A novel fractional-order discrete grey Gompertz model for analyzing the aging population in Jiangsu Province, China

Abstract

Purpose

China's population aging is gradually deepening and needs to be actively addressed. The purpose of this paper is to construct a novel model for analyzing the population aging.

Design/methodology/approach

To analyze the aging status of a region, this study has considered three major indicators: total population, aged population and the proportion of the aged population. Additionally, the authors have developed a novel grey population prediction model that incorporates the fractional-order accumulation operator and Gompertz model (GM). By combining these techniques, the authors' model provides a comprehensive and accurate prediction of population aging trends in Jiangsu Province. This research methodology has the potential to contribute to the development of effective policy solutions to address the challenges posed by the population aging.

Findings

The fractional-order discrete grey GM is suitable for predicting the aging population and has good performance. The population aging of Jiangsu Province will continue to deepen in the next few years.

Practical implications

The proposed model can be used to predict and analyze aging differences in Jiangsu Province. Based on the prediction and analysis results, identified some corresponding countermeasures are suggested to address the challenges of Jiangsu's future aging problem.

Originality/value

The fractional-order discrete grey GM is firstly proposed in this paper and this model is a novel grey population prediction model with good performance.

Citation

Zhang, W., Liu, S., Liu, L., Rathnayaka, R.M.K.T., Xie, N. and Du, J. (2023), "A novel fractional-order discrete grey Gompertz model for analyzing the aging population in Jiangsu Province, China", Grey Systems: Theory and Application, Vol. 13 No. 3, pp. 544-557. https://doi.org/10.1108/GS-01-2023-0005

1. Introduction

1.1 Background and motivation

The population is a critical component that directly contributes to a nation's strength, playing an essential role in decision-making processes related to national economy, social development and the progress of human civilization. Since China's aging society status began in 1999, the number of elderly individuals and their proportion relative to the total population has been steadily increasing, surpassing that of many other countries worldwide. Recent statistics indicate that China's aging population is experiencing accelerated development and has entered an era of rapid growth (Liu et al., 2022; Zhang et al., 2022). Moreover, the aging process in China is characterized by large-scale, rapid growth, long duration and unbalanced regional development (Guo et al., 2022). As a result, responding proactively to population aging has become a critical national strategy. It is imperative to address the challenges posed by an aging population to ensure sustainable economic and social development. Effective policy solutions must be developed to mitigate the negative impacts of population aging, such as a shrinking workforce, rising healthcare costs and increased demand for social services.

Jiangsu Province entered an aging society as early as 1986, 13 years ahead of the rest of the country. The aging problem in Jiangsu Province has become more prominent than in other developed provinces in China, with a large scale, high speed and unbalanced distribution. Recent findings show that the aging process of Jiangsu's population will continue to accelerate in the future. To address the challenges posed by this demographic shift and optimize the formulation of relevant policies, it is crucial to introduce an appropriate prediction methodology for Jiangsu's population aging today.

1.2 Overview of population prediction algorithms

Choosing an appropriate forecasting method based on the specific characteristics of the population and the research objectives is essential to ensure accurate results. The acquisition of relevant data, identification of influencing factors and choice of forecasting methods significantly affect the accuracy of results. The available population prediction models mainly depend on the Malthus model (Kosobud and O'neill, 1981; Ehrlich and Lui, 1997), the Logistic model (Wilson et al., 2022) and Geopertz model (Vaghi et al., 2020). In addition, Bass et al. (1994) proposed a diffusion model, which can predict the system without decision variables. Verhulst differential equation model is also a reliable method to describe population growth behavior (Brilhante et al., 2012). Linear regression model (Sulaiman et al., 2019), neural network model (Xiang and Liu, 2018) and the grey system model (Fan et al., 2019) are also commonly used methods. Various models have been developed based on their unique assumptions, characteristics and conditions (Tu and Chen, 2021). For instance, the grey system model is used to tackle the issue of poor information and uncertainty. It can extract useful information from its own time series to build a model and has shown promising prediction performance for time series with limited data (Qian and Sui, 2021; Zhou et al., 2022; Li et al., 2022b; Liu et al., 2022).

