SOFM is an unsupervised learning method [

24,

25,

26] for clustering multi-dimensional data into two-dimensional neurons which are directly connected with each other through a pre-defined topology. SOFM is based on competitive learning, similar to winner-take-all networks. The additional feature of SOFM different than winner-take-all is the winner neuron, also called the Best Matching Unit (BMU). BMU is not only updated to be closer to the input coordinates, but also connected neurons of the BMU are updated as well. Therefore BMU and its neighbors are closer to the input coordinates than the other neurons. Here, clustering implies that each neuron in SOFM can cover a set of individuals. Thus, the number of clusters is equal to the number of neurons in SOFM. After applying constraints to SOFM, the population size can be found in that region which is covered by the corresponding SOFM neuron.

This process is repeated for all inputs for each epoch, and finally, the maximum number of epochs is used as the stopping condition for SOFM.

The number of neurons are matched with number of regions in selected area for this study. That is why determining of this neuron number is calculated in terms of two criteria. The first criterion is based on both latitude and longitude intervals of the selected area and second one is directly effected from a uniform grid-based distribution of these neurons. The uniform grid-based distribution is assumed at the beginning of the process about how to find the total number of neurons in order to calculate how many number of neurons are required for latitude axis and longitude axis. The aim of this process is to find the number of neurons according to geometric shape of the selected area. After choosing the neuron distance parameter in Equations (

1) and (

2) as depicted in

Figure 3, each neuron is far from not only other neurons but also minimum and maximum of each coordinate. Neuron numbers for latitude axis in Equation (

1) and longitude axis in Equation (

2) are multiplied by each other to find the total number of neurons before initialization of each neuron in SOFM training process. The neuron distance parameter

$Neuron\_dist$ can be chosen as coordinate unit only.

Random initialization for SOFM neurons is selected to initialize neuron positions which place along inside the

$Lat\_dist$ and

$Lat\_dist$ as depicted in

Figure 3. The training process in SOFM is very similar to the winner-take-all algorithm, which utilizes vectorial distance-based competitive learning. Once each training sample is compared to the weights of each neuron on the 2D hexagonal structure, the Euclidean distance coming out of each comparison is computed separately. As seen in

Figure 3, neuron

${y}_{1}$ can be chosen as BMU through its Euclidean distance less than others for a input coordinate (Lat, Lon) which is depicted as a red square in

Figure 3.

The weights of the BMU and its neighboring neurons in the SOFM 2D-grid are adjusted to be closer to the input vector. The magnitude of the change decreases with both time and the distance from the BMU. Distance function for 2 dimensional inputs (Lat, Lon) and weight vectors can be formulated as shown in Equation (

3).

where

X is the current 2D input vector and

W is 2D node’s weight vector as shown in

Figure 3.

$Dist$ is utilized to determine how close

w is to

x leading to the information about the output layer neuron that is the best to represent the input features.

where BMU indicates best matching neuron

y to the input

x as shown in

Figure 3.

where

W indicates BMU for

ith input

X. During the learning phase of SOFM, the area of the neighborhood that is represented by the radius in Equation (

6) shrinks according to the number of epochs. Here,

${\sigma}_{0}$ denotes the initial value of radius and

$\lambda $ denotes a time constant while

i is the current time-step.

where

$Dist$ which is the distance from a node to the BMU can be calculated from Equations (

3) and

$\sigma $, is the width of the neighborhood function as calculated by Equation (

6). Weights are updated using Equations (

7)–(

9). Learning rate in Equation (

9) is adaptively changed by the iteration steps of SOFM and reduced gradually starting from

${L}_{0}$.

After the training phase of SOFM, each neuron position is used for the center of each region as the first usage of SOFM technique in the general methodology as indicated in

Figure 1 and

Figure 2. Normally, SOFM covers 100% to all individuals through neuron clusters. After applying the radius constraint to each neuron used for representing the region of a mobile assessment center, distance-based coverage can be calculated and used for performance metric for comparing with random deployment-based region selection.

SOFM neurons also provide stops for route planning during the testing process to all individuals as fast as possible to detect any infected case in the early-stage before spreading to all healthy regions. Therefore, it is possible to isolate the infected regions from other regions without having infected cases.