中国人口增长预测模型-唐芳等

中国人口增长预测模型

唐芳 何志红 周泽兴

摘要

人口问题是世界上人们最关心的问题之一, 也是制约我国发展的关键因素之一. 随着人口的增长, 人类对自然资源的利用不断扩大和增强, 对生态系统的干扰也日益加深, 人类活动的增长影响着生态系统的平衡, 因此, 弄清人类本身的增长规律, 限制自身的增长速度是十分重要的, 而认识人口数量的变化规律, 建立人口模型, 做出较准确的预测, 是控制人口增长的前提. 所以本文针对我国近年来人口出现的新特点: 老龄化进程加速, 出生人口性别比持续升高, 乡村人口城镇化, 建立相关的人口增长预测模型.

问题1: 我们采用人口预测离散模型[1]即:

(t)(t)ki(t)hi(t)xi(t),

ir1

r2

x0(t)[1u00(t)](t), x1(t1)[1u0(t)]x0(t), ...

xm(t1)[1um1(t)]xm1(t)fm1(t),

其中(t)为t年度中全体育龄妇女生育婴儿总数, 其它符号意义见正文. 该模型选用总和生育率方法进行人口预测, 以2005年作为基础数据, 2020年为预测末年. 用Matlab求解模型可得到预测结果: 2006年、2010年、2020年、2025年的总人口分别为13.148亿、13.466亿、14.058亿、14.101亿. （具体结果见正文表1和表2）

问题2: 我们继续用问题1的模型, 通过调整预测参数, 即人口死亡率和按龄生育率来做长期的人口预测, 我们预测到2055年, 预测结果: 2006年、2010年、2020年、2030年、2040年、2050年的人口总量分别如下: 13.733亿、15.036亿、15.562亿、15.766亿、15.364亿. （具体数据见正文表3和表4） 模型的优点在于: （1）可以知道各年龄阶段人口结构状况, 便于计算负担系数、老少比、少年儿童抚养比、老年人口抚养比、老龄化比率, 乡村人口城镇化比率等. （2）模型还充分考虑到各年龄阶段人口结构, 为国家人口的发展计划及有关人口政策调整提供一个恰当的咨询作用. 缺点在于: 模型假定妇女总和生育率和妇女生育模式不变, 只适用于预测短中期的人口增长数, 预测长期人口总数时, 需要调整预测参数, 预测不够精确.

关键词: 人口发展离散模型 人口预测分析 人口增长模型 总和生育率

一、问题重述

人口是社会经济活动的主体, 人口的发展变动趋势, 对社会经济发展的影响关系极大, 因此人口预测在社会经济实践中占有十分重要的地位. 我国是一个人口大国, 人口问题始终是制约我国发展的关键因素之一. 根据已有数据, 运用数学建模的方法, 对中国人口做出分析和预测是一个重要问题.

近年来我国的人口发展出现了一些新的特点, 例如, 老龄化进程加速、出生人口性别比持续升高, 以及乡村人口城镇化等因素, 这些都影响着我国人口的增长. 2007年初发布的《国家人口发展战略研究报告》还做出了进一步的分析. 关于我国人口问题已有多方面的研究, 并积累了大量数据资料. 附录2就是从《中国人口统计年鉴》上收集到的部分数据. 我们需要解决的问题是: 从中国的实际情况和人口增长的上述特点出发, 参考题目附录中的相关数据（也可以搜索相关文献和补充新的数据）, 建立中国人口增长的数学模型, 并由此对中国人口增长的中短期和长期趋势做出预测;特别要指出模型中的优点与不足之处.

二、问题分析

我国是一个发展中国家, 又是世界上人口最多的国家, 人口问题一直是制约我国经济和社会发展的首要因素, 因此, 能否对人口增长做出比较准确的预测, 对于加速推进我国现代化建设有着极为重要的现实意义. 对于人口增长预测有非常多的方法, 如: Logistic模型, 灰色系统模型[2], 费氏人口预测模型, BP神经网络模型以及本文用到的预测离散模型等等. 该模型选用总和生育率方法进行人口预测, 模型中涉及到人口预测参数的问题, 预测模型一经选定, 对预测参数的科学认定就成为直接关系到预测成果的质量优劣乃至预测成败的关键因素[3], 可见预测参数在人口预测中占有极为重要的地位．我们由离散模型求得预测年份的婴儿出生人数和各年龄的总人数, 对它们求和, 可以算出预测年份的总人口数量. 也可以按市、镇、乡或者性别来分别考虑.

三、模型假设及说明

1、农村人口一旦迁入城镇或城镇化, 其人口行为和特征即与城镇人口 相同, 即忽略城镇人口与迁入城镇人口或城镇化的差别 2、假定预测期内我国的计划生育政策不会有太大变动 3、未来人口的死亡模式和生育模式保持不变 4、妇女总和生育率和妇女生育模式不变

5、迁出和迁入的人口数量一样, 即模型可以不考虑迁入迁出人口

四、符号说明

s(t), z(t), x(t): 市、镇、乡t年度一年中全体育龄妇女生育婴儿总数

s(t), z(t), x(t): 市、镇、乡在t年度育龄妇女的总和生育率

ksi(t), kzi(t), kxi(t): 市、镇、乡i岁人口中女性所占的比例 hsi(t), hzi(t), hxi(t): 市、镇、乡在t年度的妇女生育模式 xsi(t), xzi(t), xxi(t): 市、镇、乡t年度i岁人口数 us00(t), uz00(t), ux00(t): 市、镇、乡第t年的婴儿死亡率

usi(t), uzi(t), uxi(t): 市、镇、乡t年度i岁按龄死亡率

D(t): t年度总死亡人数

r1, r2: 生育年龄的最低和最高年龄

五、模型的建立与求解

5.1问题一的求解（建立我国增长数学模型和短中期预测）

5.1.1预测参数的设定

模型中涉及到的总和生育率(t)hi(t)和生育模式h(t)b(t)/(t)将育龄妇女按龄

ir1r2

生育率规格化, 根据2005年人口普查资料, 先模拟出自然按龄妇女生育模式, 从而得出我国每年的出生人口．2005年的生育模式来源于2005年人口普查数据, 假设未来20年保持这一生育模式不变. 5.1.2 模型的建立

首先我们对附录2的数据用SPSS13.0和Excel处理得到模型中所涉及的参量, 由于题目所给数据仅仅是全国人数的抽样调查, 我们从人口调查报告中得知2005年的全国总人口为13.0756亿, 而抽样的总人口为13985767人, 那么人口抽样比为: =16987567/（13.0756*10^8）= 1.30%, 所以相应的其它参量我们都除以抽样比就得到相应的各年龄的全国人数. 具体模型如下: ⑴ 城市人口预测的离散模型即:

s(t)s(t)ksi(t)hsi(t)xsi(t),

ir1r2

xs0(t)[1us00(t)]s(t), xs1(t1)[1us0(t)]xs0(t), ...

xsm(t1)[1us_m1(t)]xs_m1(t),

其中总和生育率s(t)hsi(t), 生育模式hs(t)bs(t)/Bs(t), 城市死亡人口预测模型为:

ir1

r2

Ds(t)usi(t)xsi(t)(10.1%)t2005us00(t)xs0(t)(10.1%)t2005那么城市总人口的模型为:

i1m

m

Zs(t)xsi(t1)Ds(t1)

i0

⑵ 城镇人口离散预测模型即:

)kztih()tzix()tzi (), z(t)zt(

ir1r2

xz0(t)[1uz00(t)]z(t), xz1(t1)[1uz0(t)]xz0(t), ...

xzm(t1)[1uz_m1(t)]xz_m1(t),

同样的总和生育率z(t)hzi(t), 生育模式hz(t)bz(t)/z(t), 死亡率模型为:

ir1

r2

Dz(t)uzi(t)xzi(t)(10.1%)t2005uz00(t)xz0(t)(10.1%)t2005全国城镇总人数模型为:

i1m

m

Zs(t)xsi(t1)Ds(t1)

i0

⑶ 乡村人口离散预测模型:

)kxtih()txix()tx x(t)xt(i(),

ir1r2

xx0(t)[1ux00(t)]x(t), xx1(t1)[1ux0(t)]xx0(t), ...

xxm(t1)[1ux_m1(t)]xx_m1(t),

类似的总和生育率x(t)hxi(t), 生育模式hx(t)bx(t)/x(t), 死亡率模型为:

ir1

r2

Dx(t)uxi(t)xxi(t)(10.1%)t2005ux00(t)xx0(t)(10.1%)t2005全国城镇总人数模型为:

i1m

m

Zx(t)xxi(t1)Dx(t1)

i0

⑷ 所以第t年全国总人数的预测模型就是（1）（2）（3）模型的和, 即:

W(t)ZS(t)ZZ(t)ZX(t)

5.1.3模型的求解和预测结果分析

我们对附录2的数据用SPSS13.0和Excel处理得到: s(t), z(t), x(t), ksi(t), kzi(t),

kxi(t), hsi(t), hzi(t), hxi(t), xsi(t), xzi(t), xxi(t), us00(t), uz00(t), ux00(t), usi(t), uzi(t), uxi(t)这些参量的具体数据（见附录1）, 用MATLAB求解模型（程序见附录2）预测未

来20年老龄人所占比例和城镇乡村比（见表1, 表2）.

