On the Stabilization of the Spread of the Coronavirus (COVID-19) Pandemic in the World

At the end of December 2019, a novel coronavirus (2019-Cov) emerged in China specifically it appeared in the city of Wuhan, and it has spread to the entire world very fast and in a very short time


INTRODUCTION
It has spread COVID-19 to the entire world very fast and in a very short time, that caused thousands of people infected in the world by the respiratory tract infection, his first appearance was at the end of 2019 in the city of Wuhan in the province of Hubei in the People's Republic of China.
The diagnosis of COVID-19 is made by reverse transcriptionpolymerase chain reaction (RT-PCR) tests.With the help of Chinese scientists sharing genomic information on the virus fast with the entire world, it has become possible to conduct RT-PCR analysis without the virus completely spreading in the world.COVID-19 RT-PCR analysis has provided a significant opportunity in fighting with the pandemic that has emerged, which led to a marked decrease in the infections.
However, according to mutual opinions from the laboratories, some patients may have negative RT-PCR analysis results but positive clinical findings.In these cases, the RT-PCR analysis results may turn out positive in the following days.
Therefore, in the diagnosis of the disease, clinical findings are as important as laboratory findings.Additionally, in routine biochemical analysis, increases in the serum Creactive protein (CRP), lactate dehydrogenase (LDH), European Journal of Medical and Educational Technologies 2020; 13(1): em2004 erythrocyte sedimentation rate (ESR), monocyte volume distribution width (MDW) and D-dimer levels and a decrease in albumin concentration may be expected (see [8,7,6,9,10] and references therein).
Due to the gravity of the COVID-19 pandemic, the necessity of mathematical modeling is clear in terms of being able to make medical planning and see the long-term course of the pandemic.
In the literature, for a region with a population of  (city, country), it is stated that "the time-dependent change (spreading) rate of the , number of individuals who have caught a contagious disease is proportional to the multiplication of the numbers of those who have caught the disease and those who have not" ( [4]).The mathematical model of this situation is given by the following differential equation, The model above was revised specifically for the "corona pandemic" in [3], were the authors discussed it in the form of "the spreading rate of the disease ( This research concerns to investigate the course of the pandemic by mathematical modeling based on the information that the time-dependent change (spreading) rate of the  number of individuals who have caught a contagious disease is proportional to the multiplication of the numbers of those who have caught the disease in time delay and those who have not.
The purpose of the study is, in the model given in (Eq.1), to take "the time-varying delay-dependent change (spreading) rate of the , number of individuals who have caught a contagious disease is proportional to the multiplication of the numbers of those who have caught the disease in time delay and those who have not, we are proposing the following initial value problem related to the issue as where,  1 ,  2 are fixed real constants, with  1 > 0,  2 ≠ 0, () > 0 represents the time delay.
: Independent time variable in units of days.It is well-known that the above model is not stable in absence of delay, that is if () = 0, see [3] The main objective of the present work is to establish a decay result of the (spreading) rate of the .
Furthermore, let the initial ( = 0) number of patients be (0) =  0 , the number of patients at a time  = −(0) (the past time), given as additional information be (−(0)) =  0 and the number of individuals who are open to the disease be .
By computation, we have Therefore, problem ( 2)-( 4) can be transformed into

DECAY OF SOLUTION
In this section, we shall investigate the asymptotic behavior of the energy function .For this, we construct a Lyapunov functional L equivalent to , with which we can show the desired result given by Theorem 3.3.Let define the modified energy functional  with problem ( 7)- (10) by (−)  2 ()d.
Note that, from (6), such a constant  exists.
The following lemma shows that the associated energy of the problem under the condition Lemma 2.1 Let (, ) be the solution of ( 7)- (10).Then, for some two positive constants  1 and  2 , we have Proof.Multiplying the third equation ( 9) by () =  −() , and integrating over (0,1) with respect to , we have Thus Where () = Now, multiplying the first equation of ( 2) by , integrating over Ω and exploiting the third equation ( 9) in above system, and multiplying the third equation by (), and integrating over (0,1) with respect to , making use of ( 12) we get By young's inequality, we have and By (5), we get therefore, from (13) we get > 0, this completes the proof of Lemma 2.1.

GENERAL DECAY
In this section, we shall investigate the asymptotic behavior of the energy function .For this, we construct a Lyapunov functional L equivalent to , which we can show the desired result.First, we define some functionals and establish several lemmas.Let European Journal of Medical and Educational Technologies 2020; 13(1): em2004 where  is positive constant to be choose later, and The functional L is equivalent to the energy function  by the following lemma. Therefore then choose  > 0 so large that we complete the proof of Lemma 3.2.Now, we are ready to give and prove the main theorem.( 4 + ) 2 .
2. We can give exactly the values of all constants.
multiplication of the numbers of those who have caught the disease and those who have not, and inversely proportional with the square root of time-variable, were   derivative corresponding to the time-dependent change (spreading) rate of the disease,  is a parameter that covers all factors that influence the spreading rate, when the authors take  = () =  2√ dependent on the -time variable instead of the constant on the right-hand side of (1), and they give an expect the number of patients in the near time.

:
(): Dependent variable expressing the number of patients at the time , � − ()�:Dependent variable expressing the number of patients at the paste time  − (),   Derivative corresponding to the time-dependent change (spreading) rate of the disease,  1 : A parameter of delay,  2 : A parameter that covers all factors that influence the spreading rate.