Artificial neural network-based heuristic to solve COVID-19 model including government strategies and individual responses.

The current work aims to design a computational framework based on artificial neural networks (ANNs) and the optimization procedures of global and local search approach to solve the nonlinear dynamics of the spread of COVID-19, i.e., the SEIR-NDC model. The combination of the Genetic algorithm (GA) and active-set approach (ASA), i.e., GA-ASA, works as a global-local search scheme to solve the SEIR-NDC model. An error-based fitness function is optimized through the hybrid combination of the GA-ASA by using the differential SEIR-NDC model and its initial conditions. The numerical performances of the SEIR-NDC nonlinear model are presented through the procedures of ANNs along with GA-ASA by taking ten neurons. The correctness of the designed scheme is observed by comparing the obtained results based on the SEIR-NDC model and the reference Adams method. The absolute error performances are performed in suitable ranges for each dynamic of the SEIR-NDC model. The statistical analysis is provided to authenticate the reliability of the proposed scheme. Moreover, performance indices graphs and convergence measures are provided to authenticate the exactness and constancy of the proposed stochastic scheme.

Introduction

The progress of science based on technology and material has been highly demanded in recent years. However, there are various obstacles to encounter hardly in daily life. For example, a novel coronavirus is one of the most dangerous viruses humans have attacked since 2020. This virus transmits rapidly from human to human, and now it has spread worldwide. Millions of people have died from this virus, and many positive cases have been reported with high recovery rates [[1], [2], [3], [4], [5]]. The common symptoms of coronavirus are sore throats, coughs, fever, and headaches, as well as some other respiratory indications, like high temperature, even bleeding, phlegm, chest pain, and breath shortness [[6], [7], [8], [9]].

Recently, the researcher's community has taken a keen interest in investigating the dynamics of COVID-19. Donders et al. [10] presented the ISIDOG recommendations related to COVID-19 and pregnancy. Wang [11] designed a model for coronavirus using the applications, confines, and capacities. Rhodes et al. [12] presented a mathematical model of public troubles in COVID-19 infection control. Jewell et al. [13] showed the potential effects of disruption to HIV programs in sub-Saharan Africa caused by COVID-19. Khrapov et al. [14] expressed the comparative analysis of the mathematical models based on the dynamics of the coronavirus COVID-19 epidemic development in different countries. Thompson [15] constructed epidemiological models, considered essential tools for guiding COVID-19 interventions. Sánchez et al. [16] proposed a SITR fractal nonlinear dynamics of a novel COVID-19. Elsonbaty et al. [17] analyzed the discrete fractional SITRs model for COVID-19. Umer et al. [18,19] presented the numerical solutions using the heuristic and swarming approaches based on the SITR model.

Various areas involve mathematical models, like health, biology, physics, chemistry, civil/mechanical engineering, and economics. Kharis and Kholisoh stated that mathematical science plays a crucial role in preventing the spread of illness [20]. A mathematical form of the model can be implemented to investigate the spread of viruses. Yulida stated that mathematics had a significant part in exploring the disease outbreak, spreading, and predicting designs known as epidemiology [21]. For the numerical outcomes of these models, the stochastic solvers based on the artificial neural networks (ANNs) together with the optimization procedures of global/local search approaches based on genetic algorithm (GA) and active-set approach (ASA), i.e., GA-ASA has been implemented for solving the spreading of coronavirus, i.e., SEIR-NDC model. The stochastic approaches of ANNs under the optimization procedures of the GA-ASA have never been implemented to solve the SEIR-NDC model. Some well-known applications of the stochastic procedures are the delay differential singular systems [22,23], prey-predator system [24], fractional singular models [[25], [26], [27]], HIV infection system [28], corneal shape models [29,30], functional differential models [31], Thomas-Fermi differential model [32], mosquito dispersal system [33], heat conduction model [34] and periodic form of the singular models [35,36].

