空间异质环境中SIS传染病模型若干问题研究
发布时间：2018-07-30 06:22
【摘要】：传染病一直伴随着人类社会的发展.历史上,传染病的不断爆发和传播给人类带来了巨大的灾难.尽管当今社会科学技术持续发展、医疗条件也得到了很大的改善,然而世界卫生组织(WHO)宣称传染病仍是人类健康的最大威胁.因此,人们有必要了解疾病的分布情况、时空传播规律并采取适当的控制策略.自1927年美国数学家Kermack和苏格兰医学家、流行病学家McKendrick构造了著名的SIR“仓室”模型以来,数学模型成为研究疾病传播规律、评估感染风险、优化控制策略的重要工具.早期,研究人员主要研究与空间无关的常微分系统,仅反应随时间推移的动力学特征.为了更加真实地描述现实,研究人员发现空间扩散是影响疾病传播的重要因素.为此许多学者建立了一系列偏微分方程模型分析传染病的动力学性态.近年来,随着研究的更加深入,研究人员逐渐意识到空间扩散和环境异质性在一些传染病传播过程中产生了重要影响,例如:流感、疟疾、西尼罗河病毒等.除此之外,周期性、对流、媒体报道和有限医疗资源配置等在传染病传播中的作用也引起了广泛关注.这篇博士论文主要围绕空间异质性、周期性、对流、非线性恢复率以及非线性发生率对于SIS传染病模型蔓延和消退的影响展开的.本文的主要研究工作组织如下.第一章介绍本文研究主题的一些背景知识和已经取得的最新进展.第二章主要研究异质环境中的一类具自由边界和对流项影响的传染病模型.首先利用抛物方程初边值问题的Lp理论、Zorn引理、压缩映像原理得到全局解的存在唯一性和正性性.接着给出反应扩散系统的基本再生数及其性质,引进自由边界问题风险指标R0F(t)的定义和讨论了其解析性质.借助于风险指标RF(t),通过构造精细上解、下解得到了疾病蔓延和消退的二择一定理,给出了蔓延和消退的判据.并且用半波方法得到了当疾病蔓延时受对流影响的渐近扩张速度.数值模拟给出了对流强度和扩张能力对于染病区域边沿的影响.这些结果与固定区域上传染病模型的动力学性质完全不同.第三章深入探讨了周期异质环境下具自由边界的传染病模型.首先引入基本再生数,并且给出了两种特殊情形下显式表达式.再借助谱半径的方法给出自由边界问题的风险指标R0F(τ),该指标与相应的周期抛物特征问题的主特征值密切相关.利用最大模原理、上下解方法、谱分析以及偏微分方程其它多种技巧证明了疾病蔓延和消退的充分条件.当疾病蔓延发生时,估计了受对流影响的左右不同的渐近扩张速度.最后利用数值模拟给出对流强度、扩散率和扩张能力对疾病传播机理的影响.第四章提出一个异质环境中具媒体报道影响的SIS传染病反应扩散模型.在该模型中,我们用媒体报道影响的因子体现疾病的非线性接触率.首先利用变分法给出异质环境中具媒体报道和扩散影响的基本再生数的定义及其解析性质.接着给出无病平衡点和染病平衡点的存在性,再利用上下解方法、单调迭代序列、经典的半群理论和强极值原理证明了当R0D1时,无病平衡点全局渐近稳定;而当R0D1时,证明了当ds=dI时染病平衡点的全局渐近稳定性.数值模拟表明如果增加媒体报道强度,疾病的感染风险会降低,从而传染病能够得以快速有效地控制.第五章考虑了空间异质环境中一类受有限医疗资源配置影响的具非线性恢复率的SIS传染病模型.探讨了环境异质性、有限医疗资源配置等对于疾病蔓延和消退的影响.首先利用变分方法给出与最大、最小恢复率有关的阈值R0*和R0*及其性质.借助于这两个阈值,以及上下解方法、单调迭代动力学、乘乘减积技巧等证明了无病平衡点和染病平衡点的存在性、唯一性和稳定性.数值模拟表明适当的病床数的配置对于疾病的控制是非常关键的.我们的理论结果对于公共卫生管理部门优化有限医疗资源的配置提供了理论依据.第六章中,我们总结了本文的主要工作,并且在此基础上对今后的研究工作作了进一步的规划.
