Mean Latin Hypercube Runge-Kutta Method to Solve the Influenza Model

In this study, we propose a suitable solution for a non-linear system of ordinary differential equations (ODE) of the first order with the initial value problems (IVP) that contains multi variables and multi-parameters with missing real data. To solve the mentioned system, a new modified numerical simulation method is created for the first time which is called Mean Latin Hypercube Runge-Kutta (MLHRK). This method can be obtained by combining the Runge-Kutta (RK) method with the statistical simulation procedure which is the Latin Hypercube Sampling (LHS) method. The present work is applied to the influenza epidemic model in Australia in 1919 for a previous study. The comparison between the numerical and numerical simulation results is done, discussed and tabulated. The behavior of subpopulations is shown graphically. MLHRK method can reduce the number of numerical iterations of RK, and the number of LHS simulations, thus it saves time, effort, and cost. As well as it is a faster simulation over the distribution of the LHS. The MLHRK method has been proven to be effective, reliable, and convergent to solve a wide range of linear and nonlinear problems. The proposed method can predict the future behavior of the population under study in analyzing the behavior of some epidemiological models.


Introduction:
The differential equation is very useful for modeling, simulating phenomena and understanding how the natural problem can be formulated not just a value or a set of values, but a function or a class of functions.
The problem of the present study, multi-parameters of the non-linear system of ordinary differential equations (ODE) with missing their real data is that matter needs to solve the system and to estimate their parameters in the same time. Epidemic refers to an increasing in the number of disease cases in the population of a particular region such as influenza, cholera, measles, malaria and others. The epidemic may be in one place, however if it spreads to other countries and it affects a large number of people, it may be called a pandemic. Two kinds of epidemic models social and biology epidemic models. The study of epidemiology has spread significantly; the problem was studied by several authors such as study the fitting curve of the influenza epidemic in Australia in 1919 [1]. 50-100 million deaths people were recorded who have been affected by the influenza epidemic that means the influenza epidemic is a global epidemic that is spreading very quickly across the world and infecting much of the population. Estimation of outstanding claims liability and sensitity analysis. Herlambang and Tampubolon in 2008 studied estimation of outstanding claim liability and sensitivity analysis. probabilistic trend family (PTE) model [2]. Mutaqin, et al. in 2008 studied generating claim data of general insurance based on collective risk model and claim process [3]. Asmawati and Juliana in 2008 studied Combining individual learning ,and group discussion in calculus course [4]. Kinyanjui, et al. in 2015 studied Vaccine induced herd immunity for control of respiratory syncytial virus disease in a low-income country setting [5]. Beauparlant in 2016 studied a metapopulation model for the spread of in correctional facilities [6]. Leung, et al. in 2016 studied periodic solutions in a model with immune boosting and crossimmunity [7]. Pei, et al. in 2019 studied predictability in process-based ensemble forecast of influenza [8]. The social epidemic problems attracted a lot of attention and was studied by several authors such as Sabaa, et al. studied approximate solutions for alcohol consumption model in Spain in 2019 [9]. Many authors proposed Runge-Kutta numerical method to solve the social epidemic models. Mohammed et al. discussed numerical solution for weight reduction model due to health campaigns in Spain in 2015 [10]. Sabaa and Mohammed in 2020 studied approximate solutions of nonlinear smoking habit model [11].
To produce data for the parameters of a real model random numbers are usually created in a sequence by a specific code for some programs in computer to generate them, this process is called simulation. In our study simulations process (LHS) is used. Elsevier and AB McBratney studied a conditioned Latin hypercube method for sampling in the presence of ancillary information in 2006 [12]. One of the statistical tasks is to determine the appropriate sample size for the study because this statistic largely depends on the sample size [10]. Inevitable epidemiological models are represented as a non-linear system of differential equations that have parameters missing their real data. These models are solved using a simulation approach to evaluate random parameters for deterministic epidemiological models that have specific distribution and approximate their solutions. The statistical simulations process (LHS) run multiple times to simulate the parameter values, then estimated parameters are combined with a reliable numerical method to be a numerical simulation method to simulate the parameter values to solve the influenza epidemic problem under study. Some authors solved the social epidemic models by numerical simulation methods such that Mohammed, et al. in 2018 discussed the non-conventional hybrid numerical approach with multi-dimentional random sampling for cocaine abuse in Spain [13]. Mohammed, et al. in 2019 discussed Mean Monte Carlo finite difference method for random sampling of a nonlinear epidemic system [14]. Sabaa in 2019 proposed modified numerical simulation methods Mean Monte Carlo Runge-Kutta that was applied to two the social epidemic models, namely nonlinear smoking habit, and alcohol consumption model [15].
This research is arranged as follows: in Section 2, the mathematical model of an influenza epidemic is descried. Statistical technique: Latin hypercube sampling to simulate model's parameters is introduced in Section 3. In Section 4, numerical methods (Runge-Kutta) are presented. In Section 5, the modified process which is Mean Latin Hypercube Runge-Kutta method is constructed as a sutible numerical simulation method to applied on the influenza epidemic model. In Section 6, the results of the current methods are discussed and shown tabulary and graphically the influenza epidemic model under study, the version (9.0) for MATLAB R2016a is used to obtain the results of the influenza epidemic model which is the present application. Finally, some conclusions for outcomes are mentioned in Section 7.

