Comparative analysis of practical identifiability methods for an SEIR model

Omar Saucedo; Amanda Laubmeier; Tingting Tang; Benjamin Levy; Lale Asik; Tim Pollington; Olivia Prosper Feldman; Omar Saucedo; Amanda Laubmeier; Tingting Tang; Benjamin Levy; Lale Asik; Tim Pollington; Olivia Prosper Feldman

doi:10.3934/math.20241204

AIMS Mathematics

2024, Volume 9, Issue 9: 24722-24761. doi: 10.3934/math.20241204

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Research article

Comparative analysis of practical identifiability methods for an SEIR model

1.
Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA
2.
Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA
3.
Department of Mathematics and Statistics, San Diego State University, San Diego, CA 92182, USA
4.
Division of Mathematics, Analytics, Science, and Technology, Babson College, Wellesley, MA 02481, USA
5.
Department of Mathematics and Statistics, University of the Incarnate Word, San Antonio TX 78209, USA
6.
Big Data Institute, University of Oxford, OX1 2JD, Oxford, UK
7.
Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA

Received: 27 April 2024 Revised: 16 July 2024 Accepted: 12 August 2024 Published: 23 August 2024
MSC : 92-10, 92D30, 34C60

Identifiability of a mathematical model plays a crucial role in the parameterization of the model. In this study, we established the structural identifiability of a susceptible-exposed-infected-recovered (SEIR) model given different combinations of input data and investigated practical identifiability with respect to different observable data, data frequency, and noise distributions. The practical identifiability was explored by both Monte Carlo simulations and a correlation matrix approach. Our results showed that practical identifiability benefits from higher data frequency and data from the peak of an outbreak. The incidence data gave the best practical identifiability results compared to prevalence and cumulative data. In addition, we compared and distinguished the practical identifiability by Monte Carlo simulations and a correlation matrix approach, providing insights into when to use which method for other applications.

Keywords:

Citation: Omar Saucedo, Amanda Laubmeier, Tingting Tang, Benjamin Levy, Lale Asik, Tim Pollington, Olivia Prosper Feldman. Comparative analysis of practical identifiability methods for an SEIR model[J]. AIMS Mathematics, 2024, 9(9): 24722-24761. doi: 10.3934/math.20241204

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Abstract

1. Introduction

The relationship between industrial structure and economic growth has been a hotspot in economy for a long time, which generates some remarkable achievement known as Political Arithmetic from William Petty, Classification for three industries by Fisher, “Petty-Clark” law proposed by Clark, national income growth theory from Kuznets and so on. Ding and Knight (2009) explored the remarkable economic growth puzzle of China after the reform and opening-up policy and revealed some indirect determinants including the degree of openness, institutional change and sectoral change, based on a cross-province dataset. However, according to development experience in common industrialized countries, the so-called structural deceleration will turn out as the economic structure develops to some high stage. Structural deceleration, namely economic growth rate is decline with economic structure change. Since industrialization advances toward urbanization, labor was allocated once again and move from industrial sector with high productivity growth rate to service sector with low growth rate, leading to economic growth rate slowed down. This is so-called structural deceleration. There are several origins results in structural deceleration. The first one is represented by the declining efficiency of resource allocation. The structure of resource allocation evolves following the industrial structure. Second, the efficiency of factor supply declines, which is captured by the declining investment yield rate and increasing labour cost. Third, technological innovation capacity needs to be improved. Forth, constraints on environmental resources are going stronger. When we consider environmental constraints, the negative factor of constraints on environmental resources should be added endogenously into the economic growth function of China.

In most existing literatures, the GDP plays the key role in measuring the economic growth. Meanwhile, they neglect the environmental consumption accompanied with the economic growth. In fact, economic growth quality involves two important problems. First, we should coordinate the contradiction among economy, society and environment in the process of economic development, where the economy and environment address the most prominent contradiction. This contradiction between the economy and environment should be coordinated through environmental regulation. However, the environmental regulation results in a larger production cost for the enterprises and then lower production efficiency potentially. This goes against the economic growth. Another problem is the promotion for total factor productivity, who deducts undesirable outputs. The economic growth quality is promoted followed by the promotion of production efficiency. In addition, both the economic development and resource nature of China show heterogeneous characteristic. Accordingly, it should be considered whether the structural deceleration stage has been attained by each district in China.

The primary problem is the measurement for economic growth quality. Green total factor productivity (GTFP) is widely accepted to assess the economic growth quality, which not only takes environment factor but also natural resource consumption, such as energy, into consideration (Feng et al., 2018). A great deal of existing literatures concern about the relationship between industrial structure and GTFP. They focus on the improvement of economic growth quality through the way how industrial structure affects GTFP. First, industrial structure mostly improves GTFP by promoting the development of clean and pollution free industries (Cheng et al., 2018), especially the tertiary industries (Zhang et al., 2014; Tripathy, 2019). Second, different technical inputs and equipment updates demonstrate technical improvement reduces pollutant discharge and then promotes the GTFP (Ambec et al., 2013; Li and Wu, 2017). Furthermore, industrial structure can affect GTFP through optimization and upgrading for industrial structure. Optimization and upgrade of industrial structure improve allocation efficiency as well as competitiveness of professional departments (Zhou et al., 2013; Adom et al., 2012), which then promote the GTFP. These ways result in a great popularity for the impacts of industrial structure on the economic growth quality.

The change of industrial structure is essentially the process of reallocating micro production factors, which will change the structure of resource allocation and the usage efficiency of factor, therefor affecting economic growth quality. As a perpetual theme of economic growth quality, industrial structure has a close relation with economic growth quality, and the reasonable industrial structure has a profound impact on economic growth quality. However, there is dissent with regard to the channels through which industrial structure fosters the economic growth quality. Theoretical models identify multiple channels through which reasonable industrial structure should enhance the economic growth quality, measured by the GTFP, such as human capital, innovation and economic development level. First, the human capital affects the relationship between industrial structure and GTFP through improving not only labor allocation efficiency, but also advanced technology and so on. Second, innovation promotes the technological progress (Corradini et al., 2014; Hashmi and Alam, 2019; Li et al., 2018; Li et al., 2019; Kanamura, 2019), which is beneficial to reduce the pollutant discharge and drive industrial restructuring. This results in the enhancement of GTFP and then the economic growth quality. Third, economic development level promotes the evolution of industrial structure and further affects the economic growth quality (Kwakwa et al., 2018).

Besides the measurement for economic growth quality, in this paper, we devote to exploring the problems on structural deceleration in China through the relation between industrial structure and economic growth quality. First, we concern on whether the structural deceleration happens in China. This conjecture may hold after a high-speed economic growth of China since 1978, when the reform and opening-up policy issued. Second, if structural deceleration happened in China, then we consider the difference of structural deceleration in different districts of China, involving the east, the middle and the west. Economic development of China shows remarkably imbalanced in different districts. It is known that the east addresses a better economic development then the west and the middle. This imbalance motivates us to focus on the heterogeneity of structural deceleration in the east, the middle and the west of China. Third, we investigate the intervening variables playing the intermediate roles between industrial structure and economic growth quality, which may illustrate the influencing mechanism.

Hence, the highlight of the current paper is the structural deceleration and its heterogeneity dominated by districts in China, addressed by the relationship between industrial structure and economic growth quality.

The paper is organized as follows. In Section 2, we study the impact of industrial structure on economic growth quality, followed by the test for structural deceleration in Section 3. In Section 4, we investigate the influencing mechanism of industrial structure on economic growth quality through moderating effect. In Section 5, we conclude the paper with policy implications. See Figure 1 for the logic organization of the paper.

Figure 1. The logical organization of this paper.

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2. The impact of industrial structure on economic growth quality

Our goal in this section is to test the structural deceleration of China through analyzing the impact of industrial structure on economic growth quality. We should set up appropriate model to measure relevant variables in the model.

2.1. Panel regression model

The relation between the industrial structure and economic growth quality has been the subject of considerable debate. First, industrial structure exerts positive effect on economic growth quality. Compared with the secondary industry, the tertiary sector is much less relying on resources and creates fewer pollutants. Therefore, a higher ratio of the service sector may lead to lower pressure on environment, which can, in turn, lead to a better economic growth quality. Second, human capital is an important driving force of change of industrial structure and therefore increases economic growth quality indirectly by accelerating technological change. In addition, innovation can promote the improvement of production technology, and reduce the input of raw materials and energy consumption of unit product (Cheng et al., 2019). Moreover, as the economic development level rises, the industrial structure will also be varied, thus improve economic growth quality. At the same time, we attempt to better investigate the impact of industrial structure on economic growth quality from the perspective of province and time. The panel data framework makes it possible to allow for differences of time in the form of unobservable individual “province effects”.

