前言

今天研究一个简单一点的问题，预计2023.1.18正式结题做PPT展示。

定义

老龄人: 60岁≤ 的人
老龄化：老年人占比增加

建模

用昨天get到的数据就行了

假设
去年的总人口数：$x$
去年的老龄人口数：$y$
今年相比去年增加老龄人口数:$v$
今年相比去年增加总人口数:$u$

如果老龄化，那$v$和$u$有具体的比例的取值范围

假设今年相比去年老龄化

则：
$y/x<(y+v)/(x+u)$
$y(x+u)<x(y+v)$
$xy+yu<xy+xv$
$yu<xv$
$u/v<x/y$

∴这个比率是$u/v$（右手就行

综上：
若$u/v<x/y$
∴人口老龄化

否则，人口未老龄化

模型细节

1. 每年死亡人口平分至各个年龄段，如果某年龄段当年人口不够减，优先减至年龄大的人口数中。
2. 1949年前的数据没有get到，所以用1949年总人口分成1-99岁（0岁以给出），死亡人口按1算

代码实现

#include<bits/stdc++.h>
using namespace std;

const int maxn=1e5+10;
double b[maxn],d[maxn],tot[maxn],cnt[maxn];
int main(){
	freopen("old.in","r",stdin); 
	freopen("old.out","w",stdout);
	//1949年前的数据处理 
	for(int i=1850;i<=1948;i++)cnt[i]=538.38715; 
	//1949-2021的数据求 
	for(int i=1949;i<=2021;i++){
		scanf("%lf%lf",&b[i],&d[i]);//出生和死亡人口 
		double pr_all=0,pr_old=0;//去年数据统计
		for(int j=1850;j<i;j++){
			pr_all+=cnt[j];
			if(i-j>=59)pr_old+=cnt[j];
		}
		//printf("pr = %.5lf %.5lf\n",pr_all,pr_old);
		double pr=pr_all/pr_old;
		cnt[i]=b[i];
		double c=0,u=0;//u指差多少 
		double add_all=cnt[i]-d[i],add_old=cnt[i-60];//前年的数据
		for(int j=1850;j<=i-60;j++){
			if(cnt[j]>0)add_old-=c;
		}
		bool flag=false;
		for(int j=1850;j<=i;j++){
			//最老人口年龄枚举 
			if(cnt[j]>0&&!flag){
				c=d[i]*1.0/(i-j+1);//平均每个年龄的死亡人数计算 
				flag=true;
			}
			//人口死亡人数减少 
			if(flag){
				if(cnt[j]>=c+u){
					cnt[j]=cnt[j]-c-u;
					u=0;
				}
				else{
					u=c+u-cnt[j];
					cnt[j]=0;
				}
			}
		}
		//打印 
		printf("%d = %.5lf ",i,add_all*1.0/add_old);
		bool yes=false;;
		if(add_all*1.0/add_old<pr)yes=true;
		if(yes)printf("< %.5lf => 老龄化\n",pr);
		else printf("≥%.5lf => 未老龄化\n",pr);
	}
	return 0;
} 

.in文件

//出生人口 和 死亡人口
1950.012	1083.34
2042.252	993.528
2128.14	1002.14
2126.834	977.194
2175.452	823.144
2288.30002	794.30588
2003.759	754.7902
2004.2132	716.2392
2200.14159	698.2524
1928.34468	790.60812
1665.38946	980.55013
1381.07802	1683.64401
1194.02367	943.75947
2504.75712	678.34368
3015.8992	698.6372
2773.43066	814.96844
2756.444	689.111
2624.62382	661.18754
2605.67616	646.83696
2807.5905	647.9055
2762.98175	650.20826
2787.70128	634.05888
2619.93946	625.58086
2608.33584	666.90405
2504.15277	631.61388
2266.93205	670.53942
2137.6746	680.2112
1875.27717	683.19693
1807.35522	656.27034
1756.72675	601.61875
1738.19844	605.73582
1797.41805	625.7897
2092.50552	636.45792
2264.85112	670.9164
2079.73152	710.7552
2076.7043	711.71474
2227.10504	717.66978
2411.38201	737.49802
2549.969	734.496
2483.65162	737.21264
2432.15232	737.08416
2407.85298	762.60111
2279.39664	776.0141
2137.19904	778.01544
2143.97253	786.95288
2121.345	777.8265
2073.59152	795.76497
2078.16522	802.87184
2048.48282	804.80526
1951.26204	810.9465
1841.50704	812.57756
1778.20429	817.49235
1707.64926	820.64161
1651.90558	823.38373
1603.70707	827.0528
1597.55252	834.52296
1621.3744	851.22156
1589.20632	895.16088
1598.7609	915.65397
1612.21628	937.58212
1594.7275	944.826
1595.6829	953.38701
1790.33532	963.30024
1980.38354	969.12386
1781.53978	974.85638
1903.64418	980.03952
1658.52874	977.96482
1889.37824	980.19328
1769.73904	988.47766
1526.27526	995.03028
1467.89328	999.74672
1203.12624	998.36884
1062.2752	1014.2468