Mathematical Model:
Modeling is an important and powerful tool for understanding the effects and transmission of epidemic diseases [16]. Mathematical models are used to plan, implementation, comparision, evaluation, prevention, treatment, and other matters [17]. The mathematical models describe different processes in mathematics, physics, and biology [18]. The modeling requires an initial effort to reduce complexity. Examining the model is important to prevent the spread of the epidemic. The system consists four equations, and four variables which are represent healthy persons, infected persons that does not cause infection, the person is in the incubation period, infected persons and the cause infection after in the incubation period the persons who recover or die. So that our model is made with eight parameters that connect variables. The model is used to study disease progression and to obtain some estimates and comparisons [19]. The influenza model was introduced as a system of the first order nonlinear ordinary differential equations in equations (1-4): , (3)

Variables Description
Proportion of susceptible population Proportion of exposed population

Proportion of infective population
Proportion of recovered population

Methodology
In this section, statistical method is sued to simulate the model's parameters, numerical methods and modified numerical simulation method is used to solve the system numerically and approximately.

Latin Hypercube Sampling
Technique Latin Hypercube Sampling is one of the known stratified statistical methods and type of stratified sampling that used to generate random samples for a multidimensional random distribution of a multidimensional model over the certain period in present study. It is fast method, and it is efficiency method that saves effort and time with a small number of sample elements. The results can be accurately reached. The sampling methods are different in the time and the results obtained.
is an appropriate method to solve any system because the system is made up of multiple variables and parameters to solve, therefore a stratif sampling becomes necessary.
provides more accurate, better simulation results and closer to the statistical values even if the number of iterations is few because it is often faster to reach a good representation of the probability distribution.
The name is Latin squares sampling that arrays contain various elements. Each element in a Latin square exactly occurs once in each row and column. The elements occur once in each hyperplane when the concept is applied to a multidimensional setting to make up a Latin hypercube. Each layer is represented by the elements in the hypercube, the layer is divided into equal intervals based on the number of samples required [20]. Before using this method we must create the random numbers then we create a sample that has a uniform distribution, after that we specify the number of simulations [21]. In addition LHS remember in which row and column the specific sample was taken, this is called a system mentioned in the approach compared with the rest of the methods such as MC it needs a large number of iterations in order to provide accurate simulation results because the sample comes from anywhere to distribute the inputs so this is without memory in the approach. Therefore it is sometimes inactive. LHS distributes the sample values even in small areas, while MC distributes the samples in large areas, and without cutting so that the LHS method is better than MC method [9,19,20].

