點(diǎn)的動(dòng)態(tài)穩(wěn)定性,利用譜半徑的方法求出基本再生數(shù)R_0。當(dāng)R_0≤1時(shí),通過(guò)構(gòu)造Lyapunov函數(shù),利用Lassalle不變性原理驗(yàn)證了系統(tǒng)無(wú)病平衡點(diǎn)的局部穩(wěn)定性;當(dāng)R_0＞1時(shí),研究分析了系統(tǒng)地方病平衡點(diǎn)的局部漸近穩(wěn)定性。再通過(guò)選取恰當(dāng)?shù)膮��?shù)進(jìn)行數(shù)值模擬驗(yàn)證了理論結(jié)果。其次,研究了一類具有時(shí)滯和飽和發(fā)生率的乙肝病毒模型,考慮到感染細(xì)胞的恢復(fù)率,分析確定了疾病是否流行的閾值R_0,通過(guò)構(gòu)造Lyapunov函數(shù),利用Lassalle不變集原理證明了當(dāng)R_01時(shí),對(duì)于任意時(shí)滯,系統(tǒng)在無(wú)病平衡點(diǎn)處是全局漸近穩(wěn)定的;當(dāng)R_01時(shí),分析了地方病平衡點(diǎn)的局部漸近穩(wěn)定性。再通過(guò)選取恰當(dāng)?shù)膮��?shù)進(jìn)行數(shù)值模擬驗(yàn)證了理論結(jié)果。最后,研究了一類帶有接種的非線性發(fā)生率的傳染病模型,分析了該模型的平衡點(diǎn)的動(dòng)態(tài)穩(wěn)定性,得到了疾病流行與否的閾值R_0。假設(shè)所有輸入者都是易感者,當(dāng)R_01時(shí),通過(guò)構(gòu)造Lyapunov函數(shù),驗(yàn)證了無(wú)病平衡點(diǎn)的全局漸近穩(wěn)定性;當(dāng)R_01時(shí),利用Huwitz判據(jù)證明了地方病平衡點(diǎn)的局部漸近穩(wěn)定性。再通過(guò)選取恰當(dāng)?shù)膮��?shù)進(jìn)行數(shù)值模擬驗(yàn)證了理論結(jié)果。
[Abstract]:The infectious disease model is the main content of the biological mathematics research. By using the knowledge of infectious disease dynamics, the mathematical model of the infectious disease is established, and the numerical simulation is carried out to obtain the law of the spread of the infectious disease, and the main reasons for the outbreak and epidemic of the infectious disease are analyzed. To find the best way to prevent infectious diseases. The main contents of this paper are as follows: firstly, a hepatitis B virus model with CTL immunity is established, and the dynamic stability of the equilibrium point of the model is analyzed. The basic regenerative number R _ S _ 0 is obtained by the method of spectral radius. The local stability of the disease-free equilibrium point of the system is verified by constructing the Lyapunov function when R _ S _ 0 鈮，

本文編號(hào)：2294677