【作者】 王霞；

【导师】 袁朝晖；

【作者基本信息】 湖南大学 ， 应用数学， 2014， 硕士

【摘要】 本学位论文主要研究了有干预措施的年龄结构传染病模型.论文假设对人群采取隔离措施,同时假设处于潜伏期、隔离期以及染病期的患者均具有恢复能力,建立了一类具有年龄结构的传染病模型;其次,考虑到药物治疗对病毒传播的影响,我们建立了一类相应的HIV年龄结构模型.本论文共有三章组成:第一章主要介绍了传染病发展的历史、目前研究现状及研究传染病的意义,并简单介绍了本文所要研究的问题.第二章建立了处在潜伏期、隔离期及染病期的患者均具有恢复能力和传染能力的MSEIQR年龄结构模型,证明了地方病平衡点的存在性及唯一性,并对模型的无病平衡点和地方病平衡点的稳定性进行了分析,得出无病平衡点全局稳定的充分条件和地方病平衡点局部稳定的充分条件.将人口分为几部分建立宏观的传染病模型,进行分析研究疾病的传播和发展.对传染病的防止起到积极的指导作用.第三章研究了一类具有进入酶抑制剂和蛋白酶抑制剂治疗的Holling-Ⅱ型感染率HIV年龄结构模型,得出基本再生数的表达式及其平衡点,证明了当R0>1时,感染平衡点的唯一性,并通过构造两个李雅普诺夫泛函分别对无感染平衡点和感染平衡点的稳定性进行了分析.具体地,当R0<1时无感染平衡点E0全局渐近稳定,当R0>1时E0不稳定,此时感染平衡点E全局渐近稳定.根据病毒与免疫细胞之间的相互作用,建立微观传染病动力学模型,来研究随时间的改变系统状态的变化,可以有效地反映病毒与免疫系统的动态行为.为获得最佳治疗时间提供帮助.更多还原

【Abstract】 In this paper we mainly study the age-structured epidemic model with theintervention. Assuming the isolation measures are taken and the patients haverecovery capability during the incubation, quarantine and infected period, we es-tablish a class of age-structured epidemic models. Secondly, we consider the infu-ence of drug therapy on the spread of the virus and a dynamic model of HIV withage-structured is established.The thesis is composed of three chapters:In the frst chapter, we mainly introduces the history of the development ofinfectious disease and progresses in this feld so far, as well as the signifcance ofstudying the infectious disease, and simply introduce the problems we study in thispaper.In the second chapter, we established the age-structured MSEIQR model withrecovery capability and infectious ability of the patient in latent period, isolationperiod, and infected period, and proved the existence and uniqueness of the endemicequilibrium, we also study the global dynamics of the disease-free equilibrium andthe endemic equilibrium, the sufcient conditions for the global stability of thedisease-free equilibrium and for the local stability of the infected equilibrium areobtained. We build the macroscopic epidemic model by dividing the populationinto several parts, and analysis the spread and development of diseases, that playa positive role in the prevention of infections diseases.In the third chapter, we studied a class of age-structured HIV model withHolling-Ⅱinfection rate, that takes into account the efects of both entry inhibitorsand protease inhibitors is considered, and the basic reproduction number expres-sion and the existence of equilibrium are obtained, we also proved that if R0>1,the infection equilibrium is only, and the infection-free equilibrium and the infec-tion equilibrium are studied by constructing two Lyapunov function respectively.Specifcally, when R0<1, the infection-free equilibrium E0is global asymptotical-ly stable; when R0>1, E0is unstable, and the infection equilibrium E is globalasymptotically stable. According to the interaction between the virus and immunecells, the microscopic dynamic model of infections disease are built to study thechanges of the system state with time changing, the dynamic behavior of the virusand the immune system can be efectively refect. Helping get the best treatment time.更多还原