Mathematical Modelling of Index System of Economic Vitality During The COVID-19 Epidemic System of Economic Vitality During The COVID-19

contradiction between the development of become the main social contradiction, and the unbalanced economic development between different regions is the concentrated reflection of imbalance is not fully developed; To accelerate the narrowing of the gap in regional economic development, promote the vitality of regional economic development, and promote the coordinated development of regional economy is the basis and key to solve the main social contradictions in the new age, and is also the driving force of economic and social development axis. Regional economic vitality is an important part of regional comprehensive competitiveness. In recent years, in ABSTRACT Economic vitality is an important indicator to measure the level and potential of economic development. The paper puts forward three social problems about economic vitality and establishes a model to solve them. We build panel data model to analyze the influencing factors of economic vitality. Based on the section data of Beijing, the VAR-VEC model is established to analyze the long-term and short-term effects of economic policies on economic vitality. The development strategy of ORT is put forward, and the scheme to promote the growth of economic vitality is given. For the first problem, the paper preprocess panel data, and test its independence, and find that each factor is not independent of each other. Through the correlation analysis, we found that there is a strong correlation between the various elements. After Random Effect Test and Fixed Effect Test combined with Hausman Test, the data panel conforms to fixed effect model. Population change and enterprise vitality have a positive impact on economic vitality, the influencing factors are 0.01 and 0.07 respectively. We put forward the strategy of adjusting the overall structure of enterprises to improve economic vitality. For the second problem, the paper selects the section data of Beijing city and construct the VAR-VEC model. Based on ADF unit root test and Johansen cointegration test, we find that there are at least three cointegration relationships between time series. We use Ais-Sc Criterion to determine the order of delay as the third order. We use OLS estimation method to get the coefficients of VEC Model. Through the IRF response, we find that the long-term impact of economic policy on economic vitality is positive correlation effect. Due to the effect of experience accumulation, the economic vitality presents a W-shaped trend. For the third problem, the paper uses the minimum average deviation method to preprocess the index data and get 9 representative indexes. We extract two main factors by factor analysis and build an index system of economic vitality. The economic vitality of each city from 2009 to 2017 is calculated according to the index system. Beijing, Shanghai, Guangzhou and Shenzhen often rank first, while Kunming and Dongguan often rank last. Based on the same data, the panel data model test results are similar to index system. For the fourth problem, we review the previous conclusions and put forward the ORT development strategy to improve economic vitality based on the established model. 33(4)-2021.


Introduction
Under the background of new age, China's economic, social, cultural, ecological, political and other fields are coruscate gives new vigor and vitality, at the same time the good life is people's increasing need to inadequate and imbalance of the contradiction between the development of become the main social contradiction, and the unbalanced economic development between different regions is the concentrated reflection of imbalance is not fully developed; To accelerate the narrowing of the gap in regional economic development, promote the vitality of regional economic development, and promote the coordinated development of regional economy is the basis and key to solve the main social contradictions in the new age, and is also the driving force of economic and social development axis. Regional economic vitality is an important part of regional comprehensive competitiveness. In recent years, in