In this paper, we introduce a panel date model that captures variation of economic growth quality both over time and across provinces. There are three kinds of panel data regression models-the fixed effects model, the random effects model, and the mixed effects model. Firstly, we identify if either mixed or fixed effects are preferred for the regression specification with F-test. The null hypothesis of the F-test which accept mixed effects model is rejected at 1% level. Then, the fixed effects model is used to the regression specification instead of the random effects model through Hausman test (Hausman, 1978). Finally, we applied a fixed effects model to analyze the impact of industrial structure on economic growth quality. To assess the extent to which industrial structure affects economic growth quality, we estimate the following regression specification:

$eg{q_{it}} = {\beta _0} + {\beta _1}\ln ind{u_{it}} + {\beta _2}\ln h{c_{it}} + {\beta _3}\ln inn{o_{it}} + {\beta _4}perGD{P_{it}} + {\mu _i} + {\gamma _t} + {\varepsilon _{it}}$

(1)

where i and t represent province and time respectively; $eg{q_{it}}$ is a dependent variable refers to economic growth quality, which is measured by GTFP; $\ln ind{u_{it}}$ is an independent variable denotes industrial structure; $\ln h{c_{it}}$ , $\ln inn{o_{it}}$ , $perGD{P_{it}}$ are control variables denote human capital, innovation capacity and the level of economic development respectively. The specification also includes province-fixed effects, ${\mu _{\rm{i}}}$ , and year-fixed effects, ${\gamma _{\rm{t}}}$ , ${\varepsilon _{it}}$ is a stochastic disturbance term.

2.2. Variable and Data

2.2.1. Measurement of economic growth quality

This section measures economic growth quality of China’s 30 provinces (excluding Tibet) from 1998 to 2015. The Global Malmquist–Luenberger index developed by and SBM-based global directional distance function could be regarded as an effective tool to measure GTFP and then the economic growth quality. A province i is represented as $DM{U_i}$ . Under a panel of $i = 1, ..., I$ provinces and $t = 1, ..., T$ time periods, the production technology for each province obtains $m$ kinds of desired output ( $y$ , $y \in {R^{{m_ + }}}$ ) and $n$ kinds of undesired output ( $b$ , $b \in {R^{{n_ + }}}$ ) by using $p$ inputs ( $x$ , $x \in {R^{{p_ + }}}$ ). In order to emphasis consistency and comparability of the production frontier, constructed the global production possibility set ${P^G}(x)$ . Thus, we construct the global production possibility set ${P^G}(x)$ as follows:

${P^G}(x) = \left\{ \begin{gathered} \left({{y^t}, {b^t}} \right):\sum\nolimits_{t = 1}^T { \sum\nolimits_{i = 1}^I {z_i^ty_{im}^t \geqslant y_{im}^t, \forall m;\sum\nolimits_{t = 1}^T { \sum\nolimits_{i = 1}^I {z_i^tb_{in}^t = b_{in}^t, \forall n;} } } } \\ \sum\nolimits_{t = 1}^T { \sum\nolimits_{i = 1}^I {z_i^tx_{ip}^t \leqslant x_{ip}^t, \forall p;\sum\nolimits_{t = 1}^T { \sum\nolimits_{i = 1}^I {Z_i^t = 1, z_i^t \geqslant 0, \forall i;} } } } \\ \end{gathered} \right\},$

(2)

where $z_i^t$ represents the weight of each cross-section.

Taking the influence of input and output slack variables on efficiency into consideration, we apply the global SBM directional function considering undesired output according the research of Fukuyama and Weber (2009) as well as Liu and Xin (2019). The global SBM directional function is constructed:

$\begin{gathered} \vec S_{}^G\left({{x^{t, {i'}}}, {y^{t, {i'}}}, {b^{t, {i'}}}, {g^x}, {g^y}, {g^b}} \right) = \mathop {\max }\limits_{{s^x}, {s^y}, {s^b}} \frac{{\frac{1}{p}\sum\limits_{p = 1}^P {\frac{{S_p^x}}{{g_p^x}} + \frac{1}{{M + N}}\left({\sum\limits_{m = 1}^M {\frac{{S_m^y}}{{g_m^y}} + \sum\limits_{n = 1}^N {\frac{{S_n^b}}{{g_n^b}}} } } \right)} }}{2} \\ s.t. \sum\limits_{t = 1}^T {\sum\limits_{i = 1}^I {z_i^t} } x_{ip}^t + s_p^x = x_{i'p}^t, \forall p;\sum\limits_{t = 1}^T {\sum\limits_{i = 1}^I {z_i^t} } y_{im}^t - s_m^y = y_{i'm}^t, \forall m; \\ \sum\limits_{t = 1}^T {\sum\limits_{i = 1}^I {z_i^t} } b_{in}^t + s_n^b = b_{i'p}^t, \forall n;\sum\limits_{i = 1}^I {z_i^t} = 1, z_i^t \geqslant 0, \forall i;s_m^y \geqslant 0, \forall m;s_n^b \geqslant 0, \forall n \\ \end{gathered},$

(3)

where ( ${g^x}$ , ${g^y}$ , ${g^b}$ ) refers to the direction vectors for decreasing inputs, increasing desirable outputs and decreasing undesirable outputs, respectively; ( $s_p^x$ , $s_m^y$ , $s_n^b$ ) denotes the slack vectors for redundant inputs, inadequate desirable outputs and redundant undesirable outputs, respectively.

Drawing from the study of Oh (2010), we construct the Global Malmquist–Luenberger index based on SBM-based directional distance function as follows:

$GML_t^{t + 1} = \frac{{1 + {{\vec S}^G}\left({{x^t}, {y^t}, {b^t};{g^x}, {g^y}, {g^b}} \right)}}{{1 + {{\vec S}^G}\left({{x^{t + 1}}, {y^{t + 1}}, {b^{t + 1}};{g^x}, {g^y}, {g^b}} \right)}},$

(4)

where, the GML index denotes the change from time t to time t + 1.

Then, the $egq$ in 1998 would be $egq$ in 1997 multiplied by the GML index. At the same time, $egq$ of each province in 1997 is 1. Therefore, the $egq$ in 1998 can be expressed as: $eg{q_{1998}}{\rm{ = }}eg{q_{1997}}$ $\times GML_{1997}^{1998}$ . The $egq$ of other years can be calculated similarly.

2.2.2. Explanatory variables and descriptive statistics

In our model, we concentrate on the role played by the industrial structure. In the empirical study, we controlled for possible confounding effects by including various relevant control variables, such as human capital, innovation capacity, the level of economic development. However, certain variables should be calculated with its original data. The GTFP, who assesses the economic growth quality, is measured through input (Li and Lin, 2016), desired output (Long et al., 2018) and undesired output (). Input includes labor input, capital input and energy input. Labor input is measured by the number of year-end of employed persons in each province, where a unit is 10 thousand persons (). Capital input is calculated through capital stock. Capital stock was estimated through perpetual inventory method () in this paper, and the formula is ${K_{it}} = {K_{i(t - 1)}}\left({1 - \delta } \right) + {I_{it}}$ , where ${K_{it}}$ , ${I_{it}}$ denote the actual capital stock and the gross fixed capital formation in province i at time t, respectively. $\delta$ refers to the depreciation rate. According to the research of Zhang and Tan (2016), we adopt the value of 9.6% as depreciation rate. Year 1997 is the base period. Energy input is measure by total energy consumption which is used to measure energy input. It can be express that GDP divided by GDP per unit of energy consumption. The unit of energy input is 10000tce. We used the real GDP to measure desired output. Real GDP is converted based on the year 1997 to ensure data are comparable. Undesired output mainly includes waste water, waste gas as well as solid waste. Because provincial solid waste is unavailable, we exclude it from this paper. Consequently, we consider industrial waste discharge and total industrial waste gas discharge as undesirable outputs. The unit of waste water and waste gas are 10000 tons and 100 million cube meters, respectively.