Runge-Kutta
Numerical Method Runge-Kutta is one of the most famous multi-stage methods. Runge-Kutta techniques had been introduced around 1900 by C. Runge and M. W. Kutta. Runge-Kutta is an effective, iterative numerical method consisting of several stages, the arrangement of the method is determined by the number of stages, this method can be applied to the ordinary, partial, explicit, implicit, delaying differential equations etc [22]. It is easy to understand and implement. Runge-Kutta generates a series of approximate solutions converging to the exact solution. Runge-Kutta plays a major role in many branch of science, engineering and economics that frequently contain mathematical models of ordinary differential equations [23]. In our study, method is used to solve the nonlinear system of an influenza model of the first order for initial value problems. Two types of Runge-Kutta numerical methods have been used.

Runge-Kutta of order four
: Runge-Kutta of order four is suggested an approximation for solving the system of non-linear ordinary differential equations of the first order with the initial value problem . The is a numerical method that gives more accurate results than some other numerical methods such as Finite Difference method. The general form of the first order for ordinary differential equation can be written as follows: , , with initial value , where is nonlinear function, is the dependent variable, is the independent variable while and are positive real constant values. The most common form of the four-stage of the explicit method [23] is: The general formula of for the method is (9) where are unknown function, is a time and is a step size The most common form of the four-stage of the implicit method [23] is: The general formula of for the method is (14) where are unknown function, is a time and is a step size. Now, to find and , we following the next steps in equations (10)(11)(12)(13)(14).
. (18) where . In the same way, and can be found to obtain the second step in equations (15)(16)(17)(18) where are unknown functions, is a time and is a step size: where . In the third stage, we try to find and by substituting in the system (1) as below in equations (19)(20)(21)(22) where are unknown functions, is a time and is a step size: where . The fourth stage needs to find and as below, in equations (23)(24)(25)(26) where are unknown functions, is a time and is a step size:

Runge-Kutta of order four
: Runge-Kutta of order four is presented an approximation for solving the system of non-linear ordinary differential equations of the first order with the initial value problem . It is a suitable method to obtain stable results. The most common form of the six-stage of the explicit method is: (36) The general formula of for the method is (37) where are unknown function, is a time and is a step size. Now, To find and , we following the next steps in equations (38-41) where are unknown functions, is a time and is a step size: (41) where . In the same way, and can be found to obtain the second step in equations (42-45) where are unknown functions, is a time and is a step size: where . In the third stage, try to find and by substituting in the system (1) as below in equations (46-49) where are unknown functions, is a time and is a step size: where . The fourth stage needs to find and as below, in equations (50-53) where are unknown functions, is a time and is a step size: where . The fifth stage needs to find and As below, in equations (54-57) where are unknown functions, is a time and is a step size: where .

