NEW TECHNIQUES TO ESTIMATE THE SOLUTION OF AUTONOMOUS SYSTEM

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INTRODUCTION
Throughout history, many epidemic 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.These epidemic may be contagious or transmitted in other ways.Among these disease are the Black Death, which spread widely in Europe, malaria, the plague in Africa, SARS in China 2002-2003, AIDS and cancer, etc. [1][2][3].At the beginning of 2019, the Corona virus 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].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 [5][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 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 [5,9].The mathematical system in our study is an epidemiological model formulated in the form of a system of first order nonlinear differential equations.These epidemiological models deals with rapidly spreading diseases that occupy large areas, and this epidemic model is considered as stochastic-deterministic models [1,5,10], see also [11,12].SIR epidemic model was also studied by Temimi-Ansari method, Daftardar-Jafari method, and Banach contraction method, [13].The stochasticity in COVID -19 for SIR epidemic model was discussed in Iraq to die out the epidemic in [14], see also [11].Shafeeq, et al., studied Bifurcation analysis of a vaccination mathematical model with application to COVID-19 pandemic in 2022 [15].For the first time, LTAM was discussed to solve the nonlinear epidemic model, this method is combine Laplace transform with Tamimi and Ansari iterative method, [16].Yaseen, et al., in 2023 discussed stability and Hopf bifurcation of an epidemiological model with effect of delay the awareness programs and vaccination [17].
Many ways can be solved the epidemiological models, like semi-analytic methods.Mahdi and Maha in 2020 discussed some like semi analytic methods for nonlinear Smoking Habit Model [18].As well, the numerical method Runge-kutta for the 4 th order (4), which is one of the reliable methods for solving differential equations of different orders high accuracy [19].Some authors created modified numerical simulation approaches to get good results for epidemiology models, such as MMC_FD was discussed by Maha, et al. in 2019 the new approach that mixed two different methods which are the Mean Monte Carlo simulation technique and numerical iteration method which is finite difference method to sample randomly from a nonlinear epidemic model [20] and MLH_FD was studied by Mohammed, et al. in 2018 a nonconventional hybrid numerical approach with multi-dimensional random sampling for cocaine abuse in Spain [21].Shatha and Maha discussed Runge-kutta numerical method for solving nonlinear Influenza Model in 2021.Emad and Maha studied nonlinear COVID-19 mathematical model using a reliable 4 numerical method, in 2022 [22,23].Mahdi and Maha in 2019 discussed the modified numerical simulation technique for solving nonlinear epidemic models [24] which is Mean Monte Carlo Runge-Kutta method (MMC_RK) which is an efficient numerical simulation technique mixed two methods of different natures together that are Monte Carlo simulation process (MC) and Runge-Kutta numerical method (RK) that used to find the solution for system of equations.Shatha and Maha in 2022 studied the other numerical simulation method is Mean Latin Hypercube Runge-Kutta (MLH_RK) which is hybrid of a Latin Hypercube sampling (LHS) simulation method and a numerical Runge-Kutta (RK) method, to solve the influenza model, [25,26].MLH_RK is one of the reliable method to solve such NEW TECHNIQUES TO ESTIMATE THE SOLUTION OF AUTONOMOUS SYSTEM systems.Emad and Maha discussed applying a suitable approximate-simulation techniques of an epidemic model with random parameters in 2022 [25,26].
In this study, many methods are used, the first one, the numerical method Runge-kutta for the 4 th order ( 4) is used for solving the system under study.Addition that, There are two numerical simulation echniques in our study, one of them by Mean Monte Carlo Runge-Kutta method (MMC_RK) and the other Mean Latin Hypercube Runge-Kutta method (MLH_RK).
The new approaches which called Approximate Shrunken Methods denoted by (ASM_MMCRK and ASM_MLHRK) are applied on COVID-19 mathematical model.Approximate Shrunken Methods are hybrid between classic numerical method which is Runge-Kutta 4 th (RK4) and numerical simulation techniques which are Mean Monte Carlo Runge-Kutta method (MMC_RK) or Mean Latin Hypercube Runge-Kutta method (MLH_RK) in the statistical form which is shrinkage estimation method.These a new proposed methods are more accurate and reliable than other numerical simulation methods in solving such nonlinear mathematical system that is used under study.This research is divided into the following: in Section 2, the mathematical model of COVID-19 is presented; Section 3, obtains the deriving of the numerical method 4 and Section 4, contains numerical simulation methods MMC_RK, MLH_RK and in Section 5, the new approach approximate shrunken methods (ASM_MLHRK and ASM_MMCRK) that used for solving the epidemic model under study.Section 6, contains the discussion and tables of the methods used, as well as their graphic representation.Finally, Section 7 explains the final conclusion of the research.

