A Stochastic Intelligent Computing with Neuro-Evolution Heuristics for Nonlinear SITR System of Novel COVID-19 Dynamics

: The present study aims to design stochastic intelligent computational heuristics for the numerical treatment of a nonlinear SITR system representing the dynamics of novel coronavirus disease 2019 (COVID-19). The mathematical SITR system using fractal parameters for COVID-19 dynamics is divided into four classes; that is, susceptible ( S ), infected ( I ), treatment ( T ), and recovered ( R ). The comprehensive details of each class along with the explanation of every parameter are provided, and the dynamics of novel COVID-19 are represented by calculating the solution of the mathematical SITR system using feed-forward artiﬁcial neural networks (FF-ANNs) trained with global search genetic algorithms (GAs) and speedy ﬁne tuning by sequential quadratic programming (SQP)—that is, an FF-ANN-GASQP scheme. In the proposed FF-ANN-GASQP method, the objective function is formulated in the mean squared error sense using the approximate di ﬀ erential mapping of FF-ANNs for the SITR model, and learning of the networks is proﬁciently conducted with the integrated capabilities of GA and SQP. The correctness, stability, and potential of the proposed FF-ANN-GASQP scheme for the four di ﬀ erent cases are established through comparative assessment study from the results of numerical computing with Adams solver for single as well as multiple autonomous trials. The results of statistical evaluations further authenticate the convergence and prospective accuracy of the FF-ANN-GASQP method.


Introduction
Human life has faced various challenges and obstacles throughout time in the form of floods, earthquakes, and diseases, among others. We first mention some widespread diseases such as dengue fever, which is a mosquito-borne disease produced by the dengue virus. This disease spread extensively disease, T(χ) is the treatment class, and the last class R(χ) is the recovered class from COVID-19 disease. The SITR model governing the novel COVID-19 dynamics is given as follows [34]: S 2 (0) = A 2 , I (χ) = −µI(χ) + βI(χ)(S 1 (χ) + S 2 (χ)) − αI(χ) + βδT(χ) + σI(χ), I(0) = A 3 , T (χ) = µI(χ) − ρT(χ) − αT(χ) + ψT(χ) + εT(χ), T(0) = A 4 , R (χ) = −αR(χ) + ρT(χ), The objective of the proposed study is to find the solution of the system of equations given in set (1) by integrating intelligent neuro-evolution computing heuristics via feed-forward artificial neural networks (FF-ANNs), and its adjustable parameters are tuned with an optimization procedure of genetic algorithms (GAs) integrated with sequential quadratic programming (SQP), i.e., FF-ANN-GASQP solver. The parameter of interest with appropriate settings for particular descriptions of the dynamics are provided in Tables 1 and 2, while the block diagram of the proposed study is given in Figure 1, which presents the system model, mathematical formulation, proposed methodology, evaluation criteria, and main results in pictorial form.
The major features of the suggested FF-ANN-GASQP scheme are briefly given as: • The solution of the mathematical expression for the nonlinear SITR model for novel COVID-19 dynamics is calculated viably by using the novel application of the intelligent neuro-evolution-based integrated computing paradigm, i.e., FF-ANN-GASQP. In addition to the precise and accurate solutions for the SITR-based mathematical model of the COVID-19 pandemic, other valuable perks are that it is easy to understand the concepts, and it also has smooth operation, exhaustive applicability, consistency, and extendibility.
The remaining sections of the current study are organized as follows. The designed framework of FF-ANN-GASQP is given in Section 2, measures of the performance are listed in Section 3, a description of the results with necessary interpretations are provided in Section 4, and concluding remarks with related potential studies are given in the last section.  FF-ANN-GASQP: feed-forward artificial neural networks trained with global search genetic algorithms and speedy fine tuning by sequential quadratic programming.

Proposed Methodology
ANN is an artificial intelligence-based computing approach that is used widely to effectively solve a variety of challenging stiff linear/nonlinear problems in terms of reliability, stability, and robustness. In this study, the strength of feed-forward ANN is exploited to solve the SITR model representing novel COVID-19. The weights of ANN are optimized with a genetic algorithms (GA)-based evolutionary metaheuristic along with sequential quadratic programming (SQP). The proposed FF-ANN-GASQP structure for getting the numerical solutions of the nonlinear SITR model based on COVID-19 is described in two phases:

•
To exploit the FF-ANN models, an error-based objective function is introduced.

