几类分数阶传染病模型的动力学分析

发布时间：2018-06-13 15:42

本文选题：分数阶向量比较原理 + Lyapunov函数　；参考：《东南大学》2017年硕士论文

【摘要】：本论文主要研究Caputo意义下的分数阶传染病模型。第一章介绍了研究分数阶传染病模型的重要意义以及国内外的研究概况,并列出了本文研究所需要的预备知识。第二章首先给出了分数阶向量比较原理及相关的稳定性理论的证明;其次,给出了判断分数阶稳定性的新充分条件。在第二节,在前人的基础上,把分数阶标量比较原理往向量比较原理进行推广,该方法在研究分数阶微分系统的稳定性上有着重要的意义。第三节,主要研究了当0α1和1α2时,非线性分数阶微分系统平衡点稳定性的一些结论;当0α1时,利用广义的Granwall不等式和相关引理给出了非线性分数阶微分系统平衡点全局渐近稳定的充分条件;当1α2时,给出了非线性分数阶微分系统平衡点全局渐近稳定的新充分条件。第三章主要研究了一类具有CTL免疫响应的分数阶HIV-1模型的动力学行为。首先,在整数阶HIV-1模型基础上,建立了相应的分数阶模型;其次,得到了模型的阈值参数,并利用Lyapunov函数和相关的稳定性理论来分析分数阶模型平衡点的稳定性。当R_01时,无病平衡点全局渐近稳定;当R1R_0时,CTL无免疫激活平衡点全局渐近稳定;R1时,CTL免疫激活平衡点全局渐近稳定。最后,研究了分数阶传染病模型的最优控制问题。第四章主要研究了疫苗具有自然减弱和不完全免疫的SVIRS模型。首先,讨论了后向分支产生的原因,并给出了疾病灭绝的阈值Rvc;其次,在讨论原系统平衡点的稳定性时,根据原系统建立了相应的标准系统和极限系统,通过构造一系列合适的Lyapunov函数,给出了平衡点稳定性的相关结论。最后,利用数值模拟验证了理论结果。第五章对本文的研究工作作出总结与展望。
[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 伪 1 and 1 伪 2, when 0 伪 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 伪 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.
【学位授予单位】：东南大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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