Soft Computing Paradigms to Find the Numerical Solutions of a Nonlinear Influenza Disease Model

The aim of this work is to present the numerical results of the influenza disease nonlinear system using the feed forward artificial neural networks (ANNs) along with the optimization of the combination of global and local search schemes. The genetic algorithm (GA) and active-set method (ASM), i.e., GA-ASM, are implemented as global and local search schemes. The mathematical nonlinear influenza disease system is dependent of four classes, susceptible S(u), infected I(u), recovered R(u) and cross-immune individuals C(u). For the solutions of these classes based on influenza disease system, the design of an objective function is presented using these differential system equations and its corresponding initial conditions. The optimization of this objective function is using the hybrid computing combination of GA-ASM for solving all classes of the influenza disease nonlinear system. The obtained numerical results will be compared by the Adams numerical results to check the authenticity of the designed ANN-GA-ASM. In addition, the designed approach through statistical based operators shows the consistency and stability for solving the influenza disease nonlinear system.


Introduction
There are various serious diseases produced by viruses, of which influenza is one of them that primarily attacks the upper respiratory portions, bronchi, nose, throat and sometimes disturbs the lungs. The influenza is not a fatal illness, and most people recover within one to two weeks without medical care. This disease is a serious risk to older people or those with serious illnesses such as cancer, diabetes, heart, kidney problems and lung disease. Among these people, infection can lead to serious problems of primary diseases, such as pneumonia causing death. The epidemic rate of influenza is reported as between 5% and 15% per year of the population, which is affected by upper respiratory tract infections. Worldwide, the annual epidemics are witnessed between 3 and 5 million cases of serious illness and the number of deaths is reported to be around 250,000 and 500,000 [1]. Many mathematical epidemiological models are illustrated by the ordinary nonlinear autonomous differential systems, which designate the assumptions of the model that the parameters are time independent. In such systems, variables refer to recovered, infected, transmitted and susceptible disease vectors.
Astuti et al. [2] suggested a step-by-step differential transformation approach to solving the disease-resistant influenza virus model. Erdem et al. [3] presented mathematical investigations of a susceptible-infectious-quarantine-recovered (SIQR) influenza model with imperfect quarantine. Alzahrani and Khan [4] introduced a numerical technique to solve a fractional pandemic influenza model. Sun et al. [5] presented multi-objective optimization models for allocating patients during a pandemic influenza outbreak. Ghanbari et al. [6] provided an analysis of two models of avian influenza outbreaks relating fractal-fractional derivatives with power and memorabilia from Mittag-Leffler. González-Parra et al. [7] designed and discussed a fractional epidemiological model for simulating influenza A outbreak. Tchuenche et al. [8] researched the impact of media coverage on human influenza transmission dynamics. Schulze-Horsel et al. [9] discussed the dynamics of infection and virus-induced apoptosis in the production of influenza vaccines in cellular culture. Hovav et al. [10] presented a network flow system for managing inventory and distributing influenza vaccines in a healthcare supply chain. Patel et al. [11] used genetic algorithms to discuss the optimal vaccination plans for pandemic influenza. Kanyiri et al. [12] introduced the optimum control applications for influenza and pulmonary congestion with antiviral resistance.
The influenza disease nonlinear system has four categories, susceptible (S(u)), infectious (I(u)), recovered (R(u)) and cross-immune (C(u)). The mathematical design of the nonlinear influenza disease system is written as follows [13]: where β shows the transmission rate for the susceptible to the infected individual, and a 1 , a 2 , a 3 and a 4 are the initial conditions. The infected, infectious and cross-immune are signified as γ −1 , δ −1 and α −1 , respectively. σ shows the exposed cross-immune individuals, who are shifted in a unit time to transmittable subpopulations [14]. The recently reported studies addressing the different aspect of influenza nonlinear modelling can be seen in [15][16][17][18][19]. The aim of this work is to solve the above nonlinear influenza disease model using the stochastic capabilities of artificial neural networks (ANNs), genetic algorithms (GA) and the active set method (ASM), i.e., ANN-GA-ASM. Stochastic numerical methods have been implemented to solve a great number of applications, such as singular fractional models, COVID-19-based susceptible-infectious-treatment-recovered (SITR) dynamics, the delay singular functional model, the prey-predator model, singular higher order nonlinear models, the dengue fever nonlinear system, multi-singular differential models and the mosquito release nonlinear system based on the heterogeneous environment (please see [20,21] and citation therein). Based on these well-known applications, the authors are interested in solving the nonlinear influenza system using the ANN-GA-ASM.
Few key factors of the ANN-GA-ASM are briefly given as:

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The proposed ANN-GA-ASM provides effective solutions of the nonlinear influenza system. • Consistent, stable and reliable outcomes from the nonlinear influenza system validate the value of the proposed ANN-GA-ASM.

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The absolute error (AE) values are in the good agreements, indicating the reliability of the proposed ANN-GA-ASM.

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The performance is certified through ANN-GA-ASM using different statistical observations to solve the nonlinear influenza system for thirty independent trials.

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The designed ANN-GA-ASM is effortlessly implemented to solve the nonlinear influenza system with smooth operations, inclusive and easy to understand. The other paper parts are as follows. Section 2 shows the methodology and statistical measures. Section 3 demonstrates the simulation of the outcomes. Section 4 provides the final remarks and future research remarks.

Methodology
The methodology of the proposed ANN-GA-ASM structure to solve the nonlinear influenza system is defined in two steps: an objective function is designed to solve the ANN parameters and some crucial settings are provided to improve the objective function based on the GA-ASM.

ANN Structure
In this section, the mathematical design is presented to solve each category of the susceptible (S), infectious (I), recovered (R) and cross-immune (C) groups based on the influenza disease system. The proposed outcomes of these classes areŜ,Î,R andĈ given as: where m is number of neurons, and W is the unknown weight vector, with its components for S, I, R and C are defined as W S , W I , W R and W C , respectively, and are given mathematically as follows: The updated form by applying the log-sigmoid function v(µ) = (1 + exp(−µ)) −1 is provided in Appendix A for the interested readers. To find the optimization measures, an error based objective function becomes: whereŜ k = S(u k ),Î k = I(u k ),R k = R(u k ) andĈ k = C(µ k ). In the above network E 1 , E 2 , E 3 and E 4 indicate the objective functions related to system (1), while E 5 is designed on the basis of the initial conditions of system (1).

Optimization Performances: GA-ASM
In this section, the ANN-GA-ASM performance is presented for solving the influenza disease nonlinear system. The designed ANN structure through GA-ASM for solving the influenza disease nonlinear system is depicted in Figure 1. The block structure of the proposed methodologies is illustrated by defining the problem, mathematical modeling, the formulation of the fitness function on mean square error sense, the workflow of GA and ASM.
In this study, the global search GA is one of optimization process that is executed to solve the influenza disease nonlinear system. GA is pragmatic to standardize the specific population for solving the various complicated models using the optimal training. To achieve the best system outcomes, GA operates through a selection operator, a crossover process, reproduction practice and a mutation procedure. Recently, GA has been applied in the hospitalization expenditure system [22], for feature assortment in cancer microarray [23], brain tumor images [24], air blast prediction [25], monorail vehicle dynamics [26], cloud service optimization [27] and liver disease prediction [28].
The active-set method is one of the rapid local search optimization approaches that works to solve the constrained/unconstrained models broadly. ASM is executed in numerous optimizations models of numerous complex and non-stiff systems. Recently, ASM is applied to execute the real-time optimal control [29], the pricing of American better-of option on two assets [30], the pressure-dependent model of water distribution systems [31], overcurrent relays in microgrid optimization [32], embedded model predictive controls [33] and elastodynamic frictional contact problems [34]. To control the slowness of GA, the process of hybridization into GA-ASM is provided in Table 1 for training or learning of the decision variables, i.e., the unknown weights of ANNs, while the parameter settings of GA and ASM are handled using the 'optimset' routine of the Matlab optimization toolbox. The setting of the parameters, i.e., fitness function tolerance (TolFun), constraints tolerance (TolCon), population size (PopSiz), generations, Stall generation limits (StallLimit), decision variable tolerance (TolX), iteration and maximum number of fitness function evalutions (MaxFunEvals), is done with care to avoid the premature convergence of the optimization mechanism.

