A Prediction Model of China Population Growth

In this article, we develop a prediction model of China population growth, and notice that the modified index curve is a sort of curve that possesses the growth limit K in the statistics, which is the same with the problem of China population growth. Considering influences of urbanization, population aging and sex proportion, we adopt and improve the modified index model, and add some coefficients to adjust the mathematical equation according to the data. The computation results show that in a short period, the population quantity will increase slowly and approach a fixed value, and in a long term, with the influence of population aging, some factors will put up periodic fluctuations.


Introduction
The population growth prediction mainly includes the prediction of total population and birthrate, death rate, sex proportion.In China, many special problems still exist in the prediction of population growth, such as national population policy, education degree, economic environment and human ideas.All population predictions are implemented based on certain data, and available data generally include sex proportion, death rate and birthrate.But these data are always incomplete and random, so in this article, we select and deal with these data and introduce some authorized data, and overcome the difficulties of incomplete information and accuracy.
The population growth is influenced by many aspects, in this article, we mainly consider following aspects.
(1) The influence of sex proportion.The difference of sex proportion is larger and the negative influence to the total population is larger.
(2) The influence of village population urbanization.The urbanization of village population cannot but induces the change of village and urban population structure and influences the development tendency of total population.
(3) The influence of birthrate to the total population.The national policies have large influences to the birthrate, and influence the change tendency of the total population.
(4) In the long-term population growth prediction, some influencing factors put up tendency fluctuation, which makes the population growth bring the element of tendency fluctuation, and some factors present periodic fluctuation, which makes the population growth bring the element of periodic fluctuation, and some factors present occasional fluctuation, which make the population growth bring the element of random fluctuation.
In this article, we establish the model based on above factors which are presented in the equation through the mathematical expression, and approximated and weakened to a certain extent in the middle and short term model.

Symbol explanations
t: The year needed to be predicted (In this article, the year of 1984 represents t=1 which is the initial value.). it e : The proportion of the influence factor i in the t'th year (i=1 represents the city, i=2 represents the town, and i=3 represents the village.). it p : The sex proportion in the t'th year.i k : The association degree of city population, town population and village population with the total population.

The model of middle and short term
According to the actuality of China and the development of population model, we develop the equation based on the modified index, and the equation is as follows: .
Supposed that every part has m periods, and the sums of various observation value are respectively S 1 , S 2 and S 3 , i.e.

S mK ab ab ab mK ab b b b
.
And we can obtain solutions.
We respectively consider the city population, town population and village population, and In the following text, we will concretely explain the function of every item and their influences to the development of the population.

Explanations of it e and its computation method
it e represents the weight of the influence of the population in the structure i occupying the total influences of city, town and village to the development of the population, i.e. we can confirm the influencing degree of city, town and village in every year to the development tendency of total population according to the available data.For example, in a certain year, e 1t , e 2t and e 3t are respectively 0.4, 0.3 and 0.4.
= it e the sum of female and male in i in the t'th year/ the sum of female and male in the t'th year.
The urbanization of village population is also embodied in e i because in the urbanization process of village population, changes of various weight e 1 , e 2 and e 3 will certainly occur.For example, above various weights are respectively changed to 0.4, 0.4 and 0.2, which indicates that when village population enters into the city and town, the proportion of city and town population will increase, i.e. the influence of city and town to the development tendency of the total population increase.
So e i represents the influence of city, town and village to the total population development tendency of China, and fully embodies the influence of village population urbanization to the development tendency of the total population in China.
Because of the middle and short term model, the urbanization of village population can not achieve saturation, so we approximate e i as the linear function.Because the sex will influence the total population, so we introduce the influencing factor 2 ( 1) p it − which represents the difference among sex proportion in every year, and the sex proportion difference are larger, the negative influence to the total population is more obvious and the sex proportion is more closed to 1, so the influence of the sex proportion to the total population is smaller.We select   The computation method is as follows: 1 2 8 4 5 3 1 2 8 5 1 3 .7 9 7 7 0 .26 2 0 .00 0 3 4 2 0 3 0 .1 2 6 0 .00 0 1 2 0 6 0 .6 1 3 0 .00 3 0 5 129988 130237.19330.258 0.001132724 0.154 0.000266555 0.588 0.001306106 130756 131044.0880.005592994 0.277 0.171 0.000448304 0.000303747 0.552 From above equation group, we can obtain

Confirmations of parameter k, a and b
Because we suppose the China population policy in middle and short term is stable, so its influence to the population growth tendency is stable, for example, the family planning policy and other factors are embodied in the coefficient a and b.The method to compute parameters is as follows: .
From the equation

The model of long term
The population model of long term is got based on the modification and perfection to the model of middle and short term which is as follows: Where, e i goes to the balance in the long-term process, and according the status of developed country and the development tendency of China, we thought e i can be approximated as a constant, we take that e 1 is 0.4, e 2 is 0.2 and e 3 is 0.4.
P it is the sex proportion, and we find that it shakes in a certain value through large of data, and we take it as 1.03 in the long-term prediction.
In the long-term prediction, the influence of replacement rate to the population is more obvious.So we add e -0.09t to embody its influence in the equation.We confirm that the value of c is 200000 according to the influence of the aging population.
Therefore, we put forward the long-term model as follows.

The solution of the Model
The computation results are seen in Table 3.
The prediction data are seen in Table 4.
We compute the standard deviation S and the mean absolute percent error (MAPE) utilizing the data in above tables and review the prediction effect of the model.
the sex population of city, town and village in turn and the influence of village population urbanization to the distribution of the population.
deviation degree of sex proportion of city, town and village with 1.

k
We use the parameter i k to denote the association degree of city, town and village with the total population, which is relative with human education degree, concept, living level and other factors.i k is computed through the statistical data from 2001 to 2005 of China Statistics Bureau and the equation.The rapid change occurred in the data in 2003 because of the large sized SARS and the reason that Chinese thought the Yang year went against the birth.So we select data of 2002, 2003 and 2005 to compute i k .
figure the Figure 1 and Figure 2, and the Figure 1 denotes the tendency of population growth when t<23, and the Figure 2 denotes the tendency of population growth when t>=23.

∑
In the long-term prediction, some influencing factors put up tendency fluctuation, which makes the population growth bring the element of tendency fluctuation, and some factors present periodic fluctuation, which makes the population growth bring the element of periodic fluctuation, and some factors present occasional fluctuation, which make the population growth bring the element of random fluctuation.

Figure 1 .
Figure 1.The Tendency of Population Growth When t<23 represents the sex proportion in city, town and village.According to the bearing age and other information embodied in the date, we can confirm values of it p from 2001 to 2005, and fit the development tendency in the middle and short term, and we also fit these changes to the linear function.But up to 2001, the sex proportion has gone to be stable, so we can take it as the constant which is the value fitted on the curve in 2010.We use the tool of Mathematica to fit the data, and the fitted equations are as follows: According to the data from 2001 to 2005 in China population sampling data, we fit the data to the function which takes t as the variable, and the fitted equations are: it p

Table 2 .
Data from China Statistical Yearbook