ARTICLE INFO ABSTRACT
Economic vitality is an important indicator to measure the level and potential of economic development. The paper puts forward three social problems about economic vitality and establishes a model to solve them. We build panel data model to analyze the influencing factors of economic vitality. Based on the section data of Beijing, the VAR-VEC model is established to analyze the long-term and short-term effects of economic policies on economic vitality. The development strategy of ORT is put forward, and the scheme to promote the growth of economic vitality is given. For the first problem, the paper preprocess panel data, and test its independence, and find that each factor is not independent of each other. Through the correlation analysis, we found that there is a strong correlation between the various elements. After Random Effect Test and Fixed Effect Test combined with Hausman Test, the data panel conforms to fixed effect model. Population change and enterprise vitality have a positive impact on economic vitality, the influencing factors are 0.01 and 0.07 respectively. We put forward the strategy of adjusting the overall structure of enterprises to improve economic vitality.
For the second problem, the paper selects the section data of Beijing city and construct the VAR-VEC model. Based on ADF unit root test and Johansen cointegration test, we find that there are at least three cointegration relationships between time series. We use Ais-Sc Criterion to determine the order of delay as the third order. We use OLS estimation method to get the coefficients of VEC Model. Through the IRF response, we find that the long-term impact of economic policy on economic vitality is positive correlation effect. Due to the effect of experience accumulation, the economic vitality presents a W-shaped trend. For the third problem, the paper uses the minimum average deviation method to preprocess the index data and get 9 representative indexes. We extract two main factors by factor analysis and build an index system of economic vitality. The economic vitality of each city from 2009 to 2017 is calculated according to the index system. Beijing, Shanghai, Guangzhou and Shenzhen often rank first, while Kunming and Dongguan often rank last. Based on the same data, the panel data model test results are similar to index system. For the fourth problem, we review the previous conclusions and put forward the ORT development strategy to improve economic vitality based on the established model. order to improve economic vitality, some regions have introduced a lot of preferential policies to stimulate economic vitality, such as reducing the approval steps for investment, providing financial support for entrepreneurship, and lowering the threshold for settling down in order to attract talents [1].
However, due to different resource endowments, these policies have different effects in different regions. How to grasp the key factors and effectively improve the regional economic vitality is a worthy research topic. In order to study how to improve regional economic vitality, given some data. Based on these data and my own survey data, this paper established an appropriate model to solve the following problems: 1.
Problem 1, it is necessary to take a certain region (or city or province) as an example, and combine the data collected in the attachment to establish the appropriate relationship model of the influencing factors of economic vitality and give the action plan to improve the regional economic vitality. The influence of population changes trend and enterprise vitality change on regional economic vitality change is analyzed.

2.
Problem 2, selecting a region (or city or province) and investigating the appropriate data analyze the short-term and long-term impact of economic policy transformation on the economic vitality of the region (or city or province). 3. Problem 3, this paper collects relevant data, selects appropriate indicator system, establishes mathematical model to analyze and measure regional (or city or provincial) economic vitality, and ranks urban economic vitality [2].

The Model of Problem 1
This section, based on the panel data model, collects data from various provinces and cities, performs correlation test and principal component analysis on the data. The fixed effect test and random effect test were carried out for the obtained factors, and the influence of policy and enterprise vitality on economic vitality was analyzed based on the established relationship model between each factor and economic vitality [3][4][5][6][7].
Data Analysis and Processing: Based on the collected data has certain error and deficiencies, in order to reduce the invalid, the influence of the error data of the following model, improve the reliability of data, need to collect the data pretreatment, firstly the filtered data, remove abnormal data, secondly, proper supplement of incomplete data, finally, has strong correlation data linear regression analysis forecasting and slight fluctuation data using the moving average method to fill the missing value, to further improve the accuracy and the integrity of the data [8,9]. Data Selection Principle: This paper needs to collect various indicator data describing economic vitality and influencing economic vitality, and the following classical indicators can be obtained according to the expert method and the literature [10][11][12][13][14]. Dependent variable. In the existing economic vitality research and analysis, more choose Gross Domestic Product (GDP) as a measure of regional economic vitality. In this paper, in order to measure regional economic vitality, main factors from the effects of the economic vitality, that reflects the GDP growth rate as the level of economic development during the period of change degree of dynamic indexes, namely whether a national economic basic index of the dynamic and USES the linear regression analysis and panel data model analysis, the main measures for regional economic vitality. Independent variables. Based on the existing literature research results and the aforementioned analysis, this paper selects 9 aspects including population growth rate, fiscal expenditure and employment rate (mainly used to reflect the main influencing factors of regional economic vitality and its growth trend).
The employment rate is expressed by the number of unemployed; At the same time, in the establishment of the model, for the negative value of population growth rate, in order to reduce the error in the large number region, dummy variables can be used instead of the original statistical samples, which are reset to zero in this paper.
Control variables. Based on the analysis of the comprehensive evaluation index system of urban economy, and considering the availability of data, this paper introduces independent innovation ability, per capita length of education, professional and technical talent inflow and other irrelevant variables as control variables.
Through certain analysis, the variables other than independent variables that can affect the change of dependent variables should be well controlled and regarded as constants, so as to obtain appropriate causal relationship and obtain the most true and accurate value.

