A Spatial Accessibility Study of Public Hospitals: A Multi-Mode Gravity-Based Two-Step Floating Catchment Area Method

Abstract

The multi-modal two-step floating catchment area (MM-2SFCA) method is an extension of the two-step floating catchment area (2SFCA) method that incorporates the impact of different transportation modes, thereby facilitating more accurate calculations of the spatial accessibility of public facilities in urban areas. However, the MM-2SFCA method does not account for the impact of distance within the search radius on supply–demand capacities, and it assumes an idealized supply–demand relationship. This paper introduces the gravity model into the MM-2SFCA method, proposing a multi-modal gravity-based 2SFCA (MM-G2SFCA) method to better account for distance decay and supply–demand relationships. Furthermore, a standardized gravity model is proposed based on the traditional gravity model. This model imposes constraints on upper and lower limits for distance decay weights without compromising the fundamental curve characteristics of the gravity model, thereby avoiding extreme weight scenarios. The accessibility of public hospitals in Shenzhen is evaluated through the integration of basic geographic information data, resident travel data, and official statistical data. The findings demonstrate that the standardized gravity model effectively addresses the issue of excessively high local distance weights in the traditional gravity model, making it more suitable as a distance decay function. The MM-G2SFCA method improves the consideration of distance and supply–demand relationships, thereby facilitating a more rational distribution of accessibility on a global scale. This study discovers differences in the spatial allocation of public hospital resources across the Shenzhen’s districts. Accessibility within the metropolitan core is significantly higher than that outside the core. Additionally, there is a notable difference in the level of accessibility among the districts. Accessibility is found to be better in district centers and along the main traffic arteries.

1. Introduction

Spatial accessibility is a critical metric for evaluating the spatial distribution of public resources and their utilization efficiency, holding significant importance in various fields such as land use, urban planning, and facility siting analysis [1,2,3]. The study of spatial accessibility primarily stems from Hansen’s concept of location accessibility, defined as the magnitude of interaction opportunities between nodes within a transportation network [4]. In the 1950s, the concept of spatial accessibility began to garner attention from Western scholars [5]. Since the 1980s, China has employed computer-aided assessments to evaluate travel accessibility, and by 2024, the volume of related research achieved a leading position internationally. Compared to traditional “per capita indicators,” spatial accessibility offers a more comprehensive and intuitive approach to evaluating the rationality of public facility spatial layouts, thereby fostering the development of various methods for assessing the accessibility of public service facilities. Among these methods, the gravity model [6] and the two-step floating catchment area (2SFCA) method [7,8] have garnered significant attention due to their similar theoretical foundations, broad application scenarios, and relatively comprehensive considerations. The gravity model, also known as the gravitational or potential model, is one of the most classic spatial interaction models that calculates the reachable area by setting a continuous scale of travel impedance. The 2SFCA method determines the accessible facilities within the search radius of the demand points by sequentially searching for supply points and demand points. The 2SFCA method is easy to understand, simple to operate, and highly interpretable, making it a foundational framework for various extended forms of research [9].

The 2SFCA method has been extensively employed to quantify the spatial accessibility of diverse facilities. Researchers have applied it to assess spatial disparities in residents’ access to medical resources, informing rational resource allocation [10,11,12,13]; to explore the accessibility of urban park green spaces, advancing urban ecological restoration and green space functionality [14,15]; and to measure the scarcity of elderly care facilities, leading to policy recommendations for optimized facility placement [16]. As the research scope broadens and methodological advancements occur, scholars have addressed the limitations of the traditional 2SFCA method, leading to the development of a robust 2SFCA methodology cluster. For instance, Wang [17] introduced the concept of distance decay functions to refine accessibility estimates based on spatial distance relationships, addressing inaccuracies stemming from simple distance dichotomies. Luo [18] integrated the classical Huff model [19] with the 2SFCA method, considering both travel costs and facility service capabilities to enhance the method’s rationality. Furthermore, with the diversification of travel options, research on multi-mode accessibility has gained momentum. Mao [20] proposed the multi-mode 2SFCA (MM-2SFCA), distinguishing between driving and public transportation modes to modify the traditional single-mode approach. Ni [21] introduced an enhanced multi-mode 2SFCA method for estimating healthcare accessibility. Guo [22] leveraged map APIs to obtain realistic travel time cost matrices, considering multi-mode transportation combinations. Langford [23] incorporated segmented distance functions into the MM-2SFCA, segmenting the population into public and private transportation users. Wang [24] accounted for facility attractiveness and supply-driven selection behavior, enhancing the MM-2SFCA with the Huff model, considering variations in travel modes and facility appeal. A significant portion of the research on 2SFCA methods focuses on the exploration of the distance, with many improved 2SFCA methods incorporating distance decay functions. The gravity model within the distance decay function can more accurately characterize the intrinsic nature of distance effects [25], although it suffers from the issue of excessive local distance weighting, which hampers further expansion.

