Senegal covers an area of 196,712 km² and is bordered to the west by a coastline of over 700 km. It is bordered to the north by Mauritania, to the east by Mali, to the south by Guinea-Bissau and the Republic of Guinea, and finally by Gambia, which is almost entirely landlocked within its territory ( Figure 1 ). Senegal is divided into 14 regions, 45 departments, 123 districts, and 557 communes. According

to the National Agency of Statistics and Demography, the population of Senegal is expected to be 18,126,390 in 2023 [18] . This population was 13,508,715 in 2013. It therefore experienced an average annual intercensal growth rate of 2.9%, which remained constant between the two intercensal periods.

Northern Senegal has a desert climate, while the south has a tropical savannah climate. There are two main seasons: a dry season (November to June) and a rainy season (June to October), marked by monsoons. The terrain is essentially flat, consisting of plains and plateaus, with some hills and sandy coastal areas. The plains and plateaus are located in the center, the hills in the southeast, and the Senegal River valley in the north.

The data used come from the ANSD and the National Agency for Territorial Planning (ANAT). The ANSD provided us with data on the population and spatial location (in latitude and longitude) of health facilities collected during the 2023 general population and housing census. The administrative division maps used were collected from the ANAT in 2023.

The objective is to measure the overall spatial dispersion of the health post pattern and clearly visualize its orientation. To do this, we use the mean point, the standard distance, and the standard deviation (or dispersion) ellipse. These centrographic measurements allow us to globally display the spatial distribution patterns of a phenomenon [19] [20] , providing information on the degree of spatial concentration or dispersion and its orientation.

The mean point was used to measure and visualize the center of gravity of the health post pattern, as it allows the spatial distribution to be summarized. Its $x_g$ and $y_g$ coordinates are calculated as follows:

$x_g=1/N{\displaystyle \underset{i=1}{\overset{N}{\sum }}{x}_i}$ $y_g=1/N{\displaystyle \underset{i=1}{\overset{N}{\sum }}{y}_i}$ (1)

with N: number of points, $x_i$: longitude of given point i, and $y_i$: latitude of given point i.

To determine the overall spatial dispersion of health centers around the mean point, we used standard distance. This is a statistic that measures the average variability of point positions around the center of gravity. Its formula is as follows:

${\sigma }_{x,y}=\sqrt{1/N{\displaystyle \sum {\left(x_i-x_g\right)}^2+{\left(y_i-y_g\right)}^2}}=\sqrt{{\displaystyle \sum 1/N\left(d_{iG}^2\right)}}$ (2)

with N: number of points, $x_i$: longitude of point i and $y_i$: latitude of point i; $x_g$: longitude of the mean point and $y_g$: latitude of the mean point; d: distance between point i and the mean point.

The use of the dispersion ellipse made it possible to clearly visualize the orientation of the spatial distribution of health posts, which is not readily apparent on the map. This ellipse represents the intensity of the minimum and maximum dispersion of a series of points relative to their mean center [21] . The two standard deviations ( ${\sigma }_x$ and ${\sigma }_y$), in the X and Y directions, are orthogonal and, by construction, form an ellipse showing the distribution orientation of the phenomenon. These are calculated, one along the transposed X axis and the other along the transposed Y axis [22] as follows:

${\sigma }_x=\frac{\sqrt{{\displaystyle \underset{I=1}{\overset{n}{\sum }}{\left(\left(X_i-\bar{X}\right)\cos \vartheta -\left(Y_i-\bar{Y}\right)\sin \vartheta \right)}^2}}}{N-2}$ ${\sigma }_y=\frac{\sqrt{{\displaystyle \underset{I=1}{\overset{n}{\sum }}{\left(\left(X_i-\bar{X}\right)\sin \vartheta -\left(Y_i-\bar{Y}\right)\cos \vartheta \right)}^2}}}{N-2}$(3)

where N is the number of points, θ is the rotation angle of the Y axis relative to the horizontal (X axis), $X_i$: longitude of the given point i and $Y_i$: latitude of the given point $\bar{X}$: longitude of the mean point and $\bar{Y}$: latitude of the mean point.

