### field of study

Ecuador is located in South America, bordering Colombia (north), Peru (southeast) and the Pacific Ocean (west). Politically, it is divided into 24 provinces and 221 counties, which correspond to municipalities or communes (second political-administrative level after the provinces). It has four natural regions: Coast, Highlands, Amazon, and Galapagos Islands. Only continental Ecuador was considered for this study.

### Data Source

The secondary databases on live births and all-cause deaths are downloaded from the INEC website [24, 25]. The period under review covers ten years from 2010 to 2019. The birth database for the period under review includes all live births reported on birth certificates [24] and the death record includes all deaths of children under the age of 1 year listed on death certificates [25] collected by each municipal registry office from physical and digital forms of the National Civil Registration System.

### data extraction

To apply a spatial study, the level of municipality (canton) is selected for which the registered records are counted to obtain the number of live births by canton of residence of the mother and the number of deaths of children under 1 year of age by canton of death (from For reasons of confidentiality, the place of residence does not appear in these databases). The records of nonresidents in Ecuador are discarded as they are not charted.

### infant mortality rate

The formula used is as follows:

$$IMR=1000\times \frac{{Deaths}_{<1 year}}{Live\Births}$$

The annual tables of IMR per 1000 live births by municipality allow thematic maps to be drawn up.

### time trend

The Mann-Kendall nonparametric statistical test is used to determine the time trend over a period of the annualized IMR. The data does not have to conform to any particular distribution to apply this test [26]. The statistic forms combinations of each observed pair of values over time, i.e. it checks whether *IMR*_{j}> *IMR*_{I} or *IMR*_{j}< *IMR*_{I} and counts the number of pairs increasing or decreasing over time. It expresses the relative frequency of increases minus the relative frequency of decreases and is calculated for each spatial unit as [27]:

$$S=\frac{2\left(t-2\right)!}{t!}\sum_{i=1}^{t-1}\;\sum_{j=i+1}^tsign\ left({IMR}_j-{IMR}_i\right)$$

where the sign function is given by

$$sign\left({IMR}_{j}-{IMR}_{i}\right)=\left\{\begin{array}{c}1\ if \left({IMR}_{j} -{IMR}_{i}\right)>0\\ 0\ if\left({IMR}_{j}-{IMR}_{i}\right)=0\\ -1\ if \left( {IMR}_{j}-{IMR}_{i}\right)<0\end{array}\right\}$$

IMR_{I} is the IMR per year \(i\in \left\{\mathrm{1,2},\dots ,t-1\right\}\) With \(t\) B. the number of years available and IMR_{j} is the IMR per year \(j=(i+1) \in \left\{\mathrm{1,2},\dots ,t\right\}\).

Mann-Kendall scores range from -1 to +1. When a value approaches +1 it means there is a monotonous uptrend, when it approaches -1 the trend is down and a value of 0 indicates no trend [28].

The Terrset software [28] was used to apply this calculus.

### Spatial trend

The observed variable, in this case the IMR in the study area, is presented both globally and locally using maps and using the spatial statistical technique for cluster detection using the Moran indicator. The goal is to observe the spatial dependency that may or may not exist between nearby locations.

Considering a number of *N* Spatial units in a region, spatial autocorrelation represents the relationship between the IMR in a spatial unit and the IMR values of its n neighbors, which can be visualized by a connectivity map. To quantify the proximity between two spatial units, a positive *nxn* matrix *W* is used consists of *n(n-1)* called spatial weights *wi, j* defined based on binary contiguity, as here [29]:

$${w}_{i,j}=\left\{\begin{array}{c}{w}_{i,j}=1\ if\ j\ne i,neighbouring\ space\ units\\ {w}_{i,j}=0\ opposite\ case\end{array}\right\}$$

The Moran Index (I) is the test considered to be the most commonly used and statistically most powerful to detect spatial independence of debris, as it is a summary measure of the intensity of spatial association between units [29, 30]. Its range of values is based on the weight matrix, which usually varies between -1 and +1, but in contrast to a correlation coefficient is not necessarily limited by it [31]. If its neighboring municipalities tend to have similar values in their IMR, *I* will be positive, indicating the pattern is clustered if they are different, *I* be negative, that is, the pattern is regular, and if spatial randomness is present, the expected value of *I* is given by *E[I]*= *(-1)/(n-1),* how *n* increases, *E[I]* approaches [31].

given *I* and *j* in {1,2,…,n} the index is defined by:

\(I=\frac{n}{{\sum }_{i=1}^{n}\sum _{j=1}^{n}{w}_{i,j}}\frac{{\ Sum }_{i=1}^{n}\sum _{j=1}^{n}{w}_{i,j}\left({x}_{i}-\overline{X}\right )\left({x}_{j}-\overline{X}\right)}{{\sum }_{i=1}^{n}{\left({x}_{i}-\overline {X }\right)}^{2}}\) to the \(y\ne I\),

Where *n* is the sum of the communes, *x*_{I} the IMR in the community *i, x*_{j} the IMR in another municipality *y,* \(\overline{X}\) the average of the IMR and *w*_{i, j} the elements of the contiguity matrix *W* that connects the community*I* to*j*.

Because there are spatial effects such as heterogeneity related to the indistinct behavior of the variables observed in each of the units, local patterns can be recognized that were ignored with the global measure, hence local measures are introduced as Local Spatial Association Indicators (LISA). . is calculated as [32]:

$$I_i=\left(x_i-\overline X\right)\overset n{\underset{j=1}{\sum\;}}w_{i,j}\left(x_j-\overline X\right) forj\neq i$$

With this analysis using Moran’s calculation*I*_{I} and the scatterplot, four categories of groupings can be identified based on the type of spatial allocation: the hotspots, which are communities with an above-average rate and the rate of their neighbors as well, the category high-high, or otherwise the average rate below, the low-low category and the outliers or atypical values, i [33]. To see whether these groupings were not randomly created, a Moran statistical test is applied, in which the null hypothesis of randomness is contrasted with the alternative of clustering, and significance is obtained using a permutation approach. These techniques are available in the GeoDa software [33].

### Prioritization criteria to identify spatio-temporal critical areas

Different types of criteria can be developed and implemented depending on the prioritization needs of the study.

The methodology was designed according to logical criteria. To eliminate inconsistent rates, communities with fewer than 2 deaths were initially excluded. The districts with a higher last year IMR were selected using the 90% percentile threshold. The frequency of belonging to a high-high or hotspot cluster in years is used for prioritization. The third factor considered is the higher positive trend over the entire period examined.

Finally, years of hotspot repetition can be evaluated more strictly using the logical AND operator instead of the OR operator (Fig. 1).