In this paper, a multi-mode gravity-based two-step floating catchment area method (MM-G2SFCA) is proposed to enhance the MM-2SFCA method in terms of the distance decay and supply–demand relationships. Additionally, a standardized gravity model is proposed to mitigate the shortcomings of the traditional gravity model. To validate the effectiveness of the proposed methods, this study targets the accessibility of public hospitals in Shenzhen. Accessibility is calculated using three methods: the MM-2SFCA, the traditional gravity model MM-G2SFCA, and the standardized gravity model MM-G2SFCA. The spatial accessibility results of public hospitals obtained from these methods are compared and analyzed. The spatial distribution characteristics of public hospital accessibility in Shenzhen are examined in depth to provide a scientific basis for the rational allocation of medical resources.

2. Related Work

2.1. Two-Step Floating Catchment Method

The core of the 2SFCA method lies in evaluating the spatial alignment between service supply and demand through a two-step process. The first step involves determining the service area $d_0$ for each supply point $j$ (i.e., service facility). This $d_0$, also known as the search radius, is typically defined based on either distance or time. Within this radius, the supply–demand ratio $R$ for each supply point is calculated based on the potential demand:

$$R_j={\displaystyle \frac{S_j}{\sum _{k\in \{d_{kj}\le d_0\}}{D}_k}},$$

(1)

where $S_j$ denotes the scale or capacity of supply point $j$, $D_k$ represents the demand at demand point $k$, usually measured by the population size, and $d_{kj}$ is the distance from the demand point $k$ to the supply point $j$.

In the second step, for each demand point $i$, the accessibility $A_i$ is calculated by summing the supply–demand ratios $R_j$ of all supply points $j$ within its search radius $d_0$. Initially,

$$A_i={\sum }_{j\in \left\{d_{ij}\le d_0\right\}}{R}_j={\sum }_{j\in \left\{d_{ij}\le d_0\right\}}\left({\displaystyle \frac{S_j}{{\sum }_{k\in \left\{d_{kj}\le d_0\right\}}{D}_k}}\right).$$

(2)

The method sets a fixed search radius around the supply point, considering all residents within this radius as potential users to compute the supply–demand ratio. Subsequently, it evaluates accessibility by summing the supply–demand ratios of supply points within the search radius centered on the demand point. The supply–demand ratio reflects the capacity of service facilities to meet the potential user demand within the service area, while accessibility indicates the cost and convenience for users in obtaining these services.

2.2. Multi-Mode Two-Step Floating Catchment Area Method

Considering the practical scenario where individuals utilize different transportation modes, the traditional 2SFCA method’s assumptions are limited. For instance, residents may prefer public transportation due to traffic congestion and parking difficulties, and low-income groups may use cars less frequently. Ignoring these differences can introduce errors in accessibility analysis. Based on this, Mao [20] proposed the MM-2SFCA method, which categorizes the population at the demand points and calculates the supply–demand ratio $R_j$ and accessibility $A_i$ for each demand point according to different population categories. The supply–demand ratio $R_j$ for supply point $j$ is calculated as follows:

$$R_j={\displaystyle \frac{S_j}{{\sum }_{k\in \{d_{kj}\left(M_1\right)\le d_0\left(M_1\right)\}}{P}_{k,M_1}+{\sum }_{k\in \{d_{kj}\left(M_2\right)\le d_0\left(M_2\right)\}}{P}_{k,M_2}+\cdots +{\sum }_{k\in \{d_{kj}\left(M_n\right)\le d_0\left(M_n\right)\}}{P}_{k,M_n}}},$$

(3)

where $M$ represents the transportation mode, and the population $P_k$ at demand point $k$ is divided into $n$ subgroups according to $n$ transportation modes $\{M_1,M_2,\dots ,M_n\}$; $d_{kj}\left(M_n\right)$ is the time from demand point $k$ to supply point $j$ under mode $M_n$; and $d_0\left(M_n\right)$ is the search radius under mode $M_n$. That is, multiple search radius areas are drawn according to the transportation modes, and the sum of the travel population within the corresponding area is considered the demand for service $S_j$.

The overall accessibility $A_i$ of demand point $i$ under multiple transportation modes is calculated as follows:

$$A_i={\displaystyle \frac{P_{i,M_1}{\sum }_{j\in d_{ij}\left(M_1\right)\le d_0\left(M_1\right)}{R}_j+P_{i,M_2}{\sum }_{j\in d_{ij}\left(M_2\right)\le d_0\left(M_2\right)}{R}_j+\cdots +P_{i,M_n}{\sum }_{j\in d_{ij}\left(M_n\right)\le d_0\left(M_n\right)}{R}_j}{{\sum }_{v=1}^n{P}_{i,M_v}}}.$$

(4)

$A_i$ is not simply the sum of all $R_j$ within supply point $i$ but rather a weighted sum based on the size of each $P_k$ subgroup within the search radius of each facility, which is then summed to calculate the overall accessibility of demand point $i$.

