## 1 Introduction  
There is no life without disease. Therefore, over the years, people have spent a lot of effort and financial and moral support to control the spread of diseases or avoid their side effects. The side effects of the spread of diseases vary from health damage to psychological impacts, social impacts, and economic impacts. With the scientific and technological developments of humankind, studies of problems and setting up scenarios for dealing with them have become one of the most important strategies. Accordingly, nations can avoid the serious side effects of such problems. Among the most prominent of these problems is the spread of infectious diseases, whether they are bacterial or viral. The category of epidemic diseases is one of the most difficult and dangerous disease types that civilization has faced. The dynamics of transmission of almost all infectious diseases are known, so the establishment of good mathematical models for the spread of infectious diseases are reachable tasks [1, 2]. Mathematical models are considered among the most major pre-studies that can be used in developing strategies for dealing with epidemics. Mathematical models, in principle, fall into two categories:  
First, the statistical models are those that monitor reality in all its details, but their future role is restricted, their accuracy is difficult to be estimated, and they are judged by realistic measurements, which usually suffer from exactness. Therefore, the role of statistical models is limited in strategic planes despite their essential responsibility in determining the model parameters for any model type [2, 3]. Second, mathematical models are based on fixed, unquestionably physical, mathematical laws and facts. Mostly, they are related to rates of change, and therefore they are often differential or integrodifferential equations in different forms. Now there are many mathematical models. All the models can give acceptable results, provided that are used in an appropriate manner, and their parameters are rightly determined [1,2]. Epidemiological mathematical models are a remarkable type of population model, especially the well-known prey-predator model, which represents the struggle of survival. The concepts of prey and predator are the cornerstone of the mathematical models of many biological studies. The following system of two nonlinear ordinary differential equations (ODEs), is known as [PERSON]'s prey-predator model. \[ \begin{array}{rcl}\frac{dx}{dt}&=&\mu x-vxy;\ \ \mu,\
u>0,\\ \frac{dy}{dt}&=&-\rho y+\sigma xy;\ \ \rho,\sigma>0,\end{array} x(0)=c_{1}\ \ and\ \ y(0)=c_{2}. \tag{1}\]  
To discuss the solution of such systems, one must determine the parameters of the model \(\mu,\
u,\rho,\sigma,c_{1}\) and \(c_{2}\). Determination of the parameters of the model is the most important step in the mathematical treatment; wrong values of even only one of the parameters can destroy all the results. Although this model is one of the elementary models described in calculus books, it contains all the characteristics of the most complicated models (ODE models). One can see, from the first equation, the exponential behavior of \(x(t)\) in the case of \(\
u=0\), also the role of \(\
u\) in controlling the exponential increase in \(x(t)\). Furthermore, one can see the exponential decay in \(y(t)\) from the second equation, and the role of \(\sigma\) to control the exponential decay in \(y(t)\)[2,4,5]. Now we consider one of the simplest, oldest mathematical models used in the epidemic diseases cited in many publications and can be considered as the extension of the prey-predator model, the SIR model introduced by [PERSON] and [PERSON] in 1927-1933. The model consists of a system of three coupled nonlinear ordinary differential equations for the unknowns \(S(t)\) (susceptible), \(I(t)\) (infective), and \(R(t)\) (removable) Figure 1, with \(N=S(t)+I(t)+R(t)\) a fixed closed population under consideration, as follows:  
\[