Independence Test:
In the analysis of the relationship between the factors affecting economic vitality, in order to fully understand whether there is an internal relationship between the factors, according to the processed data, this paper carries out an independence test for each factor. The data source is the national bureau of statistics, and the independence test is conducted on the pre-processed data. See the appendix for the specific data. Make the following assumptions about the research hypothesis: Null Hypothesis: The factors that influence positive energy are independent of each other.
Alternative Hypothesis: The factors influencing economic vitality are not independent.
Firstly, chi-square independence test was conducted, and SPSS was used to conduct independent test for each influencing factor to observe whether there was any correlation between each factor Table 1. The test results are as follows: It can be seen from Table 2 that the cross relation between each factor and the year, and the cross  Table 3. It can be seen from Table 3 that the degree of freedom is the probability of Person chi-square, which is less than 0.05, so the null hypothesis is rejected, that is, the influencing factors are not independent of each other.    Correlation Analysis: Each factor in the collection is the indicator data of each city in the country, which belongs to the panel data. There may be a certain correlation between the data. Con-sidering the correlation among various factors, the linear strength relationship diagram of each factor is obtained based on the data as follows: As can be seen from the observation in Figure 1 Table 4. Correlation coefficients can quantitatively describe the closeness of linear relationships among factors, and SPSS is used for correlation analysis to obtain the correlation coefficients among the influencing factors, as shown in Table 5. According to the above correlation analysis Table 5, there is a correlation among all factors, and the positive correlation coefficient is distributed between 0.5 and 1, reflecting a strong correlation; And then according to the significance test of the correlation coefficient, the significance values are all less than 0.05, indicating that the correlation coefficient has reached a high level of significance. Therefore, there is a strong correlation between various factors influencing economic vitality ( Figure 2).    Table 6. Based on the data and problem in this question, it is obvious that the panel data model is a better choice. The panel data model includes both the cross-section and the time dimension.
Here, the factors affecting economic vitality are taken as the crosssection, and the year is taken as the time dimension. Among them, i(i=1…8)represents the following linear model set for the year: y α λ β ε = + + + . The panel data model can be further divided into fixed effect model and random effect model.

a. Fixed Effect Model
The individual effect is regarded as a fixed factor that does not

b. Random Effect Model
The individual effect i α is regarded as a random factor that changes with time. By using the random effect model, the long-term factors and short-term factors in the variance can be separated. The basic setting of the model is as follows:

Model Determination Based on Hausman Test
Because the missing related variables are not excluded, there will be dependent variable-local GDP will change with the same patents and local GDP. The inconsistency of the order of magnitude of each part will cause trouble to the model fitting. According to the statistical yearbook, the city is divided into 1-31, and the distribution of various data is shown in Figure 3. Figure 3 for example, standardize it first. Assume that the original data is x m , after standardization is X m . After obtaining standardized data, it is shown as follows. It can be seen from the observation Figure 4 that after the standardization, the feature expression is clearer, which is conducive to the next model inspection work.  Hypothesis 1: The ξ in Hypothesis one is the independent variable interference term. Hypothesis 1: Assume that thex has no effect on the observed value, unobserved value and post observed value.  Table 7. Among them, the F value is very close to 0, indicating that the fixed effect is very significant. In this case. Among the seven independent variables, the consumer index and unemployment rate are not significant within the 95% confidence interval. Local government expenditure, total tourism income, total profits of enterprises, resident population and tertiary industry income all have strong statistical significance.
The statistics are shown in Table 8. Among them, the third industry has the most significant impact on GDP, and the consumer index has the least impact on GDP. We can know that all the selected indicators have positive significance for GDP growth within the sta-tistical range. It shows that this test has passed hypothesis one and hypothesis two for panel data, and both of them are true Figure 5.
Cov a x = Hypothesis 5: According to the above assumption, suppose that the distribution of each independent variable is constrained in a specific case, and the effect of each independent variable obeys the mean value of 0.
The second is the description of random interference, which has no correlation with explanatory variables. The third term makes the two coefficients independent of each other ( Figure 6).