The MM-2SFCA method enhances the rationality of accessibility modeling by supporting multiple transportation modes through the classification of resident populations at demand points.

3. Materials and Methods

3.1. Design of Multi-Mode Gravity-Based Two-Step Floating Catchment Area Method

In the traditional 2SFCA method, the treatment of distance affecting the supply–demand ratio and accessibility calculations is dichotomy: there is no decay within the search radius, and the intensity is zero outside the search radius, as shown in Figure 1. This means that the intensity at any point within the search radius is considered to be the same. This characteristic is also present in the MM-2SFCA method. For instance, as shown in Figure 2, assuming that the demand points have the same population, the four demand points will have equal weights in the calculation of the supply–demand ratio in car mode, and $P_1$, $P_2$, and $P_3$ will have equal weights in the calculation of the supply–demand ratio in bicycle mode. The roles of $P_3$ and $P_4$ in the final supply–demand ratio calculation will be identical, even though $P_3$ is closer to the supply point $S_j$ than $P_4$. This does not align with the objective principle, which suggests that $P_1$ should be given more weight than $P_4$ in the supply–demand ratio calculation in car mode, and, similarly, $P_3$ should be given more weight than $P_4$ in the final supply–demand ratio calculation. The same issue arises in the accessibility calculation.

To address this issue, the gravity model is introduced to add support for distance decay to the MM-2SFCA method, and a multi-mode gravity-based two-step floating catchment area method (MM-G2SFCA) is proposed. The gravity model is not only well-developed in the field of spatial interactions [6] but is also widely used in the improvement of the 2SFCA method [9]. It is, therefore, well suited for the improvement of the MM-2SFCA method. The calculation methods for the supply–demand ratio $R_j$ of supply point $j$ and the accessibility $A_i$ of demand point $i$ are as follows:

$$R_j={\displaystyle \frac{S_j}{\sum _{v=1}^n\sum _{\begin{array}{c}k\in \left\{d_{kj}\right(M_v)\le d_0(M_v\left)\right\}\end{array}} t_{n,k}{D}_{k}f(d_{kj})}},$$

(5)

$$A_i=\sum _{v=1}^n{\displaystyle \sum _{j\in \left\{d_{ij}\left(M_v\right)\le d_0\left(M_v\right)\right\}}}{t_{n,k}R}_j{\left[f\left(d_{ij}\right)\right]}^2,$$

(6)

where $t_{n,k}$ represents the travel proportion of residents at demand point $k$ under transportation mode $M_n$. When $t_{n,k}$ data are lacking, $t_{n,k}={\displaystyle \frac{M_{k,n}}{M_k}}$, where $M$ is the transportation mode, $M_k$ is the total travel volume at demand point $k$, and $M_{k,n}$ is the travel volume under the $n$-th transportation mode; the meanings of $d_{kj}\left(M_n\right)$ and $d_0\left(M_n\right)$ are the same as in Equation (3). $f\left(d_{kj}\right)$ is the distance decay function, which increases the supply–demand ratio within the search radius as the distance shortens, aligning more closely with the First Law of Geography [26]. Liu [25] found that the power-law form of the distance decay function better characterizes the intrinsic nature of the distance effects compared to exponential and other forms, and its corresponding interaction model is the widely used gravity model in geography and regional economics. Therefore, this work adopts the gravity model as the distance decay function:

$$f\left(d\right)=\left\{\begin{array}{l}{d}^{-\beta },d\le d_0\\ 0,d>d_0\end{array}\right.,$$

(7)

where $d$ represents the cost of movement, calculated as the time under multiple transportation modes; $\beta$ ($\beta >0$) represents the distance decay coefficient, also referred to as the impedance coefficient. An increase in $\beta$ indicates a more pronounced effect of distance on the supply–demand ratio and accessibility.

The introduction of the gravity model serves to incorporate a distance decay property into the MM-2SFCA method. This allows the supply–demand ratio and accessibility to decrease with increasing distance within the search radius, rather than remaining constant. As shown in Figure 2b,c, as the distance cost from the supply point increases, the weight of the demand point becomes relatively smaller. $P_1$ will have a significantly larger weight than $P_4$ in the supply–demand ratio calculation in the driving mode, and, similarly, the weights of $P_3$ and $P_4$ will no longer be equal in the final supply–demand ratio calculation.