Population prediction involves forecasting the size, structure and distribution of a population. The population system can be seen as a grey system with both known and unknown information. Grey system theory, as represented by the grey forecasting model, can effectively extract the variation law of population time series and has strong applicability to population forecasting with limited information (Li et al., 2022a). Xu et al. (2019) applied the grey prediction model and radial function network to develop a combined model with distributed weights to predict the short-term total population from 2015 to 2025. Gao et al. (2017) predicted the population age structure of Anhui province in China using a similar approach.

1.3 Literature review of the grey prediction model

Grey prediction models have gained popularity due to their superior capability of generating highly accurate forecast results using sparse data samples, compared to traditional time series methods (Xiao and Duan, 2020). Additionally, differential equation modeling suggests that grey models can effectively simulate and analyze uneven data with high accuracy (Wei and Xie, 2020).

The Gompertz model (GM) (1, 1) model is represented by a first-order ordinary differential equation (Liu et al., 2021). Over the past few decades, researchers worldwide have made significant advancements in the development of grey models. To avoid systematic errors resulting from integral estimation, some improved structures of background values have been introduced (Wang et al., 2007).

Zeng et al. (2018) proposed a multivariable grey model based on a dynamic background value coefficient and utilized a particle swarm optimization algorithm to optimize the structure parameter. Meanwhile, Liu and Zeng (2020) presented a three-parameter background value method. Moreover, to avoid errors in background values, discrete grey models are often employed (Xie and Liu, 2009). In terms of data preprocessing, the grey buffer operator is an important tool for data conversion that reduces the impact of system disturbances (Zeng et al., 2018; Liu and Wu, 2021). In order to make full use of the information value of the new data, Wu et al. (2013) proposed the fractional-order accumulation operator. Liu et al. (2023) used a recursive estimation to estimate the dynamic parameters of the model. Then, some residual correction techniques are also used to improve the prediction performance of the model (Nguyen et al., 2019; Zhou et al., 2019).

1.4 Contribution and structure

The grey system prediction method is suitable for analyzing systems with poor information, sparse data and fuzzy structures that follow certain systematic rules and its effectiveness is unparalleled. This method weakens the randomness of the original data sequence and uncovers the true operational law of the system by accumulating and generating the original data sequence, ultimately leading to accurate predictions. The population development can be seen as an unclear system, and the grey prediction method is suitable for population prediction. Due to its unique advantages, an increasing number of scholars worldwide are using grey forecasting models to conduct population forecasting research. Based on the proposed Fractional-order Discrete Grey Gompertz Model (FDGGM), this study forecasts the total population, aged population and proportion of aged population of Jiangsu Province from 2021 to 2025 to analyze the regional differences in population aging in Jiangsu Province.

The remainder of this paper is organized as follows. In Section 2, we explain the modeling steps of the FDGGM as well as its properties and parameter optimization. Section 3 analyzes the population aging of Jiangsu Province in China by using the proposed model. Conclusions and suggestions are presented in Section 4.

2. Methodology

In this section, the traditional definition of Gompertz population model is proposed. Based on the grey accumulation method, an improved grey GM is introduced. Furthermore, the relevant properties of the model and parameter optimization methods are discussed.

2.1 Gompertz population model

The GM is named after British statistician B. Gompertz. It was originally used to describe the natural development of a biological population from germination to saturation (Gompertz, 1825). The basic form of the model is as follows:

(1)dxdt=αx⁡lnNx

Among the formula, α is the natural growth rate per unit time and N is the maximum norm modulus of the race under the constraints of resources and environment. Obviously, when x<N, the number of population increases continuously with time until saturation. Then the model and the coefficient of formula (1) change as in formula (2).

(2){p=−α⁡ln⁡Nq=−α

Then, the famous Gompertz differential equation can be expressed as follows:

(3)dxdt+px=qx⁡ln⁡x

The general solution of formula (1) can be estimated as x^(t)=N⋅e−ec1−αt, simplified as x^(t)=vsht. The Gompertz curve varies depending on its structural parameters. Gompertz curve can roughly describe the development trend of population growth, which is shown as an S-shaped curve. However, most of the original sequences are not smooth enough to meet the modeling requirements of Gompertz differential equation. Therefore, the original sequence usually needs to be smoothed. In this paper, the fractional accumulation grey generation operator is used to smooth the original data.

2.2 Fractional-order discrete grey Gompertz model

GM can be used to estimate the development trend of the population. However, when the population data obtained is limited or unclear, Gompertz differential equation cannot perform accurate parameter estimation. In contrast, the grey prediction method can deal with the small sample non-smooth modeling sequence. Therefore, this paper uses the fractional grey accumulation operator to process the data and proposes a discrete population prediction model.