MATHEMATICAL MODEL OF COVID-19
This model is used successfully to study the people vaccinated against [27].The population consists of five types of individuals  ,  ,  ,  and  represent susceptible, vaccinated, asymptomatic, symptomatic and the recovery respectively.They are functions of time.The governing equations for the epidemic under study by non-linear ordinary differential equation of first order [27].

APPROXIMATE SIMULATION METHODS FOR SOLVING COVID-19 MODEL
Some of the modified numerical simulation methods that are used in this study will be discussed in this section formulated in our model.

MEAN MONTE CARLO RUNGE-KUTTA (MMC_RK) METHOD
Mean Monte Carlo Runge-Kutta (MMC_RK) is an efficient numerical simulation method for solving such mathematical models.This method consists of mixing two different methods; one numerical is Runge-Kutta method 4 th (4) and the other Monte Carlo simulation process (MC) is called the Mean Monte Carlo Runge-Kutta (MMC_RK), see [24].MC estimates the model coefficients that are random variables while RK is used to solve the model numerically.The average of the last RK iteration results with each MC repetition is considered the estimated approximate solution for the model under study.The MMC_RK method is implemented by using MATLAB software, more details are shown in [24].

MEAN LATIN HYPERCUBE RUNGE-KUTTA (MLH_RK) METHOD
Mean Latin Hypercube Runge-Kutta (MLH_RK), see [25] is numerical simulation method that is a mixture of a simulation method which is Latin Hypercube Sampling (LHS) and a numerical method is Runge-Kutta (RK).It is considered one of the reliable methods for solving a system of nonlinear ordinary differential equations of the first order.LHS estimates the coefficients of the model that are considered random variable while RK is used to solve the model numerically.The average of the last RK iteration results with each LHS repetition is considered the estimated approximate solution for the model under study.
The process of MLH_RK is similar to MMC_RK which was talked about before.In addition to that MLH_RK is more accurate and faster than the MMC_RK method because it simulates model parameters at once whereas this integrated method is implemented using the MATLAB program, see [25].

APPROXIMATE SHRUNKEN METHODS
In this section, two new techniques are created to solve such models under study, especially epidemic models.These techniques have proven their efficiency and effectiveness in obtaining more accurate results than the modified numerical simulation methods in previous studies, and they are considered a new approach between statistics and numerical simulation.

APPROXIMATE SHRUNKEN METHOD (ASM_MMCRK)
Approximate Shrunken Method the form called ASM_MMCRK, is a new approach that is a hybrid between the classic approximate method which is RK4, and numerical simulation NEW TECHNIQUES TO ESTIMATE THE SOLUTION OF AUTONOMOUS SYSTEM techniques which is MMC_RK in the shrinkage estimation statistical form.This newly proposed method is a more accurate and reliable method than other numerical simulation MMC_RK and MLH_RK methods in solving such nonlinear mathematical systems that are used under study.
ASM_MMCRK gives alternative estimation values between statistical and approximate methods.This method has been calculated using the MATLAB program and as shown in the following algorithm: • Step 1: The parameters of a model have been simulated by MC for n times.

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Step 2: One value is specified from Step 1 and transformed into a specific distribution, to replace Its value in the system, for each random parameter.

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Step 3: Solve the system m-times iterations numerically by RK to get the numerical solutions, the last iterative result is the final solution which is selected.

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Step 5: To find a solution of the system under study by MMC_RK, calculating the mean of final solutions from Step 4.

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Step 6: Using the proposed algorithm as follow: Since the solution of the system for ASM_MMCRK is called  ̂ASM_MMCRK and  is weight function, where 0 ≤  ≤ 1.

APPROXIMATE SHRUNKEN METHOD (ASM_MLHRK)
Approximate Shrunken Method the form named ASM_MLHRK; is another proposed method which is mixture of a classical numerical method which is RK4 and, and another numerical simulation techniques which is MLH_RK to produce a new algorithm in the statistical form which is shrinkage estimation form.This proposed algorithm is more accurate and efficient compared to other approximate simulation methods for solving such mathematical models.
ASM_MLHRK is promising to create alternative estimation values between statistical and approximate methods.This method is implemented using the MATLAB program and as shown in the following algorithm: • Step 1: All model parameters have been simulated by LHS for n times at ones.

