【摘要】： 傳染病模型的漸近行為已經(jīng)被很多人研究。通常情況下,基本再生數(shù)是決定疾病流行與否的閾值。如果它小于1,無病平衡點(diǎn)是全局穩(wěn)定的且疾病滅絕;如果它大于1,正平衡點(diǎn)是全局穩(wěn)定的且發(fā)展為地方病。在這種情況下,從無病平衡點(diǎn)到正平衡點(diǎn)引起的分支是向前的。近年來,由于社群具有不同的感染性、非線性發(fā)生率和年齡結(jié)構(gòu)等原因,許多關(guān)于傳染病模型的論文發(fā)現(xiàn)了后向分支。在這種情況下,基本再生數(shù)不能完全描述疾病消除的效應(yīng),而這種效應(yīng)能被轉(zhuǎn)向點(diǎn)的關(guān)鍵參數(shù)描述,得到控制疾病的閾值對(duì)于確認(rèn)后向分支是重要的�；诖�,本文中我們研究了具有飽和治療函數(shù)的傳染病模型,通過數(shù)學(xué)分析和數(shù)值模擬主要得到以下結(jié)論: 1.當(dāng)感染者治療延滯的效應(yīng)弱時(shí),基本再生數(shù)是控制疾病的強(qiáng)閾值。當(dāng)感染者治療延滯的效應(yīng)強(qiáng)時(shí),后向分支將發(fā)生,對(duì)于消除疾病來說基本再生數(shù)小于1是不足的。 2.當(dāng)后向分支發(fā)生時(shí),轉(zhuǎn)向點(diǎn)關(guān)鍵值是控制疾病的新閾值。 3.當(dāng)基本再生數(shù)減少到一定程度時(shí),無病平衡點(diǎn)是全局穩(wěn)定的。 4.數(shù)學(xué)結(jié)果表明給病人及時(shí)的治療、提高治療率,和減少傳染的協(xié)同因素對(duì)控制疾病是有效的。 最后,我們結(jié)合前人的一些工作,提出了今后努力的方向。
[Abstract]:The asymptotic behavior of infectious disease models has been studied by many people. In general, the basic number of reproduction is the threshold to determine the prevalence of the disease. If it is less than 1, the disease-free equilibrium is globally stable and the disease is extinct; if it is greater than 1, the positive equilibrium is globally stable and develops into endemic disease. In this case, the branches from the disease-free equilibrium to the positive equilibrium are forward. In recent years, due to the different infectivity, nonlinear incidence and age structure of community, many papers on infectious disease models have found backward branches. In this case, the basic regeneration number can not completely describe the effect of disease elimination, but this effect can be described by the key parameters of the turning point. It is important to obtain the threshold of disease control for confirming the backward branch. Based on this, we study the infectious disease model with saturation treatment function, and get the following conclusions by mathematical analysis and numerical simulation: 1. When the effect of treatment delay is weak, the number of basic regeneration is a strong threshold for disease control. When the effect of treatment delay is strong, the backward branch will occur, and the number of basic regeneration less than 1 is insufficient to eliminate the disease. 2. When the backward branch occurs, the critical value of the turning point is a new threshold for disease control. 3. When the number of basic regenerations is reduced to a certain extent, the disease-free equilibrium is globally stable. 4. The mathematical results show that timely treatment, improved treatment rate, and reduction of infectious co-factors are effective in disease control. Finally, combined with some previous work, we put forward the direction of future efforts.
【學(xué)位授予單位】：蘭州大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2009
【分類號(hào)】：R181.3

【參考文獻(xiàn)】

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1 李建全;馬知恩;周義倉;;GLOBAL ANALYSIS OF SIS EPIDEMIC MODEL WITH A SIMPLE VACCINATION AND MULTIPLE ENDEMIC EQUILIBRIA[J];Acta Mathematica Scientia;2006年01期

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本文編號(hào)：2331530