Definition 1.

Set the original sequence as {x(0)(i)}i=1,2,⋯,n, its fractional-order accumulation generated sequence can be expressed as follows:

(4)x(r)(k)=∑i=1kCk−i+r−1k−ix(0)(i),i=1,2,⋯,n.

where, Ck−i+r−1k−i=Γ(r+k−i)Γ(k−i+1)Γ(r) and Γ(⋅) is gamma function and r∈[0,1] refers to the fractional order. Based on fractional-order accumulation (Wu et al., 2013) and Gompertz differential equation (Gao et al., 2022), the grey GM can be expressed as follows:

(5)dx(r)(t)dt+px(r)(t)=qx(r)(t)ln⁡x(r)(t)

For a given starting point x(r)(1)=x(0)(1), the solution of formula (5) is as follows:

(6)x^(r)(t)=exp((ln⁡x(0)(1)−pq)eq(t−1)+pq)

Based on the grey information coverage principle, dx(r)(t)dt can be instead of forward difference x(r)(t)−x(r)(t−1). Then its approximate discretized form (Cai and Wu, 2022) can be given as follows:

(7)x(r)(t)−x(r)(t−1)+pz(r)(t)=qz(r)(t)ln⁡z(r)(t)

where, z(r)(t)=x(r)(t)|[t−1,t]=12(x(r)(t−1)+x(r)(t)) is the background value of x(r)(t) in integral interval [t−1,t]. However, the general solution of the model is solved by formula (5) and there is the estimation error in the discretization process of differential equation. In order to eliminate the conversion error, this paper proposes a discrete form of grey GM.

Definition 2.

For the accumulation value x(r)(t), its growth rate in tth can be approximately estimated as dx(r)(t)dt=limΔt→0x(r)(t+Δt)−x(r)(t)Δt≈x(r)(t+1)−x(r)(t). Then, in order to reduce the estimation error, a constant correction term c is introduced into formula (5). A modified FDGGM can be expressed as follows:

(8)x(r)(t+1)=ax(r)(t)+bx(r)(t)ln⁡x(r)(t)+c

It is noteworthy noting that FDGGM is a modified recursive discretization form of GM. According to the Gompertz differential equation, a is used to estimate the environmental carrying capacity, b is used to adjust the natural growth rate of the population and c is a constant to modify the model results. Let U=[a,b,c]T coefficient vector of formula (8), the least squares estimation of U is as follows:

(9)U=(BTB)−1BTG

where,

B=[x(r)(1)x(r)(1)ln⁡x(r)(1)1x(r)(2)x(r)(2)ln⁡x(r)(2)1⋮⋮⋮x(r)(n−1)x(r)(n−1)ln⁡x(r)(n−1)1],G=[x(r)(2)x(r)(3)⋮x(r)(n)].

Finally, the prediction results of FDGGM model can be obtained as follows:

(10)x^(r)(t+1)=ax^(r)(t)+bx^(r)(t)ln⁡x^(r)(t)+c,t=1,2,⋯

(11)x^(0)(t)=∑i=1tCt−i−r−1t−ix^(r)(i)

where, the initial value can be set to x^(r)(1)=x(r)(1).

2.3 Model properties

As an improved GM, it is obvious that the proposed grey GM meets the properties as follows:

Property 1.

When fractional order r=0, x(r)(k)=x(0)(k). At this time, the grey GM (formula (5)) degenerates to the Gompertz population model (formula (3)).

Property 2.

The prediction curve (formula (6)) of grey GM has only one inflection point (tm,x(r)(tm)) and that can be estimated as tm=1−1qln(pq−ln(x(0)(1))).

Property 3.

From formula (6) and formula (2), we can get that limt→∞x^(r)(t)=epq and N=epq. ep/q refers to the environmental carrying capacity of natural growth of the population.

Property 4.

Due to the smoothing effect of fractional order on data, the smaller the fractional-order r is, the higher the priority of new information of the model (Wu et al., 2013).

Property 5.

When b=0, FDGGM degenerates to the traditional discrete grey model.

2.4 Parameter optimization

Reasonable parameter setting is an important guarantee to ensure the stability of model prediction. Using the mean absolute percentage error (MAPE) as the evaluation index of the model performance, the fractional-order parameters can be solved by the following optimization problem.