The SEIR-NDC nonlinear mathematical model depends upon seven categories, susceptible (S), exposed population (E), infected people (I), removed individuals (R), total population (N), public observation (D), and cumulative (C), which are given as [37]:

iwhere , , , , , and d indicate the relocation rate, latent, pr mary transmission rate, transmission rate at time , infected, public response and severe case values. The initial conditions (ICs) are i 1, i 2, i 3, i 4, i 5, i 6 and i 7. Some novel features of the ANNs by using the optimization procedures of GA-ASA are provided:

The solution of the SEIR-NDC-based COVID-19 model is effectively presented by using the stochastic computational heuristic-based ANNs along with the optimization procedures of GA-ASA.

The performances through ANNs with the optimization-based stochastic procedures are testified by matching the proposed and reference results.

The absolute error (AE) values lie in suitable ranges to solve the SEIR-NDC model, which presents the worth of the stochastic computational performances.

The consistency of the proposed computational heuristic-based ANNs, along with the optimization procedures of GA-ASA, is realistic by applying the statistical measures on multiple runs to solve the nonlinear SEIR-NDC model.

The statistical performances based on Theil's inequality coefficient (TIC), variance account for (VAF), and mean absolute deviation (MAD) have been used to perform the solutions of the SEIR-NDC model.

The paper is organized as follows: Sect 2 performs the proposed results. Sect 3 provides the performance catalogs. Section 4 presents the results and detailed discussions of the nonlinear SEIR-NDC model. The final remarks are listed in the final Sect.

Methodology

In this section, the proposed ANNs and the optimization procedures of GA-ASA are presented in two phases to find the numerical results of the nonlinear SEIR-NDC model. In addition, the introduction of an error-based fitness function and the optimization-based procedural steps of GA-ASA are also presented.

ANN modeling

The mathematical formulations given in the system (1), along with the derivative performances, are provided: Where, the unknown weight vector is , which is given:

An activation log-sigmoid function is applied in this study [38,39], the mathematical form of the log-sigmoid function is given: .

The error function is given: where and . The proposed solutions of the nonlinear SEIR-NDC model are and . Likewise, , , , , , and are the merit functions based differential SEIR-NDC model and its ICs.

Optimization performance: GA-ASA

This section provides the details of the global and local search methods based on the GA and ASA for the SEIR-NDC system based on COVID-19. The global search GA is an optimization scheme to solve the constrained/unconstrained equations. GA is an optimization procedure that is used to apply the “crossover,” “selection,” “mutation,” and “elitism” procedures. Recently, GA has been used in cancer, lung prediction [40], pupils' academic performance prediction [41], wellhead back based pressure control model [42], mutation for the test data generation systems [43], bearing accountability analysis of induction motors [44], optimization of human resources in sanitorium emergency [45], sensor acquisition frequency [46], singular nonlinear model [47] and automated manufacturing networks [48].

The hybridization of the GA is performed with the local search approach by taking GA best values as an initial input to find the rapid performances. In this regard, an operative ASA is executed to regulate the consequences and GA procedure using optimization. In recent decades, ASA has been implemented in large-scale optimization systems, including box constraints [49], cardiac defibrillation [50], regularized monotonic regression [51], predictive switch for a beam and ball system [52], and multivariate integration through the uncertain error demand [53]. The current work is linked to solving the nonlinear SEIR-NDC system through the GA-ASA optimization procedures. The inclusive descriptions of the stochastic procedures are listed in Table 1 .

Table 1

Description of the stochastic procedure for the nonlinear SEIR-NDC system.

Performance catalogues

In this section, the mathematical performances of VAF, TIC and MAD are provided for the SEIR-NDC model that is written as:

Results and discussion

This section provides detailed results and discussions for the nonlinear SEIR-NDC model. The comparative investigations using the reference solutions label the accuracy of the ANNs along with the optimization procedures of GA-ASA. Moreover, statistical valuations prompt the accuracy of the ANNs along with the stochastic performances of the SEIR-NDC model.

SEIR-NDC mathematical model

The simplified form of the nonlinear SEIR-NDC model using the suitable parameter values is given:

The error function is given:

The optimization is performed to solve the SEIR-NDC system for 20 independent executions together with 30 or ten neurons (see Fig. 1). The best weight set values through ANNs using the hybrid optimization procedures are plotted in Fig. 2 . These weights are proficient in finding the solutions of the SEIR-NDC system, given as [17].

Fig. 1

The proposed scheme GA-ASA structure for the SEIR-NDC system.

Fig. 2

Best weight vectors for the nonlinear SEIR-NDC model.