Industrial structure is measured by the coefficient of industrial structure, which can reflect the industrial distribution. According to the principle of industrial evolution from “Petty-Clark” law, we adopt the sum of the added value proportion of the three industries accounting for the added value of the primary industry to calculate the coefficient of industrial structure, so as to reflect the allocation of factors in the three industries and reflect the change of industrial structure. Industrial structure of province i at time t is measured by:

$ind{u_{it}} = \frac{{GDP_{it}^1}}{{GDP_{it}^1}} + \frac{{GDP_{it}^2}}{{GDP_{it}^1}} + \frac{{GDP_{it}^3}}{{GDP_{it}^1}},$

(5)

where $GDP_{it}^1$ , $GDP_{it}^2$ , $GDP_{it}^3$ refer to the gross regional product of the primary industry, the secondary industry and the tertiary industry, respectively.

Additionally, human capital is the years of education of the labors and measured by millions of persons. Innovation capacity denotes the natural logarithmic forms of patent applications per million people. The level of economic development is measured by per capita gross regional product, where a unit of is one thousand yuan.

In our empirical analysis, we employ original data from different sources. In measuring economic growth quality, we make use of the data of the labor of China’s 30 provinces during 1997–2015, total year consumption of every area and data on waste gas and waste water from EPS macro database. Meanwhile, we also obtain the data on the gross fixed capital formation for 1997–2015 and real GDP from National Bureau of Statistic of China. In addition, we calculate structural transformation, human capital and innovation capacity on the basis of data from EPS macro database.

The descriptive statistics of variables are presented in . Descriptive statistics are presented to summarize the basic characteristics of data in this study concerning on China’s 30 provinces during the period 1990–2014. For each variable, we present the mean, standard deviation (Std. Dev.), minimum (Min), 0.25, 0.5 and 0.75 quantiles and maximum (Max). As shown in the first row, the average economic growth quality of China’s 30 provinces is 1.012 with the minimum value 0.320 and the maximum 3.000, while there is no significant divergence on the mean of economic growth quality and the 0.5 quantile 0.75. The second row reports the variation of industrial structure. The logarithm of coefficient of industrial structure ranges from 1.009 to 5.433, while the average is 2.239 which signally higher than the 0.5 quantile 2.052. Besides, different quantiles can describe different distribution trends. According to last three rows, the $\ln hc$ ranges from 5.789 to 10.800, and the $\ln inno$ ranges from 4.820 to 13.131. And we find that the average per capita GDP is 25.239, which is significantly higher than 0.5 quantile 19.056, and there is great difference between the minimum 2.364 and the maximum 107.960.

Table 1. Descriptive statistics.

Variable	Mean	Std.Dev.	Min	0.25	0.5	0.75	Max
egq	1.012	0.431	0.320	0.788	0.950	1.121	3.000
lnindu	2.239	0.818	1.009	1.743	2.052	2.426	5.433
lnhc	8.487	0.848	5.789	8.053	8.548	8.996	10.800
lninno	8.981	1.655	4.820	7.794	8.887	10.176	13.131
perGDP	25.239	21.292	2.364	8.732	19.056	35.170	107.960
Notes: “ln” means the variable in natural logarithms.

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2.3. Testing the impact of industrial structure on economic growth quality

The unit root test is performed to check whether the variables are stationarity before we test the impact of industrial structure on economic growth quality (). reports the results for a battery of unit root tests for $egq$ , $\ln indu$ , $\ln hc$ , $\ln inno$ and $perGDP$ . In particular, we report results from two tests of the null hypothesis that each series contains a unit root. The first is the Levin-Lin-Chu unit-root test (Levin et al., 2002); the second is the standard Augmented Dickey-Fuller t-test (Dickey and Fuller, 1981). In all two cases, the tests reject the null hypothesis of contain unit roots at the 5% level of significance. Consequently, we can conclude that all variables are stationary.

Table 2. Unit root test.

variables	LLC	ADF
variables	LLC	Inverse chi-squared(60)	Inverse normal	Inverse logit t(154)	Modified inv. chi-squared
egq	0.000 (-5.579)	0.000 (144.664)	0.000 (-5.262)	0.000 (-5.690)	0.000 (7.729)
lnindu	0.0001 (-3.732)	0.000 (147.489)	0.000 (-6.350)	0.000 (-6.453)	0.000 (7.987)
lnhc	0.000 (-4.1523)	0.000 (157.881)	0.000 (-6.051)	0.000 (-6.353)	0.000 (8.935)
lninno	0.023 (-2.0044)	0.000 (171.526)	0.000 (-7.915)	0.000 (-8.174)	0.000 (10.181)
perGDP	0.000 (-7.405)	0.000 (120.700)	0.000 (-3.461)	0.000 (-3.515)	0.000 (5.541)
Notes: (1) Standard errors are in parentheses. (2) *, , and * indicate significance at the 1%, 5%, and 10% levels, respectively.

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In order to test the impact of industrial structure on economic growth quality, we employ the panel regression model to estimate the parameter for the full sample, subsample of the east, subsample of the middle and subsample of the west. A summary of the estimation results for fixed effects model is presented in Table 3.

Table 3. The result of impact of industrial structure on economic growth quality.

	FE	GMM	2SLS	East	Middle	West
lnindu	0.1669**	0.2635***	0.2635***	0.8696***	-0.1314	-0.2713***
lnindu	(0.0684)	(0.0316)	(0.0292)	(0.1176)	(0.1163)	(0.0564)
lnhc	-0.2509***	0.1186**	0.1186**	-0.3649***	-0.2264	0.1376**
lnhc	(0.0660)	(0.0488)	(0.0482)	(0.1020)	(0.1405)	(0.0547)
lninno	-0.1210***	-0.1380***	-0.1380***	-0.3522***	0.0048	0.0236
lninno	(0.0241)	(0.0314)	(0.0295)	(0.0495)	(0.0450)	(0.0201)
perGDP	0.0147***	0.0074***	0.0074***	0.0200***	0.0057	-0.0021
perGDP	(0.0014)	(0.0015)	(0.0013)	(0.0022)	(0.0040)	(0.0013)
Constant	3.4827***	0.4639**	0.4639**	4.7238***	3.0289***	0.1848
Constant	(0.4573)	(0.2119)	(0.2212)	(0.7375)	(1.1053)	(0.3452)
N	540	510	510	198	144	198
R-squared	0.2820	0.3675	0.3675	0.6710	0.0427	0.1838
F-statistic	35.57	-	-	21.20	70.84	62.48
Notes: (1) Standard errors are in parentheses. (2) *, , and * indicate significance at the 1%, 5%, and 10% levels, respectively.

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In general, industrial structure has a positive correlation with economic growth quality. As seen in the Table 3, the results imply this positive correlation, which is significant at 5% level. The coefficient indicates that one percent improvement in the coefficient of industrial structure increases leads to a 0.167 increase in economic growth quality. This result may be closely related to these reasons. First, the development of clean and pollution-free industries, especially tertiary industries, are beneficial to the improvement of GTFP and then economic growth quality. Second, because of the upgrade of technological level, the economic growth quality is promoted by reducing pollutant emissions. Moreover, the promotion of allocation efficiency and specialization can not only improve the competitiveness of the industrial department (Feldman and Audretsch, 1999), but also help raise the quality and level of the service industry (Miles, 2008). And this cooperation mechanism can further improve the economic growth quality. The industrial structure in the east of China shows positive correlation with the economic growth quality. On the contrary, the industrial structure in the west of China shows negative correlation with the economic growth quality. Due to imbalance of economic development in the east and west of China, the different industrial structure levels result in different roles in improving the economic growth quality. A reasonable industrial structure is a significant factor on economic growth quality. On the one hand, the west of China with a low industrial development, on the other hand, there are many constraints on the upgrading industrial structure. Base on above, the west of China with an unreasonable industrial structure. Therefore, a low industrial structure level in the west of China indeed restrains the economic growth quality of China and goes against the sustainable development. The economic development in the east addresses a much larger weight than that in the west of China. This results in the fact that the signs of correlation between industrial structure and economic growth quality are consistent for the full sample and the subsample of the east.

3. Structural deceleration by the impact of industrial structure on economic growth quality

The impact of industrial structure on economic growth quality provides some information for the structural deceleration of China. This is investigated in this section through a panel quantile regression analysis in this section.

