本文選題：分?jǐn)?shù)階向量比較原理 + Lyapunov函數(shù)��； 參考：《東南大學(xué)》2017年碩士論文

【摘要】：本論文主要研究Caputo意義下的分?jǐn)?shù)階傳染病模型。第一章介紹了研究分?jǐn)?shù)階傳染病模型的重要意義以及國(guó)內(nèi)外的研究概況,并列出了本文研究所需要的預(yù)備知識(shí)。第二章首先給出了分?jǐn)?shù)階向量比較原理及相關(guān)的穩(wěn)定性理論的證明;其次,給出了判斷分?jǐn)?shù)階穩(wěn)定性的新充分條件。在第二節(jié),在前人的基礎(chǔ)上,把分?jǐn)?shù)階標(biāo)量比較原理往向量比較原理進(jìn)行推廣,該方法在研究分?jǐn)?shù)階微分系統(tǒng)的穩(wěn)定性上有著重要的意義。第三節(jié),主要研究了當(dāng)0α1和1α2時(shí),非線性分?jǐn)?shù)階微分系統(tǒng)平衡點(diǎn)穩(wěn)定性的一些結(jié)論;當(dāng)0α1時(shí),利用廣義的Granwall不等式和相關(guān)引理給出了非線性分?jǐn)?shù)階微分系統(tǒng)平衡點(diǎn)全局漸近穩(wěn)定的充分條件;當(dāng)1α2時(shí),給出了非線性分?jǐn)?shù)階微分系統(tǒng)平衡點(diǎn)全局漸近穩(wěn)定的新充分條件。第三章主要研究了一類具有CTL免疫響應(yīng)的分?jǐn)?shù)階HIV-1模型的動(dòng)力學(xué)行為。首先,在整數(shù)階HIV-1模型基礎(chǔ)上,建立了相應(yīng)的分?jǐn)?shù)階模型;其次,得到了模型的閾值參數(shù),并利用Lyapunov函數(shù)和相關(guān)的穩(wěn)定性理論來(lái)分析分?jǐn)?shù)階模型平衡點(diǎn)的穩(wěn)定性。當(dāng)R_01時(shí),無(wú)病平衡點(diǎn)全局漸近穩(wěn)定;當(dāng)R1R_0時(shí),CTL無(wú)免疫激活平衡點(diǎn)全局漸近穩(wěn)定;R1時(shí),CTL免疫激活平衡點(diǎn)全局漸近穩(wěn)定。最后,研究了分?jǐn)?shù)階傳染病模型的最優(yōu)控制問(wèn)題。第四章主要研究了疫苗具有自然減弱和不完全免疫的SVIRS模型。首先,討論了后向分支產(chǎn)生的原因,并給出了疾病滅絕的閾值Rvc;其次,在討論原系統(tǒng)平衡點(diǎn)的穩(wěn)定性時(shí),根據(jù)原系統(tǒng)建立了相應(yīng)的標(biāo)準(zhǔn)系統(tǒng)和極限系統(tǒng),通過(guò)構(gòu)造一系列合適的Lyapunov函數(shù),給出了平衡點(diǎn)穩(wěn)定性的相關(guān)結(jié)論。最后,利用數(shù)值模擬驗(yàn)證了理論結(jié)果。第五章對(duì)本文的研究工作作出總結(jié)與展望。
[Abstract]:In this paper, the fractional infectious disease model in Caputo sense is studied. The first chapter introduces the significance of the fractional infectious disease model and the research situation at home and abroad, and lists the preparatory knowledge needed in this paper. In the second chapter, the comparison principle of fractional order vectors and the related stability theory are proved, and a new sufficient condition for judging fractional order stability is given. In the second section, the principle of fractional scalar comparison is extended to the principle of vector comparison on the basis of predecessors. This method is of great significance in studying the stability of fractional differential systems. In the third section, we mainly study some conclusions on the stability of equilibrium point of nonlinear fractional differential systems when 0 偽 1 and 1 偽 2, when 0 偽 1, The sufficient conditions for the global asymptotic stability of the equilibrium point of nonlinear fractional differential systems are given by using the generalized Granwall inequality and the relevant Lemma, and a new sufficient condition for the global asymptotic stability of the equilibrium points of nonlinear fractional differential systems is given when 1 偽 2. In chapter 3, the kinetic behavior of a fractional HIV-1 model with CTL immune response is studied. Firstly, based on the integral HIV-1 model, the fractional order model is established. Secondly, the threshold parameters of the model are obtained, and the stability of the equilibrium point of the fractional model is analyzed by using Lyapunov function and the relevant stability theory. It is found that the disease-free equilibrium is globally asymptotically stable when R _ S _ 1 is present, and that when R _ 1R _ 0 is zero, the global asymptotic stability of CTL is not immune activation equilibrium and that of CTL is globally asymptotically stable at R _ 1. Finally, the optimal control problem of fractional infectious disease model is studied. In chapter 4, the SVIRS model of vaccine with natural weakening and incomplete immunity was studied. Firstly, the cause of backward bifurcation is discussed, and the threshold of disease extinction is given. Secondly, when the stability of the equilibrium point of the original system is discussed, the corresponding standard system and limit system are established according to the original system. By constructing a series of suitable Lyapunov functions, some conclusions on the stability of the equilibrium point are given. Finally, the theoretical results are verified by numerical simulation. Chapter five summarizes and prospects the research work of this paper.
【學(xué)位授予單位】：東南大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175

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本文編號(hào)：2014529