一类具有性传播的艾滋病模型的研究

发布时间：2018-04-07 00:22

本文选题：HIV/AIDS模型　切入点：地方病平衡点　出处：《新疆大学》2017年硕士论文

【摘要】：传染病对人类的健康和社会发展危害巨大.其中,艾滋病是危害极大的传染病之一,它的传播经常涉及到一些特定群体和一些特定行为.近几年,艾滋病在我国的流行呈现出一些新的变化,性传播已成为主要的传播途径,经男男性接触感染艾滋病的比例显著上升.(参见文献[1])由于目前,我国感染艾滋病的男女比例不同,并且在一次性接触中男性和女性被传染艾滋病的概率也不同.将男女分开考虑将男女分开考虑,研究异性和同性(男男)间的性接触对艾滋病传播的影响,从而有针对性的制定控制策略是很有必要的.第一部分是引言,介绍了艾滋病模型的研究现状、研究背景、研究目的和研究意义,第二部分是预备知识,第三部分研究了三个艾滋病模型,模型的主要内容如下:首先,我们考虑一类具有性传播的艾滋病模型,其中我们把健康人口和染病人口根据其性别分为不同的仓室,其目的在于研究异性和同性(男男)之间性接触对艾滋病动力学的影响.接下来对模型的一些基本性质进行研究,包括模型解的正性、有界性和平衡点的存在性.其次利用谱半径的方法给出了疾病消亡或持续存在的阈值-基本再生数R0.模型无病平衡点和唯一正平衡点的全局渐近稳定性分别利用比较方法和几何方法讨论.得到了当R1时,无病平衡点是全局渐近稳定;当R1时,地方病平衡点是全局渐近稳定的.最后利用MATLAB对模型进行数值模拟,并且给出一些对比图来验证结论的正确性.第二个模型是具有染病者输入的艾滋病模型,这个模型是我们在第一个模型的基础上考虑了染病人口迁移的情况.首先给出了模型的一些基本假设,研究了模型的一些基本性质,包括模型解的正性、有界性和地方病平衡点的存在性.其次利用谱半径的方法给出了疾病消亡或持续存在的阈值-基本再生数R0.利用比较方法和几何方法分别研究模型无病平衡点和地方病平衡点的全局稳定性.之后通过数值模拟仿真来验证结论的正确性.最后我们把模型推广为有限的经济成本下的最优控制模型.引入控制模型的目的是,为了制定一系列预防投入策略,优化资源分配.其中我们把宣传教育、检测筛选作为控制艾滋病的主要控制策略.利用Pontriagon最大原理证明了最优控制的存在性,最后用四阶的龙格-库塔方法对控制模型进行了数值模拟,对比分析不同投入策略下的模拟效果图.第三个模型研究了一类带有一般发生率的HIV/AIDS模型,模型具有不同潜伏阶段.首先我们研究了模型的基本性质,证明了模型解的正性和有界性,从而给出了系统解的正向不变集.其次利用零点定理和下代矩阵的方法,分别证明了地方病平衡点的存在性,给出了疾病消亡或持续存在的阈值-基本再生数R0.然后构造了一个正定的Liapunov函数,用LaSalle不变原理证明了地方病平衡点的全局渐近稳定性,最后取发生率为双线性发生率,使用MATLAB对模型进行数值模拟,验证了理论的正确性。
[Abstract]:Infectious disease to human health and social development of great harm. Among them, AIDS is infectious disease of great harm, it spread often involves some specific groups and some specific behavior. In recent years, showing some new changes of AIDS epidemic in China, sexual transmission has become the main route of transmission was the rise of the MSM HIV infection ratio. (cf. [1]) due to the current our country, different HIV infection probability and the proportion of men and women, men and women are in sexual contact in the spread of AIDS is also different. Men and women will be considered separately for men and women will be considered separately, of heterosexual and homosexual (male) of the the contact and influence on the spread of AIDS, so as to develop control strategy is very necessary. The first part is the introduction, introduces the research status, the AIDS model research background, research purpose and The significance of the research, the second part is the preliminary knowledge, the third part of the three AIDS model, the main content are as follows: firstly, we consider a class of models with the spread of AIDS, which we put healthy population and infected population is divided into different bins according to their gender, the aim is to study (male heterosexual and homosexual male) between the contact effect on HIV dynamics. Then research on some basic properties of the model, including the positive solutions of the model, boundedness of solutions and existence of equilibrium point. Secondly the persistent disease. Using the method of the spectral radius of the threshold - the basic reproduction number R0. model of the disease-free equilibrium and the the only positive equilibrium point of the global asymptotic stability using the comparison method and the geometric method has been discussed. When R1, the disease-free equilibrium is globally asymptotically stable; when R1, the endemic equilibrium is globally Asymptotically stable. Finally the use of MATLAB to simulate the correctness of the model, and some comparison chart to verify the conclusion. The second model is a model of AIDS infected input, the model we consider with patient migration in the first model is based on the first model was given some. The basic assumption of the model, some basic properties, including positive solutions of the model, boundedness of solutions and existence of the endemic equilibrium. The global stability and then a persistent disease. Using the method of the spectral radius of the threshold - the basic research model respectively the disease-free equilibrium and the endemic equilibrium point comparison method the number of R0. and geometric methods. Through numerical simulation to verify the correctness of the conclusion. Finally we extend our model for the optimal cost control model under the limited lead. The control model is designed in order to develop a series of preventive investment strategy, optimize the allocation of resources. We put the publicity and education, screening as the main control strategy for AIDS control. By using the Pontriagon maximum principle to prove the existence of the optimal control, finally using Runge Kutta method of four order control model by numerical simulation comparative analysis of the simulation, effects of different investment strategies under the map. The third model is studied for a class of general incidence rate of HIV/AIDS model with models with different latent phase. First we study the basic properties of the model, the model is proved and the boundedness of the solution, and gives the solution of the system positive invariant set using the zero point theorem and method. Then the next generation matrix, we obtain the existence of endemic equilibrium, given disease. Persistent threshold - the basic reproduction number R0. then A positive definite Liapunov function is constructed. The global asymptotic stability of the endemic equilibrium is proved by the LaSalle invariance principle. Finally, the bilinear incidence rate is taken, and the MATLAB is used to simulate the model, which verifies the correctness of the theory.

【学位授予单位】：新疆大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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