3.1. Panel quantile regression model

The impact of industrial structure on economic growth quality has been attributed, in no small part, to differences at quantiles of economic growth quality. On one hand, as a developing country, China is in a pivotal transformation stage, where the economic growth quality is transferring from low to high level. On the other hand, when the economic growth quality is at a low quantile, optimization and upgrading of industrial structure can be more capable to improve allocation efficiency, environmental quality and productivity, thus improves the economic growth quality much more easily. In short, the impact of industrial structure on economic growth quality at different quantiles shows different degrees. Therefore, we employ panel quantile regression model to further investigate the impact of industrial structure on economic growth quality at different quantiles.

Panel quantile regression model is a powerful tool to provide a more complete picture for the relationship between industrial structure and economic growth quality. Compared with some conventional regression, panel quantile regression may be preferable. While usual regressions, like OLS, only consider the mean, quantile regression can evaluate the different points on the full conditional distribution of the dependent variable (Canay, 2011). Therefore, this method can avoid heterogeneity problems effectively in the data distribution, and it is robust to the presence of heteroscedasticity, outliers and structural change. It allows us to draw conclusion on the impacts of industrial structure on economic growth quality in different provinces at different quantiles of the economic growth quality distribution.

We specify the following panel quantiles function:

${Q_{eg{q_{it}}}}\left({\tau |{\theta _i}, {\delta _t}, {X_{it}}} \right) = {\theta _i} + {\delta _t} + {\beta _{1\tau }}\ln indu + {\beta _{2\tau }}\ln h{c_{it}} + {\beta _{3\tau }}\ln inn{o_{it}} + {\beta _{4\tau }}perGD{P_{it}} + {\varepsilon _{it}},$

(6)

where i and t represent province and time respectively; $eg{q_{it}}$ is a dependent variable refers to economic growth quality; $\ln ind{u_{it}}$ is an independent variable denotes industrial structure; $\ln h{c_{it}}$ , $\ln inn{o_{it}}$ , $perGD{P_{it}}$ are control variables denote human capital, innovation capacity and the level of economic development respectively. The specification also includes province-fixed effects ${\theta _i}$ , and year-fixed effects ${\delta _t}$ , ${\varepsilon _{it}}$ is a stochastic disturbance term.

3.2. Empirical results: structural deceleration

The empirical results are presented in Table 4, 5, 6 and 7. Table 4 shows the panel quantile regression results for full sample, while Table 5, 6 and 7 show the panel quantile regression results for subsamples in the east, middle and west of China respectively. As shown in Table 4, the impact of 1% increase taken by the industrial structure on the economic growth quality is 0.263 at the 10% percentile, in comparison to only 0.105 at the 90% percentile. It is evident that this positive effect of industrial structure on economic growth quality decreases gradually with increased quantiles. We conclude that the structural deceleration has happened in China. Although the industrial structure still plays a positive role in the economic growth quality, the effect is decreasing. Since China was experiencing its transformation stage at the low quantile of economic growth quality, it is easier to optimize and upgrade industrial structure. Meanwhile, the industrial structure could show notably positive effect on the environment quality as well as productivity, which improve the GTFP and then the economic growth quality, recalling that GTFP as the measurement assessing the economic growth quality. However, we note that the structural deceleration does not happen in the east of China. The industrial structure still plays an important role in improving the economic growth quality. One can see from Table 5 that there is jump between the contributions of industrial structure to the economic growth quality at the quantile 75% and 90%, where the corresponding contributions are given as 0.2593 and 0.4047 respectively. This might be due to a relatively rational industrial structure in the east of China and it keeps a notable effect on the economic growth quality at each stage. The industrial structure in the middle shows negative effect on the economic growth quality. In addition, one can see from Table 6 that the negative effect is anabatic with the increase of economic growth quality. In other words, in the middle of China, the industrial structure absolutely restrains the improvement of economic growth quality. The panel quantile regression results for subsample of the west are not remarkable. The level of industrial structure is relatively low, such that it cannot contribute clearly to the economic growth quality at each stage.

Table 4. Panel quantile regression results for full sample.

	QR_10	QR_25	QR_50	QR_75	QR_90
lnindu	0.263***	0.198***	0.171***	0.102***	0.105**
lnindu	(0.044)	(0.027)	(0.033)	(0.034)	(0.046)
lnhc	-0.016	0.091**	0.098*	0.099*	0.275***
lnhc	(0.072)	(0.044)	(0.054)	(0.055)	(0.074)
lninno	-0.065	-0.081***	-0.132***	-0.125***	-0.211***
lninno	(0.044)	(0.027)	(0.033)	(0.034)	(0.046)
perGDP	0.001	0.0004	0.007***	0.013***	0.020***
perGDP	(0.002)	(0.001)	(0.002)	(0.002)	(0.002)
constant	0.693**	0.300	0.784***	0.933***	0.221
constant	(0.332)	(0.202)	(0.248)	(0.255)	(0.343)
N	540	540	540	540	540
Notes: (1) Standard errors are in parentheses. (2) *, , and * indicate significance at the 1%, 5%, and 10% levels, respectively.

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Table 5. Panel quantile regression results for subsample of the east.

	QR_10	QR_25	QR_50	QR_75	QR_90
lnindu	0.2676***	0.2254***	0.2431***	0.2593***	0.4047***
lnindu	(0.0596)	(0.0310)	(0.0310)	(0.0462)	(0.0502)
lnhc	0.2624	0.1928**	0.2301***	0.3251***	0.2067
lnhc	(0.1609)	(0.0836)	(0.0837)	(0.1247)	(0.1354)
lninno	-0.2109**	-0.3255***	-0.3275***	-0.3663***	-0.3308***
lninno	(0.1011)	(0.0525)	(0.0526)	(0.0783)	(0.0850)
perGDP	0.0040	0.0146***	0.0150***	0.0174***	0.0167***
perGDP	(0.0031)	(0.0016)	(0.0016)	(0.0024)	(0.0026)
Constant	-0.2954	1.3383***	1.1044***	0.6800	1.1977**
Constant	(0.6678)	(0.3467)	(0.3474)	(0.5173)	(0.5619)
N	198	198	198	198	198
Notes: (1) Standard errors are in parentheses. (2) *, , and * indicate significance at the 1%, 5%, and 10% levels, respectively.

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Table 6. Panel quantile regression results for subsample of the middle.

	QR_10	QR_25	QR_50	QR_75	QR_90
lnindu	-0.1503***	-0.2196**	-0.2855***	-0.3152**	0.3705
lnindu	(0.0361)	(0.1084)	(0.0828)	(0.1562)	(0.5746)
lnhc	-0.3847***	0.1972	-0.1596	-0.5573**	0.0549
lnhc	(0.0556)	(0.1673)	(0.1278)	(0.2410)	(0.8865)
lninno	-0.1485***	-0.2070***	-0.0927*	0.1015	-0.1770
lninno	(0.0243)	(0.0729)	(0.0557)	(0.1050)	(0.3864)
perGDP	0.0204***	0.0107**	0.0123***	0.0104	0.0252
perGDP	(0.0017)	(0.0052)	(0.0040)	(0.0075)	(0.0278)
Constant	5.1399***	1.0890	3.3879***	5.3879***	1.3618
Constant	(0.3746)	(1.1261)	(0.8599)	(1.6222)	(5.9669)
N	144	144	144	144	144
Notes: (1) Standard errors are in parentheses. (2) *, , and * indicate significance at the 1%, 5%, and 10% levels, respectively.

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Table 7. Panel quantile regression results for subsample of the west.

	QR_10	QR_25	QR_50	QR_75	QR_90
lnindu	0.4960**	-0.1024	0.0102	0.0765	0.1296
lnindu	(0.2113)	(0.1543)	(0.0624)	(0.0552)	(0.1183)
lnhc	0.0435	0.0633	0.1255***	0.1096***	0.1593**
lnhc	(0.1393)	(0.1017)	(0.0411)	(0.0364)	(0.0780)
lninno	-0.0510	-0.0442	-0.0430*	-0.0045	-0.0527
lninno	(0.0756)	(0.0552)	(0.0223)	(0.0197)	(0.0423)
perGDP	-0.0080*	-0.0002	-0.0026*	-0.0037***	-0.0002
perGDP	(0.0047)	(0.0034)	(0.0014)	(0.0012)	(0.0026)
Constant	-0.1518	0.9156	0.3185	0.1109	0.0052
Constant	(0.8721)	(0.6368)	(0.2573)	(0.2277)	(0.4884)
N	198	198	198	198	198
Notes: (1) Standard errors are in parentheses. (2) *, , and * indicate significance at the 1%, 5%, and 10% levels, respectively.