根据上面两表的分析显示我国人口预测具有以下特点:

1．我国人口变化普遍呈现出缓慢上升的趋势（有个别年份出现下降趋势）. 2015、2025年, 我国人口分别达到13.826亿人和14.101亿人.

2．2017年人口出现负增长系数比较大. 2025年也出现了人口负增长.

3．随着我国人口老龄化加剧, 由表数据和（图-1）都可以看出生育妇女急剧的减少, 导致人口增长率下降, 人口将进入负增长阶段.

4. 2005-2025年我国老年人口将增加1.7349亿. 由此可见我国人口老龄化非常严重, 老年人口占总人口的比重将由2005年的13.395％ 分别上升到2015年的19.384％ 和2025年的24.724％.

5.到2015年以后, 人口的老龄化将极其严峻, 且持续时间较长.

6.由此离散预测模型得到的数据, 农村人口城镇化不是很明显, 比率基本上保持在0.8左右, 相对比较稳定.

7.我国人口抚养比呈上升趋势, 劳动力负担系数增加, 劳动力的抚养压力加重

图-1 未来我国育龄妇女（15-49岁）人数预测

图- 2未来我国生育旺盛妇女（20-29）人数预测

通过图-2可知道我国未来生育旺盛妇女人数呈现先上升后下降的趋势. 这也是导致我国人口在20年后出现负增长的原因.

5.2问题二的求解（对我国人口做长期预测）

问题2要求对我国人口做长期预测, 我们继续利用问题1的离散预测模型, 但是随着未来我国医疗卫生和社会福利事业的发展, 婴童死亡率和老年死亡率将会递减, 假定其死亡概率每年递减0．1％ , 以此确定预测期内每一年按年龄死亡的概率, 假设青壮年人口死亡概率衰减为0．调整死亡人口预测模型为:

D(t)ui(t)xi(t)(10.1%)t2005u00(t)x0(t)(10.1%)t2005

i1m

由于要做长期预测, 所以我们在原来的模型中重新设定人口预测参数, 即根据所给数据对模型1的预测参数即死亡率和按龄生育率做调整变化, 用Matlab（程序见2只是数据稍有改动, 模型还是没变化的）求得结果（见表3表4）

图-3我国未来50年总人口预测

图-4未来我国育龄妇女15到49人数预测

由于数据太多, 我们只把部分数据显示在表3和4中（每隔5年的数据）我国总人口数呈先增长后减少的趋势. 老年人口占的比率越来越大, 我国抚养比也呈现上升趋势, 随劳动力负担系数的增加, 劳动力的抚养压力加重, 人口年龄由青年型向壮年型过渡, 由此带来劳动力年龄老化问题, 所以人口快速老龄化是我国人口的最主要的问题, 也是在人口快速增长后控制人口出生过程中必然出现的一个现象, 是一时无法调和的矛盾．而且,

人口惯性是自然现象, 所以这个矛盾将长期存在[5] 因此, 在控制人口数量的同时, 应密切关注人口老龄化带来的问题, 积极研究并寻求延缓和解决之策, 这是今后人口管理工作的重大课题, 也是一个全社会应该关注的系统工程. 5.3模型的检验

此模型可以通过计算2001到2004年的预测数据和实际数据相比较, 得出正确率, 但是本文可以不用此方法. 因为题目给了国家人口发展战略研究报告, 其中也有预测数据, 它的预测结果是总人口在2010年、2020年将分别达到13.6亿人和14.5亿人. 而我们得到的短期预测总人数在2010年、2020年分别达到13.446亿人和14.058亿人, 数据非常的相近;2005年人口总数为13.076亿, 2030年人口数为15.567亿, 30年的时间净增2亿多, 也与报告的预测相吻合;2033年前后人口数为15.5亿左右, 也符合报告预测的人口峰值15亿人左右;由表三可知, 2016年劳动年龄人口为9.9亿, 与题目预测的10.1非常的相近;综上, 说明这个人口离散预测模型是比较有可行性的, 而且可以比较精确的预测短期人口数[6].

六、模型的优缺点分析

6.1模型的优缺点:

离散预测模型的优点在于:

(1)可以知道各年龄阶段人口结构状况, 便于计算负担系数、老少比、少年儿童抚养比、老年人口抚养比、老龄化比率、乡村人口城镇化比率等.

(2)充分考虑到各年龄阶段人口结构, 为国家人口的发展计划及有关人口政策调整提供一个恰当的咨询作用.

(3)对于人口预测本模型比其他模型更注意人口本分的结构,模型中采用到人口的死亡率、出生率、婴儿出生死亡率、妇女生育率等人口的特点去预测未来人口数量, 考虑得比较全面. (4)短中期预测出的人口增长数比较精确. 缺点在于:

(1)模型假定妇女总和生育率和妇女生育模式不变, 只能预测短中期的人口增长数, 对于长期预测还要调整预测参数, 比较麻烦.

(2)这个方程假定人口出生育、各年龄段人口存活率不变. 6.2模型的推广

此模型不仅仅可以预测未来人口的数量, 还可以应用到其他领域, 比如生物领域, 化学领域等等, 例如, 用它可以预测某一区域内某一动物种群的未来某时期的数量, 它可以做为预测我国未来稀有种群动物的数量的一种参考的方法, 从而采取有效措施, 保护我国珍惜动物

参考文献

[1] 王浣尘．人口系统工程[M]．上海: 上海交通大学出版社, 1985

[2] 拉腾图雅, 金良．人I=I预测模型[J]．内蒙古科技与经济, 1999(4): 21—27．

[3] 蔡防．我国人口总量增长与人口结构变化的趋势[J]．中国经贸导刊．2004(13): 29． [4] 求是科技.MATLAB7.0从入门都精通 人民邮电出版社出版发行, 2006年3月. [5] 黄荣清．关于人口预测问题的思考 [J]．人口研究．2003．28(1): 88—90．

[6] 马小红, 侯亚非．北京市未来5O年人口变动趋势预测研究[J]．市场与人门分析, 2004．10(2): 46—49

附录

附录1: （模型所须用到的参量）

2005年各岁人口中女性所占的人口比例（表1）

ksi(2005)

kzi(2005)

kxi(2005)

0.4 0.39 0.34 0.39 0.42 0.43 0.42 0.46 0.47 0.48 0.53 0.52 0.54 0.55 0.58 0.69 0.74 0.79 0.92 0.81 0.74 0.76 0.78 0.94 0.81 0.79 0.85 0.84 0.8 0.88 0.9 0.96 1.02 1.04 1.06 1.13

0.48 0.47 0.44 0.47 0.51 0.54 0.52 0.58 0.59 0.61 0.69 0.66 0.73 0.76 0.83 0.96 0.89 0.82 0.8 0.61 0.51 0.53 0.57 0.72 0.67 0.67 0.74 0.74 0.72 0.81 0.86 0.94 0.97 1.01 1.06 1.14

0.54 0.51 0.51 0.51 0.56 0.58 0.58 0.64 0.66 0.68 0.78 0.74 0.85 0.89 0.98 1.1 0.95 0.78 0.74 0.58 0.53 0.52 0.54 0.62 0.56 0.57 0.57 0.57 0.54 0.61 0.65 0.74 0.77 0.82 0.86 0.98

1.12 0.84 0.94 0.99 1.01 1.22 0.87 0.52 0.68 0.61 0.77 0.83 0.76 0.79 0.8 0.71 0.7 0.59 0.59 0.57 0.49 0.47 0.43 0.4 0.38 0.35 0.36 0.36 0.37 0.32 0.35 0.32 0.32 0.32 0.28 0.28 0.24 0.2 0.21 0.17 0.17

1.15 0.91 1.06 1.06 1.01 1.19 0.87 0.47 0.59 0.53 0.68 0.78 0.71 0.76 0.74 0.67 0.69 0.58 0.57 0.54 0.46 0.46 0.42 0.41 0.36 0.34 0.34 0.32 0.35 0.27 0.3 0.29 0.29 0.29 0.25 0.26 0.22 0.19 0.2 0.17 0.17

1.03 0.87 0.97 0.97 0.91 1.04 0.82 0.45 0.55 0.5 0.67 0.78 0.74 0.8 0.77 0.74 0.74 0.63 0.63 0.59 0.52 0.51 0.47 0.46 0.41 0.39 0.37 0.36 0.38 0.3 0.33 0.32 0.32 0.33 0.29 0.3 0.26 0.24 0.25 0.2 0.2

0.12 0.12 0.1 0.09 0.08 0.07 0.06 0.04 0.12 0.13 0.1 0.09 0.08 0.07 0.06 0.05 0.15 0.16 0.13 0.11 0.1 0.08 0.07 0.05 0.03 0.03 0.02 0.07

hsi(2005)