[Abstract]:Infectious diseases have been accompanied by the development of human society. In history, the continuous outbreak and spread of infectious diseases have brought great disasters to human beings. Although today's social science and technology continue to develop and medical conditions have been greatly improved, the WHO (WHO) claims that infectious diseases are still the greatest threat to human health. It is necessary to understand the distribution of the disease, the law of space-time transmission and the appropriate control strategy. Since the 1927 American mathematician Kermack and the Scotland medical scientist, the epidemiologist McKendrick constructed the famous SIR "warehouse room" model, the mathematical model has become a study of the law of disease transmission, the assessment of the risk of infection, and the optimization of the control strategy. In the early stage, the researchers mainly studied the space independent ordinary differential system, which only responded to the dynamic characteristics of the time lapse. In order to describe the reality more truly, the researchers found that space diffusion was an important factor affecting the spread of disease. In recent years, with the further research, researchers have gradually realized that space diffusion and environmental heterogeneity have played an important role in the transmission of some infectious diseases, such as influenza, malaria, West Nile virus and so on. In addition, periodicity, convection, media coverage, and the allocation of limited medical resources in the transmission of infectious diseases This thesis mainly focuses on the effects of spatial heterogeneity, periodic, convection, nonlinear recovery and nonlinear incidence on the spread and decline of SIS infectious disease model. The main research work of this paper is as follows. In the second chapter, the second chapter mainly studies an infectious disease model with free boundary and convective effects in the heterogeneous environment. First, the existence and uniqueness and the positive nature of the global solution are obtained by using the Lp theory of the initial boundary value problem of the parabolic equation, the Zorn lemma and the compression mapping principle. It introduces the definition and the analytic properties of the risk index R0F (T) of the free boundary problem. By means of the risk index RF (T), by constructing the fine upper solution and the lower solution, the two alternative theorem of the spread and decline of the disease is obtained, and the criterion of the spread and regression is given. Near expansion speed. The numerical simulation gives the effect of convection intensity and expansion ability on the edge of the infected region. These results are completely different from the kinetic properties of the fixed area. The third chapter discusses the infectious disease model with free boundary in the periodic heterogeneous environment. First, the basic regeneration number is introduced, and two is given. An explicit expression under special circumstances. The risk index R0F (tau) of the free boundary problem is given by means of the aid spectrum radius. The index is closely related to the principal eigenvalues of the corresponding periodic parabolic problem. The maximum modulus principle, the upper and lower solutions, the spectral analysis and the other techniques of the partial differential equation prove the spread and decline of the disease. In the fourth chapter, a diffusion model of SIS infectious disease with the influence of media coverage in a heterogeneous environment is proposed. In the model, we use the factor of media coverage to reflect the nonlinear contact rate of the disease. First, we use the variational method to give the definition and the analytic properties of the basic regeneration number with media coverage and diffusion in the heterogeneous environment. Then we give the existence of the disease-free equilibrium point and the equilibrium point of the disease, and then use the upper and lower solutions and the monotone iterative sequence. The classical semigroup theory and the strong extremum principle prove that the disease free equilibrium point is globally asymptotically stable when R0D1, and when R0D1, the global asymptotic stability of the equilibrium point of the disease is proved. The numerical simulation shows that the risk of infection of the disease will be reduced if the media coverage is increased, thus the infectious disease can be controlled quickly and effectively. In the fifth chapter, the SIS infectious disease model, which is affected by the allocation of limited medical resources in the spatial heterogeneity environment, is considered. The effects of environmental heterogeneity and the allocation of limited medical resources on the spread and regression of the disease are discussed. First, the threshold R0* and R0* related to the maximum and minimum recovery rate are given by the variational method. By means of these two thresholds, as well as the method of the upper and lower solutions, the monotone iterative dynamics, the multiplication and multiplication technique, the existence, uniqueness and stability of the disease free equilibrium point and the equilibrium point of the disease are proved. The numerical simulation shows that the allocation of the appropriate number of beds is not essential for the control of the disease. Our theoretical results are for public health. The management department provides a theoretical basis for the optimization of the allocation of limited medical resources. In the sixth chapter, we have summarized the main work of this paper, and on this basis, we made further plans for the future research work.