•
Optimize the objective function for system (1) using the hybrid GA-SQ programming approach.

ANN Modeling
The mathematical measures using system (1) are given by the differential mapping of FF-ANN for approximate solutionsŜ 1 (χ),Ŝ 2 (χ),Î(χ),T(χ) andR(χ) along with their nth order derivatives, which are given as: where the unknown weight vector is W and given as: Using the log-sigmoid, i.e., an function activation P(χ) = (1 + e −χ ) −1 , the updated form of system (2) becomes: Using the values of system (3), an error-based objective function e is given as: where The approximate results for both sub-classes of susceptible S 1 and S 2 , infected class I, treatment class T, and recovered class R are denoted asŜ 1 ,Ŝ 2 ,Î,T, andR. Accordingly, e 1 , e 2 , e 3 ,e 4 ,e 5 , and e 6 are the error-based functions for system dynamics and initial conditions of the SITR model representing COVID-19 as given in the set of Equation in (1). The proposed results are achieved from the adjustable weights for which the objective function as shown in system (4) tends to zero, i.e., mathematically e → 0 . Hence, approximate solutions

Optimization Technique: Hybrid of GA with SQP
In this section, a brief overview of GASQP is given to optimize networks for system (4) representing COVID-19 dynamics.
GAs is an efficient global search optimization approach, which was introduced by Holand [30,31]. GA is one of the mathematical models of human genetics, which is implemented as a global search tool for linear/nonlinear optimization processes in assorted domains of applied sciences. Population in GAs represents the set of possible solutions as per the obligation of the optimizing systems, while the populations are modified through the exploitation of selection, crossover, elitism, and mutation such that an appropriate outcome is achieved. Some well-known applications of GA in various fields include multilayer piezoelectric transducer [32][33][34][35][36][37], Hammerstein controlled autoregressive models [38], nonlinear circuits [39], economic load dispatch [40], active noise control system [41], and power signal modeling [42].
The GA optimization procedure is further enhanced normally using the hybridization process with the speedy local search approach, so in this study, SQP is applied for a fine-tuning mechanism. The GA optimization procedure is further enhanced normally using the hybridization process with the speedy local search approach, so in this study, SQP is applied for nonlinear models in combustion theory [43], Bagley-Torvik systems [44], and system identification [45]. In the presented investigation, the integrated metaheuristics through GASQP is exploited to optimize the unknown weights of FF-ANN models, which are demonstrating the SITR model based on COVID-19. The pseudocode detail of FF-ANN-GASQP for a nonlinear SITR system is given in Table 3. Table 3. Pseudocode of FF-ANN-GASQP for nonlinear SITR system for COVID-19 dynamics.
Start the GA process Inputs: The individuals with genes equally representing the decision values of FF-ANN as: as per the details provided in the system (3). Population: Number of chromosomes in a set define a population as: The best global decision variables/trained weights of the ANN-GASQ programming scheme denoted as W GA-Best Initialization: Generate chromosome W and P with pseudo random numbers.
Initialization is performed for {GA} and {gaoptimset} routines with suitable declarations and settings.  Fitness Evaluate: Compute fitness value of every W of P by using system (4) to (10). Adjustments: Fine-tune {fmincon} with SQ programming scheme to tune W and adjust again the fitness value by using systems (4) to (10). Accumulate Store fitness, time, W GASQ , function counts and generations for multiple trials of SQ programming.

End of the SQ programming scheme Data Generations
Repeat the procedure 30 times based on the GASQ programming to get a massive dataset of the optimization variables of ANNs for numerical solutions of the SITR model based on COVID-19

Performance Measures
The mathematical form of the performance grades using the root mean square error (RMSE) and Theil's inequality coefficient (TIC) for SITR model based on COVID-19 is presented as:

Results and Discussion
The detailed description of the results with necessary interpretations are provided here to solve the SITR system for COVID-19 dynamics, as given in the system of Equation (1). The numerical results are also determined with Adams method, which are used in this work as reference solutions for comparison with the designed FF-ANN-GASQP solver. Moreover, statistical observations are also listed to evaluate the performance. The stability and reliability are verified by the results obtained through 30 independent runs achieving a higher level of accuracy by means of statistical measures based on TIC and RMSE, which further validate the worth of the proposed FF-ANN-GASQP solver. The convergence analysis based on the performance measures of statistical operators TIC and RMSE is provided for reliability and robustness. The graphs of absolute error (AE) using the proposed results and numerical reference solutions are presented for the validity of the proposed scheme.