0.00002 0.00027 0.00115 0.00331 0.01061 0.03131 0.05199 0.0701 0.09085 0.1041 0.10418 0.09975 0.08969 0.07499 0.05936 0.04894 0.03824 0.02924 0.02297 0.01889 0.01345 0.00969 0.00686 0.00487 0.00364 0.00291

0.04 0.04 0.03 0.03 0.02 0.03 0.07

0.08

2005市、镇、乡的妇女生育模式值

hzi(2005)

hxi(2005)

0.00003 0.0001 0.0003 0.00056 0.00178 0.00264 0.00515 0.00821 0.01849 0.02206 0.05339 0.05745 0.08877 0.09121 0.10128 0.09952 0.10916 0.10109 0.11043 0.0951 0.09271 0.08282 0.07739 0.07093 0.06361 0.06071 0.05333 0.05384 0.04366 0.04864 0.03668 0.04184 0.03226 0.03958 0.02704 0.03093 0.02109 0.02474 0.01667 0.0185 0.01198 0.01368 0.00988 0.01029 0.0075 0.00737 0.00484 0.00516 0.00301 0.00369 0.00261

0.00226

0.00133 0.00097 0.00069 0.00082 0.00097 0.00054 0.00056 0.0008

xsi(2005)3114462 3078231 2716077 3042000 3295538 3404154 3259308 3585231 3621462 3766308 4092231 3983615 4164692 4237077 4454385 5214923 5540846 5758154 6518615 5721923 5178692 5251154 5468385 6518615 5685692 5504615 5939231 5903000 5649462 6228923

0.00103 0.00115 0.00103 0.00082 0.00092 0.00101 0.00068 0.00057 0.00036 0.00059 0.00044 0.00042 0.00076 0.00051 0.00049 0.00048

2005年度各年龄对应的人口数

xzi(2005)

xxi(2005)

2327154 8576154 2349538 8359923 2215308 8359923 2349538 8359923 2506231 9080615 2685231 9296846 2550923 9224769 2864231 10233692 2931385 10377846 2976077 10738154 3356538 12179538 3199846 11530923 3535538 13116462 3669769 13620923 3983077 15062308 4564846 16791923 4184462 14485769 3804077 11819231 3692154 11242692 2797077 8720231 2304769 7639231 2237692 7278923 2416692 7423077 2998462 8648231 2797077 7855462 2819462 7855462 3177538 8071692 3177538 7999615 3110385 7711308 3513154

8648231

6880769 7315385 7532615 7749923 8256923 7496462 8075846 6084077 6880769 7315385 7424000 9053692 6446231 3802538 5033846 4490615 5649462 6084077 5468385 5758154 5758154 5070077 4997615 4273308 4237077 4092231 3476615 3367923 3078231 2860923 2679846 2498846 2535000 2571231 2571231 2245308 2462615 2317769 2281538 2245308 1991769

4072538 4274000 4453000 4632000 5079538 4676769 5124308 4027846 4632000 4766231 4475385 5303308 3893538 2058692 2595692 2416692 3088000 3513154 3155154 3446000 3289385 3020846 3065615 2573308 2550923 2461462 2058692 2081077 1902000 1857308 1655846 1544000 1521615 1499231 1544000 1208308 1365000 1320231 1253077 1275462 1096462

10377846 10882308 11530923 12179538 13693000 12828154 14413692 12179538 13620923 13693000 12756077 14774000 11530923 6269923 7855462 7206846 9729231 11386769 10522000 11747154 11314692 10810231 10882308 9224769 9296846 8936462 7711308 7639231 6918538 6846462 6269923 5837538 5549231 5549231 5693385 4540308 5044769 4828615 4684462 4828615 4107923

1702077 1484769 1521000 1195077 1195077 1014000 832923 796692 651846 579462 507000 434538 362154 253538 181077 181077 108615 362154

usi(2005)

6.24047 0.80329 0.33973 0.47179 0.26462 0.28947 0.22467 0.3102 0.2295 0.28308 0.16124 0.19473 0.23548 0.1394 0.35528 0.40333 0.22647 0.26572 0.39311 0.35209

962231 3675462 827923 3315154 872692 3459308 716077 2666538 693692 2666538 604154 2378231 492308 1945846 514692 2017923 402769 1657538 358000 1369308 290923 1225154 268538 936923 201385 792769 156615 576538 134231 432385 89538 360308 67154 288308 223769

792769

2005年度的按龄死亡率

uzi(2005)

uxi(2005)

9.88 16.04832 0.75543 1.67422 0.5 1.22595 0.50286 0.69922 0.34911 0.63222 0.517 0.63798 0.33789 0.5425 0.40672 0.33282 0.17595 0.45167 0.26699 0.54027 0.3474 0.50615 0.24846 0.53513 0.39057 0.48258 0.40122 0.35582 0.20601 0.44684 0.26 0.55086 0.48294 0.78896 0.45929 0.86073 0.66758 1.16821 0.69592

1.3405

0.3409 0.43305 0.31033 0.44783 0.31401 0.36 0.48411 0.46256 0.52953 0.45559 0.49779 0.63307 0.675 0.79271 0.7275 0.9114 0.89865 1.04 1.14968 1.29619 1.09571 1.4856 1.69972 1.90705 1.91791 2.46815 2.14712 2.92804 2.91629 3.14686 2.86912 3.21771 4.05304 4.525 4.37462 4.35593 5.28354 5.71817 7.02671 7.44506 7.91189

0.8383 0.89139 0.57119 0.62976 0.50175 0.66648 17.29859 0.85806 1.15707 0.77 0.72099 0.98277 0.83834 0.98623 1.24652 1.20517 1.3738 1.22322 1.31401 1.66939 2.08635 1.57219 2.195 1.81772 2.1931 2.70685 3.16478 2.95166 2.89532 4.10766 4.09034 4.1577 4.63686 5.5193 6.115 6.35345 6.56 6.84548 7.86988 8.76024 10.40054

1.25059 1.46233 1.17583 1.25706 1.54459 1.26384 1.36243 1.44963 1.72542 1.57442 1.77347 1.88338 1.82138 1.94249 1.89758 2.01213 2.08655 2.19249 2.26122 2.38426 2.24531 2.4442 2.7795 3.12759 3.39431 3.445 3.73889 4.24595 4.16575 4.34018 4.51994 5.27107 5.24172 6.56094 6.42442 7.53282 7.46617 9.01934 9.88135 11.24589 11.00253

10.37029 9.60352 12.60803 12.64258 15.12044 16.785 19.01397 20.67258 21.29655 24.475 28.1434 31.08561 35.01 38.03 40.66879 50.49179 49.99913 57.99182 63.46111 66.66375 78.18571 91.54583 92.684 117.9671 118.754 136.964 153.8 275.672 11.04 12.37358 13.27957 14.64 17.72574 20.85254 22.98518 24.74649 25.81143 28.99451 34.70535 34.35189 42.37282 42.14719 52.04161 51.33333 67.61273 71.55783 70.48778 79.25188 94.65769 102.9825 98.43 96.54429 121.4633 131.1475 148.3933 196.926 14.5174 16.75481 18.16709 18.84048 21.36586 23.69313 28.09754 31.33537 30.81982 39.475 42.04392 44.33261 49.84938 54.16703 57.75189 67.83 75.45222 80.59 86.98435 97.16632 107.8624 116.6731 136.1855 127.59 154.7033 143.666 165.3175 269.2591

附录2:

MATLAB求解程序:

clc;clear; k

=[0.4,0.39,0.34,0.39,0.42,0.43,0.42,0.46,0.47,0.48,0.53,0.52,0.54,0.55,0.58,0.69,0.74,0.79,0.92,0.81,0.74,0.76,0.78,0.94,0.81,0.79,0.85,0.84,0.8,0.88,0.9,0.96,1.02,1.04,1.06,1.13,1.03,1.12,0.84,0.94,0.99,1.01,1.22,0.87,0.52,0.68,0.61,0.77,0.83,0.76,0.79,0.8,0.71,0.7,0.59,0.59,0.57,0.49,0.47,0.43,0.4,0.38,0.35,0.36,0.36,0.37,0.32,0.35,0.32,0.32,0.32,0.28,0.28,0.24,0.2,0.21,0.17,0.17,0.15,0.12,0.12,0.1,0.09,0.08,0.07,0.06,0.04,0.03,0.03,0.02,0.07;

0.48,0.47,0.44,0.47,0.51,0.54,0.52,0.58,0.59,0.61,0.69,0.66,0.73,0.76,0.83,0.96,0.89,0.82,0.8,0.61,0.51,0.53,0.57,0.72,0.67,0.67,0.74,0.74,0.72,0.81,0.86,0.94,0.97,1.01,1.06,1.14,1.06,1.15,0.91,1.06,1.06,1.01,1.19,0.87,0.47,0.59,0.53,0.68,0.78,0.71,0.76,0.74,0.67,0.69,0.58,0.57,0.54,0.46,0.46,0.42,0.41,0.36,0.34,0.34,0.32,0.35,0.27,0.3,0.29,0.29,0.29,0.25,0.26,0.22,0.19,0.2,0.17,0.17,0.15,0.12,0.13,0.1,0.09,0.08,0.07,0.06,0.05,0.04,0.03,0.02,0.07;