(12)minrMAPE=1n∑k=1n|x^(0)(k)−x(0)(k)x(0)(k)|

{x(r)(k)=∑i=1kCk−i+r−1k−ix(0)(i)U=(BTB)−1BTGx^(r)(t+1)=ax^(r)(t)+bx^(r)(t)ln⁡x^(r)(t)+cx^(0)(t)=∑i=1tCt−i−r−1t−ix^(r)(i)

Considering that the above optimization problem is nonlinear and non-differentiable, this study uses whale optimization algorithm (WOA) to solve formula (12). Compared with other heuristic algorithms, WOA algorithm converges faster and its spiral search path can avoid falling into local optimum (Liu et al., 2022). The specific combination of FDGGM and WOA is shown in Figure 1.

3. Empirical analysis of the population aging in Jiangsu Province

3.1 Data sources and analysis

3.1.1 Data sources

As one of the more developed provinces along the east coast of China, Jiangsu Province is facing an increasingly prominent aging problem, which is expected to accelerate at a high rate in the future. Therefore, it is crucial to correctly recognize the present situation and development trend of population aging in the province, as this can play a significant role in solving population problems and realizing the coordinated development of population, economy and society in future China.

The purpose of this study is to forecast and analyze population aging in Jiangsu Province and its 13 prefecture-level cities for the next 5 years, from 2021 to 2025. The study considers total population, aged population and the proportion of the aged population in the whole province and each of its 13 cities. The data for these variables were obtained from the Jiangsu Statistical Yearbook, covering the period from 2011 to 2020. Total population refers to the number of individuals living in a country or region in a given year. The aged population is defined as those aged 65 years and above. The proportion of the aged population is the percentage of the population aged 65 and above in the total population.

3.1.2 Data analysis

Table 1 presents the total population of Jiangsu Province and its 13 prefecture-level cities from 2011 to 2020. The statistics suggested that the total population of Jiangsu Province and has reached 84.47 million in 2020. In 2020, among the 13 prefecture-level cities in Jiangsu Province, Suzhou has the largest total population, followed by Nanjing, Nantong and Wuxi, and Zhenjiang and Xuzhou have the smallest population. In the past ten years, the total population of Nantong, Lianyungang, Huaian, Yangzhou, Zhenjiang, Taizhou and Suqian increased relatively slowly. As the capital of Jiangsu Province, the total population of Nanjing shows a slight upward trend and the fluctuation range between different years is relatively small.

The aged population in Jiangsu Province and its 13 prefecture-level cities from 2011 to 2020 can be seen in Table 2. The growth rate of the aged population in Jiangsu Province is much faster than that of the total population. From 2011 to 2020, the aged population of Jiangsu Province increased by more than 5 million and it has reached up to 18.51 million.

Furthermore, in the 13 prefecture-level cities, Nanjing and Suzhou have a large number of elderly people, both exceedingly more than 2 million in 2020. In addition, Wuxi, Nantong and Yancheng also have a large elderly population, nearly 2 million. However, the aged population in Xuzhou, Changzhou and Zhenjiang is relatively small.

The proportion of aged population in Jiangsu Province from 2011 to 2020 can be seen in Table 3. In 2011, the proportion of aged population is 16.7%, while the proportion of aged population in 2020 is nearly 22%. Since 2016, the proportion of the aged population in Jiangsu Province has exceeded 20%, which indicates that the degree of aging in Jiangsu Province has been very high.

3.2 Data prediction and analysis

3.2.1 Model validation

In this section, the prediction performance of even grey model (EGM), non-homogeneous discrete grey model (NDGM), fractional grey model (FGM) (Wu et al., 2013), grey power model (GPM) (Liu, 2021), Bass model Bass et al. (1994) and Verhulst model (Brilhante et al., 2012) are used to compared with FDGGM. Initially, the original data in Table 1 and Table 2 are taken into these grey models. The data collected from 2011 to 2018 are used as the training data and the data from 2019 to 2020 are used as the test to select the best model for the prediction.

1. Bring the data into the WOA-FDGGM algorithm and the best parameter is r=0.000787.

2. Set up the model x(r)(t+1)=ax(r)(t)+bx(r)(t)ln⁡x(r)(t)+c. The structural parameters of FDGGM model can be estimated as

U=[a,b,c]T=[28.92,−2.80,−21619.87]T

1. Calculate the prediction result and prediction error. The prediction results are shown in Table 4 as follows.

According to Table 4, the prediction errors of the seven models are as 1.01, 0.21, 0.07, 0.56, 0.13, 0.07 and 0.05%, respectively. The fitting error and prediction error of the proposed FDGGM model have the best prediction performance among the seven models.

