2013年数模美赛论文

地球健康的网络建模

摘要

随着当今科学技术的快速发展，地球的生态和环境压力正在变的越来越大，但是却很少有准确有信服力的数学模型来验证这个观点，所以全社会都迫切地希望建立一个模型来准确的描述和预测地球健康。

为了方便模型的建立，我们选择以EHI（earth health index）作为衡量地球压力的指标，EHI的决定因素包括无机环境和有机生物两个部分，其中无机环境部分我们选择了地球大气状况、水资源及土地资源作为研究对象，有机生物则研究地球上的物种状况。由网络的相关概念，我们选择以不同地区作为网络节点，并将全球分为12个网络节点，每个网络节点即地区选择具有代表性的国家为研究对象.通过研究12个网络节点的EHI进而得出全球的EHI。

研究12个网络节点时，我们选该地区４个决定性因素的数据做研究，通过相关关系即可得出该地区的EHI，进而即可得出全球，得到历年全球EHI的数据后，我们用Matlab做仿真模拟得出曲线图，并可找到模拟函数，通过计算出模拟函数的系数后，即可得到该模拟函数，由模拟函数，我们即可模拟出未来几年内全球EHI的情况，并且成功找到了该函数及全球压力的临界状态，另外如果某地区条件变化引起全球变化，该模型模型能为此提供预警。

最后，我们分别从自然因素和社会因素两个方面对的到的模型进行测试。分别选择了马尔代夫和中国作为测试对象，测试结果与实际数据吻合良好，表明该模型可以准确模拟全球生态和环境压力状况。

关键词：EHI，网络节点，仿真模拟

１．问题重述

我们能否利用地球健康的本地或地区性参数，建立一个全球模型，来预测潜在的状态变化，以政策对地球健康的潜在影响为基础，帮助政策制定者设计出有效的政策。尽管许多警示性的信号已经出现，没人知道地球是否已经在全球范围内逼近了临界点，以及这样一个极端的状态是否是不可避免的。

要求完成以下任务：

问题1: 通过确定影响地球健康某一方面的本土因素（网络节点），并且通过恰当的方式把这些节点关联起来（网络链接）进行关系和属性影响的跟踪分析，在此基础上建立一个动态的全球网络模型。由于这些影响是动态变化的，因此重要的一点是，该模式必须包括能够对该健康状况未来发展势头进行预测的动态时间因素。比如，你的节点可能是国家，大陆，海洋，栖息地，等因素或以上几个因素的结合，这些因素共同构成一个全球模型。你的链接可能代表随时间变化而变化的节点影响或环境影响，或物理因素（比如污染）的走势。你所参照的地球健康因子可以是包括人口、生物、环境、社会、政治、物理和（或）化学在内的地球状态的任何要素。一定要明确定义你的模型中的所有要素，并且解释你在建模过程中界定网络建构考量因素、节点实体和链接特性的科学依据。如果有足够的数据的话，确定一个方法来设定参数和解释你如何测试你所建立的模型。什么类型的数据可以用来证明或验证你所建立的模型的有效性？（注意：如果你缺乏必要的数据来确定参数或进行验证，请不要否决你的模式。你的导师确信，在这个阶段，有创意的想法和理论和有充分数据支持验证的模型同样重要。）请在模型中考虑人的因素，并解释人类的行为和政府的政策会在什么方面影响你的模型结果。

问题2：运行你的模型，观察它是如何预测未来地球的健康的。对通常从数据中中确定出的参数，你可能需要进行评估。

2.相关概念

为了建立一个合适需求的模型，首先，我们要建立一个叫做EHI（Abbreviation of earth health index）的指数来描述这个问题。

2.1 EHI的决定因素和相关指数

2.2 选择以上因素的原因

1.地球包含有机生物和无机环境，以上两部分对EHI都有非常重要的影响。

2.生物圈是地球上最大的生态系统，它有大气、水圈和岩石圈组成。前三个决定因素可以全面的代表无机环境。

3.物种是地球上有机生命存在的客观形式，因此物种状况可以最准确地衡量地球有机生命的状态。

4.为了实现EHI的定性计算，我们选择以上四个相关指数来代表EHI的决定因素。

3.全球网络模型的建立

3.1 EHI的计算方法

3.1.1特定地区EHI计算

假设我们要计算全球的EHI,我们应该先选择一个地方的EHI进行计算，然后我们可以选择多个特定地区的EHI来代表全球的EHI.