0.54,0.51,0.51,0.51,0.56,0.58,0.58,0.64,0.66,0.68,0.78,0.74,0.85,0.89,0.98,1.1,0.95,0.78,0.74,0.58,0.53,0.52,0.54,0.62,0.56,0.57,0.57,0.57,0.54,0.61,0.65,0.74,0.77,0.82,0.86,0.98,0.92,1.03,0.87,0.97,0.97,0.91,1.04,0.82,0.45,0.55,0.5,0.67,0.78,0.74,0.8,0.77,0.74,0.74,0.63,0.63,0.59,0.52,0.51,0.47,0.46,0.41,0.39,0.37,0.36,0.38,0.3,0.33,0.32,0.32,0.33,0.29,0.3,0.26,0.24,0.25,0.2,0.2,0.18,0.15,0.16,0.13,0.11,0.1,0.08,0.07,0.05,0.04,0.03,0.03,0.08]; h=[0.00002,0.00027,0.00115,0.00331,0.01061,0.03131,0.05199,0.0701,0.09085,0.1041,0.10418,0.09975,0.08969,0.07499,0.05936,0.04894,0.03824,0.02924,0.02297,0.01889,0.01345,0.00969,0.00686,0.00487,0.00364,0.00291,0.0019,0.00133,0.00097,0.00069,0.00082,0.00097,0.00054,0.00056,0.0008;

0.00003,0.0003,0.00178,0.00515,0.01849,0.05339,0.08877,0.10128,0.10916,0.11043,0.09271,0.07739,0.06361,0.05333,0.04366,0.03668,0.03226,0.02704,0.02109,0.01667,0.01198,0.00988,0.0075,0.00484,0.00301,0.00261,0.00124,0.00103,0.00103,0.00092,0.00068,0.00036,0.00044,0.00076,0.00049;

0.0001,0.00056,0.00264,0.00821,0.02206,0.05745,0.09121,0.09952,0.10109,0.0951,0.08282,0.07093,0.06071,0.05384,0.04864,0.04184,0.03958,0.03093,0.02474,0.0185,0.01368,0.01029,0.00737,0.00516,0.00369,0.00226,0.00154,0.00115,0.00082,0.00101,0.00057,0.00059,0.00042,0.00051,0.00048];

x=[3114462,3078231,2716077,3042000,3295538,3404154,3259308,3585231,3621462,3766308,4092231,3983615,4164692,4237077,4454385,5214923,5540846,5758154,6518615,5721923,5178692,5251154,5468385,6518615,5685692,5504615,5939231,5903000,5649462,6228923,6410000,6880769,7315385,7532615,7749923,8256923,7496462,8075846,6084077,6880769,7315385,7424000,9053692,6446231,3802538,5033846,4490615,5649462,6084077,5468385,5758154,5758154,5070077,4997615,4273308,4237077,4092231,3476615,3367923,3078231,2860923,2679846,2498846,2535000,2571231,2571231,2245308,2462615,2317769,2281538,2245308,1991769,2028000,1702077,1484769,1521000,1195077,1195077,1014000,832923,796692,651846,579462,507000,434538,362154,253538,181077,181077,108615,362154;

2327154,2349538,2215308,2349538,2506231,2685231,2550923,2864231,2931385,2976077,3356538,3199846,3535538,3669769,3983077,4564846,4184462,3804077,3692154,2797077,2304769,2237692,2416692,2998462,2797077,2819462,3177538,3177538,3110385,3513154,3736923,4072538,4274000,4453000,4632000,5079538,4676769,5124308,4027846,4632000,4766231,4475385,5303308,3893538,2058692,2595692,2416692,3088000,3513154,3155154,3446000,3289385,3020846,3065615,2573308,2550923,2461462,2058692,2081077,1902000,1857308,1655846,1544000,1521615,1499231,1544000,1208308,1365000,1320231,1253077,1275462,1096462,1141231,962231,827923,872692,716077,693692,604154,492308,514692,402769,358000,290923,268538,201385,156615,134231,89538,67154,223769;

8576154,8359923,8359923,8359923,9080615,9296846,9224769,10233692,10377846,10738154,12179538,11530923,13116462,13620923,15062308,16791923,14485769,11819231,11242692,8720231,7639231,7278923,7423077,8648231,7855462,7855462,8071692,7999615,7711308,8648231,9296846,10377846,10882308,11530923,12179538,13693000,12828154,14413692,12179538,13620923,13693000,12756077,14774000,11530923,6269923,7855462,7206846,9729231,11386769,10522000,11747154,11314692,10810231,10882308,9224769,9296846,8936462,7711308,7639231,6918538,6846462,6269923,5837538,5549231,5549231,5693385,4540308,5044769,4828615,4684462,4828615,4107923,4324077,3675462,3315154,3459308,2666538,2666538,2378231,1945846,2017923,1657538,1369308,1225154,936923,792769,576538,432385,360308,288308,792769];

u=[6.24047,0.80329,0.33973,0.47179,0.26462,0.28947,0.22467,0.3102,0.2295,0.28308,0.16124,0.19473,0.23548,0.1394,0.35528,0.40333,0.22647,0.26572,0.39311,0.35209,0.31573,0.3409,0.43305,0.31033,0.44783,0.31401,0.36,0.48411,0.46256,0.52953,0.45559,0.49779,0.63307,0.675,0.79271,0.7275,0.9114,0.89865,1.04,1.14968,1.2

9619,1.09571,1.4856,1.69972,1.90705,1.91791,2.46815,2.14712,2.92804,2.91629,3.14686,2.86912,3.21771,4.05304,4.525,4.37462,4.35593,5.28354,5.71817,7.02671,7.44506,7.91189,9.53014,10.37029,9.60352,12.60803,12.64258,15.12044,16.785,19.01397,20.67258,21.29655,24.475,28.1434,31.08561,35.01,38.03,40.66879,50.49179,49.99913,57.99182,63.46111,66.66375,78.18571,91.54583,92.684,117.9671,118.754,136.964,153.8,275.672;

9.88,0.75543,0.5,0.50286,0.34911,0.517,0.33789,0.40672,0.17595,0.26699,0.3474,0.24846,0.39057,0.40122,0.20601,0.26,0.48294,0.45929,0.66758,0.69592,0.50718,0.8383,0.89139,0.57119,0.62976,0.50175,0.66648,17.29859,0.85806,1.15707,0.77,0.72099,0.98277,0.83834,0.98623,1.24652,1.20517,1.3738,1.22322,1.31401,1.66939,2.08635,1.57219,2.195,1.81772,2.1931,2.70685,3.16478,2.95166,2.89532,4.1766,4.09034,4.1577,4.63686,5.5193,6.115,6.35345,6.56,6.84548,7.86988,8.76024,1.40054,11.28884,11.04,12.37358,13.27957,14.64,17.72574,20.85254,22.98518,24.74649,25.81143,28.99451,34.70535,34.35189,42.37282,42.14719,52.04161,51.33333,67.61273,71.55783,70.48778,79.25188,94.65769,12.9825,98.43,96.54429,121.4633,131.1475,148.3933,196.926;

16.04832,1.67422,1.22595,0.69922,0.63222,0.63798,0.5425,0.33282,0.45167,0.54027,0.50615,0.53513,0.48258,0.35582,0.44684,0.55086,0.78896,0.86073,1.16821,1.3405,1.365,1.25059,1.46233,1.17583,1.25706,1.54459,1.26384,1.36243,1.44963,1.72542,1.57442,1.77347,1.88338,1.82138,1.94249,1.89758,2.01213,2.08655,2.19249,2.26122,2.38426,2.24531,2.4442,2.7795,3.12759,3.39431,3.445,3.73889,4.24595,4.16575,4.34018,4.51994,5.2717,5.24172,6.56094,6.42442,7.53282,7.46617,9.01934,9.88135,11.24589,11.00253,12.72889,14.5174,16.75481,18.16709,18.84048,21.36586,23.69313,28.09754,31.33537,30.81982,39.475,42.04392,44.33261,49.84938,54.16703,57.75189,67.83,75.45222,80.59,86.98435,97.16632,17.8624,116.6731,136.1855,127.59,154.7033,143.666,165.3175,269.2591]; beta=[0.92648;1.278;1.6537]; r1=15; r2=49;

A=k(:,r1:r2).*h.*x(:,r1:r2); pha=beta.*sum(A,2); x(:,1)=(1-u(1)/1000)*pha; y=zeros(3,50); for i=1:50

A=k(:,r1:r2).*h.*x(:,r1:r2); pha=beta.*sum(A,2); x(:,1)=(1-u(:,1)/1000).*pha;

x(:,2:end)=(1-u(:,2:end)/1000).*x(:,1:end-1); y(:,i)=sum(x,2); for l=1:50 d(l)=sum(y(:,l)); end end y; yy=sum(y); plot([2005:2054],yy)

中国人口增长预测模型

唐芳 何志红 周泽兴

摘要

人口问题是世界上人们最关心的问题之一, 也是制约我国发展的关键因素之一. 随着人口的增长, 人类对自然资源的利用不断扩大和增强, 对生态系统的干扰也日益加深, 人类活动的增长影响着生态系统的平衡, 因此, 弄清人类本身的增长规律, 限制自身的增长速度是十分重要的, 而认识人口数量的变化规律, 建立人口模型, 做出较准确的预测, 是控制人口增长的前提. 所以本文针对我国近年来人口出现的新特点: 老龄化进程加速, 出生人口性别比持续升高, 乡村人口城镇化, 建立相关的人口增长预测模型.