Nonlinear SITR Model Based on COVID-19
The different suitable values of the epidemic parameters have been given in Table 4, and the updated form of the SITR model based on COVID-19 given in system (1) using these Table 4 values is given as: The objective function of set (13) is formulated as follows: Optimization of the SITR system for COVID-19 dynamics is performed by the hybrid-computing structure of GA and SQP for 30 numbers of independent trials to achieve the parameters of FF-ANN having five neurons. The set of the weight vectors of FF-ANN by GASQP are provided in Figure 2 for the best fitness, and these weights are applied to achieve the numerical outcomes of the FF-ANN-GASQP solver for the SITR system. Values of different parameters against four cases are given in Table 5. The mathematical formulations of the approximate numerical outcomes of FF-ANN-GASQP is given as follows:  Figure 2a depicts that the rate of contact is the source of a significant increase in the number of susceptible persons primarily, but with the passage of time, it starts declining. This happens because the higher contact rate causes most of the people to get infected and transfer to an infected class; therefore, the number of people in the susceptible class is reduced. Figure 2b clearly reflects that the number of infected persons increases with the increase of contact rate. Reduction in the contact rate results in a slight rise in the number of infected persons, while an increase in the contact rate suddenly enhances the number of infected persons. Figure 2c shows the growing behavior of susceptible persons with higher recovery rate values. Figure 2d depicts that as the recovery rate upsurges, so does the number of persons in the infectious class. It can be noticed that when the recovery rate is low, then less persons are recovered from the virus, because of the death of infected persons. Figure 3a demonstrates the increasing behavior of susceptible persons with various death rates. Figure 3b shows the variation of recovered persons with different death rate values. Figure 3 depicts that with a high death rate, the number of recovered persons are reduced abruptly. This is because when the death rate is high, then a large number of persons from the infected and recovered class die, which results in the reduction of persons in all of these classes. As the death rate becomes very high, the infectious and recovered persons are approximately wiped out.   The set of Equations (15)- (19) are used to get the solution of the nonlinear SITR model based on COVID-19 as given in system (13) using the FF-ANN-GASQP, and numerical results are provided in Figures 2-4 using five neurons. The trained weights set for the five neurons-based FF-ANN model is shown in Figure 4a-e for all the classes of a nonlinear SITR system for COVID-19 dynamics. The graphs of absolute error (AE) using the proposed results and numerical Adams scheme are plotted in Figure 5. It is seen that most of the AE values lie around 10 −4 to 10 −5 for all the parameters of the nonlinear SITR model using five numbers of neurons.
The convergence analysis based on the performance measures of the statistical operators TIC and RMSE values is drawn in Figure 6. The first part of Figure 6 shows the performance measures of the TIC values, while the second part of Figure 6 indicates the performance of the RMSE values for all the parameters of the nonlinear SITR model of COVID-19. One may conclude from these obtainable results that most of independent runs achieved a higher level of accuracy.

Conclusions
A new computational intelligent solver FF-ANN-GASQP is designed for the solution of a nonlinear SITR system for COVID-19 dynamics using the approximation capability of FF-ANN optimized globally with GAs and locally with SQP. The dynamics of the nonlinear SITR model of COVID-19 are proficiently assessed by the designed FF-ANN-GASQP approach with the single hidden layer formulation of FF-ANNs using five neurons. The exactness of the designed FF-ANN-GASQP technique is endorsed by matching of the outcomes with reasonable precision from the reference Adams numerical scheme to solve a nonlinear SITR system based on COVID-19. Moreover, statistical clarifications through 30 autonomous nuns by means of TIC and RMSE indices further authenticate the worth and value of the proposed FF-ANN-GASQP solver.

Conflicts of Interest:
All the authors of the manuscript have no conflict of interest and have worked in an equal way to obtain the results presented in this paper.