Performance Measures
The mathematical representations using the statistical operators containing "variance account for (VAF)", "mean absolute deviation (MAD)", "semi-interquartile (S.I.R)" and "Theil's inequality coefficient (TIC)" along with their Global representations are presented to solve the influenza disease nonlinear model and are provided in Appendix B. Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 17

Local search: ASM Start point, Best individual, Bounds as defined in Optimset routine
The Problem

GA process starts
Inputs: Select the chromosomes with the same numbers as decision variable of ANN models for solving influenza disease system: The chromosomes set is given as:

Simulations and Results
The relative presentations of the results obtained using the Adams method are used to certify or analyze the accuracy of the ANN-GA-ASM. Moreover, the statistical performance is testified for the precision, reliability and accuracy of the proposed structure. The updated form of the nonlinear model of influenza illness using the appropriate parameters as defined in reported study [13]: The fitness function for the influenza disease system (10) becomes: Performance by optimization is demonstrated for the nonlinear influenza disease model using the proposed ANN-GA-ASM for 30 independent runs, i.e., sufficiently large multiple autonomous executions, having 30 numbers of hidden neurons in ANN models in set (3). The routine for the fitness function as given in Equations (4)-(9) is developed in the Matlab software package while the optimization is conducted as per procedure of pseudocode in Table 1 The proposed outputs are obtained using the above systems ( (12)-(15) The global performances of (G-TIC), (G-MAD) and (G-EVAF) operators for 30 tri to solve the proposed ANN-GA-ASM is shown in Table 6 to solve the nonlinear influen disease model. These global MAD, TIC and EVAF performances based on Min lie arou 10 −02 -10 −03 , 10 −06 -10 −07 and 10 −01 -10 −02 , respectively, while the global performances based S.I.R lie around 10 −02 to 10 −03 , 10 −07 -10 −08 and 10 −01 -10 −02 for each category of the influen disease nonlinear model. These close ideal values obtained through global measures in cate the precision, correctness and accuracy of the designed ANN-GA-ASM.    The graphical representations based on the statistical procedures to authenticate the convergence performance are given in Figure 5 to solve the nonlinear influenza disease model. In the performance through TIC values using thirty trials to solve the nonlinear influenza disease model, it is seen that most of the trials based on the susceptible, infectious, recovered and cross-immune for the TIC values lie around 10 −5 -10 −7 , 10 −6 -10 −8 , 10 −5 -10 −8 and 10 −5 -10 −7 , respectively. For the EVAF values, the performances of the susceptible, infectious, recovered and cross-immune lie around 10 −1 -10 −3 . For the MAD values, the performances of the susceptible, infectious, recovered and cross-immune lie around 10 −2 -10 −4 . These best presentations of the executions using the ANN-GA-ASM are calculated as suitable for the TIC, EVAF and MAD operators.         The global performances of (G-TIC), (G-MAD) and (G-EVAF) operators for 30 trials to solve the proposed ANN-GA-ASM is shown in Table 6

Conclusions
This study is associated with the submission of numerical studies of the non-linear influenza disease system. The influenza disease nonlinear system is dependent on four categories named as susceptible, infected, recovered and cross-immune. Artificial neural networks, as well as global and local research approaches, i.e., ANN-GA-ASM, are proposed to address each category of the influenza illness model. For finding numeric results, an objective function based on the differential system and initial conditions is optimized by the proposed ANN-GA-ASM. The log-sigmoid works as an activation function and 30 numbers of variables have been proposed to solve the influenza disease nonlinear model. In the future, the ANN proposed with the GA-ASM is capable of solving nonlinear biological models, singular systems of higher order and fluid dynamics models. Moreover, the provided ANN-GA-ASM can be implemented to those problems which are still considered to be stiff for traditional deterministic computing schemes.

Appendix A
The updated form of the networks as given in set of Equation (2) using the log-sigmoid function are given as follows:

Appendix C
The proposed solutions of ANN-GA-ASM are for reproduction of the results.  (A9)