问题1: 我们采用人口预测离散模型[1]即:

(t)(t)ki(t)hi(t)xi(t),

ir1

r2

x0(t)[1u00(t)](t), x1(t1)[1u0(t)]x0(t), ...

xm(t1)[1um1(t)]xm1(t)fm1(t),

其中(t)为t年度中全体育龄妇女生育婴儿总数, 其它符号意义见正文. 该模型选用总和生育率方法进行人口预测, 以2005年作为基础数据, 2020年为预测末年. 用Matlab求解模型可得到预测结果: 2006年、2010年、2020年、2025年的总人口分别为13.148亿、13.466亿、14.058亿、14.101亿. （具体结果见正文表1和表2）

问题2: 我们继续用问题1的模型, 通过调整预测参数, 即人口死亡率和按龄生育率来做长期的人口预测, 我们预测到2055年, 预测结果: 2006年、2010年、2020年、2030年、2040年、2050年的人口总量分别如下: 13.733亿、15.036亿、15.562亿、15.766亿、15.364亿. （具体数据见正文表3和表4） 模型的优点在于: （1）可以知道各年龄阶段人口结构状况, 便于计算负担系数、老少比、少年儿童抚养比、老年人口抚养比、老龄化比率, 乡村人口城镇化比率等. （2）模型还充分考虑到各年龄阶段人口结构, 为国家人口的发展计划及有关人口政策调整提供一个恰当的咨询作用. 缺点在于: 模型假定妇女总和生育率和妇女生育模式不变, 只适用于预测短中期的人口增长数, 预测长期人口总数时, 需要调整预测参数, 预测不够精确.

关键词: 人口发展离散模型 人口预测分析 人口增长模型 总和生育率

一、问题重述

人口是社会经济活动的主体, 人口的发展变动趋势, 对社会经济发展的影响关系极大, 因此人口预测在社会经济实践中占有十分重要的地位. 我国是一个人口大国, 人口问题始终是制约我国发展的关键因素之一. 根据已有数据, 运用数学建模的方法, 对中国人口做出分析和预测是一个重要问题.

近年来我国的人口发展出现了一些新的特点, 例如, 老龄化进程加速、出生人口性别比持续升高, 以及乡村人口城镇化等因素, 这些都影响着我国人口的增长. 2007年初发布的《国家人口发展战略研究报告》还做出了进一步的分析. 关于我国人口问题已有多方面的研究, 并积累了大量数据资料. 附录2就是从《中国人口统计年鉴》上收集到的部分数据. 我们需要解决的问题是: 从中国的实际情况和人口增长的上述特点出发, 参考题目附录中的相关数据（也可以搜索相关文献和补充新的数据）, 建立中国人口增长的数学模型, 并由此对中国人口增长的中短期和长期趋势做出预测;特别要指出模型中的优点与不足之处.

二、问题分析

我国是一个发展中国家, 又是世界上人口最多的国家, 人口问题一直是制约我国经济和社会发展的首要因素, 因此, 能否对人口增长做出比较准确的预测, 对于加速推进我国现代化建设有着极为重要的现实意义. 对于人口增长预测有非常多的方法, 如: Logistic模型, 灰色系统模型[2], 费氏人口预测模型, BP神经网络模型以及本文用到的预测离散模型等等. 该模型选用总和生育率方法进行人口预测, 模型中涉及到人口预测参数的问题, 预测模型一经选定, 对预测参数的科学认定就成为直接关系到预测成果的质量优劣乃至预测成败的关键因素[3], 可见预测参数在人口预测中占有极为重要的地位．我们由离散模型求得预测年份的婴儿出生人数和各年龄的总人数, 对它们求和, 可以算出预测年份的总人口数量. 也可以按市、镇、乡或者性别来分别考虑.

三、模型假设及说明

1、农村人口一旦迁入城镇或城镇化, 其人口行为和特征即与城镇人口 相同, 即忽略城镇人口与迁入城镇人口或城镇化的差别 2、假定预测期内我国的计划生育政策不会有太大变动 3、未来人口的死亡模式和生育模式保持不变 4、妇女总和生育率和妇女生育模式不变

5、迁出和迁入的人口数量一样, 即模型可以不考虑迁入迁出人口

四、符号说明

s(t), z(t), x(t): 市、镇、乡t年度一年中全体育龄妇女生育婴儿总数

s(t), z(t), x(t): 市、镇、乡在t年度育龄妇女的总和生育率

ksi(t), kzi(t), kxi(t): 市、镇、乡i岁人口中女性所占的比例 hsi(t), hzi(t), hxi(t): 市、镇、乡在t年度的妇女生育模式 xsi(t), xzi(t), xxi(t): 市、镇、乡t年度i岁人口数 us00(t), uz00(t), ux00(t): 市、镇、乡第t年的婴儿死亡率

usi(t), uzi(t), uxi(t): 市、镇、乡t年度i岁按龄死亡率

D(t): t年度总死亡人数

r1, r2: 生育年龄的最低和最高年龄

五、模型的建立与求解

5.1问题一的求解（建立我国增长数学模型和短中期预测）

5.1.1预测参数的设定

模型中涉及到的总和生育率(t)hi(t)和生育模式h(t)b(t)/(t)将育龄妇女按龄

ir1r2

生育率规格化, 根据2005年人口普查资料, 先模拟出自然按龄妇女生育模式, 从而得出我国每年的出生人口．2005年的生育模式来源于2005年人口普查数据, 假设未来20年保持这一生育模式不变. 5.1.2 模型的建立

首先我们对附录2的数据用SPSS13.0和Excel处理得到模型中所涉及的参量, 由于题目所给数据仅仅是全国人数的抽样调查, 我们从人口调查报告中得知2005年的全国总人口为13.0756亿, 而抽样的总人口为13985767人, 那么人口抽样比为: =16987567/（13.0756*10^8）= 1.30%, 所以相应的其它参量我们都除以抽样比就得到相应的各年龄的全国人数. 具体模型如下: ⑴ 城市人口预测的离散模型即:

s(t)s(t)ksi(t)hsi(t)xsi(t),

ir1r2

xs0(t)[1us00(t)]s(t), xs1(t1)[1us0(t)]xs0(t), ...

xsm(t1)[1us_m1(t)]xs_m1(t),

其中总和生育率s(t)hsi(t), 生育模式hs(t)bs(t)/Bs(t), 城市死亡人口预测模型为:

ir1

r2

Ds(t)usi(t)xsi(t)(10.1%)t2005us00(t)xs0(t)(10.1%)t2005那么城市总人口的模型为:

i1m

m

Zs(t)xsi(t1)Ds(t1)

i0

⑵ 城镇人口离散预测模型即:

)kztih()tzix()tzi (), z(t)zt(

ir1r2

xz0(t)[1uz00(t)]z(t), xz1(t1)[1uz0(t)]xz0(t), ...

xzm(t1)[1uz_m1(t)]xz_m1(t),

同样的总和生育率z(t)hzi(t), 生育模式hz(t)bz(t)/z(t), 死亡率模型为:

ir1

r2

Dz(t)uzi(t)xzi(t)(10.1%)t2005uz00(t)xz0(t)(10.1%)t2005全国城镇总人数模型为:

i1m

m

Zs(t)xsi(t1)Ds(t1)

i0

⑶ 乡村人口离散预测模型:

)kxtih()txix()tx x(t)xt(i(),

ir1r2

xx0(t)[1ux00(t)]x(t), xx1(t1)[1ux0(t)]xx0(t), ...

xxm(t1)[1ux_m1(t)]xx_m1(t),

类似的总和生育率x(t)hxi(t), 生育模式hx(t)bx(t)/x(t), 死亡率模型为:

ir1

r2

Dx(t)uxi(t)xxi(t)(10.1%)t2005ux00(t)xx0(t)(10.1%)t2005全国城镇总人数模型为:

i1m

m

Zx(t)xxi(t1)Dx(t1)

i0

⑷ 所以第t年全国总人数的预测模型就是（1）（2）（3）模型的和, 即:

W(t)ZS(t)ZZ(t)ZX(t)

5.1.3模型的求解和预测结果分析

我们对附录2的数据用SPSS13.0和Excel处理得到: s(t), z(t), x(t), ksi(t), kzi(t),

kxi(t), hsi(t), hzi(t), hxi(t), xsi(t), xzi(t), xxi(t), us00(t), uz00(t), ux00(t), usi(t), uzi(t), uxi(t)这些参量的具体数据（见附录1）, 用MATLAB求解模型（程序见附录2）预测未

来20年老龄人所占比例和城镇乡村比（见表1, 表2）.