3.2. Standardized Gravity Model

Although the gravity model, characterized by a power-law function, is extensively utilized in spatial accessibility analysis, it presents a notable issue: as the distance $d$ approaches zero, the function $f\left(d\right)$ tends to infinity, as illustrated in Figure 1b. This can result in significant discrepancies in the magnitude of $f\left(d\right)$ at varying distances, leading to an overall abnormal distribution of accessibility values. To mitigate this problem, this paper introduces a standardized gravity model:

$$f\left(d\right)=\left\{\begin{array}{l}{({\displaystyle \frac{d}{d_0}}+1)}^{-\beta },d\le d_0\\ 0,d>d_0\end{array}\right.,$$

(8)

where $d_0$ denotes the search radius. Unlike the original gravity model, the standardized version is independent of the unit of $d$, making it more appropriate for use as a decay function. Through standardization, the range of the distance decay function is confined to $(2^{-\beta },1]$, establishing the upper and lower limits of the distance decay effect, as shown in Figure 1c. The standardized gravity model prevents extreme variations while preserving the ability to adjust the strength of the distance decay function via $\beta$, thereby making it more suitable for the 2SFCA method’s application scenarios.

The term $\left[f\right(d_{ij}){]}^2$ in Equation (6) represents an enhancement of the supply–demand relationship using the standardized gravity model. Delamater [27] noted that, in both traditional and distance-decay-adjusted 2SFCA methods, the sum of the product of accessibility and demand scale for each demand point equals the total facility scale of all supply points. This suggests that in this model, the resources provided by all facilities perfectly align with the demand at the demand points, which is an idealized scenario that deviates from reality. Therefore, introducing a certain loss to the accessibility of demand points, making the supply scale less than the demand scale, aligns better with real-world conditions.

The MM-G2SFCA method is compatible with the MM-2SFCA method. When the search radius $f\left(d_{ij}\right)$ is fixed at 1, $M_k$ represents the number of residents at point $k$, and $M_{k,n}$ represents the number of residents traveling by the $n$-th mode of transportation at point $k$, Equations (5) and (6) are equivalent to Equations (2) and (3).

With advancements in information technology, spatiotemporal big data related to residents’ travel, such as OD data, travel data, and passenger volume data, as shown in

Table 1. Accessibility relevance of three types of data in multiple transportation modes.

Data Type	Data Description	Data Characteristics	Accessibility Relevance
OD Data	Data on the origin and destination of individual trips	Directly includes time and spatial costs, and its distribution can directly reflect the accessibility of different regions under corresponding transportation modes	High
Travel Data	Proportion of residents’ travel modes within a region obtained through statistics, interviews, or calculations	Provides an intuitive reflection of residents’ choice of transportation modes but does not include destination information	Medium
Passenger Volume Data	Data on the passenger volume carried by various transportation modes within a region or on roads	Represents the passenger capacity of each transportation mode, reflecting residents’ travel modes to a certain extent	Low

, are continuously being enriched. These data can reflect residents’ choices of travel modes to varying degrees and hold potential for application in accessibility analysis, providing data support for the MM-G2SFCA method beyond the residents’ transportation population structure data [20].

3.3. Study Area

Shenzhen is a special economic zone approved by the State Council, a national economic center, and an innovative city. Located in the core area of the Pearl River Delta economic zone and adjacent to Hong Kong, it boasts abundant natural resources and a strategic geographical position. By the end of 2022, Shenzhen’s permanent population reached 17.6618 million [28]. Despite its large population and rapid urban development, the former special zone management system led to the formation of Shenzhen’s central area, with Luohu District, Futian District, and Nanshan District at its core. The 14th Five-Year Plan for National Economic and Social Development and the Long-Range Objectives through the Year 2035 of Shenzhen has expanded the original central area to include parts of Bao’an District, Longhua District, and Longgang District as the new metropolitan core area, as shown in Figure 3.

The development level, environment, and management system in Shenzhen exhibit significant spatial imbalances, which not only present high research value but also impose higher requirements for spatial accessibility research. Furthermore, Shenzhen’s official establishment of multiple data release platforms, with a high degree of data openness, provides ample data support for research.

3.4. Data Collection and Preprocessing

According to the health statistics summary released by the Shenzhen Municipal Health Commission [29], in 2022, medical institutions in the city conducted 103.6719 million outpatient visits, with public hospitals accounting for 91.3814 million visits, representing 88.14%. This indicates that public hospitals play a central role in the medical service acquisition of urban residents.

The number of beds is commonly used as an indicator to measure the supply capacity of medical facilities [8] and is widely used in availability analysis studies. However, in daily practice, hospitalized patients only represent a portion of all patients, so the number of beds can only partially reflect the supply capacity of medical facilities. In contrast, outpatient registrations are a necessary prerequisite for inpatient treatment, so outpatient visits can more comprehensively demonstrate the ability of medical facilities to receive patients. Therefore, this study uses the monthly outpatient visits to public hospitals to represent the service capacity of hospitals.

3.4.1. Population Data

The population data used in this study came from the 2020 WorldPop [30] top-down constrained 100 m resolution total population data. These data are based on top-down decomposed population and housing census data and use settlement data as a mask to filter out uninhabited areas, making the spatial distribution of the population more accurate than the general data. On this basis, the population data of the study area were corrected according to the total population given in the 2023 Shenzhen Statistical Yearbook [28].