Model Determination Based on Hausman Test: According
to the reference [13], the difference between the random effect model and the fixed effect model is that it is very difficult to try to distinguish them in a high degree in the description of individuals.
The fixed effect will consume a large degree of freedom, while the random effect is more universal on this basis

Activation Scheme Proposed Based on Fixed Effect
Model: According to the fixed effect model shown in Figure 4, the explanation degree of each factor to economic vitality has been given, and the following Suggestions are given according to the influence degree. 2) In the process of development, the region should combine its resource endowment and industrial foundation to find the optimal ratio of enterprise structure, complete the adjustment of enterprise structure as soon as possible, and develop appropriate leading industries to promote economic growth.
Will be conducive to a steady increase in economic vitality ( Figure 8).      Table 13. It can be seen from     Carry out the second-order difference differentiation on original data and continue the ADF test on the data after the difference, and the results are shown in Table 15. It can be seen from Table 15 that all the data after the second-order difference have passed the ADF test, that is to say, this group of data is zero in the second order, and then the inter group cointegration test is carried out. Figure 12 shows the visual information of three points of each variable under three tests. The confidence intervals of the middle three levels are 1%, 5% and 10% respectively. After the first-order difference, only lnpopulation passed the test; After the second-order difference, all the data pass the test, that is, the group of data is the second-order zero integer data.  Figure 12: impulse response.

Johansen Co Integration Test of Variables:
According to ADF test, the original variable is a second-order zero integer sequence, that is to say, the original variable is an unstable sequence. First, Johansen co integration test is carried out to find out whether there is a co integration relationship between its combinations. The test method is to calculate the trace statistics trace and the maximum eigenvalue Max eigenvalue (Figures 13 & 14). Using the cyclic statistical hypothesis, the existence of cointegration logarithm is assumed. Table 16 shows the Johansen co integration test results.
From the trace statistics trace in Table 16, it is assumed that none is the sequence without cointegration. Under this assumption, the trajectory value is 255.6213, which is greater than the critical value of 95.7537, if the original hypothesis is rejected, there is at least one co integration relationship. In the case of 5% confidence level, the original assumption is that there are at least four sets of co integration relationship whose trajectory value is less than the critical value, and the determination of the fourth set of co integration relationship is rejected by the assumption. There are at least three cointegration relations in the linear combination of time series with surface instability.   it is necessary to determine the optimal lag order of the model. The stability of the model is explained by AR root graph and Roland causality analysis. Finally, the impulse response chart is given, and the long-term and short-term effects of policy implementation on economic vitality are analyzed.

Determination of Lag Period Based on AIS-SC Minimization
Criterion: When the model is not integrated and stable, multiple in Table 17 are calculated by Eviews software. It can be seen from  (3). According to AIS information standard and SC standard, the optimal lag time is selected as the third-order lag time.  Through the co integration relationship, we can see that the long-term equilibrium relationship between economic vitality and local government expenditure, local tourism revenue and local resident population is positive; There is a long-term negative correlation between economic vitality and local residents' living index and local unemployment rate. According to the test results (see Appendix 1), write the VEC model as The specific coefficients are described as follows: In the formula: LY t =(LY1 t +LY2 t +LY3 t )  circumstances. Figure 15 is the AR root test. The absolute value of the root is less than one, that is, all the roots are in the plane of the unit circle, and the stability test of the model is passed. The impulse function is applied to the model to observe the long-term and shortterm effects of economic policies on economic vitality.
It can be seen from Figure 16 that the promotion effect of economic policies on economic vitality gradually declines after 1-3 periods, and the economic vitality has increased since the third period, Because the experience of implementation after the implementation of economic policies can be applied, which has a secondary effect. After the fourth period, the promoting effect gradually decreased, the decreasing trend was relatively slow, and the long-term positive correlation effect continued.

The Model of Problem 3
This section aims at question 3. Firstly, we establish a scientific economic vitality index system as the standard of data selection.
Secondly, the minimum average difference method is used to screen the data, and the index is initially extracted; Further, the factor analysis method is used to select the main influencing factors, and finally the comprehensive score of each factor is weighted to give the ranking of urban economic vitality.

The Construction Principle of Index System of Economic
Vitality: In order to select effective data to measure the economic vitality of each city, the following five principles are given in this paper, and the general process is as follows:  Where is the total factor score, Fi is the score of the first influencing factor; Bi is the contribution of the first factor, and factor contribution = variance contribution rate / total variance interpretation after the factor rotation.

