摘要
本文根据一些关于健康的理论研究成果及目前人们的健康状况,建立了健康状况系统模型,并研究了健康状况随时问以一定概率而发生的各种转化,通过对系统模型的分析并结合特定人群的健康状况及转化情况,不仅可以具体说明对采取相应措施建议的有效程度,还可以预测一段时间后的健康状况,因此,具有重大的应用价值和实践意义。本文的主要工作如下:
建立了关于不同人群健康状况系统的数学模型。根据某一地区或某一特定人群的健康状况,将整个人群分为健康人群、亚健康人群、患轻病人群和患重病人群四种类型。假设每一类型人群以一定概率向其他类型转化,且满足一定的转化关系,在此基础上,建立了以随机现象为基本点,基于随机变化,且随时间演化的特定人群健康状况系统的数学模型。
研究了所建系统模型解的适定性以及非负解的存在性。由于该系统模型中既含有常微分方程,又含有偏微分方程,并含有积分项,因此求解方程比较困难。为此,在将其转化为抽象发展方程的基础上,利用线性算子半群理论,证明了该系统模型解的存在性和唯一性。
研究了所建系统模型解的正保守性与稳定性。利用实Banach格理论及半群生成理论证明了系统正解的存在性及正保守性质;通过算子的特征值,相应特征子空间的维数,相应根子空间的维数,证明了系统定态解的存在性;通过算子的谱分析以及预解式估计,特别是预解式沿虚轴的估计,证明了在一定条件下,系统的动态解渐近收敛于系统的定态解。
研究了所建系统模型的定态解及其随各参数的变化情况。给出了定态解的求解过程,并逐一分析了系统定态解随各参数的变化情况,得出了健康人群、亚健康人群、患轻疾病人群、患重疾病人群这四种人群的分布概率,以及与各个参数之间的关系。
研究了特定人群的健康状况及转化分析。对具有典型代表的白领人群、农民工人群进行了现状分析,给出相应的改进措施和建议。
Taking into account the sort of given public health status, in this dissertation, proposes a mathematical model with time for all types of health status and their conversion among the types.
In this dissertation, one aspect of the public health problem-the status of healthy distribution is studied. According to a particular region or a particular state of health of the crowd under consideration, the people are divided into crowd-four types: healthful group; sub-healthful group; insignificant illness group and serious illness group. With these separated groups, it discusses the conversion of all types of relations among the crowd group, and establishes a mathematical model for this system under certain assumptions, which is described by coupled equations of ordinary and partial differential equations. Using linear operator semigroup theory, the dissertation shows that this system denoted by the model established is well-posed and associates with a positive C_0 semigroup in a Hilbert sate space. Hence the positive solution of system exists.
The dissertation analyzes the stability of this system and conservatism of its solution. According to the theory of Banach lattice and semigroup, and shows that the stable solution of system exists and give the expression of it, and the dynamic solution to system converges asymptotically to the stable one under some conditions.
The dissertation studies the stable solution to the system. First of all, it gives process of calculating the solution to system. Then from mathematical theorems, it analyzes the state of the system with the changes of parameters so as to obtain the probability distribution of these four crowds given above, as well as the relationship between the distribution of crowds and the various parameters.
Finally, the dissertation indicates some examples (officers' crowd and workers' crowd) and have an analysis about them so as to study the specific conditions in our life. By analyzing some correlative data, it gets some conclusion of them and gives some advices in order to improve the status of these crowds. Furthermore, the dissertation shows that these advices can be effective to improving them.
引文
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