The Systems (18–24) are derived to assess the outcomes of the SEIR-NDC system using the optimization procedures of the ANNs together with GA-ASA. The results are drawn in Fig. 2, Fig. 3, Fig. 4 based on the 30 variables or ten neurons. Fig. 2(a–g) depicts the best weights for the nonlinear SEIR-NDC model. The mean and best results comparison for solving the SEIR-NDC model are presented in Fig. 3(a–g). It is observed that the proposed, best, and mean solutions are overlapped for each category of the nonlinear SEIR-NDC system. Fig. 4 is drawn based on AE in terms of best and mean outcomes for the nonlinear SEIR-NDC model.One can observe that the best AE for the and classes of the SEIR-NDC system lie as 10−06-10−08, 10−05-10−07, 10−02-10−04, 10−04-10−05, 10−06-10−08, 10−05-10−07 and 10−06 to 10−08, respectively. The mean values for these respective dynamics of the SEIR-NDC system are calculated as 10−05-10−07, 10−05-10−06, 10−02-10−06, 10−05-10−07, 10−05-10−06, 10−04-10−07 and 10−04-10−06. It is seen that the AE through the designed computational scheme is accurate in solving the nonlinear SEIR-NDC model. One can accomplish that the designed numerical procedure is accurate in terms of the AE.

Fig. 3

Result comparison for the nonlinear SEIR-NDC model.

Fig. 4

Best/Mean AE for the nonlinear SEIR-NDC model.

The performances through the statistical TIC, EVAF and MAD operators are demonstrated in Fig. 5 . It is indicated that the performances of TIC for the and lie around 10−08-10−10, 10−09-10−10, 10−07-10−08, 10−08-10−10, 10−11-10−12, 10−09-10−10 and 10−10-10−11. The performance of EVAF for the respective dynamics of the SEIR-NDC model found 10−06-10−07, 10−07-10−08, 10−03-10−04, 10−04-10−05, 10−11-10−12, 10−04-10−05 and 10−09-10−10, respectively. The performance of MAD and also found in good ranges. These optimal small measures improve the worth and accuracy of the stochastic scheme.

Fig. 5

Performance measures for the nonlinear SEIR-NDC model.

Fig. 6 shows the analysis of TIC, EVAF and MAD operator for the nonlinear SEIR-NDC model. It is observed through these analyses that most of the runs achieve good fitness levels. One can accomplish that the designed approach is stable and reliable.

Fig. 6

Convergence of TIC, EVAF and MAD for the nonlinear SEIR-NDC model.

To check the precision, accuracy, and correctness of the designed scheme, the statistical depictions based on the SEIR-NDC system are derived in Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 for the and . The statistical outcomes for the operators Maximum (Max), Minimum (Min), semi-interquartile range (SIR), Median, Mean and standard deviation (STD) are presented. The Min and Max values represent the best and bad trials, while S.I.R is one-half of the difference between the 3rd and first quartiles. One can observe that the Min operator values of and found as 10−06-10−11, 10−06-10−10, 10−04-10−10, 10−05-10−11, 10−07-10−11 and 10−06-10−10. The Max operator values that indicate the worst trials were found around 10−05-10−06 for each category of the SEIR-NDC system. The S.I.R, Mean and STD and Median values of the statistics lie as 10−06-10−08 for each dynamic of the SEIR-NDC system. These calculated values indicate the reliability of the proposed scheme to solve the SEIR-NDC model.

Table 2

Statistics results of the nonlinear SEIR-NDC model for.