The prediction curves of the seven models are shown in Figure 2. The existing grey model often overestimates the development trend of population, and its prediction result is greater than the true value. The proposed FDGGM model is an optimized form of the Gompertz population model, which can adaptively estimate the environmental carrying capacity and natural growth rate. Due to the improvement of the structure, the FDGGM model has better adaptability to the population prediction problem. At the same time, fractional accumulation gives the FDGGM model the ability to deal with nonlinear trends with the priority of new information.

3.2.2 Forecasted data analysis

According to the FDGGM model in Section 2, the number of the total population in Jiangsu province and its 13 prefecture-level cities from 2021 to 2025 can be forecasted with the original data from 2011 to 2020. Table 5 presents the forecast number of total population in Jiangsu Province and its prefecture-levels.

From Table 5, it can be seen that the total population of most prefecture-level cities in Jiangsu Province will not change much in the next five years from 2021 to 2025. However, the forecast data show that the total population of Changzhou and Lianyungang will grow rapidly. The same model and method can be used to predict the number of aged population in Jiangsu Province and its prefecture level cities from 2021–2025.

For specific data in Table 6. From 2023 to 2025, the aged population of Jiangsu Province will decline slightly. The aged population in Nanjing, Suzhou, Wuxi and Nantong is relatively large. In 2025, the aged population in these four prefecture-level cities will reach 2.14 million, 2.85 million, 1.70 million and 1.89 million. The aged population of Xuzhou, Changzhou and Zhenjiang will still be less than 1 million in the next five years.

According to the predicted number of the total population and aged population in Jiangsu Province and its 13 prefecture-level cities from 2021 to 2025, the proportion of aged population can be estimated as shown in Table 7. According to Table 7, it can be seen that the proportion of the aged population in Jiangsu Province will remain at about 23% in the next five years. The proportion of aged population of Nanjing, Xuzhou, Yancheng and Yangzhou will exceed the provincial average. Furthermore, result shows that the aging degree of these four prefecture-level cities is relatively high. In 2025, the proportion of the aged population in Xuzhou is the highest among the 13 prefecture-level cities, reaching 23.3%, and the proportion of the aged population in Huaian is the lowest among the 13 prefecture-level cities, at only 19.9%. Besides, the proportion of the aged population in the Jiangsu Province and its 13 prefecture-level cities will basically increase first and then decrease. This, to some extent, shows that the growth rate of population aging in Jiangsu Province will be curbed.

4. Conclusions and suggestions

Accurately predicting the development trend of population aging is crucial for effectively addressing the challenges posed by an aging population, such as allocating medical resources and formulating aging policies. In this study, a new model (FDGGM) based on grey system theory is developed to predict and analyze the aging differences in Jiangsu Province, China from 2021 to 2025. The model provides insights into the characteristics of population aging in the province and its 13 prefecture-level cities during the next few years. The findings of this study are as follows:

1. The aging problem in Jiangsu Province is becoming increasingly severe, with the aged population projected to reach 19.47 million by 2025. Among the 13 prefecture-level cities in Jiangsu Province, Suzhou, Nanjing, Wuxi, and Nantong have relatively large aged populations.

2. The population aging rates in the 13 prefecture-level cities of Jiangsu Province are unbalanced. The proportion of the aged population in Nanjing, Xuzhou, Yancheng, and Yangzhou is relatively high, while the aging rates in Wuxi, Changzhou, Suzhou, Nantong, Lianyungang, Huaian, Zhenjiang, Taizhou and Suqian are projected to decrease over the next five years.

3. The forecast results show that the aging problem in relatively developed regions will be even more severe in the next five years. People in economically developed areas have a higher living standard and longer life expectancy than those in underdeveloped areas, which contributes to a higher degree of aging in these regions. In contrast, life expectancy is relatively low in economically underdeveloped areas and medical care conditions are relatively weak, leading to higher death rates.

According to the forecasting results of the FDGGM model and relevant analysis, several management inspirations are suggested as follows.