【学位授予单位】：扬州大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175
本文编号：2154089
[Abstract]:Infectious diseases have been accompanied by the development of human society. In history, the continuous outbreak and spread of infectious diseases have brought great disasters to human beings. Although today's social science and technology continue to develop and medical conditions have been greatly improved, the WHO (WHO) claims that infectious diseases are still the greatest threat to human health. It is necessary to understand the distribution of the disease, the law of space-time transmission and the appropriate control strategy. Since the 1927 American mathematician Kermack and the Scotland medical scientist, the epidemiologist McKendrick constructed the famous SIR "warehouse room" model, the mathematical model has become a study of the law of disease transmission, the assessment of the risk of infection, and the optimization of the control strategy. In the early stage, the researchers mainly studied the space independent ordinary differential system, which only responded to the dynamic characteristics of the time lapse. In order to describe the reality more truly, the researchers found that space diffusion was an important factor affecting the spread of disease. In recent years, with the further research, researchers have gradually realized that space diffusion and environmental heterogeneity have played an important role in the transmission of some infectious diseases, such as influenza, malaria, West Nile virus and so on. In addition, periodicity, convection, media coverage, and the allocation of limited medical resources in the transmission of infectious diseases This thesis mainly focuses on the effects of spatial heterogeneity, periodic, convection, nonlinear recovery and nonlinear incidence on the spread and decline of SIS infectious disease model. The main research work of this paper is as follows. In the second chapter, the second chapter mainly studies an infectious disease model with free boundary and convective effects in the heterogeneous environment. First, the existence and uniqueness and the positive nature of the global solution are obtained by using the Lp theory of the initial boundary value problem of the parabolic equation, the Zorn lemma and the compression mapping principle. It introduces the definition and the analytic properties of the risk index R0F (T) of the free boundary problem. By means of the risk index RF (T), by constructing the fine upper solution and the lower solution, the two alternative theorem of the spread and decline of the disease is obtained, and the criterion of the spread and regression is given. Near expansion speed. The numerical simulation gives the effect of convection intensity and expansion ability on the edge of the infected region. These results are completely different from the kinetic properties of the fixed area. The third chapter discusses the infectious disease model with free boundary in the periodic heterogeneous environment. First, the basic regeneration number is introduced, and two is given. An explicit expression under special circumstances. The risk index R0F (tau) of the free boundary problem is given by means of the aid spectrum radius. The index is closely related to the principal eigenvalues of the corresponding periodic parabolic problem. The maximum modulus principle, the upper and lower solutions, the spectral analysis and the other techniques of the partial differential equation prove the spread and decline of the disease. In the fourth chapter, a diffusion model of SIS infectious disease with the influence of media coverage in a heterogeneous environment is proposed. In the model, we use the factor of media coverage to reflect the nonlinear contact rate of the disease. First, we use the variational method to give the definition and the analytic properties of the basic regeneration number with media coverage and diffusion in the heterogeneous environment. Then we give the existence of the disease-free equilibrium point and the equilibrium point of the disease, and then use the upper and lower solutions and the monotone iterative sequence. The classical semigroup theory and the strong extremum principle prove that the disease free equilibrium point is globally asymptotically stable when R0D1, and when R0D1, the global asymptotic stability of the equilibrium point of the disease is proved. The numerical simulation shows that the risk of infection of the disease will be reduced if the media coverage is increased, thus the infectious disease can be controlled quickly and effectively. In the fifth chapter, the SIS infectious disease model, which is affected by the allocation of limited medical resources in the spatial heterogeneity environment, is considered. The effects of environmental heterogeneity and the allocation of limited medical resources on the spread and regression of the disease are discussed. First, the threshold R0* and R0* related to the maximum and minimum recovery rate are given by the variational method. By means of these two thresholds, as well as the method of the upper and lower solutions, the monotone iterative dynamics, the multiplication and multiplication technique, the existence, uniqueness and stability of the disease free equilibrium point and the equilibrium point of the disease are proved. The numerical simulation shows that the allocation of the appropriate number of beds is not essential for the control of the disease. Our theoretical results are for public health. The management department provides a theoretical basis for the optimization of the allocation of limited medical resources. In the sixth chapter, we have summarized the main work of this paper, and on this basis, we made further plans for the future research work.
【学位授予单位】：扬州大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175
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