根据上面两表的分析显示我国人口预测具有以下特点:

1．我国人口变化普遍呈现出缓慢上升的趋势（有个别年份出现下降趋势）. 2015、2025年, 我国人口分别达到13.826亿人和14.101亿人.

2．2017年人口出现负增长系数比较大. 2025年也出现了人口负增长.

3．随着我国人口老龄化加剧, 由表数据和（图-1）都可以看出生育妇女急剧的减少, 导致人口增长率下降, 人口将进入负增长阶段.

4. 2005-2025年我国老年人口将增加1.7349亿. 由此可见我国人口老龄化非常严重, 老年人口占总人口的比重将由2005年的13.395％ 分别上升到2015年的19.384％ 和2025年的24.724％.

5.到2015年以后, 人口的老龄化将极其严峻, 且持续时间较长.

6.由此离散预测模型得到的数据, 农村人口城镇化不是很明显, 比率基本上保持在0.8左右, 相对比较稳定.

7.我国人口抚养比呈上升趋势, 劳动力负担系数增加, 劳动力的抚养压力加重

图-1 未来我国育龄妇女（15-49岁）人数预测

图- 2未来我国生育旺盛妇女（20-29）人数预测

通过图-2可知道我国未来生育旺盛妇女人数呈现先上升后下降的趋势. 这也是导致我国人口在20年后出现负增长的原因.

5.2问题二的求解（对我国人口做长期预测）

问题2要求对我国人口做长期预测, 我们继续利用问题1的离散预测模型, 但是随着未来我国医疗卫生和社会福利事业的发展, 婴童死亡率和老年死亡率将会递减, 假定其死亡概率每年递减0．1％ , 以此确定预测期内每一年按年龄死亡的概率, 假设青壮年人口死亡概率衰减为0．调整死亡人口预测模型为:

D(t)ui(t)xi(t)(10.1%)t2005u00(t)x0(t)(10.1%)t2005

i1m

由于要做长期预测, 所以我们在原来的模型中重新设定人口预测参数, 即根据所给数据对模型1的预测参数即死亡率和按龄生育率做调整变化, 用Matlab（程序见2只是数据稍有改动, 模型还是没变化的）求得结果（见表3表4）

图-3我国未来50年总人口预测

图-4未来我国育龄妇女15到49人数预测

由于数据太多, 我们只把部分数据显示在表3和4中（每隔5年的数据）我国总人口数呈先增长后减少的趋势. 老年人口占的比率越来越大, 我国抚养比也呈现上升趋势, 随劳动力负担系数的增加, 劳动力的抚养压力加重, 人口年龄由青年型向壮年型过渡, 由此带来劳动力年龄老化问题, 所以人口快速老龄化是我国人口的最主要的问题, 也是在人口快速增长后控制人口出生过程中必然出现的一个现象, 是一时无法调和的矛盾．而且,

人口惯性是自然现象, 所以这个矛盾将长期存在[5] 因此, 在控制人口数量的同时, 应密切关注人口老龄化带来的问题, 积极研究并寻求延缓和解决之策, 这是今后人口管理工作的重大课题, 也是一个全社会应该关注的系统工程. 5.3模型的检验

此模型可以通过计算2001到2004年的预测数据和实际数据相比较, 得出正确率, 但是本文可以不用此方法. 因为题目给了国家人口发展战略研究报告, 其中也有预测数据, 它的预测结果是总人口在2010年、2020年将分别达到13.6亿人和14.5亿人. 而我们得到的短期预测总人数在2010年、2020年分别达到13.446亿人和14.058亿人, 数据非常的相近;2005年人口总数为13.076亿, 2030年人口数为15.567亿, 30年的时间净增2亿多, 也与报告的预测相吻合;2033年前后人口数为15.5亿左右, 也符合报告预测的人口峰值15亿人左右;由表三可知, 2016年劳动年龄人口为9.9亿, 与题目预测的10.1非常的相近;综上, 说明这个人口离散预测模型是比较有可行性的, 而且可以比较精确的预测短期人口数[6].

六、模型的优缺点分析

6.1模型的优缺点:

离散预测模型的优点在于:

(1)可以知道各年龄阶段人口结构状况, 便于计算负担系数、老少比、少年儿童抚养比、老年人口抚养比、老龄化比率、乡村人口城镇化比率等.

(2)充分考虑到各年龄阶段人口结构, 为国家人口的发展计划及有关人口政策调整提供一个恰当的咨询作用.

(3)对于人口预测本模型比其他模型更注意人口本分的结构,模型中采用到人口的死亡率、出生率、婴儿出生死亡率、妇女生育率等人口的特点去预测未来人口数量, 考虑得比较全面. (4)短中期预测出的人口增长数比较精确. 缺点在于:

(1)模型假定妇女总和生育率和妇女生育模式不变, 只能预测短中期的人口增长数, 对于长期预测还要调整预测参数, 比较麻烦.

(2)这个方程假定人口出生育、各年龄段人口存活率不变. 6.2模型的推广

此模型不仅仅可以预测未来人口的数量, 还可以应用到其他领域, 比如生物领域, 化学领域等等, 例如, 用它可以预测某一区域内某一动物种群的未来某时期的数量, 它可以做为预测我国未来稀有种群动物的数量的一种参考的方法, 从而采取有效措施, 保护我国珍惜动物

参考文献

[1] 王浣尘．人口系统工程[M]．上海: 上海交通大学出版社, 1985

[2] 拉腾图雅, 金良．人I=I预测模型[J]．内蒙古科技与经济, 1999(4): 21—27．

[3] 蔡防．我国人口总量增长与人口结构变化的趋势[J]．中国经贸导刊．2004(13): 29． [4] 求是科技.MATLAB7.0从入门都精通 人民邮电出版社出版发行, 2006年3月. [5] 黄荣清．关于人口预测问题的思考 [J]．人口研究．2003．28(1): 88—90．

[6] 马小红, 侯亚非．北京市未来5O年人口变动趋势预测研究[J]．市场与人门分析, 2004．10(2): 46—49

附录

附录1: （模型所须用到的参量）

2005年各岁人口中女性所占的人口比例（表1）

ksi(2005)

kzi(2005)

kxi(2005)

0.4 0.39 0.34 0.39 0.42 0.43 0.42 0.46 0.47 0.48 0.53 0.52 0.54 0.55 0.58 0.69 0.74 0.79 0.92 0.81 0.74 0.76 0.78 0.94 0.81 0.79 0.85 0.84 0.8 0.88 0.9 0.96 1.02 1.04 1.06 1.13

0.48 0.47 0.44 0.47 0.51 0.54 0.52 0.58 0.59 0.61 0.69 0.66 0.73 0.76 0.83 0.96 0.89 0.82 0.8 0.61 0.51 0.53 0.57 0.72 0.67 0.67 0.74 0.74 0.72 0.81 0.86 0.94 0.97 1.01 1.06 1.14

0.54 0.51 0.51 0.51 0.56 0.58 0.58 0.64 0.66 0.68 0.78 0.74 0.85 0.89 0.98 1.1 0.95 0.78 0.74 0.58 0.53 0.52 0.54 0.62 0.56 0.57 0.57 0.57 0.54 0.61 0.65 0.74 0.77 0.82 0.86 0.98

1.12 0.84 0.94 0.99 1.01 1.22 0.87 0.52 0.68 0.61 0.77 0.83 0.76 0.79 0.8 0.71 0.7 0.59 0.59 0.57 0.49 0.47 0.43 0.4 0.38 0.35 0.36 0.36 0.37 0.32 0.35 0.32 0.32 0.32 0.28 0.28 0.24 0.2 0.21 0.17 0.17

1.15 0.91 1.06 1.06 1.01 1.19 0.87 0.47 0.59 0.53 0.68 0.78 0.71 0.76 0.74 0.67 0.69 0.58 0.57 0.54 0.46 0.46 0.42 0.41 0.36 0.34 0.34 0.32 0.35 0.27 0.3 0.29 0.29 0.29 0.25 0.26 0.22 0.19 0.2 0.17 0.17

1.03 0.87 0.97 0.97 0.91 1.04 0.82 0.45 0.55 0.5 0.67 0.78 0.74 0.8 0.77 0.74 0.74 0.63 0.63 0.59 0.52 0.51 0.47 0.46 0.41 0.39 0.37 0.36 0.38 0.3 0.33 0.32 0.32 0.33 0.29 0.3 0.26 0.24 0.25 0.2 0.2

0.12 0.12 0.1 0.09 0.08 0.07 0.06 0.04 0.12 0.13 0.1 0.09 0.08 0.07 0.06 0.05 0.15 0.16 0.13 0.11 0.1 0.08 0.07 0.05 0.03 0.03 0.02 0.07

hsi(2005)