3.4.2. Public Hospital Data

The public hospital data included the POI points and outpatient visits of 71 public hospitals in Shenzhen. The POI points came from Amap, and the outpatient visits came from the monthly public hospital appointment registration work briefings published on the official website of the Shenzhen Municipal Health Commission [31].

3.4.3. Residential Travel Data

The resident travel data were based on the 2019 Shenzhen Resident Travel Behavior and Willingness Survey Report released by the Shenzhen Urban Transport Planning and Design Research Center [32]. The travel modes of residents within Shenzhen can be divided into seven types, as shown in

Table 2. Mode of Shenzhen residents’ travel.

Travel Mode	Walking	Bicycle	Electric Bicycle	Rail Transit	Bus	Taxi	Private Car
Percentage	46.7%	4.0%	7.1%	8.4%	12.5%	4.0%	17.3%

. Based on the types of transportation tools and combined with the route planning travel modes provided by mainstream map service providers, this study simplified the travel modes into four types: walking, cycling (bicycle, electric bicycle), public transit (rail transit, conventional bus), and driving (taxi, private car), accounting for 46.7%, 11.1%, 20.9%, and 21.3%, respectively.

3.4.4. Basic Geographic Information Data

The administrative division data came from the National Geographic Information Public Service Platform of Tianditu, and the road data came from the OSM (Open Street Map) official website. The OD cost data for the four types of route planning between the study units and hospitals were obtained by calling the Amap API route planning interface to ensure the practicality and accuracy of the data.

This study selected highways, main roads, and major roads from the OSM data combined with administrative division data for the division of study units. Within the city, the units were divided based on roads, while in the suburban and sparsely populated areas, the units were divided based on the visible residential areas and population data. The division process was further refined using population data and remote sensing images, splitting residential areas in parks and suburban plots and merging non-residential areas to ensure that the study units highlighted population attributes.

3.5. Overall Workflow

The technical workflow for the research in GIS is shown in Figure 4. Initially, input layers such as road data, administrative division data, population data, and hospital location point data are imported into ArcGIS. During the data processing phase, study units are divided based on road and administrative data. Population-weighted centroids are then extracted and assigned population data. Subsequently, OD cost matrices are created using the NumPy library to build travel cost matrices for various travel modes, and actual travel distances and times are retrieved via a Python-based data crawler using the Amap API.

In the calculation phase, three main steps are performed: calculating filter matrices, calculating the supply–demand ratio, and calculating accessibility. Filter matrices are generated based on the required time cost or search radius for each OD cost matrix. These matrices are then used to filter OD cost matrices to select demand points within the search radius, which allows for the calculation of supply–demand ratio matrices. Finally, accessibility is calculated by filtering the supply–demand ratio matrices to select supply points within the search radius, and the results are computed according to the specified formula.

4. Results

4.1. Coefficient Test of the MM-G2SFCA Method

We chose a series of search radii and distance decay coefficients for experimentation to verify the roles of these two coefficients in the MM-G2SFCA method. Unless otherwise specified, the MM-G2SFCA method discussed in the following section uses the standardized gravity model by default.

Fixed-duration isochrones, such as 10 min community life circles and 30 min travel time circles, are crucial tools for assessing accessibility. They are commonly used for tasks related to accessibility analysis. Isochrones with 10 to 30 min search radii are often chosen for community-wide accessibility analysis [23], while city-wide search radii may extend to hours [22]. To test the performance of the MM-G2SFCA method with different search radii, 10 to 60 min intervals at 10 min increments, along with supplementary 90 and 120 min intervals, are selected in our experiments. Figure 5 shows the accessibility of the study area as the search radius increases from 10 min to 60 min, as well as at 90 min and 120 min, with the distance decay coefficient set at 1.

The search radius represents two interrelated concepts: the range within which medical facilities provide services and the maximum time that residents are willing to spend (cost) to access these facilities. At a very low search radius, the accessibility values are concentrated in a few areas with a high density of medical facilities, leaving vast regions with an accessibility value of zero. As the search radius expands, the spatial distribution of accessibility results across the region becomes more uniform, leading to a more rational representation of the service capacity of the facilities.

Unlike common public facilities such as parks, dining, and shopping, medical service facilities are irreplaceable, leading residents to generally accept higher time costs. Therefore, taking into account the typical time cost, the search radius for the experiment in the following section is set at 60 min.

The distance decay coefficient regulates the degree to which distance (cost) influences the supply–demand ratio and accessibility. The selection of distance decay coefficients is guided by the existing literature and empirical observations. For individual movement at the urban scale, distance decay coefficients typically ranges from 1.0 to 2.0 [25]. In accessibility studies using the gravity model, distance decay coefficients typically range from 0.9 to 2.29 [7,8,9]. These values, derived from empirical studies analyzing travel behavior and accessibility patterns in various urban settings, serve as a benchmark for understanding the sensitivity of accessibility measures to distance. Additionally, the modifications to the supply–demand relationship in the MM-G2SFCA method (Equation (6)) on the distance decay coefficient can be regarded as 2β. Considering the aforementioned points, an initial distance decay coefficient range of 0.5 to 2.0 is selected for the experiment. However, upon analyzing the experimental results, it is evident that the relatively gradual distance decay curve of the standardized gravity model suggests that an increase in the distance decay coefficient has a relatively minor impact on accessibility. Consequently, a distance decay coefficient range of 0.5 to 4.0 is selected, as illustrated in Figure 6.