（一个特定的地方，特定的时间）

}

d GEF

c API

b IWS

a FAR

= =

通过这些因素之间的相互关系，局部地区的EHI可以有上图中的有四条边组成的四边形的面积描述，每条边代表一种指数。指数的大小可以由边的长短来衡量。

通过原理图的图解

)/2

m(P

)

EHI

(local)

3.1.2 全球EHI计算

了解每个地方的EHI，我们开始连接节点来构建E HI的全球网络模型，我们从全球选择12个节点相连。对节点的描述如下：

选择网络连接节点的原则

1.通过地理划分来选择节点

2.每个节点必须准确地描述出特定地区的EHI

3.所选地区要保持相对平衡

4.组成的网络要能够最大化的覆盖全球

设定

1.用字母i来标示每一个节点

2.每个地区的面积=Ai

3.每个节点的局部EHI=Si；

4.全球的EHI的三维图表如下

通过计算原则，全球的EHI 可以用上图的体积来描述。

∑∑==?=12112

)

/()(i i Ai Ai Si S

用最终结果“S ”来表示，全球EHI 就很明显了。人们使用这个指数就可以来判断全球的健康状况。

全球网络模型可以处理各种各样的变化因素，包括自然因素和社会因素。不管那种因素变化，都可能造成这四种决定因素的变化。当基本因素变化时，EHI 的最终结果会变成一个新的数据，由此就能反应地球健康水平的变化。

4.全球网络模型的求解

全球网络模型可以用来预测未来地球健康状况的发展。使用这个模型可以了解EHI 的发展趋势，步骤如下：

我们用一些国家的数据来描述12个节点的数据.结果如下：

FAR

IWS

用以上方法,我们可以计算每个网络节点的EHI，计算结果如下：

利用以上数据，我们可以绘制每个网络节点的EHI图，结果如下：

这是每个网络节点的局部EHI图形，由此，我们可以很容易的计算出每年全球的EHI，相关的数据和图表如下：

数据

图形

为了预测EHI的趋势，我们用MATLAB软件来做仿真模拟。

我们用现有的数据来找到一个合适的函数，由软件仿真结果可知，这个函数如下：

))(E/(t )

/(2Csin B At S(t)F M D t +?+++=π

注释：

１．在公式中，字母A 、B 、C 、D 、E 、F 、M 都是系数。２．所有的系数都由具体节点的复杂程度来描述。３．12个系数可以由一个确定的矩阵来描述。 A : p1=[ 0.002329 0.002236 0.001331 0.001339 -0.003669 -0.006524

0.000973 0.001474 0.001561 0.0004794 0.001311 0.005169 ];

B : p2=

[ -4.167 -4.043 -2.604

-2.358 7.922 13.64

-1.176 -2.478 -2.65

-0.4775 -2.162 -10.11 ];

C: p3=[ -0.002182 0.001647 -2.91e-006 0.009545 0.007509 -0.001384

-0.0005095 0.0007418 5.447e-005

0.001176 -0.005093 -0.008302 ];

D: p4= [ 2.8 2.8 3.4

3 2.8 2 .8

2.4

3.6 3.6

2.8 2.4 2.8 ];

M: p5=[ 1.549 1.59 0.9853

1.57 1.413 1.744

1.705

2.012 1.168

1.672

2.32 1.749 ];

E: p6=[ 5531 3.08 -9.827

1.46 1

2.05 4334

-5.738 4670 4507

13.09 1.558 4732 ];

F: p7=[ 3257 -2001 0.3918

0.8031 -1993 2584

-2001 2009 1991

-2000 -2003 3357 ];

趋势图如下：

我们可以得到全球EHI的函数系数

通过全球EHI函数，我们可以得到其未来发展趋势图。

从上述图表，我们得出：

1．这个曲线可以看做一种特殊的阻尼振荡。

2．纺锤式振荡的曲线可以用一个时间函数来描述它的解析公式为：

p(x)=0.0006675x - 0.8887

公式中(x +E) /F 可以用来描述函数的振幅，由以上图表可知，这个因素基

本保持不变，并且

))

(

)

sin(F

C+

几乎不不减弱，所以很

明显振幅基本上保持不变。

由函数的特征，我们可以预测全球健康的未来趋势。

1.在很长的时间内，EHI缓慢增长，所以我们可以预测地球的健康状况基本保持在一个比较稳定的水平，保持振荡平衡。

2.如果负面的因素变得更加严重，那么系数Ａ将会变成负数，同时EHI将震荡衰减。当A=0时，它将达到临界状态。假如这种情况发生，振幅会突然增加。如果函数图像接近纺锤形，将会达到严重的临界状态。如果超过临界状态，任何措施都将无济于事，地球必然走向灭亡。

5.模型测试

为了测试这个模型，我们需要分析每个节点的连接情况。当地球上某个地区的自然环境或社会环境发生变化，EHI 将会受到影响。我们需要讨论局部因素和全球EHI 的关系，结果如下：

当局部因素发生变化时，它首先会影响到四个决定因素的变化。

决定因素：i P

决定因素的变化量：i P ? EHI 的初始值：0S EHI 的瞬时值：Ｓ

)(i i P P m S ?+=

0S S S -=?

由于每一种影响因素都是离散的，这些因素的影响将被分为两种变量，即为时间增量和空间增量。

通过微观物质理论和量子力学理论，我们可以对模型做如下讨论：

起点Ａ的坐标：ξ

随机点B 的坐标：x Ａ和Ｂ间距离：ξ-x

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