0.00002 0.00027 0.00115 0.00331 0.01061 0.03131 0.05199 0.0701 0.09085 0.1041 0.10418 0.09975 0.08969 0.07499 0.05936 0.04894 0.03824 0.02924 0.02297 0.01889 0.01345 0.00969 0.00686 0.00487 0.00364 0.00291

0.04 0.04 0.03 0.03 0.02 0.03 0.07

0.08

2005市、镇、乡的妇女生育模式值

hzi(2005)

hxi(2005)

0.00003 0.0001 0.0003 0.00056 0.00178 0.00264 0.00515 0.00821 0.01849 0.02206 0.05339 0.05745 0.08877 0.09121 0.10128 0.09952 0.10916 0.10109 0.11043 0.0951 0.09271 0.08282 0.07739 0.07093 0.06361 0.06071 0.05333 0.05384 0.04366 0.04864 0.03668 0.04184 0.03226 0.03958 0.02704 0.03093 0.02109 0.02474 0.01667 0.0185 0.01198 0.01368 0.00988 0.01029 0.0075 0.00737 0.00484 0.00516 0.00301 0.00369 0.00261

0.00226

0.00133 0.00097 0.00069 0.00082 0.00097 0.00054 0.00056 0.0008

xsi(2005)3114462 3078231 2716077 3042000 3295538 3404154 3259308 3585231 3621462 3766308 4092231 3983615 4164692 4237077 4454385 5214923 5540846 5758154 6518615 5721923 5178692 5251154 5468385 6518615 5685692 5504615 5939231 5903000 5649462 6228923

0.00103 0.00115 0.00103 0.00082 0.00092 0.00101 0.00068 0.00057 0.00036 0.00059 0.00044 0.00042 0.00076 0.00051 0.00049 0.00048

2005年度各年龄对应的人口数

xzi(2005)

xxi(2005)

2327154 8576154 2349538 8359923 2215308 8359923 2349538 8359923 2506231 9080615 2685231 9296846 2550923 9224769 2864231 10233692 2931385 10377846 2976077 10738154 3356538 12179538 3199846 11530923 3535538 13116462 3669769 13620923 3983077 15062308 4564846 16791923 4184462 14485769 3804077 11819231 3692154 11242692 2797077 8720231 2304769 7639231 2237692 7278923 2416692 7423077 2998462 8648231 2797077 7855462 2819462 7855462 3177538 8071692 3177538 7999615 3110385 7711308 3513154

8648231

6880769 7315385 7532615 7749923 8256923 7496462 8075846 6084077 6880769 7315385 7424000 9053692 6446231 3802538 5033846 4490615 5649462 6084077 5468385 5758154 5758154 5070077 4997615 4273308 4237077 4092231 3476615 3367923 3078231 2860923 2679846 2498846 2535000 2571231 2571231 2245308 2462615 2317769 2281538 2245308 1991769

4072538 4274000 4453000 4632000 5079538 4676769 5124308 4027846 4632000 4766231 4475385 5303308 3893538 2058692 2595692 2416692 3088000 3513154 3155154 3446000 3289385 3020846 3065615 2573308 2550923 2461462 2058692 2081077 1902000 1857308 1655846 1544000 1521615 1499231 1544000 1208308 1365000 1320231 1253077 1275462 1096462

10377846 10882308 11530923 12179538 13693000 12828154 14413692 12179538 13620923 13693000 12756077 14774000 11530923 6269923 7855462 7206846 9729231 11386769 10522000 11747154 11314692 10810231 10882308 9224769 9296846 8936462 7711308 7639231 6918538 6846462 6269923 5837538 5549231 5549231 5693385 4540308 5044769 4828615 4684462 4828615 4107923

1702077 1484769 1521000 1195077 1195077 1014000 832923 796692 651846 579462 507000 434538 362154 253538 181077 181077 108615 362154

usi(2005)

6.24047 0.80329 0.33973 0.47179 0.26462 0.28947 0.22467 0.3102 0.2295 0.28308 0.16124 0.19473 0.23548 0.1394 0.35528 0.40333 0.22647 0.26572 0.39311 0.35209

962231 3675462 827923 3315154 872692 3459308 716077 2666538 693692 2666538 604154 2378231 492308 1945846 514692 2017923 402769 1657538 358000 1369308 290923 1225154 268538 936923 201385 792769 156615 576538 134231 432385 89538 360308 67154 288308 223769

792769

2005年度的按龄死亡率

uzi(2005)

uxi(2005)

9.88 16.04832 0.75543 1.67422 0.5 1.22595 0.50286 0.69922 0.34911 0.63222 0.517 0.63798 0.33789 0.5425 0.40672 0.33282 0.17595 0.45167 0.26699 0.54027 0.3474 0.50615 0.24846 0.53513 0.39057 0.48258 0.40122 0.35582 0.20601 0.44684 0.26 0.55086 0.48294 0.78896 0.45929 0.86073 0.66758 1.16821 0.69592

1.3405

0.3409 0.43305 0.31033 0.44783 0.31401 0.36 0.48411 0.46256 0.52953 0.45559 0.49779 0.63307 0.675 0.79271 0.7275 0.9114 0.89865 1.04 1.14968 1.29619 1.09571 1.4856 1.69972 1.90705 1.91791 2.46815 2.14712 2.92804 2.91629 3.14686 2.86912 3.21771 4.05304 4.525 4.37462 4.35593 5.28354 5.71817 7.02671 7.44506 7.91189

0.8383 0.89139 0.57119 0.62976 0.50175 0.66648 17.29859 0.85806 1.15707 0.77 0.72099 0.98277 0.83834 0.98623 1.24652 1.20517 1.3738 1.22322 1.31401 1.66939 2.08635 1.57219 2.195 1.81772 2.1931 2.70685 3.16478 2.95166 2.89532 4.10766 4.09034 4.1577 4.63686 5.5193 6.115 6.35345 6.56 6.84548 7.86988 8.76024 10.40054

1.25059 1.46233 1.17583 1.25706 1.54459 1.26384 1.36243 1.44963 1.72542 1.57442 1.77347 1.88338 1.82138 1.94249 1.89758 2.01213 2.08655 2.19249 2.26122 2.38426 2.24531 2.4442 2.7795 3.12759 3.39431 3.445 3.73889 4.24595 4.16575 4.34018 4.51994 5.27107 5.24172 6.56094 6.42442 7.53282 7.46617 9.01934 9.88135 11.24589 11.00253

10.37029 9.60352 12.60803 12.64258 15.12044 16.785 19.01397 20.67258 21.29655 24.475 28.1434 31.08561 35.01 38.03 40.66879 50.49179 49.99913 57.99182 63.46111 66.66375 78.18571 91.54583 92.684 117.9671 118.754 136.964 153.8 275.672 11.04 12.37358 13.27957 14.64 17.72574 20.85254 22.98518 24.74649 25.81143 28.99451 34.70535 34.35189 42.37282 42.14719 52.04161 51.33333 67.61273 71.55783 70.48778 79.25188 94.65769 102.9825 98.43 96.54429 121.4633 131.1475 148.3933 196.926 14.5174 16.75481 18.16709 18.84048 21.36586 23.69313 28.09754 31.33537 30.81982 39.475 42.04392 44.33261 49.84938 54.16703 57.75189 67.83 75.45222 80.59 86.98435 97.16632 107.8624 116.6731 136.1855 127.59 154.7033 143.666 165.3175 269.2591

附录2:

MATLAB求解程序:

clc;clear; k

=[0.4,0.39,0.34,0.39,0.42,0.43,0.42,0.46,0.47,0.48,0.53,0.52,0.54,0.55,0.58,0.69,0.74,0.79,0.92,0.81,0.74,0.76,0.78,0.94,0.81,0.79,0.85,0.84,0.8,0.88,0.9,0.96,1.02,1.04,1.06,1.13,1.03,1.12,0.84,0.94,0.99,1.01,1.22,0.87,0.52,0.68,0.61,0.77,0.83,0.76,0.79,0.8,0.71,0.7,0.59,0.59,0.57,0.49,0.47,0.43,0.4,0.38,0.35,0.36,0.36,0.37,0.32,0.35,0.32,0.32,0.32,0.28,0.28,0.24,0.2,0.21,0.17,0.17,0.15,0.12,0.12,0.1,0.09,0.08,0.07,0.06,0.04,0.03,0.03,0.02,0.07;

0.48,0.47,0.44,0.47,0.51,0.54,0.52,0.58,0.59,0.61,0.69,0.66,0.73,0.76,0.83,0.96,0.89,0.82,0.8,0.61,0.51,0.53,0.57,0.72,0.67,0.67,0.74,0.74,0.72,0.81,0.86,0.94,0.97,1.01,1.06,1.14,1.06,1.15,0.91,1.06,1.06,1.01,1.19,0.87,0.47,0.59,0.53,0.68,0.78,0.71,0.76,0.74,0.67,0.69,0.58,0.57,0.54,0.46,0.46,0.42,0.41,0.36,0.34,0.34,0.32,0.35,0.27,0.3,0.29,0.29,0.29,0.25,0.26,0.22,0.19,0.2,0.17,0.17,0.15,0.12,0.13,0.1,0.09,0.08,0.07,0.06,0.05,0.04,0.03,0.02,0.07;