It can be observed that only significant changes in the distance decay coefficient result in noticeable differences in accessibility. The experiment demonstrates that the distance decay coefficient can control the distance decay effect in the MM-G2SFCA method using the standardized gravity model and is relatively insensitive. Based on the experimental results and drawing on the experience of distance decay coefficient parameter settings from the aforementioned literature, the distance decay coefficient in the subsequent experiments is set to 1.

4.2. Comparative Analysis of the Traditional Gravity Model MM-G2SFCA and the Standardized Gravity Model MM-G2SFCA

To validate the effectiveness of the standardized gravity model, the accessibility of public hospitals in Shenzhen was calculated using three main methods: the MM-2SFCA, the traditional gravity model MM-G2SFCA, and the standardized gravity model MM-G2SFCA.

The accessibility results obtained using the traditional gravity model MM-G2SFCA method exhibited an order of magnitude difference in numerical range compared to other methods. Specifically, the accessibility of one area was significantly higher than all the other areas, as shown in Figure 7. This is primarily due to the extremely low time cost (very short distance) between the demand point and the supply point in that area, resulting in an excessively high weight in the gravity model. This phenomenon was not isolated; the issue of excessively high accessibility due to very short distances between supply and demand points was widespread in the traditional gravity model.

The visual effect of the accessibility shown in Figure 7b is extreme because the MM-G2SFCA method amplified the effect of the distance weight in Equation (6) on the supply–demand relationship. Figure 7c shows the one-hour accessibility results using the gravity-based 2SFCA (G2SFCA) method for the driving mode. The G2SFCA method provides an improvement over the original 2SFCA formula via the gravity model. Although the accessibility results were relatively normal, the uneven spatial distribution of high accessibility areas persists. In fact, this problem of the traditional gravity model is common in various gravity-based model 2SFCA methods, although it is relatively less pronounced.

The scatter density plot in Figure 8 shows the distribution characteristics of accessibility using the traditional gravity model MM-G2SFCA method and comparing with the standardized gravity model MM-G2SFCA method. The data distribution of the traditional gravity model MM-G2SFCA method was found to fit the gravity model curve to a certain extent. However, individual extremely high values stretched the overall numerical range, causing the accessibility distribution in other areas to fall within a relatively narrow range. In contrast, the accessibility of the standardized gravity model MM-G2SFCA method was observed to be more continuous in spatial distribution, without extreme value points. The accessibility results were proportionally reduced, while the numerical distribution was more uniform.

Therefore, the standardized gravity model effectively addresses the issue of excessively high local weights commonly found in traditional gravity models while maintaining the role of the distance decay function, making it suitable for use as the distance decay function for the 2SFCA method.

4.3. Improved Effectiveness of the MM-G2SFCA Method

Scatter plots were used to characterize the distribution of the accessibility results, and the normalized difference between the two results visually expresses the difference in their spatial distribution. Local spatial autocorrelation analyses were conducted on the MM-2SFCA and MM-G2SFCA methods’ accessibility using the Getis-Ord Gi* statistics to identify hot spot and cold spot areas. This was carried out to gain a deeper understanding of the spatial differences and clustering characteristics of the two accessibility results.

As illustrated in Figure 9, the scatter density plot reveals that the accessibility results for the MM-G2SFCA method exhibit a relatively higher degree of concentration and a lower overall level compared to the MM-2SFCA method. This can be more clearly verified through the use of a difference plot of the normalized results of the two models, as illustrated in Figure 10. The red color in the plot represents a more prominent accessibility for the MM-G2SFCA method, while the orange color and below represent a lower accessibility than the MM-2SFCA method. It is evident that the number of areas with reduced levels of accessibility is significant and widespread.

The difference between the normalized values reveals the accessibility differences between the MM-G2SFCA method and the MM-2SFCA method. The MM-G2SFCA method increased the limitation of the time cost on the expression of accessibility, emphasizing the attraction of the supply points to the surrounding demand points, making the two high-accessibility centers in Luohu District and Nanshan District in the southwest of Shenzhen more prominent. Conversely, the accessibility in the central and western regions significantly declined. This is confirmed in Figure 11, where the MM-G2SFCA method shows a slight reduction in the extent of the hot spot areas and that the cold spot areas are more prominent in the central and western regions. Combined with the distribution analysis of public hospitals, shown in Figure 12, the number of public hospitals in this region is relatively small, and the original high accessibility mainly comes from the complete radiation of a large number of public hospitals in the south. That is, once the hospital is accessible under a particular transportation mode, the MM-2SFCA method will fully include the hospital’s supply capacity under the corresponding transportation mode in the region’s accessibility, while the MM-G2SFCA method corrects this issue.

