A RELIABLE NUMERICAL SIMULATION TECHNIQUE FOR SOLVING COVID-19 MODEL

. The nature of epidemiological models is characterized by randomness in their coefficients, while the classical or analytical and numerical methods deal with systems with fixed coefficients, which makes these methods inappropriate for solutions of epidemiological systems that have coefficients that change with time. For that, the numerical simulation methods that deal with time change are more appropriate than other ways. The aim of the research is to apply some of these methods to the COVID-19 system. Two efficient methods used for previous studies are used to solve this system, which are Monte Carlo Finite Difference Method and Mean Latin Hypercube Finite Difference Method. For the sake of comparison, a numerical method, the finite difference method, is used to solve this system. We have reached good results that give an analysis and impression of the behavior of the Covid 19 epidemic since its inception and predict its behavior for the next years. All results have been written in graphs and tabulated.


INTRODUCTION
Throughout history, many epidemics appeared and posed a real threat to the world, as well as greatly affected economic and population growth, and caused trips to stop in some cities. This epidemic may be contagious or transmitted in other ways. Among these diseases is the Black Death, which spread widely in Europe, malaria, the plague in Africa, SARS in China from 2002 to 2003, AIDS and cancer, etc. [1][2][3]. At the beginning of 2019, the Coronavirus appeared, specifically in the Chinese city of Wuhan, and this epidemic is considered one of the most dangerous and fastest spreading epidemics, and it is of the SARS-CV type [4,5]. In the year 2020, on March 29, the epidemic spread significantly and rapidly throughout the world, which led to the suspension of flights through airports, land transport between countries, schools and universities, and most jobs with direct mixing [6,7]. The World Health Organization declared this epidemic to be a pandemic after it infected 199 countries around the world and caused the death of thousands of people [8]. The emergence of the epidemic coincided with the period of spring festivals and celebrations in Asia, and this helps to spread the epidemic due to the mixing of many people, especially on flights with all countries of the world. This is considered one of the reasons for the spread of the virus to the rest of the world [9]. As a result of the lack of health facilities in some countries, including developing countries, and the severity and speed of the virus's spread, the virus turned into a global pandemic that caused the death of thousands of people around the world because they did not receive appropriate treatment is social distancing and adherence to health prevention ways and the directive of the World Health Organization [6,9].
One of the most prominent epidemics that researchers have been interested in is Covid 19 since 2020. Among these researches that have been formulated in the form of a system of differential equations to study the behavior of the spread of the epidemic are [10][11][12][13][14][15][16] and see [17,18]. As well, there are those who are interested in predicting the behavior of the epidemic among them; the stochasticity in COVID -19 SIR epidemic model was discussed in Iraq to die out the epidemic in [19,20], see also [21,22].
In general, from the research that focused on studying the approximate methods (analytical and numerical) for solving epidemiological systems: SIR epidemic model was studied by Temimi-Ansari method, Daftardar-Jafari method, and Banach contraction method, [23]. For the first time, LTAM was discussed to solve the nonlinear epidemic model, this method is combine 3 NUMERICAL SIMULATION TECHNIQUE FOR SOLVING COVID-19 MODEL Laplace transform with Tamimi and Ansari iterative method, [24]. Shurowq K. Shafeeq, S.K., et al., discussed Bifurcation analysis of a vaccination mathematical model with application to COVID-19 pandemic in [25].Sabaa and Mohammed discussed in 2020 the approximate solutions of the nonlinear smoking habit model [26]. Shatha and Maha discussed Runge-Kutta numerical method for solving nonlinear influenza model in 2021 [27]. Emad and Maha studied COVID-19 model using Runge-Kutta numerical method in 2022 [10,11].
On the other hand, there is interest in numerical simulation approach for the behavior of epidemics and estimating the behavior of epidemics for the future. Among these researchers who developed a new approach linking transaction simulation of the epidemic system with methods for solving these systems, where the simulation method was used Monte Carlo Process (MC) with numerical method which is Finite Difference Mehod (FD), and other to use a more efficient simulation method, which is Latin Hypercub Sampling (LHS) with numerical methods for both.
The importance of our study is to find easy, fast, effective, and suitable ways to solve specific models that have some difficulties in solving, since these systems by their nature, are nonlinear, as well as have random coefficients, two numerical simulation processes MLH_FD and The search division is as follows: define the mathematical model used of COVID-19 in Section 2, the numerical finite difference method and the analytical variation iteration method have applied to COVID-19 model in Section 3. Section 4 exhibits two numerical simulation methods MMC_FD and MLH_FD to solve the nonlinear COVID-19 model respectively. Section 5 discusses the findings and results of proposed methods that represent in tables and graphs.
Lastly, the summary and conclusion of the research, are in Section 6.