Measurement of Economic Vitality of Regional Cities:
Before measuring the economic vitality of each city, the relationship between variables and factor analysis is further verified through the variance of common factors. The common factor variance can effectively reflect the strength of its interpretation ability. The larger the common factor variance extracted between variables, the stronger the ability to be interpreted by the common factor. Most of the variable factors proposed by the extracted common factor variance are explained to a higher degree than 70%. Therefore, the extraction effect is better, the information of the original data loss is less, and the data extracted is more reliable. For the factor whose characteristic root is greater than 1, data analysis is carried out based on SPSS software, and two factors are finally obtained, as shown in the Table 19 below, with the explanation of total variance. From Table 20, it can be seen that the cumulative variance contribution rate is 73.174%, indicating that the first two factors contain 73.174% of all indicator information, and the extracted information is large and highly representative.  Therefore, it can be seen that factor analysis is effective in extracting original variable information. It can also be seen from the gravel map that the information contributed by the first two factors in the overall influence factors represents that the broken line is relatively steep, and the slope of the broken line is relatively gentle after that, so it can be considered that the two factors extracted are relatively reasonable. It can be seen that the primary industry, tertiary industry, college students, population and road traffic noise level are factor 1, which reflects the level of social production and security Table 21. Therefore, factor 1 can be named as social production and security factor; local GDP, financial expenditure, real estate investment and per capita wage are factor 2, which reflects the government regulation and control. Therefore, the Factor 2 is called government regulation factor. The contribution rate of factors is analyzed by the method of normalmaximization variance, and the conversion correlation coefficient is obtained, which shows the correlation of two factors. It can be seen from   It is necessary to extract the component matrix of the factor load matrix. According to the component score coefficient matrix, local GDP, fiscal expenditure, tertiary industry, tertiary industry and real estate investment have a positive impact on the ranking; the primary industry has a negative impact on the ranking. The expression of each influence factor is given according to Table 23. Taking the variance contribution rate of each factor as the weight, the weighted analysis is carried out. After weighted average, the growth index scores are as follows: The final weight value of each influencing factor is obtained by factor analysis, and the comprehensive score of each factor is obtained by factor score weighting function. The sub-factor ranking and comprehensive factor ranking of each city are shown in Table 24. It can be seen from the ranking table that the cities such as Beijing, Shanghai and Guangzhou rank the second, third and fourth respectively in the ranking, which indicates that the central economic zone of the country has high stability and is not easy to change. The highest ranking is Chongqing. Shenyang is ranked next, and the transfer of its industrial center may be one of the reasons for this result. It can be seen from Figure 15 that the ranking of Kunming and Ningbo fluctuates greatly. Considering that the local industrial structure is not obvious enough, it is necessary to strengthen the industrial structure adjustment to improve its economic vitality.
Shenyang's ranking is declining year by year, which may also be related to local policies and development strategies, so it needs to be noticed in time.  Table 25. In order to analyze the regional economic vitality more specifically, it is necessary to understand the distribution characteristics of each data. The number of post offices is on the high side in some areas, resulting in waste of resources.      In these variables, when one variable changes, the other variables remain unchanged, then the economic vitality changes in the same direction. Therefore, it can be further proved that the economic vitality index system constructed in this paper can accurately measure the economic vitality.

A Development Plan Based on the Perspective of the Decision Maker
This section first reviews questions 1 to 3 above and obtains the general universality of the model established in this paper.
Finally, according to the results, it proposes measures conducive to improving economic vitality and promoting economic development.

Conclusion Review
From question 1 to question 3, we can roughly divide Vigorously developing the tertiary industry can rapidly expand employment fields and jobs, avoid labor surplus, and improve residents' income. For modern cities, residents not only have material needs, but also pursue spiritual level. This development trend promotes the region to continuously develop new industries to meet the needs of the people, so as to improve residents and to improve the quality of life. Therefore, we vigorously develop the tertiary industry, which has a significant role in promoting the sustainable development of economic vitality.

Strengthening the Development of Primary and Secondary
Industries: For the adjustment of Beijing's economic structure and the promotion of its regional competitiveness, it is necessary to develop the tertiary industry while strengthening the primary industry and expanding the scale of the secondary industry.
The first industry is the basic industry of the national economy, strengthening the first industry, and laying the foundation for the development of the second industry and the third industry.

Advantages
The advantages and disadvantages of model factor analysis and panel data model are analyzed.

Improvements Needed
When using panel data model to study influence factors, there are some difficulties in variable design and data collection, some errors in factor prediction, and selection difficulties in influence factors; panel data analysis of time series of factors is short, which can only reflect the data characteristics in the short term, not the long-term changes of factors.