## 2.1. DATABASE

The analysis is based on data collected from the website (https://www.worldometers.info/about/) run by an international team of developers, researchers, and volunteers to collect various statistical data from around the world real to represent -time. For the COVID-19 pandemic, they manually analyze, validate and collect real-time data from thousands of sources around the world. By definition, the total number of cases is the cumulative total number of reported cases that were found to be clinically confirmed (criterion by country). Active Cases = (Total Cases) - (Total Deaths) - (Recovered). This is data on the number of people currently infected with the virus. We will implement our model using COVID-19 data in the world with a reported population of 7,953,091,060 on May 22, 2022.

## 2.2. MODIFIED SEIR EPIDEMIC MODEL

We started by introducing the SEIR model, which is one of the most widely used extensions of the standard SIR model, the epidemiological model based on Ordinary Differential Equations (ODE). In general, the SEIR models classify the population into four classes: Susceptible (S), Exposed (E), Infected (I), and Recovered or Removed (R). The model describes how different parts of the population change over time. The total population is *N = S + E + I + R*. The interaction of these dynamics is represented by recursive formulation22,23:

$${S}_{t+1}={S}_{t}-\left(\frac{r\beta {S}_{t}{I}_{t}}{N}\right)dt \left(1\right)$$

$${E}_{t+1}={E}_{t}+\left(\frac{r\beta {S}_{t}{I}_{t}}{N}-\sigma {E}_{t}\right)dt \left(2\right)$$

$${I}_{t+1}={I}_{t}+\left(\sigma {E}_{t}-\gamma {I}_{t}\right)dt \left(3\right)$$

$${R}_{t+1}={R}_{t}+\gamma {I}_{t} dt \left(4\right)$$

These four equations are called Euler formulas and can be employed to generate a set of values that reflect what is happening with the epidemic over a period of time. Where *r* is the number of contact per unit time, \(\)is the probability of disease transmission per contact which is transmission rate, \(\)is a per-capita rate of progression to infectious state which is the loss of latency rate, and\(\) is the per-capita recovery rate. This formulation assumes full population composition rather than population dynamics (i.e. natural births and deaths are considered negligible during epidemic periods)13. In order to calculate anything from these formulas we need to have explicit values for *β, σ, γ, S, E, I*, and *R*. In the standard SEIR model the parameter *β, σ*, and *γ* is a constant that does not depend on time. However, the characteristics of the epidemic show us that these parameters can change and vary over time. Note that in some models for diseases where contacts are not well defined (e.g. influenza and COVID-19), \(r\)(the number of contacts per unit time multiplied by the probability of disease transmission per contact) are combined into one parameter (often also referred to as the constant of \(\), the number of adequate contacts per unit of time)24,25. For this reason, we evaluate the SEIR model by proposing to express the number of suitable contacts per unit of time \(\) is expressed as a function of time, the Gaussian pulse26, as defined:

$$\beta \left(t\right)={\sum }_{n=1}^{\infty }{r}_{n}{e}^{-\frac{{\left(t-{t}_{n}\right)}^{2}}{{{d}_{n}}^{2}}} \left(5\right)$$

Where \({r}_{n}\) and \({d}_{n}\) are amplitude (number of contacts) and half of the pulse duration respectively. So that, equations (1) and (2) are simplified as:

\(\) \({S}_{t+1}={S}_{t}-\left(\frac{{\beta }_{t}{S}_{t}{I}_{t}}{N}\right)dt \left(6\right)\)

$${E}_{t+1}={E}_{t}+\left(\frac{{\beta }_{t}{S}_{t}{I}_{t}}{N}-\sigma {E}_{t}\right)dt \left(7\right)$$

In fact, the average number of new infections caused by a single infected person at the time of the partially susceptible population can be determined by the effective reproduction number *Re* generated throughout the infection period, which varies proportionally to the population of the susceptible population over time be changed:

$${R}_{e}\left(t\right)= \frac{{\beta }_{t}}{}\frac{S\left(t\right)}{N} \left(8\right)$$

It describes whether the infection population is increasing or not. It increases when *R**e* > 1; decreases when *R**e* < 1. When *R**e* = 1, the disease is in equilibrium19,27−30.