.out文件

1949 = 1.60976 < 2.41463 => 老龄化
1950 = 1.98790 < 2.44466 => 老龄化
1951 = 2.17493 < 2.48029 => 老龄化
1952 = 2.26355 < 2.52119 => 老龄化
1953 = 2.71327 ≥2.56335 => 未老龄化
1954 = 3.04592 ≥2.60803 => 未老龄化
1955 = 2.58626 < 2.65834 => 老龄化
1956 = 2.70694 ≥2.69660 => 未老龄化
1957 = 3.20157 ≥2.73529 => 未老龄化
1958 = 2.45920 < 2.78298 => 老龄化
1959 = 1.50385 < 2.82085 => 老龄化
1960 = -0.67767 < 2.85134 => 老龄化
1961 = 0.58024 < 2.88268 => 老龄化
1962 = 4.31895 ≥2.89664 => 未老龄化
1963 = 5.55858 ≥2.96458 => 未老龄化
1964 = 4.76799 ≥3.05583 => 未老龄化
1965 = 5.12140 ≥3.14004 => 未老龄化
1966 = 4.93667 ≥3.22161 => 未老龄化
1967 = 4.99610 ≥3.29753 => 未老龄化
1968 = 5.58647 ≥3.37300 => 未老龄化
1969 = 5.54320 ≥3.45799 => 未老龄化
1970 = 5.73191 ≥3.54198 => 未老龄化
1971 = 5.38305 ≥3.62732 => 未老龄化
1972 = 5.31374 ≥3.70605 => 未老龄化
1973 = 5.20238 ≥3.78698 => 未老龄化
1974 = 4.49884 ≥3.86310 => 未老龄化
1975 = 4.17037 ≥3.93203 => 未老龄化
1976 = 3.46452 < 3.99728 => 老龄化
1977 = 3.39851 < 4.05292 => 老龄化
1978 = 3.46280 < 4.10573 => 老龄化
1979 = 3.44305 < 4.15476 => 老龄化
1980 = 3.61332 < 4.20443 => 老龄化
1981 = 4.55762 ≥4.25908 => 未老龄化
1982 = 5.06567 ≥4.32844 => 未老龄化
1983 = 4.42162 ≥4.40896 => 未老龄化
1984 = 4.48559 < 4.48598 => 老龄化
1985 = 5.04771 ≥4.56506 => 未老龄化
1986 = 5.69820 ≥4.65336 => 未老龄化
1987 = 6.29557 ≥4.75370 => 未老龄化
1988 = 6.17005 ≥4.86273 => 未老龄化
1989 = 6.10292 ≥4.97203 => 未老龄化
1990 = 6.03802 ≥5.08216 => 未老龄化
1991 = 5.62909 ≥5.19683 => 未老龄化
1992 = 5.19548 < 5.31064 => 老龄化
1993 = 5.29738 < 5.42205 => 老龄化
1994 = 5.35900 < 5.53876 => 老龄化
1995 = 5.20841 < 5.65775 => 老龄化
1996 = 5.31619 < 5.78108 => 老龄化
1997 = 5.30519 < 5.91042 => 老龄化
1998 = 4.97979 < 6.04389 => 老龄化
1999 = 4.60276 < 6.17906 => 老龄化
2000 = 4.40433 < 6.31496 => 老龄化
2001 = 4.16993 < 6.45462 => 老龄化
2002 = 3.99642 < 6.59777 => 老龄化
2003 = 3.84607 < 6.74549 => 老龄化
2004 = 3.88183 < 6.89876 => 老龄化
2005 = 4.02841 < 7.06106 => 老龄化
2006 = 3.73698 < 7.23666 => 老龄化
2007 = 3.79458 < 7.42939 => 老龄化
2008 = 3.87216 < 7.06642 => 老龄化
2009 = 0.41134 < 6.71257 => 老龄化
2010 = 0.38297 < 6.36918 => 老龄化
2011 = 0.46806 < 6.06307 => 老龄化
2012 = 0.57149 < 5.78464 => 老龄化
2013 = 0.44283 < 5.51673 => 老龄化
2014 = 0.47695 < 5.32698 => 老龄化
2015 = 0.41157 < 5.15692 => 老龄化
2016 = 0.54928 < 4.95594 => 老龄化
2017 = 0.42185 < 4.82475 => 老龄化
2018 = 0.33606 < 4.74011 => 老龄化
2019 = 0.35487 < 4.69543 => 老龄化
2020 = 0.19728 < 4.67787 => 老龄化
2021 = 0.05583 < 4.46046 => 老龄化