一类离散SIS传染病模型的动力学性态分析

发布时间：2018-03-11 11:54

本文选题：SIS传染病模型　切入点：基本再生数　出处：《陕西科技大学》2017年硕士论文　论文类型：学位论文

【摘要】：随着传染性疾病的大肆流行,人们关于传染病的研究越来越多。传染病模型主要分为连续模型和离散模型两大类。因为传染病模型的数据大多采用离散时间,因此离散传染病模型的描述更为合理。离散传染病模型的求解问题是传染病模型研究过程中的重中之重。关于离散传染病模型的研究主要集中于模型的平衡性,持久性,分支理论。在大量传染病模型研究的基础上,本文构造出一类具有指数型发生率的离散SIS传染病模型。这类指数型发生率的离散SIS传染病模型是在仓室理论基础上建立的。所研究的人群分为易感者和染病者。通过一定的控制措施,疾病的传播速度降低,疾病的发生率可调控为指数型发生率。因为易感者是通过上一刻的染病者被传染的,因此疾病的发生率也与上一时刻的染病者数量存在密不可分的关系。根据离散SIS传染病模型的主要的研究内容和研究方向,本文主要的工作和内容如下:首先,在精确分析和合理假设的情况下,建立了一个具有指数型发生率的离散SIS传染病模型。讨论了这个具有指数型发生率的离散SIS传染病模型的平衡点的稳定性,分析了该模型的持久性,通过稳定性理论,得到了模型无病平衡点的全局渐近稳定性,以及有病平衡点的局部渐近稳定性,运用数值模拟的方法验证了理论成果,并展示了模型动力学性态的复杂性。其次,在已构造具有指数型发生率的模型的基础上,建立了一个具有时滞的离散传染病模型。讨论了这个时滞模型的平衡点的存在性和稳定性,经过分析可知,这个时滞模型在有病正平衡点不稳定情况下出现了flip分支,通过中心流形理论证明了该模型的flip分支是2周期稳定的,最后运用数值模拟验证了该flip分支是2周期稳定的。再次,分析了一个具有饱和恢复率和具有指数型发生率的离散SIS传染病模型。证明了该模型存在一个无病平衡点和两个有病平衡点,并且该模型的无病平衡点是全局渐近稳定的,因为计算的复杂性,采取数值模拟的方法证明了有病平衡点的稳定性,结果显示这个模型在其有病平衡点不稳定时会出现后向分支。最后,改进了一个具有指数型发生率的离散SIS传染病模型,讨论了该模型的无病平衡点和有病平衡点的稳定性。结果显示,其无病平衡点是全局渐近稳定的,其有病平衡点是局部渐近稳定的,当其有病平衡点不稳定时,该模型在有病平衡点处可产生neimark-sacker分支,并通过数值模拟的方法,验证了研究结果。
[Abstract]:With the prevalence of infectious diseases, more and more people are studying infectious diseases. Infectious disease models are divided into two categories: continuous model and discrete model. Therefore, the description of discrete infectious disease model is more reasonable. The solution of discrete infectious disease model is the most important in the process of studying infectious disease model. The research on discrete infectious disease model mainly focuses on the balance and persistence of the model. Branch theory. Based on a large number of infectious disease models, In this paper, a discrete SIS infectious disease model with exponential incidence is constructed. The discrete SIS infectious disease model with exponential incidence is established on the basis of storeroom theory. Through certain controls, The rate of spread of the disease is reduced, and the incidence of the disease can be regulated as an exponential incidence, because the susceptible person is transmitted through the infected person of the last moment. Therefore, the incidence of disease is closely related to the number of infected people at the last moment. According to the main research content and research direction of discrete SIS infectious disease model, the main work and contents of this paper are as follows: first, Under the condition of accurate analysis and reasonable assumption, a discrete SIS infectious disease model with exponential incidence is established. The stability of the equilibrium point of the discrete SIS infectious disease model with exponential incidence is discussed. The persistence of the model is analyzed. The global asymptotic stability of the disease-free equilibrium and the local asymptotic stability of the disease-free equilibrium are obtained by using the stability theory. It also shows the complexity of the dynamic behavior of the model. Secondly, on the basis of the established model with exponential incidence, In this paper, a discrete epidemic model with time delay is established. The existence and stability of the equilibrium point of the model are discussed. The results show that the flip bifurcation of the time-delay model appears under the unstable condition of the diseased positive equilibrium. The flip bifurcation of the model is proved to be 2-period stable by the center manifold theory. Finally, the numerical simulation is used to verify that the flip bifurcation is 2-period stable. A discrete SIS infectious disease model with saturation recovery rate and exponential incidence is analyzed. It is proved that the model has one disease-free equilibrium point and two disease-free equilibrium points, and the disease-free equilibrium point of the model is globally asymptotically stable. Because of the complexity of the calculation, the stability of the diseased equilibrium point is proved by numerical simulation. The results show that the model has backward bifurcation when the ill equilibrium point is unstable. Finally, A discrete SIS infectious disease model with exponential incidence is improved, and the stability of disease-free equilibrium and disease-free equilibrium is discussed. The results show that the disease-free equilibrium is globally asymptotically stable. The diseased equilibrium is locally asymptotically stable. When the diseased equilibrium is unstable, the model can produce neimark-sacker bifurcation at the diseased equilibrium. The results are verified by numerical simulation.
【学位授予单位】：陕西科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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