Mean Latin Hypercube Runge-Kutta
Method: The modified method is the first time mentioned that integrates the results between the statistical simulations of random sampling with the classic numerical iterative approach of Runge-Kutta formula. Whereas, the samples are estimated by LHS random process before solving the system by the RK method. These samples contain the number of LHS simulation values for each parameter that was generated by code in MATLAB [9]. This method has been suggested to calculate the numerical simulation solution obtained for each subgroup. The method is performed using the Matlab program. The method is more effective than classical numerical methods that depend on a limited time. This method is applied to solve the influenza epidemic model in the current study. The Latin Hypercube sampling is repeated times to simulate values of parameters as random sample. The modified method is called Mean Latin Hypercube Runge-Kutta method The modified process is called Mean Latin Hypercube Runge-Kutta method which is abbreviated by . The scenario of the suggested method in our study is as follows : reliable numerical methods have firstly been used for solving nonlinear ordinary differential equations of the first order to prove the solutions of the system under study. The numerical simulation solutions for the influenza epidemic in Australia [1] are obtained using a modified process that combines the numerical method and the statistical random sampling technique . The aim of this study to solve nonlinear ordinary differential equations of the first order for the influenza epidemic model by using method under initial conditions. This modified approach simulates the rate of model parameters by technology before integration with iterations of . The steps of the method are summarized: Step 1: All parameters have been simulated as sample for -times at once.
Step 2: For each random parameter, one value is specified and replaced in the system.
Step 3: Solve the system -times iterations numerically by . The last iterative solution is the final solution.
Step   Tables 3 and 5. Error is a measure of the convergence of a solution to the methods. For this, the absolute error of Step 1: All parameters have been simulated by for times at once Step 2: For each random parameter, one value is specified and replaced in the system Step 3: Solve the system -times iterations numerically by . The last iterative result is the final solution Step 4: Repeat steps 1 and 2 for -times Step 5: Calculate the mean of final solutions from step 4, as a solution system, called method is less than the absolute error of method as it is shown in the Tables 4 and 6, where the absolute error is the difference between RK numerical, and mnumerical simulation results for both orders 4,and 4 and 5. The errors of , and are less than the errors of , and respectively. Therefore, method is better than the method. The MMCRK, and MLHRK solutions for influenza model in 70 days are inside the predicted interval of numerical.
From the Tables (3-6), we conclude, with the greater number of simulations, the best results for the old method MMCRK are got. While the increasing number of simulations does not affect the accuracy of the results in our new method MLHRK. Therefore, the proposed method MLHRK is not need more repetitions to get good results as in the other numerical simulation methods like MMCRK that needs more repetitions to get accurate results. It is clear in this study, 500 repetitions at =0.04 need more time than 200 repetitions with the same step size, the MLHRK gets accurate results with 200 repetitions while MMCRK gets the accuracy with 500 repetitions under the same step size. The proposed MLHRK method gives values closer to the RK numerical solutions than the MMCRK method for both orders 4 and, 4 and 5 as it is shown in Figures (1-4).

Conclusion
The influenza model has been studied to understand, to analyze and to interpret the behavior of epidemic dynamics. Mean Latin Hypercube Runge-Kutta method is an iterative algorithm which has been presented to calculate the numerical statistical solution for the influenza epidemic problem.
The new modified approach, namely Mean Latin Hypercube Runge-Kutta method is discussed to solve influenza model wherein the sample is frequently and randomly divided into groups. The method is used to find numerical deterministic solutions to the influenza model when the model parameters are constants, while the method is used when the randomization in the model becomes necessary with parameters model are treated as random variables. There are many techniques to solve epidemiological models and evaluate performance by applying a form of segmentation like Mean Monte Carlo Finite Difference and Mean Latin Hypercube Finite Difference . The accuracy and efficiency of method have been demonstrated by studying the convergence. The error of method is less than method. Therefore, method is better than method. The advantage of MLHRK method over RK method is that it can reduce the number of numerical iterations of RK as well as it is a faster simulation over the distribution of the LHS randomized group samples. method is an upgrade to the previous numerical simulation process which is Mean Monte Carlo Runge-Kutta ( ). method is collected the last iteration of in each simulation, finally taking the average for the simulation iterations results as a final solution to the method. In the present work, the results are tabulated with the results as well as the results for the influenza pandemic model. The numerical results of Runge-kutta of order 4 and Runge-kutta of order 4 and 5 methods are compared with other statistical numerical results of Mean Latin Hypercube Runge-Kutta and Mean Monte Carlo Runge-Kutta methods in the time interval . It is found that these proposed methods are effective, reliable, and suitable in solving such problems.
considers a middle between the statistical simulation process and numerical methods. The convergence of numerical and numerical statistical methods has been discussed. method is better than the method. The MLHRK method gives values closer to the RK numerical solutions than the MMCRK method. The results have been calculated by the Matlab program and the figures have been drawn by the Magic plot program.     Table 7-Predicted interval of numerical simulation results with 500 simulations for influenza model in 70 days.

Variables
Step size