At the same time, due to the reduction in the virtual high accessibility in these areas, the accessibility of remote areas relatively improved, such as Guangming District in the northwest and Dapeng New District in the east. Accordingly, the confidence level of these two areas as cold spots in the Gi* statistics results decreases.

Therefore, the MM-G2SFCA method helps to fix the impact of local neighboring areas, making the distribution of accessibility more reasonable.

4.4. Spatial Distribution Characteristics of Public Hospital Accessibility in Shenzhen

The autocorrelation analysis of the Moran index shows that the Z-score of the accessibility of the research units in Shenzhen is 57.07, indicating that accessibility is not randomly distributed but shows significant spatial clustering characteristics. An in-depth analysis of the spatial distribution characteristics of the accessibility of public hospitals in Shenzhen reveals the following features:

(1) The accessibility of public hospitals in the metropolitan core area of Shenzhen is significantly higher than outside the core area. As shown in

Table 3. Accessibility statistics within and outside the metropolitan core of Shenzhen.

Area	Maximum	Minimum	Mean	Median	Standard Deviation
Within the metropolitan core area	0.21300	0.01582	0.08699	0.07287	0.05093
Outside the metropolitan core area	0.19086	0.00038	0.03701	0.03361	0.02393
Shenzhen citywide	0.21300	0.00038	0.06129	0.04388	0.04704

, the average and median accessibility within the metropolitan core area is almost twice those outside the core area, showing a “center–periphery” pattern in spatial distribution. The statistical data not only reflect the generally low and less varied accessibility of public hospitals outside the metropolitan core area but also indicate that while the accessibility within the metropolitan core area is generally high, there are also particularly high points. Figure 12 shows that two significant high-accessibility areas are located at the junction of Nanshan District and Bao’an District and the entire Futian District and the western part of Luohu District, all located within the metropolitan core area of Shenzhen, and most of them are within the historical special zone. The high accessibility in this area is highly related to its level of economic development and population density, highlighting the close relationship between the accessibility of public hospitals and urban economic development.

(2) Significant differences in the accessibility levels of public hospitals across different districts in Shenzhen have been observed. According to the accessibility statistics histogram of each district in Figure 13, the ten districts can be divided into four categories: (a) Luohu District and Futian District have the highest overall accessibility, with the best average accessibility level, forming the highest accessibility center in Shenzhen. (b) Nanshan District is noted for relatively high overall accessibility, with its junction with Bao’an District forming the second-highest accessibility center in Shenzhen. (c) Bao’an District, Longgang District, Longhua District, Guangming District, Pingshan District, and Yantian District have average overall accessibility, with some southern streets of Bao’an District being included in the metropolitan core area, raising the maximum accessibility value; similarly, this is observed in Longgang District near Luohu District. (d) Dapeng New District is identified as having the worst accessibility.

(3) The accessibility of regional centers has been found to be better and evenly distributed along major traffic arteries. In areas such as Bao’an, Guangming, Longhua, Longgang, and Pingshan, the accessibility of public hospitals is centered on regional centers and radiates outward along major traffic arteries, forming a relatively uniform accessibility distribution pattern. This characteristic indicates that the development of the transportation network plays an important role in improving the spatial accessibility of medical service facilities, and good transportation conditions help to improve the accessibility of medical services for residents in surrounding areas.

The Lorenz curve and the Gini coefficient are not only classical tools commonly used in economics but are also widely employed in evaluating public service facilities to assess the degree of imbalance in the spatial distribution of resources. The Gini coefficient ranges from 0 to 1, where 0 denotes perfect equality and 1 denotes perfect inequality. In this paper, based on the accessibility of public hospitals and the distribution of street populations, the Lorenz curves for Shenzhen and its metropolitan core areas, both inside and outside the city, are plotted, and the Gini coefficients for 10 districts are calculated, as shown in Figure 14 and

Table 4. Gini coefficient of accessibility of public hospital in each district.

District	Gini Coefficient	District	Gini Coefficient
Yantian	0.277	Bao’an	0.307
Nanshan	0.207	Dapeng	0.308
Luohu	0.198	Pingshan	0.200
Longhua	0.229	Futian	0.171
Longgang	0.235	Guangming	0.156

.