1. It is both important and urgent to actively respond to the aging population. The population aging of Jiangsu Province is characterized by a large scale, rapid growth and obvious regional differences, which requires a scientific and diverse response. Different cities should formulate various strategies according to their unique characteristics. Southern Jiangsu Province can take steps to improve the happiness and quality of life of the elderly and provide them with more daily activities. Northern Jiangsu can optimize elderly care facilities and improve the level of elderly care to cope with the aggravation of aging in the next few years. Moreover, Nantong, Taizhou and Yangzhou could focus on encouraging the birth of the second and third child to raise the birth rate and help deal with the aging population.

2. It is crucial to allocate elderly care and medical care resources scientifically and rationally across the 13 prefecture-level cities of Jiangsu Province. Although the population aging of southern Jiangsu is more serious, its developed economy and rich resources may support the aging response in northern Jiangsu. Relevant policies can be initiated in the northern region of Jiangsu Province, especially Suqian.

3. Creating a social environment for providing for the aged, respecting and loving the elderly is conducive to the harmonious development of China's aging society. This must be essential to build a socialist country for China.

4. The improvement of literacy and the development of elderly human resources directly affect the aging of the population. Maintaining the working state of the elderly on a voluntary basis can help their physical and mental health, especially in eastern China.

In future studies, there is room for optimizing the evaluation indicators of population aging. Specifically, the analysis of population aging can be enhanced by including the number and proportion of disabled elderly as an additional evaluation indicator. This will provide a more comprehensive understanding of the aging situation in a given population, which can be helpful in formulating effective aging policies and programs.

Figures

Figure 1

Flow chart of FDGGM parameter optimization

Figure 2

Comparison of model prediction results of the total population in Jiangsu Province

Table 1

The total population in Jiangsu Province from 2011 to 2020 (Unit: Ten thousand persons)

Year	Region
	Jiangsu	Nanjing	Wuxi	Xuzhou	Changzhou	Suzhou	Nantong
2011	8023	848	670	269	312	1111	729
2012	8120	857	691	273	329	1159	730
2013	8192	870	701	280	344	1193	730
2014	8281	889	718	286	357	1223	730
2015	8315	897	724	292	367	1231	730
2016	8381	914	731	298	378	1254	730
2017	8423	919	739	306	385	1262	731
2018	8446	924	743	310	391	1267	731
2019	8469	928	745	315	398	1271	732
2020	8477	932	746	319	407	1275	773
	Region
Year	Lianyungang	Huaian	Yancheng	Yangzhou	Zhenjiang	Taizhou	Suqian
2011	439	480	724	446	313	463	477
2012	441	480	722	447	315	463	480
2013	443	483	722	447	317	463	486
2014	445	485	722	448	317	464	490
2015	447	487	723	448	318	464	492
2016	450	489	724	449	318	465	494
2017	452	491	724	451	319	465	496
2018	452	493	720	453	320	464	497
2019	451	493	671	455	320	464	497
2020	460	456	671	456	321	452	499

Source(s): Authors' work

Table 2

The aged population in Jiangsu Province from 2011 to 2020 (unit: ten thousand persons)

Year	Region
	Jiangsu	Nanjing	Wuxi	Xuzhou	Changzhou	Suzhou	Nantong
2011	1,340	142	112	45	52	186	122
2012	1,437	152	122	48	58	205	129
2013	1,630	173	140	56	68	237	145
2014	1,648	177	143	57	71	243	145
2015	1,596	172	139	56	71	236	140
2016	1,794	196	157	64	81	268	156
2017	1,920	210	168	70	88	288	167
2018	1,959	214	172	72	91	294	170
2019	2,041	224	180	76	96	306	176
2020	1,851	204	163	70	89	278	169
	Region
Year	Lianyungang	Huaian	Yancheng	Yangzhou	Zhenjiang	Taizhou	Suqian
2011	73	80	121	75	52	77	80
2012	78	85	128	79	56	82	85
2013	88	96	144	89	63	92	97
2014	89	97	144	89	63	92	98
2015	86	94	139	86	61	89	95
2016	96	105	155	96	68	99	106
2017	103	112	165	103	73	106	113
2018	105	114	167	105	74	108	115
2019	109	119	162	110	77	112	120
2020	100	100	147	100	70	99	109

Source(s): Authors' work

Table 3

The proportion of aged population in Jiangsu Province from 2011 to 2020 (Unit: %)