0.54,0.51,0.51,0.51,0.56,0.58,0.58,0.64,0.66,0.68,0.78,0.74,0.85,0.89,0.98,1.1,0.95,0.78,0.74,0.58,0.53,0.52,0.54,0.62,0.56,0.57,0.57,0.57,0.54,0.61,0.65,0.74,0.77,0.82,0.86,0.98,0.92,1.03,0.87,0.97,0.97,0.91,1.04,0.82,0.45,0.55,0.5,0.67,0.78,0.74,0.8,0.77,0.74,0.74,0.63,0.63,0.59,0.52,0.51,0.47,0.46,0.41,0.39,0.37,0.36,0.38,0.3,0.33,0.32,0.32,0.33,0.29,0.3,0.26,0.24,0.25,0.2,0.2,0.18,0.15,0.16,0.13,0.11,0.1,0.08,0.07,0.05,0.04,0.03,0.03,0.08]; h=[0.00002,0.00027,0.00115,0.00331,0.01061,0.03131,0.05199,0.0701,0.09085,0.1041,0.10418,0.09975,0.08969,0.07499,0.05936,0.04894,0.03824,0.02924,0.02297,0.01889,0.01345,0.00969,0.00686,0.00487,0.00364,0.00291,0.0019,0.00133,0.00097,0.00069,0.00082,0.00097,0.00054,0.00056,0.0008;

0.00003,0.0003,0.00178,0.00515,0.01849,0.05339,0.08877,0.10128,0.10916,0.11043,0.09271,0.07739,0.06361,0.05333,0.04366,0.03668,0.03226,0.02704,0.02109,0.01667,0.01198,0.00988,0.0075,0.00484,0.00301,0.00261,0.00124,0.00103,0.00103,0.00092,0.00068,0.00036,0.00044,0.00076,0.00049;

0.0001,0.00056,0.00264,0.00821,0.02206,0.05745,0.09121,0.09952,0.10109,0.0951,0.08282,0.07093,0.06071,0.05384,0.04864,0.04184,0.03958,0.03093,0.02474,0.0185,0.01368,0.01029,0.00737,0.00516,0.00369,0.00226,0.00154,0.00115,0.00082,0.00101,0.00057,0.00059,0.00042,0.00051,0.00048];

x=[3114462,3078231,2716077,3042000,3295538,3404154,3259308,3585231,3621462,3766308,4092231,3983615,4164692,4237077,4454385,5214923,5540846,5758154,6518615,5721923,5178692,5251154,5468385,6518615,5685692,5504615,5939231,5903000,5649462,6228923,6410000,6880769,7315385,7532615,7749923,8256923,7496462,8075846,6084077,6880769,7315385,7424000,9053692,6446231,3802538,5033846,4490615,5649462,6084077,5468385,5758154,5758154,5070077,4997615,4273308,4237077,4092231,3476615,3367923,3078231,2860923,2679846,2498846,2535000,2571231,2571231,2245308,2462615,2317769,2281538,2245308,1991769,2028000,1702077,1484769,1521000,1195077,1195077,1014000,832923,796692,651846,579462,507000,434538,362154,253538,181077,181077,108615,362154;

2327154,2349538,2215308,2349538,2506231,2685231,2550923,2864231,2931385,2976077,3356538,3199846,3535538,3669769,3983077,4564846,4184462,3804077,3692154,2797077,2304769,2237692,2416692,2998462,2797077,2819462,3177538,3177538,3110385,3513154,3736923,4072538,4274000,4453000,4632000,5079538,4676769,5124308,4027846,4632000,4766231,4475385,5303308,3893538,2058692,2595692,2416692,3088000,3513154,3155154,3446000,3289385,3020846,3065615,2573308,2550923,2461462,2058692,2081077,1902000,1857308,1655846,1544000,1521615,1499231,1544000,1208308,1365000,1320231,1253077,1275462,1096462,1141231,962231,827923,872692,716077,693692,604154,492308,514692,402769,358000,290923,268538,201385,156615,134231,89538,67154,223769;

8576154,8359923,8359923,8359923,9080615,9296846,9224769,10233692,10377846,10738154,12179538,11530923,13116462,13620923,15062308,16791923,14485769,11819231,11242692,8720231,7639231,7278923,7423077,8648231,7855462,7855462,8071692,7999615,7711308,8648231,9296846,10377846,10882308,11530923,12179538,13693000,12828154,14413692,12179538,13620923,13693000,12756077,14774000,11530923,6269923,7855462,7206846,9729231,11386769,10522000,11747154,11314692,10810231,10882308,9224769,9296846,8936462,7711308,7639231,6918538,6846462,6269923,5837538,5549231,5549231,5693385,4540308,5044769,4828615,4684462,4828615,4107923,4324077,3675462,3315154,3459308,2666538,2666538,2378231,1945846,2017923,1657538,1369308,1225154,936923,792769,576538,432385,360308,288308,792769];

u=[6.24047,0.80329,0.33973,0.47179,0.26462,0.28947,0.22467,0.3102,0.2295,0.28308,0.16124,0.19473,0.23548,0.1394,0.35528,0.40333,0.22647,0.26572,0.39311,0.35209,0.31573,0.3409,0.43305,0.31033,0.44783,0.31401,0.36,0.48411,0.46256,0.52953,0.45559,0.49779,0.63307,0.675,0.79271,0.7275,0.9114,0.89865,1.04,1.14968,1.2

9619,1.09571,1.4856,1.69972,1.90705,1.91791,2.46815,2.14712,2.92804,2.91629,3.14686,2.86912,3.21771,4.05304,4.525,4.37462,4.35593,5.28354,5.71817,7.02671,7.44506,7.91189,9.53014,10.37029,9.60352,12.60803,12.64258,15.12044,16.785,19.01397,20.67258,21.29655,24.475,28.1434,31.08561,35.01,38.03,40.66879,50.49179,49.99913,57.99182,63.46111,66.66375,78.18571,91.54583,92.684,117.9671,118.754,136.964,153.8,275.672;

9.88,0.75543,0.5,0.50286,0.34911,0.517,0.33789,0.40672,0.17595,0.26699,0.3474,0.24846,0.39057,0.40122,0.20601,0.26,0.48294,0.45929,0.66758,0.69592,0.50718,0.8383,0.89139,0.57119,0.62976,0.50175,0.66648,17.29859,0.85806,1.15707,0.77,0.72099,0.98277,0.83834,0.98623,1.24652,1.20517,1.3738,1.22322,1.31401,1.66939,2.08635,1.57219,2.195,1.81772,2.1931,2.70685,3.16478,2.95166,2.89532,4.1766,4.09034,4.1577,4.63686,5.5193,6.115,6.35345,6.56,6.84548,7.86988,8.76024,1.40054,11.28884,11.04,12.37358,13.27957,14.64,17.72574,20.85254,22.98518,24.74649,25.81143,28.99451,34.70535,34.35189,42.37282,42.14719,52.04161,51.33333,67.61273,71.55783,70.48778,79.25188,94.65769,12.9825,98.43,96.54429,121.4633,131.1475,148.3933,196.926;

16.04832,1.67422,1.22595,0.69922,0.63222,0.63798,0.5425,0.33282,0.45167,0.54027,0.50615,0.53513,0.48258,0.35582,0.44684,0.55086,0.78896,0.86073,1.16821,1.3405,1.365,1.25059,1.46233,1.17583,1.25706,1.54459,1.26384,1.36243,1.44963,1.72542,1.57442,1.77347,1.88338,1.82138,1.94249,1.89758,2.01213,2.08655,2.19249,2.26122,2.38426,2.24531,2.4442,2.7795,3.12759,3.39431,3.445,3.73889,4.24595,4.16575,4.34018,4.51994,5.2717,5.24172,6.56094,6.42442,7.53282,7.46617,9.01934,9.88135,11.24589,11.00253,12.72889,14.5174,16.75481,18.16709,18.84048,21.36586,23.69313,28.09754,31.33537,30.81982,39.475,42.04392,44.33261,49.84938,54.16703,57.75189,67.83,75.45222,80.59,86.98435,97.16632,17.8624,116.6731,136.1855,127.59,154.7033,143.666,165.3175,269.2591]; beta=[0.92648;1.278;1.6537]; r1=15; r2=49;

A=k(:,r1:r2).*h.*x(:,r1:r2); pha=beta.*sum(A,2); x(:,1)=(1-u(1)/1000)*pha; y=zeros(3,50); for i=1:50

A=k(:,r1:r2).*h.*x(:,r1:r2); pha=beta.*sum(A,2); x(:,1)=(1-u(:,1)/1000).*pha;

x(:,2:end)=(1-u(:,2:end)/1000).*x(:,1:end-1); y(:,i)=sum(x,2); for l=1:50 d(l)=sum(y(:,l)); end end y; yy=sum(y); plot([2005:2054],yy)