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4. Influencing mechanism: moderating effect

In this section, we will take a further step to detect the influencing mechanism of industrial structure on economic growth quality.

4.1. The moderating effect of human capital

Human capital exerts a moderating effect on the impact of industrial structure on economic growth quality. On the one hand, an increase in human capital directly relates to the improvement of employees’ ability, enhancing labor productivity, as well as be conducive to optimize industrial structure. On the other hand, human capital may introduce more advanced knowledge and managerial skills to update industrial structure. In other words, the increase of human capital can be beneficial to optimize industrial structure and improve productivity, thus improve the economic growth quality. Therefore, based on model (2), we add human capital and industrial structure as interaction term to explore the moderating effect of human capital. The model can be represented by the following formulas:

$\begin{gathered} {Q_{eg{q_{it}}}}\left({\tau |{\theta _{i1}}, {\delta _{t1}}, {X_{it}}} \right) = {\theta _{i1}} + {\delta _{t1}} + {\beta _{1\tau }}c\ln indu + {\beta _{2\tau }}c\ln h{c_{it}} + {\beta _{3\tau }}c\ln indu\_c\ln h{c_{it}} + \\ {\beta _{4\tau }}c\ln inn{o_{it}} + {\beta _{5\tau }}cperGD{P_{it}} + {\varepsilon _{it}} \\ \end{gathered},$

(7)

where i and t represent province and time respectively; $eg{q_{it}}$ is a dependent variable refers to economic growth quality; $\ln ind{u_{it}}$ is an independent variable denotes industrial structure; $\ln h{c_{it}}$ , $\ln inn{o_{it}}$ , $perGD{P_{it}}$ are control variables denote human capital, innovation capacity and the level of economic development respectively; $c\ln indu$ _ $c\ln hc$ stands for interaction terms. The symbol c before variables means that the variables have been centralized (the follow-up c is the same). The specification also includes province-fixed effects ${\theta _{ij}}$ , and year-fixed effects ${\delta _{tj}}$ , ${\varepsilon _{it}}$ is a stochastic disturbance term.

The estimates presented in Table 8 focus on moderating effect of human capital on the impact of industrial structure on economic growth quality. To be more precise about the impact of industrial structure on economic growth quality, we further investigate the moderating effect of human capital under different economic growth quality levels. The results are respectively reported for the 10th, 25th, 50th, 75th, 90th quantiles of the conditional economic growth quality distribution in Columns 3–7.

Table 8. Results of moderating effect of human capital.

	FE	QR_10	QR_25	QR_50	QR_75	QR_90
clnindu	0.142**	0.263***	0.175***	0.179***	0.180***	0.135***
clnindu	(0.059)	(0.026)	(0.032)	(0.029)	(0.032)	(0.051)
clnhc	-0.210***	-0.031	0.088*	0.172***	0.117**	0.290***
clnhc	(0.057)	(0.042)	(0.052)	(0.047)	(0.051)	(0.083)
clnindu_clnhc	0.280***	0.202***	0.193***	0.201***	0.162***	0.133***
clnindu_clnhc	(0.022)	(0.025)	(0.031)	(0.028)	(0.031)	(0.050)
lninno	-0.029	-0.053**	-0.088***	-0.120***	-0.068**	-0.154***
lninno	(0.022)	(0.026)	(0.033)	(0.029)	(0.032)	(0.052)
perGDP	0.007***	0.001	-0.00003	0.002*	0.007***	0.013***
perGDP	(0.001)	(0.001)	(0.002)	(0.001)	(0.002)	(0.003)
_cons	1.029***	1.022***	1.565***	1.941***	1.585***	2.380***
_cons	(0.193)	(0.224)	(0.280)	(0.250)	(0.273)	(0.444)
N	540	540	540	540	540	540
R-squared	0.460
F-statistic	85.86
Notes: (1) Standard errors are in parentheses. (2) *, , and * indicate significance at the 1%, 5%, and 10% levels, respectively.

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Human capital plays a positive moderating effect on the impact of industrial structure on economic growth quality and it is heterogeneous at different quantiles of economic growth quality. The positive sign on coefficient of $c\ln indu$ _ $c\ln hc$ directs that human capital improves the positive effect of industrial structure on economic growth quality. The reason for this phenomenon may be that human capital improves the ability of employees and enhance labor productivity to strengthen the impact of industrial structure on economic growth quality. Besides, the significant divergence of moderating effect of human capital at different quantiles can be observed. The coefficient of interaction between industrial structure and human capital is 0.202 at the 10th quantile, in comparison to 0.133 at 90th quantile, which implies that this positive moderating effect of human capital on the impact of industrial structure on economic growth quality inclines to decrease.

4.2. The moderating effect of innovation capacity

Innovation capacity plays the moderating effect on the impact of industrial structure on economic growth quality. Innovation improves the industrial structure through both acquiring new products and improving existing varieties. Besides, increasing innovation capacity can be capable to transform useful knowledge, promote technological progress (Li et al., 2019; Wen et al., 2019). As a result, it can reduce the emission of pollutants to improve environment quality, as well as improves the industrial structure. Consequently, industrial structure raised economic growth quality. Therefore, based on model (2), we add innovation capacity and industrial structure as interaction term to explore the moderating effect of innovation capacity. The model can be represented by the following formulas:

$\begin{gathered} {Q_{eg{q_{it}}}}\left({\tau |{\theta _{i2}}, {\delta _{t2}}, {X_{it}}} \right) = {\theta _{i2}} + {\delta _{t2}} + {\beta _{1\tau }}c\ln indu + {\beta _{2\tau }}c\ln inn{o_{it}} + {\beta _{3\tau }}c\ln indu\_c\ln inn{o_{it}} + \\ {\beta _{4\tau }}c\ln h{c_{it}} + {\beta _{5\tau }}cperGD{P_{it}} + {\varepsilon _{it}} \\ \end{gathered},$

(8)

where i and t represent province and time respectively; $eg{q_{it}}$ is a dependent variable refers to green total factor productivity; $\ln ind{u_{it}}$ is an independent variable denotes industrial structure; $\ln h{c_{it}}$ , $\ln inn{o_{it}}$ , $perGD{P_{it}}$ are control variables denote human capital, innovation capacity and the level of economic development respectively; $c\ln indu$ _ $c\ln {\rm{inno}}$ stands for interaction terms. The specification also includes province-fixed effects ${\theta _{ij}}$ , and year-fixed effects ${\delta _{tj}}$ , ${\varepsilon _{it}}$ is a stochastic disturbance term.

The estimates presented in Table 9 focus on moderating effect of innovation capacity on the impact of industrial structure on economic growth quality. Besides, we also further investigate the moderating effect of innovation capacity under different economic growth quality levels. The results are respectively reported for the 10th, 25th, 50th, 75th, 90th quantiles of the conditional economic growth quality distribution in Columns 3–7.

Table 9. Results of moderating effect of innovation capacity.

	FE	QR_10	QR_25	QR_50	QR_75	QR_90
clnindu	0.120**	0.251***	0.161***	0.166***	0.115***	0.126**
clnindu	(0.056)	(0.041)	(0.031)	(0.024)	(0.026)	(0.056)
clninno	0.009	-0.063	-0.089***	-0.054**	0.016	-0.073
clninno	(0.021)	(0.041)	(0.031)	(0.024)	(0.026)	(0.056)
clnindu_clninno	0.146***	0.108***	0.121***	0.139***	0.142***	0.137***
clnindu_clninno	(0.009)	(0.020)	(0.015)	(0.012)	(0.013)	(0.027)
lnhc	-0.172***	-0.023	0.090*	0.081**	0.047	0.224**
lnhc	(0.054)	(0.065)	(0.049)	(0.038)	(0.041)	(0.089)
perGDP	0.004***	0.0008	0.0005	-0.000009	0.003**	0.005*
perGDP	(0.001)	(0.002)	(0.002)	(0.001)	(0.001)	(0.002)
_cons	2.276***	0.741	-0.023	0.187	0.591*	-0.799
_cons	(0.443)	(0.562)	(0.428)	(0.333)	(0.355)	(0.770)
N	540	540	540	540	540	540
R-squared	0.529
F-statistic	113.33
Notes: (1) Standard errors are in parentheses. (2) *, , and * indicate significance at the 1%, 5%, and 10% levels, respectively.

