Based on the previous literature (e.g., Fu et al. [10] ; Tang et al. [11] ; Li et al. [12] ) and the availability, scientificity and objectivity of data, we developed a system of indicators for health investment, resident health and economic development. The specific indicators are shown in Table 1.

Traditional coupling coordination degree model is low, can’t narrow the system reliability and validity of the gap between the problem, as a result, this study uses the adjusted coupling coordination degree model, a more reasonable measuring coupling coordination and development level of each system [13] . Medical three system, the resident health and economic growth influence each other, Assuming R 3 to R max , The calculation method is as follows:

Formulas (1) and (2) are specific formulas for calculating the comprehensive level index and coupling degree respectively.

R i = ∑ i = 1 n W i X ′ i (1)

C = [ 1 − ( R 3 − R 1 ) 2 + ( R 2 − R 1 ) 2 + ( R 3 − R 2 ) 2 3 ] × U 1 U 3 × U 2 U 3 (2)

R 1 is a comprehensive evaluation index system of health investment, R 2 is a comprehensive evaluation index system of resident health, and R 3 is a comprehensive evaluation index system of economic growth. For three system coupling, C is [0, 1], the more value it approaches to 1, the bigger degree of three system coupling will be shown. Due to the coupling can only describe the degree of internal coordination between systems, rather than measuring system in different levels of coordination degree, introducing the coupling coordination degree [14] .

T = α 1 R 1 + α 2 R 2 + α 3 R 3 (3)

D = C × T (4)

T is the comprehensive evaluation index of health investment, resident’s health and economic growth; α 1 α 2 α 3 is the contribution of the three indices,

and α 1 = α 2 = α 3 = 1 3 ; D is the coupling coordination degree, and the value’s range is from 0 to 1.

The entropy weight method was developed from the Shannon entropy, which is a quantitative measure of the uncertainty of information systems [15] . Compared to subjective human evaluation, the entropy weight method is an objective weighting method that has higher reliability, as its weighting coefficient depends entirely on the value of the indicator [16] . The calculation method is as follows:

Formula (5) Standardized calculation to eliminate data differentiation; Formula (6) is the specific calculation formula of entropy method.

X ′ i = X i − X min ( i ) X max ( i ) − X min ( i ) (5)

W i = V i ∑ i = 1 n V i (6)

where X ′ represents the standardized data for indicator in year i, X i represents the original data, while X max ( i ) and X min ( i ) indicate the maximum and minimum value of the indicator, respectively.

Grey relational analysis employs multivariate analysis to determine the degree of mutual influence or comprehensively evaluate the contribution of each index to variations in the data based on the geometric similarity of index factors [17] . Using grey correlation analysis method can be better for health investment, the resident health and economic growth three coupling in the coordinated development of various factors that affect its coordination.

Using the grey correlation method to explore the factors of the influence degree of the three system coordination development level in Sichuan province, correlation analysis and calculating the correlation. From Table 3 we can see the influence of various factors.

According to the results of 2010-2019 health investment system, the resident health and economic growth of Sichuan province’s comprehensive development level is of a steady upward trend, but the three-system overall coordination is not high, mainly because the health care investment and economic growth system lag behind the resident health. Based on this, in order to improve the medical system comprehensive development level of the health investment and economic growth system, in the health investment, increases the supply of resources at the same time, according to the need, from past vulgar to hierarchical input, setting the priority area and time preference input area, general area and input tightening area, precision type [18] ; take the way of collaboration between cities to promote the circulation of resources, to achieve the optimized allocation of resources, so as to promote the rational use of the resources of the whole. Regarding economic development, expanding domestic demand and promoting consumption upgrading is the inevitable choice to promote the development of economy; we should pay attention to improving the resident per capita income, increasing resident income channels, and giving fair and reasonable allocation policy, from the source to stimulate consumption power, to promote urban economic development.

From the overall, the advantages of regional economy in western and south of Sichuan area did not lead to the development of health investment system, coordination development level for health investment system lags behind the economic growth and resident health system, which leads to a limitation of the overall coordination development level; the east and north of Sichuan, because the economic growth lags behind the health investment and resident health system, failed to achieve a higher level of coordinated development. From the perspective of regional growth, there are differences in the coordinated development of cities (states) in the four regions. Such as the western Sichuan, Chengdu is in the western Sichuan region and completely different from Ganzi coupling coordination degree; their difference is 0.47, its health investment; resident health and economic growth system of the comprehensive development index differences were 0.717, 0.442, 0.595; significant differences led to the development of each system between regions; the coordinated development is in difference. To take different development method, therefore, for the lack of overall medical resources of the western and south of Sichuan, we should make use of the advantage of local finance, strengthen health technical personnel team construction, improve the basic public health service costs, expand health service facilities dropping from the aspects such as human, financial and increase the intensity of support, thus improve the ability of grassroots medical institutions and health services; for the east and north of Sichuan, on one hand, we should strengthen the government fiscal responsibility in grassroots medical payment and increase financial input and transfer payments [19] ; on the other hand, economic development should be taken as the key driver and multiple measures should be taken in parallel to improve the scale and economic benefits of local economy. It is imperative to formulate economic development plans that match the advance of health undertakings in Sichuan Province. By doing so, medical investment, resident health and economic growth in Sichuan Province can go hand in hand with each other.