Centrographic measurements certainly offer the possibility of producing indicators that can be projected onto a map and thus visualize the preferential direction of the spatial distribution of the phenomenon. However, they do not allow statistical confirmation of the existence of this direction. Therefore, we used directional statistics ( Figure 2 ). Using this statistic required the following steps. First, we segmented the study area into four angular sectors representing four cardinal directions, based on the average point of the health post pattern. Then, in each direction, we counted the number of health posts located there. Finally, a chi-square test [23] [24] was used to statistically confirm the existence of at least one preferred direction in the distribution of health posts. The chi-square equation is:

${\chi }^2=\frac{\left(f_i-{\bar{f}}_i\right)}{{\bar{f}}_i}$ (4)

with ${\bar{f}}_i=\frac{E}{C}$ (theoretical workforce), with E = total sample size, $f_i$ = observed workforce in the direction i and C = number of categories.

While these centrographic measurements allow for a description of the overall spatial distribution, they do not reveal the existence of a heterogeneous spatial distribution structure. The use of Ripley’s statistics was therefore necessary.

There are several methods for describing the spatial structure of a point pattern [25] - [29] . These methods have been used particularly in the field of forestry [29] [30] . These methods include those based on quadrats and those based on distances. For quadrat methods, the data are the number of points/positions in quadrats [31] , while for distance-based methods, the data are the distances between points/positions.

Distance-based methods include those that require only knowledge of the nearest neighbors of each point such as the method of [32] , used for small domains) and more expensive ones requiring a map of the entire study area. Ripley’s method is one of the latest methods and is used for large domains.

The advantage of Ripley’s method over other distance-based methods is that it allows to describe the spatial distribution structure simultaneously at multiple distances [29] , revealing the variation of spatial aggregation or dispersion of features when the neighborhood size changes. However, the Ripley function is always difficult to interpret, as the curve obtained for the null hypothesis is in the form of a parabola. In addition, the graphs are very difficult to present. Therefore, we chose to use a modified function $L\left(d\right)$proposed by Besag [33] , which is easy to use. This function is normalized and thus allows us to compare the structure of seedlings with different sizes. The equation of the function $L\left(d\right)$ is:

$L\left(d\right)=\sqrt{\frac{A{{\displaystyle \sum }}_{i=1}^N{{\displaystyle \sum }}_{i=1.j\ne i}^{N}k\left(i,j\right)}{\pi N\left(N-1\right)}}$ (5)

where A represents the area, N is the number of points, d is the distance, and $k\left(i,j\right)$ is the weight. If no boundary correction is applied, the weight is 1 when the distance between i and j is less than or equal to d; it is 0 when the distance between i and j is greater than d. If boundary correction is applied, the weight of $k\left(i,j\right)$ is slightly modified.

The kernel density method is a spatial smoothing technique that highlights spatial trends and areas of high or low concentration of the phenomenon [34] . It consists of creating a continuous density layer of the phenomenon, based on its locations. This layer (grid) is made up of small cells. The density value associated with each cell is the estimate of the phenomenon to be analyzed. This estimate is made for the center point of each grid cell. It can be made using several types of kernel functions. Determining the shape of the data’s influence, the kernel density function counts the number of events within a region called the kernel density kernel, around the location where the estimate is made. The (spatial) weights closest to the events are the most significant. According to Berlinet A. and Devroye L. (1989) [35] , the kernel density estimator $F_n\left(X\right)$ is calculated as follows:

$F_n\left(X\right)=\frac{1}{nh}{\displaystyle {\sum }_{i=1}^{n}k}\left(\frac{\left(x-X_i\right)}{h}\right)$ (6)

with n: the sample used to make the h: neighbor search radius; k: kernel function; X_i = known point, x = point to interpolate.

Centrographic measures and directional statistics are used to describe the overall spatial distribution of health posts. The kernel density method offers the possibility of revealing spatial trends. However, these measures have some limits, as they do not allow statistically significant clusters of local spatial association to be highlighted. To detect these clusters, we used the local indicators of spatial association developed by Anselin [36] . Indeed, with these measures we can determine and visualize the types of local spatial association and their levels of statistical significance.

Their formula is:

$I=\frac{{\displaystyle \underset{j}{\sum }{w}_{ij}\left(x_i-\bar{x}\right)\left(x_j-\bar{x}\right)}}{{\displaystyle \underset{i}{\sum }{\left(x_i-\bar{x}\right)}^2}}$ (7)

with, $x_i$: value taken by the observation $x_i$, $x_j$: value taken by the neighboring observation $x_j$, $\bar{x}$: mean of the variable x, $w_{ij}$ spatial weight.

The overall methodological approach is illustrated in Figure 2 .