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Innovation capacity exerts a positive moderating effect on the impact of industrial structure on economic growth quality and it is heterogeneous at different quantiles of economic growth quality. The results in reveal that the interaction of industrial structure and innovation capacity acts on economic growth quality positively. This result suggests that innovation capacity accelerates the positive effect of industrial structure on economic growth quality. Besides, innovation capacity plays heterogeneous moderating effect at different quantiles. The coefficients of interaction term $c\ln indu$ _ $c\ln {\rm{inno}}$ are 0.108, 0.121, 0.139, 0.142 and 0.137 respectively, at the 10th-90th quantile. These results indicate that as the quantiles of economic growth quality increases, the positive moderating effect of innovation capacity on the impact of industrial structure on economic growth quality increases, implying that provinces with higher economic growth quality may result in better innovation capacity to enhance the positive effect of industrial structure on economic growth quality.

4.3. The moderating effect of economic development level

Economic development level plays a moderating effect on the impact of industrial structure on economic growth quality. High economic development level promotes the evolution of industrial structure by creating more advanced technology (Li and Zhong, 2019), accelerating innovation and promoting coordinated development of industrial structural contradictions, further improves economic growth quality. That is to say, economic development level accelerates the influence of industrial structure on economic growth quality. Therefore, based on model (2), we add economic development level and industrial structure as interaction term to explore the moderating effect of economic development level. The model can be represented by the following formulas:

$\begin{gathered} {Q_{eg{q_{it}}}}\left({\tau |{\theta _{i3}}, {\delta _{t3}}, {X_{it}}} \right) = {\theta _{i3}} + {\delta _{t3}} + {\beta _{1\tau }}c\ln indu + {\beta _{2\tau }}cperGD{P_{it}} + {\beta _{3\tau }}c\ln indu\_cperGD{P_{it}} + \\ {\beta _{4\tau }}c\ln h{c_{it}} + {\beta _{5\tau }}c\ln h{c_{it}} + {\varepsilon _{it}} \\ \end{gathered},$

(9)

where i and t represent province and time respectively; $eg{q_{it}}$ is a dependent variable refers to economic growth quality; $\ln ind{u_{it}}$ is an independent variable denotes industrial structure; $\ln h{c_{it}}$ , $\ln inn{o_{it}}$ , $perGD{P_{it}}$ are control variables denote human capital, innovation capacity and the level of economic development respectively; $c\ln indu$ _ $cperGDP$ stands for interaction term of industrial structure and economic development level. The specification also includes province-fixed effects ${\theta _{ij}}$ , and year-fixed effects ${\delta _{tj}}$ , ${\varepsilon _{it}}$ is a stochastic disturbance term.

The estimates presented in Table 10 focus on moderating effect of economic development level on the impact of industrial structure on economic growth quality. Furthermore, we also investigate the moderating effect of economic development level under different economic growth quality levels. The results are respectively reported for the 10th, 25th, 50th, 75th, 90th quantiles of the conditional GTFP distribution in Columns 3–7.

Table 10. Results of moderating effect of economic development level.

	FE	QR_10	QR_25	QR_50	QR_75	QR_90
clnindu	-0.057	0.224***	0.074***	0.093***	0.074**	0.022
clnindu	(0.050)	(0.054)	(0.025)	(0.025)	(0.030)	(0.051)
cperGDP	-0.001	-0.001	-0.002	-0.0007	0.003**	0.007***
cperGDP	(0.001)	(0.003)	(0.001)	(0.001)	(0.001)	(0.002)
clnindu_cperGDP	0.008***	0.007***	0.009***	0.008***	0.007***	0.008***
clnindu_cperGDP	(0.0004)	(0.001)	(0.0005)	(0.0006)	(0.0007)	(0.001)
lnhc	-0.060	-0.080	0.014	-0.022	0.046	0.176**
lnhc	(0.048)	(0.081)	(0.037)	(0.038)	(0.044)	(0.076)
lninno	0.061***	-0.013	-0.035	-0.003	-0.026	-0.081*
lninno	(0.019)	(0.052)	(0.024)	(0.024)	(0.028)	(0.048)
_cons	0.866***	1.375***	0.923***	1.081***	0.905***	0.464
_cons	(0.376)	(0.353)	(0.162)	(0.165)	(0.195)	(0.331)
N	540	540	540	540	540	540
R-squared	0.628
F-statistic	170.55
Notes: (1) Standard errors are in parentheses. (2) *, , and * indicate significance at the 1%, 5%, and 10% levels, respectively.

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Economic development level exerts a positive moderating effect on the impact of industrial structure on economic growth quality, but there is similar at different quantiles of economic growth quality. The positive impact of the interaction between industrial structure and economic development level can be observed in . The results confirm that economic development level stimulate the positive influence of industrial structure on economic growth quality. In other words, the moderating effect of economic development level is positive. We can interpret these results as evidence that economic development level promotes the evolution of industrial structure and further improves the economic growth quality. Moreover, the coefficient of $c\ln indu$ _ $cperGDP$ is relatively stable and fluctuates between the ranges of 0.007 to 0.009. That is to say, there is no significant difference among different quantiles of economic growth quality. The moderating effects of economic development level are similar regardless of quantiles of economic growth quality.

5. Conclusions and policy implications

The article’s empirical challenge is to obtain the effect of industrial structure on economic growth quality in China. We use the Global Malmquist-Luenberger index based on SBM-based directional distance function to measure economic growth quality for 30 provinces from China over the period of 1998 to 2015. First, we use panel regression model to test the effect of industrial structure on economic growth quality. The data samples involve full sample, subsample of the east, subsample of the middle and subsample of the west. Then, we employ panel quantile regression model to investigate the impact of industrial structure on economic growth quality under different quantiles of economic growth quality. Finally, we attempt to explore the influencing mechanism by considering moderating effects of intervening variable.

The main conclusions are as follows. First, industrial structure shows a positive correlation with economic growth quality by full sample as well as subsample of the east in China. However, it is negatively correlative with economic growth quality by subsamples of the middle and west in China. This is not surprising since the east always plays the key role in the development of Chinese economy involving the economic agammaegate and economic quality. Second, structural deceleration happens in the economic growth quality of China. However, industrial structure still accelerates the economic growth quality in the east of China due to a relatively rational industrial structure. Industrial structure shows negative effect on the economic growth quality in the middle of China and meanwhile, such negative effect are anabatic, which implies an unreasonable industrial structure. The effect of industrial structure in the west of China during the increasing process of economic growth quality is not very clear. Indeed, the industrial development contributes slightly to the economy involving the economic growth quality in the west of China. Third, both human capital and innovation capacity play positive moderating effects on the impact of industrial structure on economic growth quality and it is heterogeneous at different quantiles of economic growth quality. On the other hand, although economic development level also exerts a positive moderating effect on the impact of industrial structure on economic growth quality, it is similar at different quantiles of economic growth quality.

Our conclusion leads to the following two policy implications. On the one hand, China's economy has been transitioning from a phase of rapid growth to a stage of high-quality development. It’s urgent for government to set more effective policies to optimize and upgrade of industrial structure. At the same time, they can improve the resource allocation efficiency as well as the quality of environment, thus improve the economic growth quality. On the other hand, we can also enact some polices on raising human capital, innovation capacity and economic development level to upgrade industrial structure and then improve economic growth quality. For example, increasing human capital can improve not only labor allocation efficiency, but also advanced technology. Innovation promotes the technological progress and reduces the emission of pollutants. Economic development level promotes the evolution of industrial structure. Thereby, it can further upgrade industrial structure and improve economic growth quality through these channels.

Acknowledgments

Research supported by the National Social Science Fund of China (NO. 19BTJ049).

Conflict of interest

The authors declare no conflict of interest.