Year	2011	2012	2013	2014	2015	2016	2017	2018	2019	2020
Jiangsu	16.7	17.7	19.9	19.9	19.2	21.4	22.8	23.2	24.1	21.8

Source(s): Authors' work

Table 4

The prediction of Jiangsu total population by grey models (ten thousand people)

Year	Actual value	EGM	NDGM	FGM	GPM	Verhulst	Bass	FDGGM
2011	8023	8023	8023	8023	8023	8025.83	8025.32	8023
2012	8120	8144.47	8116.86	8094.60	8120.00	8123.07	8115.90	8117.29
2013	8192	8198.46	8200.20	8208.84	8202.62	8204.53	8196.98	8200.78
2014	8281	8252.82	8269.75	8283.64	8266.18	8272.52	8268.16	8272.15
2015	8315	8307.53	8327.81	8337.96	8320.69	8329.07	8329.07	8331.26
2016	8381	8362.61	8376.26	8379.77	8369.91	8376.00	8379.40	8378.91
2017	8423	8418.05	8416.70	8413.14	8415.64	8414.85	8418.88	8416.46
2018	8446	8473.86	8450.45	8440.42	8458.94	8446.95	8447.30	8445.53
Fitting MAPE (%)		0.20	0.09	0.15	0.11	0.08	0.07	0.08
2019	8469	8530.04	8478.63	8463.10	8500.44	8473.44	8464.51	8467.70
2020	8477	8586.59	8502.14	8482.19	8540.56	8495.28	8470.43	8484.43
Forecasting MAPE (%)		1.01	0.21	0.07	0.56	0.13	0.07	0.05

Source(s): Authors' work

Table 5

The forecast number of total population in Jiangsu Province and its prefecture-level cities from 2021 to 2025 (unit: ten thousand persons)

Region	Year
	2021	2022	2023	2024	2025
Jiangsu	8489	8496	8501	8505	8508
Nanjing	933	934	934	934	933
Wuxi	748	749	748	747	746
Xuzhou	322	325	328	329	331
Changzhou	414	422	430	439	449
Suzhou	1276	1278	1279	1280	1280
Nantong	783	806	835	869	910
Lianyungang	465	471	479	489	503
Huaian	490	485	491	485	490
Yancheng	656	638	620	601	582
Yangzhou	459	461	463	466	469
Zhenjiang	322	323	324	326	328
Taizhou	465	465	464	463	462
Suqian	500	502	504	506	509

Source(s): Authors' work

Table 6

The forecast number of aged population in Jiangsu Province and its prefecture-level cities from 2021 to 2025 (unit: ten thousand persons)

Region	Year
	2021	2022	2023	2024	2025
Jiangsu	1973	1977	1974	1964	1947
Nanjing	217	218	218	216	214
Wuxi	174	174	173	172	170
Xuzhou	75	76	77	77	77
Changzhou	95	96	96	96	95
Suzhou	295	295	293	290	285
Nantong	178	182	184	187	189
Lianyungang	106	107	107	107	106
Huaian	108	106	104	101	98
Yancheng	153	149	145	140	135
Yangzhou	107	108	108	109	109
Zhenjiang	75	75	75	75	74
Taizhou	106	106	105	104	102
Suqian	116	116	115	114	113

Source(s): Authors' work

Table 7

The proportion of aged population in Jiangsu Province and its prefecture-level cities from 2021 to 2025 (unit: %)

Region	Year
	2021	2022	2023	2024	2025
Jiangsu	23.2	23.3	23.2	23.1	22.9
Nanjing	23.3	23.3	23.3	23.2	23.0
Wuxi	23.2	23.2	23.1	23.0	22.8
Xuzhou	23.3	23.4	23.4	23.4	23.3
Changzhou	22.9	22.6	22.3	21.8	21.1
Suzhou	23.1	23.1	22.9	22.6	22.3
Nantong	22.8	22.5	22.1	21.5	20.8
Lianyungang	22.9	22.7	22.3	21.8	21.1
Huaian	22.0	21.9	21.1	20.8	19.9
Yancheng	23.3	23.4	23.4	23.3	23.2
Yangzhou	23.3	23.4	23.4	23.3	23.2
Zhenjiang	23.2	23.2	23.1	22.9	22.6
Taizhou	22.8	22.7	22.6	22.5	22.2
Suqian	23.0	22.9	22.8	22.5	22.2

Source(s): Authors' work