COVID-19 MATHEMATICAL MODEL
The epidemic model in our study includes the COVID-19 of people vaccinated against the Coronavirus epidemic [32]. The population consists of five types of individuals S, V, A, I, and R represent susceptible, vaccinated, asymptomatic, symptomatic, and recovery respectively. These individuals are dependent on time. The nonlinear epidemic model under study consists of ordinary differential equations of first order [33].
where Tables 1 and 2 with the predicted parameters that are given in Table 2.

NUMERICAL METHOD FOR SOLVING COVID-19 MODEL
Epidemiological mathematical model in our study is a nonlinear system (1) of the Covid-19 with the estimated parameters that are explained in Table 2. It can be solved via the Finite Difference Method (FD) the initial conditions. The zero terms are in (2). The real step size ℎ is proposed in this study as 0.02, 0.08, and = 52, 12 refers the numbers of weeks and months respectively through one year. In order to find 1 ( ), 1 ( ), 1 ( ), 1 ( ) and 1 ( ), Backward Finite Difference (BFD) can be utilized as below: The first iteration 1 ( ), 1 ( ), 1 ( ), 1 ( ) and 1 ( ) are calculated from Eqs. Now, the Central Finite Difference (CFD) is applied to find the other terms as the follows: for all = 1,2, … , . To find 1 , 2 , …, , 1 , 2 , …, , 1 , 2 , …, , 1 , 2 , …, and 7 NUMERICAL SIMULATION TECHNIQUE FOR SOLVING COVID-19 MODEL The simulation processes MC or LH can simulate the random coefficients for the model. With each repetition, a numerical method FD is used for solving the model numerically using simulated system parameters. The average of the last FD iteration results with each MC or LH repetition is computed as the estimated approximate solution for the system under search.
The randomness in the system coefficients represents the nature of epidemic models, so, MMC_FD and MLH_FD numerical simulation methods are more suitable methods than the FD method, due to the FD method being dedicated to solving models with constant coefficients while MMC_FD and MLH_FD are dedicated to solving models with random variables. The MMC_FD and MLH_FD methods are performed using MATLAB software, more details are found in [28] and [29].

RESULT AND DISCUSSION
The results of numerical and numerical simulation methods for the nonlinear Coronavirus model are discussed and analyzed in this section. The initial conditions of the system are taken from [33]. In this study, real step size has been used such that ℎ=0.02 in a week, (52 weeks in a year, the data of the COVID-19 epidemic is taken from each week, therefore, in order to change the weeks to a months, the real step size is calculated as ℎ = 1 52 ≈ 0.02) and ℎ=0.08 in a month, (12 month in a year, the data of the COVID-19 epidemic is taken from each month, therefore, in order to change the months to a year, the real step size is calculated as ℎ = 1 12 ≈ 0.08). Table 3 Table 4 gives the expected numerical simulation results with = 1000 repetitions for variables ( ), ( ), ( ), ( ) and ( ) of COVID-19 model in the next four years, the study interval is 2021 to 2025, which is also seen in Figure 1.
In Table 5, the absolute error criterion for two years from the beging of 2021 to the end of 2022 is used to compare the numerical simulation methods proposed in this study with the numerical FD method which is considered as the exact solution for this system. The results  Table 5 show that the error of the MLH_FD method is smaller than the error of the MMC_FD method which indicates that the MLH_FD method is more efficient than the MMC_FD method because it has the lowest absolute error.
It is clear that prediction intervals (5th percentile as a minimum result, 95th percentile a maximum result) for MMC_FD and MLH_FD expected results have been accounted for in Table   6. All these MMC_FD and MLH_FD expected results to fall within these estimated intervals in Tables 6.   Table 7, explain the stability of the approximate simulation methods which are used in the study and how close the numerical simulation method (MLH_FD) is every time we reduce the step size more than MMC_FD, such that ℎ is the error between the proposed method in step size ℎ and step size ℎ/2 , = 1,2, …, to prove the convergence of the used methods [29].

Variables
Step Size, h           where we notice that the smaller step size has the greater the convergence, the convergence in Figure 2 is more between the numercal simulation methods and the numerical method Finte Difference (FD), which represented the exact solution, compared to the Figure 1 because of a small of step size, also we notice that MLH_FD method is more converge than MMC_FD to FD.

CONCLUSION
In our research paper, two reliable numerical simulation methods which are MMC_FD and MLH_FD methods have been applied to the COVID-19 epidemic model. These methods have been utilized to solve the COVID-19 model for four years from 2021 to 2025. Our work is including the numerical FD method used for comparison purposes. By a comparison tool between the numerical simulation methods which are used in our study and FD, it has been found that the MLH_FD method is more efficient than the MMC_FD because it has the lowest absolute error during the study period and the curve of MLH_FD method is more converge for curve of FD than other method MMC_FD. One of the benefits of the proposed MMC_FD and MLH_FD methods is that it gives a prediction of the future behavior of the epidemic by giving an anticipation period for approximate solutions. Good findings have been getting when using