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Step 2: For each random parameter, one value is specified and replaced in the system.

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Step 3: Solve the system m-times iterations numerically by RK to get the numerical solutions, the last iterative result is the final solution. • Step 4: Repeat Steps 1 and 2 for n repetitions.

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Step 5: Calculate the mean of final solutions from Step 4, to find a solution of the system under study by MLH_RK.

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Step 6: Using the new algorithm as follow: Since the solution of the system for ASM_MLHRK is called  ̂ASM_MLHRK and  is weight function, where 0 ≤  ≤ 1.

RESULTS AND DISCUSSION
This section discusses the approximate simulation solutions of the epidemic model under the study of the people vaccinated against COVID-19.These results are discussed and analyzed in this section.Table 3 (a) and Table 3 (   The comparison between the new approach by approximate shrunken methods; ASM_MMCRK, ASM_MLHRK and the numerical simulation methods MMC_RK, MLH_RK compared with the numerical method 4 are is achieved by absolute error criterion for (),(), (), () and () are shown numerically in Tables 4 and 6.Observe that the absolute error of the new methods ASM_MLHRK and ASM_MC, they are have error less than of the other numerical simulation methods MLH_RK and MMC_RK, for all groups () , (), (), () and () of population.ASM_MLHRK has the smallest value of absolute error and this means that the proposed method is more accurate and reliable than the other methods.
Prediction intervals that contain the minimum bound (5th percentile) and maximum bound (95th percentile) for MMC_RK and MLH_RK results in the future until 2025, MMC_RK and MLH_RK results are inside the predicted intervals, see Table 8.Table 7 discusses the convergence of the new algorithm using the residual error between every two consecutive terms for a certain number of terms of the proposed algorithm, where we notice that the error decreases as we take a larger number of iterations, and this confirms us the convergence of the proposed method.     of infection (), also note that there is an increase in the curve of this class for all methods ASM_MLHRK, ASM_MMCRK, RK, MMC_RK and MLH_RK are used under study, and to reach its highest level in the middle of the study period, specifically the 22 th month, due to mixing and lack of commitment to health prevention methods, then the curve drops until the 41 th month, then stabilizes until the end of the study period, also the curve of proposed method ASM_MLHRK and ASM_MMCRK converge to the curve of numerical method 4 than the other methods.Figure 3 (e) the curves of this group of people who have been cured or died () as a result of the epidemic, notice that there is a discrepancy in the level of rise and fall in the curve of this class and for all methods are used under study with ℎ = {0.02,0.08} through four over a period of 48 months, where we notice the rise until the 15 th month and then returns to decline in 25 th month, after which it rises very much to settle at its highest rate in the last months of the study.Whereas, the curve of the proposed algorithm remains the closest to the curve of numerical method 4 than the other numerical simulation methods.

CONCLUSION
The numerical simulation process is considered a more reliable method than the classical methods that depend on fixed period of time.Because the natural epidemic models have randomness in their coefficients.For this reason, these numerical simulation techniques is considered a more suitable method than the classical methods like 4 that solve models depending on fixed parameters.In our study, the shrinkage estimation solution represents a good estimator for the solution of the system under study.This solution is considered a link between the traditional approach of solving systems represents in numerical methods that depend on fixed coefficients for the model and the concepts of the modified simulations approach for solving these systems when dependent on random coefficients.
The results for all proposed methods; ASM_MLHRK and ASM_MMCRK are more convergence to and close to the 4 numerical result that represents a criterion solution in the current model than the other methods mentioned.Where note that the approximate shrunken method (ASM_MLHRK) is the most closely.
Studying the epidemic model under study gives an impression of the impact of this virus on society.The results show that the category () of people not infected with the epidemic began to decrease during the study period.While the category () is associated with vaccinated people, we notice an increase in this category as a result of the impact and effectiveness of the vaccine on society, also for the category () of infected people without showing symptoms, there is a gradual rise in this category of people for not adhering to health prevention methods, as well as not adhering to social distancing.However category () of infected people and the symptoms are clear to them, we notice a gradual decrease in all methods 4, ASM_MLHRK, ASM_MMCRK, MMC_RK, and MLH_RK by educating people to take the vaccine against the virus.Finally, the category () of people who have been cured or died due to disease, a clear increase for this class.
b) contain numerical simulation results through one year with step size ℎ = {0.02,0.08} weekly and monthly for the period of the beginning of 2021 to the end of 2022 under study.Also, Table 5 (a) and Table 5 (b) contain the results of the numerical simulation solution for the groups (), (), (), () and () of society, for a future period of time until 2025 in the interval [0,48].

