Analysis of Dynamical Behavior for Epidemic Disease COVID-19 with Application

: In this paper, we will use the differential equations of the SIR model as a non-linear system, by using the Runge-Kutta numerical method to calculate simulated values for known epidemiological diseases related to the time series including the epidemic disease COVID-19, to obtain hypothetical results and compare them with the dailyreal statisticals of the disease for counties of the world and to know the behavior of this disease through mathematical applications, in terms of stability as well as chaos in many applied methods. The simulated data was obtained by using Matlab programms, and compared between real data and simulated datd were well compatible and with a degree of closeness. we took the data for Italy as an application. The results shows that this disease is unstable, dissipative and chaotic, and the Kcorr of it equal (0.9621), ,also the power spectrum system was used as an indicator to clarify the chaos of the disease, these proves that it is a spread,outbreaks,chaotic and epidemic disease .


INTRODUCTION
It is necessary to model the development of an infectious disease as a dynamic system with the study of its properties.the data necessary to construct a dynamic model are often available through biological considerations and time-series data for disease cases to fit and test the models [1], the qualitative behavior of systems of ordinary differential equations describing differential solutions for a long time has been studied , which is an important issue .one of the models that can be used to explain the characteristics of epidemiological diseases is the SIR model , which is a classical model that is susceptible to infection , which was built by (Kermack and Mckendrick) in 1927 and it has been designed extensively.Many expansions and extensions of this model have been used recently [2][3].In this paper the equations of the SIR model have been used.It is a non-linear system using the numerical Runge-Kutta method of 4th order to simulate the hypothetical results of the epidemic disease that occupied the world COVID-19 and compare it with the real data given according to the statistics of the countries of the world for example we take data in Italy [4].The degree of stability of the disease was also tested in several ways, all of which led to the result that it was unstable [5][6][7][8][9], and it chaos was tested by the binary test (0-1) and the result was that it was chaotic [10][11][12][13].The Matlab program was used for the aforementioned operations to know in a time series that accompanied this disease, and we obtained the results, charts and drawings that prove the results of the solution for the theoretical side of studying this disease, also, the parameters and infected values (I) of the disease in the population were used in the power spectrum system of the SIR model and it give an indication that this disease is a chaotic [14][15].

SIR MODEL
The SIR model is a specialized mathematical model to study the behavior of epidemic diseases, where it calculate the theoretical number of people with an infectious disease in aclosed population is calculated over time , it's one of the simplest biological models and developed by Kermack Definition : is represent to the average number of new infections generated by each infected person , The high value of ( 0) mean easy to transmission the disease ,and the low value of ( 0) mean difficult to transmission the disease .( 0) is called threshold of disease , ( the value of ( 0) assumes that no pre-existign immunity , i.e it mean everyone is susceptible), where Ro = βS / γ .

RUNGE-KUTTA 4th ORDER NUMERICAL METHOD :
The Runge-Kutta numerical method of 4th order its one of the numerical classic method that it use to solve the ordinary differential equations for dynamic systems that are continuous nonlinear and time-bound with number of iterations to get the best approximation .

TRANSFORMATION OF RUNGE-KUTTAEQUATIONS FOR SIR MODEL :
From system ( (B) we notice that compatible and with a degree of closeness .

STABILITY OF SIR MODEL
Stability Analysis : there are many method to check the stability from it :

CHAOTIC ANALYSIS :
To explain the chaotic state we will take the ( 0─1) test to know if the system is regular or chaotic .In this paper the SIR model was defined and the Runge-Kutta numerical method defined, and we transformed the method with the model and used this transformation to solve non-linear problems related to the time series in which time akey factor to determine its behavior and nature and obtain simulated results after giving initial values for the epidemic disease COVID-19 .We proved in Lemma that dI/dt > 0 and through that it we proved that Ro>1 and that mean the disease is epidemic.So we have real data obtained through the daily statistics for the countries of the world and the simulated data resulting from the application of the conversion .The real and the simulated data were tested in terms of stability in several mathematical method including the characteristics ,Routh-Hurwitz criteria and Lyapunov function and it appeared to be unstable.we applied the law of dissipativity on it and we got that it dissipative, ( | J | = 0.0049 < 1 ) .The Binary test (0-1) was also used to examine its chaotion , it was found that it was chaotic ,as the value of Kcorr was nearly to one .(Korr=0.9621) ,also the power spectrum system was used as an indicator to clarify the chaos of the disease .The Matlab program has been used in all the aforementioned processes to obtain the results and the figures that illustrate this are attached to each stage of the work.We have dealt with the statistical data of Italy as an example of the application of what was in this country of dynamics and a strong impact of the disease on the population during the period of time ,and that a large part of them were exposed to this disease which is spreading in a large and rapid manner .

8. 1 Figure 1 (
Figure 1(A) : show simulated data of susceptible,infected and recovered of disease

Figure 2 (Figure 3 Figure 2 (Figure 4 :
Figure 2(A) :show the chaotic of real data for Italy by Zero-one test (a): log(M) verses (b): M verses t (c): k verses c (d) :p verses q -Mckendrick in 1927 .The model is contain three non-linear ordinary differential equations related with a time-series as the following : parameters (β= 10 -5 & γ = 0.07) were calculated from the real data of daily statistics of COVID-19 disease for Italy .and initial values of Italy (S o, I o , R o ) = ( 3×10 6 , 3500, 0 ).