References

[1]	M. Y. Li, H. L. Smith, L. Wang, Global dynamics of an SEIR epidemic model with vertical transmission, SIAM J. Appl. Math., 62 (2001), 58–69. https://doi.org/10.1137/S0036139999359860 doi: 10.1137/S0036139999359860
[2]	S. Gao, Y. Liu, J. J. Nieto, H. Andrade, Seasonality and mixed vaccination strategy in an epidemic model with vertical transmission, Math. Comput. Simul., 81 (2011), 1855–1868. https://doi.org/10.1016/j.matcom.2010.10.032 doi: 10.1016/j.matcom.2010.10.032
[3]	A. d'Onofrio, Stability properties of pulse vaccination strategy in SEIR epidemic model, Math. Biosci., 179 (2002), 57–72. https://doi.org/10.1016/S0025-5564(02)00095-0 doi: 10.1016/S0025-5564(02)00095-0
[4]	P. Yan, S. Liu, Seir epidemic model with delay, ANZIAM J., 48 (2006), 119–134. https://doi.org/10.1017/S144618110000345X doi: 10.1017/S144618110000345X
[5]	G. Li, Z. Jin, Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period, Chaos Solitons Fract., 25 (2005), 1177–1184. https://doi.org/10.1016/j.chaos.2004.11.062 doi: 10.1016/j.chaos.2004.11.062
[6]	J. Liu, Bifurcation analysis for a delayed SEIR epidemic model with saturated incidence and saturated treatment function, J. Biol. Dynam., 13 (2019), 461–480. https://doi.org/10.1080/17513758.2019.1631965 doi: 10.1080/17513758.2019.1631965
[7]	R. Engbert, M. M. Rabe, R. Kliegl, S. Reich, Sequential data assimilation of the stochastic SEIR epidemic model for regional COVID-19 dynamics, Bull. Math. Biol., 83 (2021), 1. https://doi.org/10.1007/s11538-020-00834-8 doi: 10.1007/s11538-020-00834-8
[8]	D. Efimov, R. Ushirobira, On an interval prediction of COVID-19 development based on a SEIR epidemic model, Ann. Rev. Control, 51 (2021), 477–487. https://doi.org/10.1016/j.arcontrol.2021.01.006 doi: 10.1016/j.arcontrol.2021.01.006
[9]	N. W. Ruktanonchai, J. Floyd, S. Lai, C. W. Ruktanonchai, A. Sadilek, P. Rente-Lourenco, et al., Assessing the impact of coordinated COVID-19 exit strategies across Europe, Science, 369 (2020), 1465–1470. https://doi.org/10.1126/science.abc5096 doi: 10.1126/science.abc5096
[10]	I. Borisov, E. Metelkin, Confidence intervals by constrained optimization-an algorithm and software package for practical identifiability analysis in systems biology, PLOS Comput. Biol., 16 (2020), e1008495. https://doi.org/10.1371/journal.pcbi.1008495 doi: 10.1371/journal.pcbi.1008495
[11]	R. Bellman, K. J. Åström, On structural identifiability, Math. Biosci., 7 (1970), 329–339. https://doi.org/10.1016/0025-5564(70)90132-X doi: 10.1016/0025-5564(70)90132-X
[12]	C. Cobelli, J. J. DiStefano 3rd, Parameter and structural identifiability concepts and ambiguities: a critical review and analysis, Amer. J. Physiol.-Reg. Integr. Compar. Physiol., 239 (1980), R7–R24. https://doi.org/10.1152/ajpregu.1980.239.1.R7 doi: 10.1152/ajpregu.1980.239.1.R7
[13]	E. Walter, L. Pronzato, Identification of Parametric Models: From Experimental Data, Berlin: Springer Verlag, 1997.
[14]	A. Martynenko, A. Bück, Intelligent Control in Drying, New York: CRC Press, 2018.
[15]	F. G. Wieland, A. L. Hauber, M. Rosenblatt, C. Tönsing, J. Timmer, On structural and practical identifiability, Current Opinion Syst. Biol, 25 (2021), 60–69. https://doi.org/10.1016/j.coisb.2021.03.005 doi: 10.1016/j.coisb.2021.03.005
[16]	H. Miao, X. Xia, A. S. Perelson, H. Wu, On identifiability of nonlinear ODE models and applications in viral dynamics, SIAM Rev., 53 (2011), 3–39. https://doi.org/10.1137/090757009 doi: 10.1137/090757009
[17]	N. Tuncer, T. T. Le, Structural and practical identifiability analysis of outbreak models, Math. Biosci., 299 (2018), 1–18. https://doi.org/10.1016/j.mbs.2018.02.004 doi: 10.1016/j.mbs.2018.02.004
[18]	M. Rodriguez-Fernandez, P. Mendes, J. R. Banga, A hybrid approach for efficient and robust parameter estimation in biochemical pathways, Biosystems, 83 (2006), 248–265. https://doi.org/10.1016/j.biosystems.2005.06.016 doi: 10.1016/j.biosystems.2005.06.016
[19]	M. Rodriguez-Fernandez, J. A. Egea, J. R. Banga, Novel metaheuristic for parameter estimation in nonlinear dynamic biological systems, BMC Bioinformatics, 7 (2006), 483. https://doi.org/10.1186/1471-2105-7-483 doi: 10.1186/1471-2105-7-483
[20]	H. T. Banks, S. Hu, W. C. Thompson, Modeling and Inverse Problems in the Presence of Uncertainty. New York: CRC Press, 2014.
[21]	D. Venzon, S. Moolgavkar, A method for computing profile‐likelihood‐based confidence intervals, J. Royal Stat. Soc.: Ser. C (Applied Statistics), 37 (1988), 87–94. https://doi.org/10.2307/2347496 doi: 10.2307/2347496
[22]	J. A. Jacquez, T. Perry, Parameter estimation: local identifiability of parameters, Amer. J. Physiol.-Endocrinol. Metabol., 258 (1990), E727–E736.
[23]	A. Raue, C. Kreutz, T. Maiwald, J. Bachmann, M. Schilling, U. Klingmüller, et al., Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood, Bioinformatics, 25 (2009), 1923–1929. https://doi.org/10.1093/bioinformatics/btp358 doi: 10.1093/bioinformatics/btp358
[24]	M. C. Eisenberg, M. A. Hayashi, Determining identifiable parameter combinations using subset profiling, Math. Biosci., 256 (2014), 116–126. https://doi.org/10.1016/j.mbs.2014.08.008 doi: 10.1016/j.mbs.2014.08.008
[25]	Y. H. Kao, M. C. Eisenberg, Practical unidentifiability of a simple vector-borne disease modeli: implications for parameter estimation and intervention assessment, Epidemics, 25 (2018), 89–100. https://doi.org/10.1016/j.epidem.2018.05.010 doi: 10.1016/j.epidem.2018.05.010
[26]	S. Baron, Medical Microbiology, 4 Eds., Galveston (TX): University of Texas Medical Branch at Galveston, 1996.
[27]	M. Lipsitch, T. Cohen, B. Cooper, J. M. Robins, S. Ma, L. James, et al., Transmission dynamics and control of severe acute respiratory syndrome, Science, 300 (2003), 1966–1970. https://doi.org/10.1126/science.1086616 doi: 10.1126/science.1086616
[28]	S. Riley, C. A. Donnelly, N. M. Ferguson, Robust parameter estimation techniques for stochastic within-host macroparasite models, J. Theoret. Biol., 225 (2003), 419–430. https://doi.org/10.1016/S0022-5193(03)00266-2 doi: 10.1016/S0022-5193(03)00266-2
[29]	C. Fraser, C. A. Donnelly, S. Cauchemez, W. P. Hanage, M. D. Van Kerkhove, T. D. Hollingsworth, et al., Pandemic potential of a strain of influenza A (H1N1): early findings, Science, 324 (2009), 1557–1561. https://doi.org/10.1126/science.1176062 doi: 10.1126/science.1176062
[30]	A. R. Tuite, J. Tien, M. Eisenberg, D. J. Earn, J. Ma, D. N. Fisman, Cholera epidemic in Haiti, 2010: using a transmission model to explain spatial spread of disease and identify optimal control interventions, Ann. Int. Medic., 154 (2011), 593–601. https://doi.org/10.7326/0003-4819-154-9-201105030-00334 doi: 10.7326/0003-4819-154-9-201105030-00334