The number of geolocated health posts is 2127 ( Figure 3 ), which is more than sufficient for the analysis. To determine whether the overall distribution of health posts is concentrated, dispersed, or random, we calculated centrographic indicators, including the dispersion index and the associated significance test. The dispersion index is interpreted as follows: If it is greater than 1, we say there is a tendency toward dispersion; if it is equal to or very close to 1, we say the distribution is random; and if it is less than 1, we say there is a tendency toward spatial concentration of the phenomenon.

The results of this test show a low dispersion index (0.69) and a z-value of −27.12, much higher than that of the centered-reduced normal distribution (1.96), with a risk of error of 5%; moreover, the observed average distance is much lower than the theoretical average distance ( Table 1 ). The overall spatial distribution of health posts is therefore concentrated at the national level.

We also sought to determine whether distribution patterns are the same across all administrative regions of Senegal. To do this, we measured the centrographic indicators for each region ( Table 2 ). Analysis of the results of these measurements allows us to make the following observations.

NB: *Significant at 5%.

The overall spatial distribution varies from one region to another. Three groups of regions emerge. The first consists of regions that record the highest and statistically significant levels of spatial concentration of health posts. These regions are mainly found in the west of the country (Dakar, Thiès, Fatick, Saint-Louis). The second group includes regions where levels of spatial concentration are noted, although significant, but lower. These are Louga, located in the center-west and Kaolack, located in the center of the country. Finally, the last group is made up of regions where health posts are distributed randomly in space. These regions are mainly found in the south of the country.

These results reflect the heterogeneity of the national area in terms of health center coverage. However, they do not allow us to visualize the spatial distribution of health centers. This is what we will do by analyzing the map of mean points and the standard deviation ellipse.

Figure 3 shows the distribution of health posts, mean points, and standard deviation ellipses at the national and regional levels. At the national level, the overall spatial distribution of health posts is oriented northwest/southeast, as indicated by the pink directional ellipse in Figure 3 . This figure also shows that the center of gravity of health posts is located in the center-west of the country. Furthermore, the extent of the standard deviation ellipse covers this part of the country, revealing that the majority of health posts are located in this part of the national territory. The existence of this preferential direction in the overall spatial distribution of health posts is statistically confirmed by the results of the chi-square test, as the calculated chi-square (880.5) is significantly higher than the theoretical chi-square (7.81) ( Table 3 ).

*Significant at 5%, for a ddl = 3 and a theoretical chi-square = 7, because the chi-square calculated is 880.5.

Analysis of the standard deviation and mean point ellipses at the national level conceals significant spatial disparities. To reveal these disparities, we will examine the standard deviation and mean point ellipses at the regional level.

The variation in the distribution shapes of health posts by region is summarized in Figure 3 . This figure indicates that the standard deviation (or dispersion) ellipses do not have the same size, shape, or orientation. Thus, the regions of Tambacounda and Saint-Louis have the largest ellipses, while those of Diourbel, Kaolack, Sédhiou and Ziguinchor have small ellipses. Ellipses with more elongated shapes are found in the regions of Tambacounda, Saint-Louis, and Dakar, as illustrated in Figure 3 . In addition, Table 4 shows the regions with more elongated shapes. Indeed, the long axis/short axis ratios of the standard deviation ellipse are the highest (3.97 for Saint-Louis, 2.72 for Diourbel, 2.24 for Dakar).

While the ellipses of the Tambacounda and Kédougou regions are oriented northeast/southwest, those of the Diourbel and Saint-Louis regions are oriented east/west. The ellipses of the Ziguinchor and Sédhiou regions indicate that the spatial distribution of health posts is almost unoriented, which is confirmed by values of the long axis/short axis ratios of the standard deviation ellipse very close to 1 ( Table 4 ).

*Significant at the 5% level, for a dof = 3 and a theoretical chi-square = 7.81.

Certainly, the ellipses of the standard deviation can highlight the preferential direction of the distribution of health posts. However, they do not provide the possibility of knowing whether this is statistically significant ( Table 5 ). It is the results of the chi-square tests contained in Table 5 that make it possible to know this. They show that there is no preferential direction of distribution of health posts in all regions of Senegal. Indeed, the three southern regions (Ziguinchor, Sédhiou and Kolda) do not record any preferential direction of distribution of health posts. In other words, health posts are distributed randomly in these regions.

In regions where there is a statistically significant preferential direction, the significance levels are not the same. For example, they are higher in the Matam, Thiès, Dakar, Tambacounda, and Louga regions, and lower in the Fatick and Diourbel regions. Furthermore, the preferred distribution directions for health centers vary by region. While in Matam, the preferred distribution axis is northeast, in Thiès, it is southeast.

