## 1 Introduction
\begin{array}{rcl}S^{{}^{\prime}}(t)&=&-\beta SI,\\ I^{{}^{\prime}}(t)&=&\beta SI-\alpha I,\\ R^{{}^{\prime}}(t)&=&\alpha I,\\ S(0)&=&S_{0},I(0)=I_{0},R(0)=R_{0}.\end{array}  \tag{2}\]  
Mathematically, the initial stage at \(t=0\) can be taken as any point in available trusted data. The determination of the nonnegative parameters of the model \(\beta,\alpha,S_{0},I_{0}\) and \(R_{0}\) is the most difficult and critical step in using this model. It is easy to see the increasing behavior of \(R(t)\) and the decreasing behavior of \(S(t)\) due to the sign of the first-order derivative. The behavior of \(I(t)\) is the effective part of any explanation. The peak occurs when \(\alpha=\beta S\)[1,2]. Due to the nonlinearity in such systems, methods for solution are directed towards approximate and numerical techniques [4, 5, 6, 7, 8, 9]. At the heart of the numerical methods for such systems are the Runge-Kutta (RK) methods, especially the fourth order. The parameters \(\alpha\) and \(\beta\) are known as the recovery coefficient and the transmission coefficient, respectively, [1,2]. There are many trials to upgrade the SIR model (2), but more parameters are included and other scientific concepts such as fractional calculus or stochastic behavior's must be considered [10, 11, 12]. Among them, some authors [13,14] considered the relaxation of the assumption of bilinearity of the incidence rate \(\beta SI\) to cover different dynamical behaviors. Figure 1: The transition between the components of the SIR model. In 2012, [PERSON] [14] assumed that the interaction terms that appear in the SIR model can take the form of square root and introduced a new SIR model of the form:  
\[ \begin{array}{rcl}S^{{}^{\prime}}(t)&=&-\beta^{*}\sqrt{S}I,\\ I^{{}^{\prime}}(t)&=&\beta^{*}\sqrt{S}I-\alpha^{*}\sqrt{I},\\ R^{{}^{\prime}}(t)&=&\alpha^{*}\sqrt{I},\\ S(0)&=&S_{0},I(0)=I_{0},R(0)=R_{0}.\end{array}  \tag{3}\]  
There is no doubt that the models (2) or (3) can be applied to develop comprehensive treatment scenarios and strategic plans at the beginning of an outbreak of any disease, providing a good estimation of the available data. As a result of the developments of information technology and modern methods of communication, health authorities all over the world have been able to monitor the data and make it available for use, serving humanity by developing methods for preserving human beings in general. Each model contains five parameters: three of them are just the initial values \(S_{0},I_{0},\) and \(R_{0},\) and these values can be easily obtained from the daily data available for each case study. The other two parameters \(\beta\) and \(\alpha\) for the model (2) or \(\beta^{*}\) and \(\alpha^{*}\) for the model (3) can be determined from the available data. The reliability of the obtained solutions depends on the accuracy of the determined values for the parameters as well as the method of solution [15; 16; 17]. Our main task in this work is to adopt the use of simple mathematical models and the real data available to generate good approximate values to the recorded real data so reliable prediction of the spread of the epidemic disease can be analyzed. Therefore, strategic plans of the management of situates can be introduced with high confidence. It is seen that the use of models (2) and (3) depends on the determination of accurate values of the parameters (\(\beta\) and \(\alpha\) or \(\beta^{*}\) and \(\alpha^{*}\)) and on using an efficient and reliable method of solution. In this work, we propose the use of the SIR model in a local way unlike what was used previously. Furthermore, we determine the parameter values from the actual data through mathematical treatment with high credibility (integration technique) unlike what was used previously (differentiation or fitting techniques).