(4) Shenzhen exhibits overall inequality, with significant differences in equity levels within the metropolitan core and more consistency within the districts. The highest Gini coefficient is for Shenzhen as a whole, at 0.369, followed by Bao’an and Dapeng Districts, with Gini coefficients of 0.307 and 0.308, respectively. The level of equity in the other districts is more uniform. It is worth noting that a low Gini coefficient only indicates that the difference in the level of accessibility within the district is not significant, rather than high. Combined with the analysis in Figure 13, it can be seen that Bao’an District has a high level of accessibility to public hospitals but a low level of internal equity, suggesting that relatively localized adjustments to hospital resources would be sufficient. Dapeng District has the lowest level of accessibility and insufficient internal equity, requiring both global inputs and localized adjustments. Futian, Luohu, and Nanshan Districts have high levels of accessibility and good internal equity, while Longhua, Guangming, Pingshan, and Yantian Districts require more overall resource allocation.

5. Discussion

5.1. Advantages of the MM-G2SFCA Method and the Standardized Gravity Model

In this paper, building on the MM-2SFCA method, the MM-G2SFCA method is proposed in combination with the gravity model. Meanwhile, the standardized gravity model, derived from the traditional gravity model, is proposed to overcome the shortcomings of the gravity model when applied to the 2SFCA method.

(1) In the MM-2SFCA method, the supply–demand ratio and accessibility intensity are independent of the distance between the point and the destination within each travel mode catchment area, while the MM-G2SFCA method, which introduces the gravity model as a distance decay function, solves this problem by ensuring that the supply–demand ratio and accessibility intensity decrease with the increase in distance between the point and the destination. This approach is theoretically closer to the actual situation and more in line with the First Law of Geography. (2) Example experiments show that the MM-G2SFCA method leads to a decrease in the overall accessibility level due to the increased constraint of distance cost on accessibility. However, while the accessibility level around hotspots decreases, the accessibility level of remote areas increases relatively, which demonstrates that the MM-G2SFCA method helps to correct the influence of local neighborhoods and makes the accessibility distribution more reasonable. (3) The traditional gravity model faces the issue that when the distance is close to 0, the decay function value tends to infinity. The standardized gravity model regulates the upper and lower limits of the distance decay function value range, ensuring the distance decay characteristics and addressing the problem of extremely large value points. Consequently, the results of the accessibility analysis are more reasonable and aesthetically pleasing.

5.2. Limitations and Future Research

While the research has yielded notable findings, there remains room for further enhancement. (1) Since not all official data are released every year, and some data, such as the Shenzhen travel data, were used in previous years, the experimental results may have the problem of insufficient validity. In the future, we can consider releasing questionnaires and field trips to obtain the data with good representability. (2) In choosing the distance decay factor for the experiment, more reference is made to the experience of the existing related literature. The modified standardized gravity model is slightly different from the traditional gravity model in the shape of the curve, and the distance decay coefficient deserves further study. (3) The MM-G2SFCA method proposed in this paper is based on a generalized accessibility approach, which is not limited to medical accessibility analysis; it can be extended to other areas of spatial accessibility analysis, offering the opportunity to play a broader role.

5.3. Policy Implications

The city of Shenzhen has a large population and rapid urban development. However, there is spatial imbalance in the accessibility of public hospitals due to distinct historical reasons. The Special Administrative Region Management Line was established in 1982 until 2018, when it was approved by the State Council for abolition. The line divided Shenzhen into two parts, the inner and outer parts of the city, forming the downtown area of Shenzhen with Luohu, Futian, and Nanshan Districts as the core. According to the following Shenzhen Municipal Policy directives, some streets in Bao’an District, Longhua District, and Longgang District were included in the expansion of the former Shenzhen City Center District into a new metropolitan core area. This phenomenon is evident in the case study presented in this paper, where the difference in accessibility to public hospitals between areas inside and outside the metropolitan core remains significant even when population data are considered. In light of this, the local government should consider the spatial appropriateness of public hospital locations as their own financial situation allows, pay attention to the medical needs of the residents outside the core, and ensure a balance between the supply capacity of the public hospitals and the medical needs of the residents. Given the positive role that roads play in enhancing the service capacity of public hospitals, it is also a reasonable strategy to optimize the allocation of roads and transportation routes and to improve road conditions.

6. Conclusions

In this study, a standardized gravity model was proposed, effectively addressing the issue of excessive local weights in the traditional gravity model by standardizing the upper and lower limits of the distance decay function. The MM-G2SFCA method was proposed by combining the standardized gravity model with the MM-2SFCA method, thereby making the accessibility intensity decrease with increasing distance, thus being theoretically closer to the actual situation. The MM-G2SFCA method proposed in the study represents an improved version of the MM-2SFCA method. This enhanced 2SFCA method considers multiple transportation modes and distance decay and serves as an extension of the 2SFCA method cluster.

Focusing on the accessibility of public hospitals in Shenzhen, experiments were conducted using different methods to verify the effectiveness of the proposed methods. The results revealed significant disparities in the accessibility of Shenzhen public hospitals both within and outside the core area, as well as between districts. The results of the study contribute to understanding the spatial accessibility of public hospitals in Shenzhen, offering a reference for scientific and effective planning, rational site selection, and the layout of medical facilities.