2.1 Identifying the coordinates

The coordinates of the graph were found by measuring the height of the curve at seven different horizontal intervals, as shown in Figure 2.

Figure 2: Samples taken at different space intervals.

2.2 Choosing the curve’s form

After determining the coordinates, it was necessary to choose a curve that approximates the shape of the COVID-19 sample curve. Figure 3 shows the graphs of some of the most commonly used functions for fitting curves using the LSM [3].

Figure 3: Curves of Commonly Used functions.

For this case, we chose the graph of the exponential function y = ae^bx (F1) because its mirror-image, about the horizontal axis of the graph. We will use the LSM to fit the exponential graph to the profile of the curve.

2.3 System of equations

To obtain the coefficients a and b of the exponential function, LSM requires solving the system of equations F1 shown below. Table 1 explains the meaning of the different terms depending on the type of curve selected [4].

$\begin{array}{l}\text{An+B}{\displaystyle \sum {\text{X}}_{\text{i}}}\text{=}{\displaystyle \sum {\text{Y}}_{\text{i}}}\\ \text{A}{\displaystyle \sum {\text{X}}_{\text{i}}}\text{+B}{\displaystyle \sum {\text{X}}_{{\text{i}}^{\text{2}}}}\text{=}{\displaystyle \sum {\text{X}}_{\text{i}}{\text{Y}}_{\text{i}}}\end{array}$ (F1)

Table 1: Values of the system equation elements.

2.4 Rewriting the system

Substituting the exponential values into F1 we obtain

$\begin{array}{l}\text{nLn(a)+b}{\displaystyle \sum \text{X}}\text{=}{\displaystyle \sum \text{Ln(y)}}\\ \text{Ln(a)}{\displaystyle \sum \text{X}}\text{+b}{\displaystyle \sum {\text{X}}^{\text{2}}}\text{=}{\displaystyle \sum \text{xLn(y)}}\end{array}$ (F2)

Where “n” is the number of coordinates previously obtained and whose values are shown in Table 2.

Table 2: SAMPLE COVID-19 Coordinates.

3.1 Calculating the individual terms [4]

In this example, n = 7 (the number of coordinates)

The summation of all Xs values is:

${\displaystyle \sum \text{X}}\text{=0+7+14+21+28+35+42=147}$

The summation of all the squares of the Xs coordinates is:

${\displaystyle \sum {\text{X}}^{\text{2}}}\text{=0+49+196+441+784+1225+1764=4459}$

The summation of the logarithms of the Ys coordinates of Table 2 is:

$\begin{array}{l}{\displaystyle \sum \text{Ln(y)}}\text{=Ln(100)+Ln(175)+Ln(300)+Ln(575)}\\ \text{+Ln(950)+Ln(1600)+Ln(2575)=43}\text{.91}\\ \\ {\displaystyle \sum \text{Ln(y)}}=4.61+5.16+5.7+6.35+6.86+7.38+7.85=43.91\end{array}$

The summation of all Xs coordinates multiplied by its corresponding Y logarithm is:

$\begin{array}{l}{\displaystyle \sum \text{xLn(y)}}\text{=0*4}\text{.61+7*5}\text{.16+14*5}\text{.7+21*6}\text{.35}\\ \text{+28*6}\text{.86+35*7}\text{.38+42*7}\text{.85=1029}\text{.4}\\ \\ {\displaystyle \sum \text{xLn(y)}}=0+36.12+79.8+133.35+192.1+258.3+329.7=1029.4\end{array}$

3.2 Substituting all terms in the System of equation F2

Performing the corresponding arithmetic operations, we obtain the following results:

$\begin{array}{l}\text{7*Ln(a) + 147*b = 43}\text{.91}\\ \\ \text{147*Ln(a) + 4459*b=1029}\text{.4}\end{array}$ (F3)

3.3 Calculating the b coefficient

Solving the top equation of (F3) for Ln(a) give us

$\text{Ln(a)}\approx \frac{43.91-\text{147b}}{7}$

Replacing this value in the second equation of F3 we obtain:

$147\left(\frac{43.91-\text{147b}}{7}\right)+4459*b=1029.4$

Solving for the coefficient b from the latter equation results in

$b=\frac{751.03}{9604}\approx 0.078$

3.4 Calculating the a coefficient

To find the coefficient a, we replace the value of b in the second equation of the system F3 and solve for Ln(a). That is,

$\begin{array}{l}\text{147*Ln(a)+4459*b= 1029}\text{.4 where b }\approx \text{ 0}\text{.078}\\ \text{147}\text{.Ln(a) + 4459*(0}\text{.078) = 1029}\text{.4}\\ \text{147Ln(a) = 1029}\text{.4-348}\text{.69}\\ Ln(a)\text{ = }\frac{680.71}{147}\cong \text{ 4}\text{.63}\end{array}$

Raising e to value just obtained results in:

$e^{Ln(a)}=e^{4.63}\to a=e^{4.63}\to a\approx 102.6$

Competing Interests

The authors declare that they have no competing interests.