Figure 4 shows the multi-distance spatial clusters of health posts. We note, in a way, a trend toward spatial aggregation of health posts across different distance classes (spatial scales). Indeed, the red curve, representing the observed values, is above the gray curve (confidence envelope) and the blue curve (expected/theoretical values) in almost all distance classes in most regions and at the national level. However, this overall trend hides local specificities. For example, in the Kaffrine region, we observe a random distribution over short distances. From 5 km, we note the beginning of spatial aggregation, which increases with distance. In contrast, in the Saint-Louis and Diourbel regions, we observe a concentration phenomenon that diminishes with distance. Still as an example, the regions of Kédougou, Kolda and Sédhiou are territories where no spatial aggregation of health posts is recorded, whatever the spatial scale.

The results presented so far are very interesting, as they have revealed the center of gravity of the health post distribution, the dispersal ellipses, the existence of statistically significant preferential spatial distribution directions, and the spatial aggregation clusters of health posts at several spatial scales (or over several distances).

However, they do not allow us to visualize the density of health posts or the types of local spatial aggregation clusters.

The results produced so far have highlighted the center of gravity, orientation, and presence of clusters of spatial aggregation at several scales of health posts. However, these methods have limitations, as they do not allow for a clear visualization of spatial trends in the spatial distribution of health posts. For all these reasons, we present the results obtained from smoothing using the kernel density method.

These results are shown in Figure 5 . To produce this figure, we used a smoothing radius of 30 km and a grid of square cells with sides of 3 km. Observation of Figure 6 allows the following observations. The highest densities of health posts are found in the west, particularly in the regions of Dakar, Thiès, Fatick, Diourbel, Kaolack, and Ziguinchor. Conversely, the eastern part of the territory has a low density of health posts. There is a sort of west-east gradient, with a decrease in the density of health posts from west to east.

The health post density map produced by the kernel method shows global and local spatial trends in the spatial distribution of health posts. However, it does not indicate the types of local spatial association and their level of statistical significance. This is why we produced the map of local indicators of spatial association ( Figure 6 ). To produce this map, we first generated a grid of square cells with sides of 3 km and projected the layer of health posts onto it. Then, using the spatial join method, a count of the number of health posts per square was carried out. Finally, we started calculating the local indicators of spatial association contained in Figure 7 .

This figure highlights four types of statistically significant spatial association. The first type includes red tiles with positive values (index above the average) in an environment of tiles with positive values corresponding to positive autocorrelation. This association is called strong-strong. This type of tiles is found almost exclusively in the west. The second type consists of blue tiles with negative values (index below the average) in an environment of tiles with negative values corresponding to negative autocorrelation. This association is called weak-weak. This type of tiles is found almost exclusively in the east, on a diagonal that starts from the south of the Saint-Louis region, passing through the east of the Louga region, the west and center of the Matam region, the Tambacounda region, and ends in the Kédougou region ( Figure 6 ).

The light blue tiles constitute the third type and record negative values (index below the average) in an environment of points with positive values corresponding to positive autocorrelation. This association is called weak-strong ( Figure 6 ). The fourth type consists of light red tiles with positive values (above-average index) in an environment of negative-valued points corresponding to negative autocorrelation. This association is referred to as strong-weak ( Figure 7 ). Atypical tiles comprise types three and four, but they are very weakly represented.

Gray tiles are those where the local spatial association is not significant.

Certainly, the description of centrographic indicators and clusters of local spatial association of health posts is of great interest. However, it is also important to know whether the number of health posts is locally associated with the population in the space.

To measure the local spatial association between the number of health posts and population size, we had to aggregate the number of health posts at the commune level, where data on population size are available. The results of this measurement are shown in Figure 7 , which highlights four types of local spatial association between health posts and population. The first type (red areas) shows areas with a high number of health posts and a high population size. This type is poorly represented and is located almost exclusively in the far west of the country. The second type (dark blue areas), which is more represented, describes communes where a local spatial association is noted between a low number of health posts and a low population size.

It is mainly found in the southeast and southwest. The third type (light blue areas), which is almost absent, highlights communes where a local spatial association is recorded between a low number of health posts and a large population size. It is found mainly in the southeast and southwest. Almost absent, the third type (light blue areas) highlights the municipalities where a local spatial association is recorded between a low number of health posts and a large population size. Fairly well represented, the fourth type (light red areas) shows the aggregation of municipalities where a local spatial association is observed between a high number of health posts and a small population size. The gray areas represent the municipalities where the local spatial association between the number of health posts and the population size is not statistically significant.