Figure 2
Figure 2 represents the curves of methods that are used to solve the mathematical model of COVID-19 through two years with 100 repetitions and ℎ = {0.08,0.02} step size weekly and monthly through two years.Figure 2(a) is related to the people () who are not infected with COVID-19 but are susceptible to infection.A sudden drop in the curves for all the methods used in the study after the 17 th month.It is noticeable that the sudden descent in the curve as a result of a large number of infections during the study period to still down after the end of 2022. Figure 2(b) observes the curves of group () with people who are vaccinated against COVID-19.It is noticeable that the curves rise gradually until the 15 th month, after which they increase with greater upwards until the end of the study period.Figure 2(c) is associated with the group of people () who are carriers of the virus without showing symptoms of infection and Figure 2(d)represents the infected people () with the epidemic.It is noticeable that a gradual and slight increase in the curve until the 15 th month for each () and (), after which they increase with greater upwards until the 20 th month, then a slight decline at the end of the study period as a

Figure 3
Figure 3 describes the curve of the mathematical model of the COVID-19 epidemic through 4 years from 2021 to 2025 with 100 repetitions.Figure 3 explains the convergence between the approximate simulation methods ASM_MLHRK, ASM_MMCRK, MMC_RK and MLH_RK when ℎ = 0.08 monthly with  = 1000 repetitions.

Figure 3 (
Figure3(a) represented the group of people who are not infected with COVID-19 ().Note that the gradual descent in the curve of this group of people for all methods that are used in the first months of the study with step size ℎ = {0.02,0.08} weekly and monthly through four years, and then it begins with a sudden decline due to a large number of infectious as a result of mixing and non-compliance with health prevention methods, after which it returns to stability in the last months of the study as a result of people's desire to receive the vaccine against COVID-19, also see how close the curves of proposed methods ASM_MLHRK and ASM_MMCRK with the curve of numerical method RK4 than the other numerical simulation methods.Figure3(b), this curve describes the group of people vaccinated against COVID-19 (), and as we see there is a slight rise in the curve of this class in the first months of the study period for all methods ASM_MLHRK, ASM_MMCRK, RK4, MMC_RK and MLH_RK are used with ℎ = {0.02,0.08} weekly and monthly through four years, then the curve begins to rise significantly in the middle of the study period to continue rising until the 40 th month as a result of the increase in the number of vaccinated against this epidemic and the high level of awareness the health of the people will take after that, stability until the year 2025.We also notice very clearly how close the curves of new approaches ASM_MLHRK and ASM_MMCRK of the curve of 4 are to the other approximate simulation methods.Figure3 (c), this curve of this group () , which represents people infected with the epidemic without showing symptoms of infection.Noticeable there is a rise in the highest level in the middle of the study period, specifically the 20 th month, due to mixing and lack of commitment to health prevention methods, then the curve drops until the 40 th month, then stabilizes until the end of the study period, also we notice the curve of proposed method ASM_MLHRK converge to the curve of numerical method RK4 than the other methods.Figure 3 (d), represents people infected with the epidemic without showing symptoms

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The state transformation process of individuals.

Table 4 .
Absolute error for MMC_RK, MLH_RK, ASM_MMCRK and ASM_MLHRK compared with 4 through two years
The mathematical model in our research is represented by the COVID-19 epidemic.It is formulated as a system of first-order nonlinear ordinary differential equations.The study period during 48 months from 2021 to 2025.Many methods are used for solving the model, including a numerical method, which is the Runge-Kutta method as a standard solution, and the other two modified numerical simulation methods MMC_RK and MLH_RK to solve this system.All the previous methods are formulated to create a new approach that is used for the first time, called the approximate shrunken approach represented in (ASM_MLHRK) and (ASM_MMCRK) methods .NEW TECHNIQUES TO ESTIMATE THE SOLUTION OF AUTONOMOUS SYSTEM