[31]	G. Chowell, H. Nishiura, Transmission dynamics and control of Ebola virus disease (EVD): a review, BMC Med., 12 (2014), 196. https://doi.org/10.1186/s12916-014-0196-0 doi: 10.1186/s12916-014-0196-0
[32]	D. Fisman, E. Khoo, A. Tuite, Early epidemic dynamics of the West African 2014 Ebola outbreak: estimates derived with a simple two-parameter model, PLoS Currents, 2014. https://doi.org/10.1371/2Fcurrents.outbreaks.89c0d3783f36958d96ebbae97348d571
[33]	F. Brauer, P. Van den Driessche, J. Wu, L. J. Allen, Mathematical Epidemiology, Berlin: Springer, 2008.
[34]	L. Denis-Vidal, G. Joly-Blanchard, C. Noiret, Some effective approaches to check the identifiability of uncontrolled nonlinear systems, Math. Comput. Simul., 57 (2001), 35–44. https://doi.org/10.1016/S0378-4754(01)00274-9 doi: 10.1016/S0378-4754(01)00274-9
[35]	E. Walter, I. Braems, L. Jaulin, M. Kieffer, Guaranteed numerical computation as an alternative to computer algebra for testing models for identifiability, In: Numerical Software with Result Verification. Lecture Notes in Computer Science, Berlin: Springer, 2004.
[36]	S. Vajda, K. R. Godfrey, H. Rabitz, Similarity transformation approach to identifiability analysis of nonlinear compartmental models, Math. Biosci., 93 (1989), 217–248. https://doi.org/10.1016/0025-5564(89)90024-2 doi: 10.1016/0025-5564(89)90024-2
[37]	N. D. Evans, M. J. Chapman, M. J. Chappell, K. R. Godfrey, Identifiability of uncontrolled nonlinear rational systems, Automatica, 38 (2002), 1799–1805. https://doi.org/10.1016/S0005-1098(02)00094-8 doi: 10.1016/S0005-1098(02)00094-8
[38]	J. W. Yates, N. D. Evans, M. J. Chappell, Structural identifiability analysis via symmetries of differential equations, Automatica, 45 (2009), 2585–2591. https://doi.org/10.1016/j.automatica.2009.07.009 doi: 10.1016/j.automatica.2009.07.009
[39]	H. Pohjanpalo, System identifiability based on the power series expansion of the solution, Math. Biosci., 41 (1978), 21–33. https://doi.org/10.1016/0025-5564(78)90063-9 doi: 10.1016/0025-5564(78)90063-9
[40]	G. Bellu, M. P. Saccomani, S. Audoly, L. D'Angiò, DAISY: a new software tool to test global identifiability of biological and physiological systems, Comput. Meth. Programs Biomed, 88 (2007), 52–61. https://doi.org/10.1016/j.cmpb.2007.07.002 doi: 10.1016/j.cmpb.2007.07.002
[41]	L. Ljung, T. Glad, On global identifiability for arbitrary model parametrizations, Automatica, 30 (1994), 265–276. https://doi.org/10.1016/0005-1098(94)90029-9 doi: 10.1016/0005-1098(94)90029-9
[42]	E. Walter, Y. Lecourtier, Unidentifiable compartmental models: what to do? Math. Biosci., 56 (1981), 1–25. https://doi.org/10.1016/0025-5564(81)90025-0
[43]	X. Xia, C. H. Moog, Identifiability of nonlinear systems with application to HIV/AIDS models, IEEE Trans. Automat. Control, 48 (2003), 330–336. https://doi.org/10.1109/TAC.2002.808494 doi: 10.1109/TAC.2002.808494
[44]	A. F. Villaverde, A. Barreiro, A. Papachristodoulou, Structural identifiability of dynamic systems biology models, PLoS Comput. Biol., 12 (2016), e1005153. https://doi.org/10.1371/journal.pcbi.1005153 doi: 10.1371/journal.pcbi.1005153
[45]	A. Sedoglavic, A probabilistic algorithm to test local algebraic observability in polynomial time, J. Symbol. Comput., 33 (2002), 735–755.
[46]	M. Anguelova, J. Karlsson, M. Jirstrand, Minimal output sets for identifiability, Math. Biosci., 239 (2012), 139–153. https://doi.org/10.1016/j.mbs.2012.04.005 doi: 10.1016/j.mbs.2012.04.005
[47]	N. Meshkat, C. E. Kuo, J. DiStefano III, On finding and using identifiable parameter combinations in nonlinear dynamic systems biology models and COMBOS: a novel web implementation, PLoS One, 9 (2014), e110261. https://doi.org/10.1371/journal.pone.0110261 doi: 10.1371/journal.pone.0110261
[48]	A. Raue, B. Steiert, M. Schelker, C. Kreutz, T. Maiwald, H. Hass, et al., Data2dynamics: a modeling environment tailored to parameter estimation in dynamical systems, Bioinformatics, 31 (2015), 3558–3560. https://doi.org/10.1093/bioinformatics/btv405 doi: 10.1093/bioinformatics/btv405
[49]	T. S. Ligon, F. Fröhlich, O. T. Chiş, J. R. Banga, E. Balsa-Canto, J. Hasenauer, GenSSI 2.0: multi-experiment structural identifiability analysis of SBML models, Bioinformatics, 34 (2018), 1421–1423. https://doi.org/10.1093/bioinformatics/btx735 doi: 10.1093/bioinformatics/btx735
[50]	H. Hong, A. Ovchinnikov, G. Pogudin, C. Yap, SIAN: software for structural identifiability analysis of ODE models, Bioinformatics, 35 (2019), 2873–2874. https://doi.org/10.1093/bioinformatics/bty1069 doi: 10.1093/bioinformatics/bty1069
[51]	R. Dong, C. Goodbrake, H. A. Harrington, G. Pogudin, Differential elimination for dynamical models via projections with applications to structural identifiability, SIAM J. Appl. Algebra Geometry, 7 (2023), 194–235. https://doi.org/10.1137/22M1469067 doi: 10.1137/22M1469067
[52]	X. Rey Barreiro, A. F. Villaverde, Benchmarking tools for a priori identifiability analysis, Bioinformatics, 39 (2023), btad065. https://doi.org/10.1093/bioinformatics/btad065 doi: 10.1093/bioinformatics/btad065
[53]	D. P. Lizarralde-Bejarano, D. Rojas-Díaz, S. Arboleda-Sánchez, M. E. Puerta-Yepes, Sensitivity, uncertainty and identifiability analyses to define a dengue transmission model with real data of an endemic municipality of Colombia, PLoS One, 15 (2020), e0229668. https://doi.org/10.1371/journal.pone.0229668 doi: 10.1371/journal.pone.0229668
[54]	H. Wu, H. Zhu, H. Miao, A. S. Perelson, Parameter identifiability and estimation of hiv/aids dynamic models, Bull. Math. Biol., 70 (2008), 785–799. https://doi.org/10.1007/s11538-007-9279-9 doi: 10.1007/s11538-007-9279-9
[55]	M. C. Eisenberg, S. L. Robertson, J. H. Tien, Identifiability and estimation of multiple transmission pathways in cholera and waterborne disease, J. Theor. Biol., 324 (2013), 84–102. https://doi.org/10.1016/j.jtbi.2012.12.021 doi: 10.1016/j.jtbi.2012.12.021
[56]	K. Roosa, G. Chowell, Assessing parameter identifiability in compartmental dynamic models using a computational approach: application to infectious disease transmission models, Theor. Biol. Med. Model., 16 (2019), 1. https://doi.org/10.1186/s12976-018-0097-6 doi: 10.1186/s12976-018-0097-6
[57]	N. Metropolis, S. Ulam, The monte carlo method, J. Amer. Stat. Assoc., 44 (1949), 335–341.
[58]	H. T. Banks, K. Holm, D. Robbins, Standard error computations for uncertainty quantification in inverse problems: Asymptotic theory vs. bootstrap, Math. Comput. Model., 52 (2010), 1610–1625. https://doi.org/10.1016/j.mcm.2010.06.026 doi: 10.1016/j.mcm.2010.06.026
[59]	J. A. Jacquez, P. Greif, Numerical parameter identifiability and estimability: integrating identifiability, estimability, and optimal sampling design, Math. Biosci., 77 (1985), 201–227. https://doi.org/10.1016/0025-5564(85)90098-7 doi: 10.1016/0025-5564(85)90098-7

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