本文選題：隨機(jī)SIQS傳染病模型 + 李雅普諾夫函數(shù)��； 參考：《暨南大學(xué)》2015年碩士論文

【摘要】：傳染病動(dòng)力學(xué)的研究目的是探尋疾病流行的內(nèi)在原因及影響因素,掌握疾病的傳播原理和規(guī)律,預(yù)測(cè)疾病的發(fā)展趨勢(shì),從而能對(duì)流行病的防范與消除提供理論依據(jù)。眾所周知,利用常微分方程建立傳染病的確定性模型已經(jīng)在國(guó)內(nèi)外研究了很多年,得到了大量的研究成果。但確定性傳染病模型沒(méi)有考慮到自然界中存在的隨機(jī)因素,加入白噪聲的隨機(jī)傳染病模型更能反映實(shí)際情況,得到更加準(zhǔn)確的結(jié)果。目前大部分的重點(diǎn)都在SIR,SIRS,HIV等模型的研究上,涉及到將具有隔離項(xiàng)的確定型SIRS模型隨機(jī)化的研究還很少。所以本文在原有SIQS模型基礎(chǔ)上,考慮當(dāng)系統(tǒng)受到隨機(jī)因素影響時(shí),建立隨機(jī)SIQS模型并對(duì)模型進(jìn)行動(dòng)力學(xué)行為研究。本文在確定性SIRS傳染病模型中加入隨機(jī)擾動(dòng)項(xiàng)得到了隨機(jī)傳染病模型,重點(diǎn)研究了隨機(jī)SIQS傳染病模型的動(dòng)力學(xué)行為。論文首先介紹了一類(lèi)具有隔離項(xiàng)的SIQS傳染病模型,得出了模型的疾病基本再生數(shù)??0、無(wú)病平衡點(diǎn)E0及地方病平衡點(diǎn)E?的數(shù)學(xué)表達(dá)式。再此基礎(chǔ)上研究了系統(tǒng)受噪聲影響的隨機(jī)SIQS傳染病模型解的漸近行為。通過(guò)構(gòu)造合理的李雅普諾夫函數(shù)證明了該隨機(jī)模型全局正解的存在惟一性。通常確定性SIRS傳染病模型存在無(wú)病平衡點(diǎn)和地方病平衡點(diǎn),但相應(yīng)的隨機(jī)SIQS傳染病模型不再具有上述平衡點(diǎn)。因此,論文在一定的假設(shè)條件下進(jìn)一步證明,當(dāng)基本再生數(shù)R01時(shí),隨機(jī)SIQS傳染病模型的全局正解關(guān)于確定性SIQS模型的無(wú)病平衡點(diǎn)具有漸近性質(zhì),該性質(zhì)表明疾病將最終消失;當(dāng)E01時(shí),隨機(jī)SIQS傳染病模型的全局正解關(guān)于確定性模型地方病平衡點(diǎn)具有漸近性質(zhì),該性質(zhì)意味著疾病將流行且最終形成地方病。論文最后通過(guò)數(shù)值模擬仿真例子驗(yàn)證了本文結(jié)論的正確性。
[Abstract]:The purpose of the study on dynamics of infectious diseases is to explore the internal causes and influencing factors of disease prevalence, to master the principles and laws of disease transmission, to predict the trend of disease development, and to provide theoretical basis for the prevention and elimination of epidemic diseases. It is well known that the deterministic model of infectious diseases established by ordinary differential equations has been studied for many years and a great deal of research results have been obtained. But the deterministic infectious disease model does not take into account the random factors that exist in nature. The stochastic infectious disease model with white noise can reflect the actual situation and get more accurate results. At present, most of the emphasis is on the research of SIRSIRSU HIV and other models, and there are few researches on randomization of deterministic Sirs models with isolated terms. Therefore, based on the original SIQS model, the stochastic SIQS model is established and the dynamic behavior of the model is studied when the system is affected by random factors. In this paper, the stochastic infectious disease model is obtained by adding the stochastic perturbation term into the deterministic Sirs infectious disease model, and the dynamic behavior of the stochastic SIQS infectious disease model is studied emphatically. In this paper, we first introduce a class of SIQS infectious disease models with isolation term, and obtain the disease basic regeneration number of the model, the disease-free equilibrium point E0 and the endemic equilibrium point E0? The mathematical expression of. On this basis, the asymptotic behavior of the solution of stochastic SIQS infectious disease model affected by noise is studied. The existence and uniqueness of the global positive solution of the stochastic model are proved by constructing a reasonable Lyapunov function. Usually, the deterministic Sirs epidemic model has disease-free equilibrium and endemic equilibrium, but the corresponding stochastic SIQS infectious disease model no longer has the above equilibrium. Therefore, under certain assumptions, it is further proved that the global positive solution of stochastic SIQS infectious disease model has asymptotic property on deterministic SIQS model when the basic reproduction number is R01, which indicates that the disease will eventually disappear. When E01, the global positive solution of stochastic SIQS infectious disease model has asymptotic property about the endemic equilibrium point of deterministic model, which means that the disease will be prevalent and eventually form endemic disease. Finally, a numerical simulation example is given to verify the correctness of the conclusion.
【學(xué)位授予單位】：暨南大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類(lèi)號(hào)】：O175

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