ℵ	S(ℵ)
	Min	Max	Mean	Median	S.I.R	STD
0	3.60032E-11	2.30688E-06	3.95126E-08	3.27227E-07	1.52327E-07	6.70737E-07
0.1	2.09410E-08	8.70303E-06	8.39999E-07	1.79962E-06	9.68586E-07	2.14007E-06
0.2	4.08596E-07	1.77019E-05	1.80603E-06	3.22998E-06	6.55853E-07	4.08677E-06
0.3	4.11253E-08	1.87691E-05	3.65699E-06	4.50105E-06	1.92684E-06	4.81840E-06
0.4	1.00314E-06	2.32212E-05	6.26634E-06	6.97651E-06	1.81048E-06	4.51337E-06
0.5	2.98064E-07	2.23982E-05	8.34863E-06	9.21741E-06	1.81457E-06	4.57419E-06
0.6	3.93181E-06	2.06716E-05	1.03632E-05	1.11690E-05	1.50990E-06	3.83299E-06
0.7	7.73273E-06	2.56817E-05	1.19012E-05	1.25365E-05	1.24511E-06	3.55931E-06
0.8	6.62059E-07	2.76447E-05	1.27180E-05	1.32761E-05	9.02358E-07	5.27729E-06
0.9	3.87578E-06	3.28068E-05	1.26747E-05	1.34751E-05	1.15532E-06	5.99386E-06
1	3.84973E-06	2.75718E-05	1.14109E-05	1.19712E-05	8.16100E-07	4.50341E-06

Table 3

Statistics results of the nonlinear SEIR-NDC model for.

ℵ	E(ℵ)A_h(χ)
	Min	Max	Mean	Median	S.I.R	STD
0	6.88809E-10	2.10246E-06	7.62645E-08	2.21527E-07	1.01650E-07	4.67928E-07
0.1	7.49512E-08	1.60480E-05	5.09385E-07	1.66523E-06	2.88441E-07	3.57085E-06
0.2	3.15757E-07	2.66627E-05	1.98827E-06	3.63804E-06	1.33269E-06	5.63233E-06
0.3	9.48688E-07	2.03071E-05	4.20348E-06	5.32163E-06	2.15614E-06	4.23606E-06
0.4	2.16166E-06	1.20152E-05	6.06888E-06	6.52171E-06	2.14547E-06	2.87345E-06
0.5	1.93589E-06	1.89592E-05	7.60552E-06	8.80009E-06	1.59389E-06	3.77635E-06
0.6	1.64671E-07	3.56753E-05	9.56830E-06	1.05852E-05	1.21693E-06	6.80060E-06
0.7	7.01474E-06	4.31385E-05	1.09666E-05	1.25809E-05	9.70676E-07	7.34798E-06
0.8	8.89149E-06	3.87457E-05	1.14043E-05	1.32153E-05	6.20230E-07	6.70759E-06
0.9	5.69500E-06	2.76896E-05	1.04845E-05	1.19431E-05	8.13679E-07	4.81023E-06
1	6.51409E-06	1.79725E-05	8.85306E-06	9.43398E-06	8.88891E-07	2.52373E-06

Table 4

Statistics results of the nonlinear SEIR-NDC model for.

ℵ	I(ℵ)A_h(χ)
	Min	Max	Mean	Median	S.I.R	STD
0	1.13711E-10	2.84983E-06	3.85940E-08	3.21677E-07	1.35843E-07	6.89181E-07
0.1	2.56215E-04	2.71479E-04	2.59523E-04	2.60247E-04	8.52226E-07	3.46002E-06
0.2	4.36681E-04	4.56616E-04	4.41761E-04	4.43102E-04	1.17234E-06	4.30802E-06
0.3	5.45026E-04	5.62898E-04	5.50013E-04	5.51683E-04	2.89995E-06	4.53615E-06
0.4	5.84565E-04	5.99582E-04	5.87136E-04	5.89382E-04	3.28600E-06	4.61257E-06
0.5	5.55910E-04	5.68026E-04	5.57382E-04	5.59677E-04	1.74146E-06	3.99099E-06
0.6	4.60442E-04	4.75154E-04	4.65838E-04	4.66607E-04	1.53037E-06	3.52522E-06
0.7	3.04993E-04	3.19985E-04	3.14695E-04	3.14234E-04	1.25718E-06	3.18414E-06
0.8	9.50511E-05	1.09362E-04	1.07102E-04	1.06356E-04	5.74370E-07	3.14357E-06
0.9	1.41975E-04	1.65273E-04	1.52984E-04	1.53684E-04	1.64303E-06	4.22739E-06
1	4.52250E-04	4.72026E-04	4.63375E-04	4.63114E-04	9.81812E-07	3.52136E-06

Table 5

Statistics results of the nonlinear SEIR-NDC model for.

ℵ	R(ℵ)A_h(χ)
	Min	Max	Mean	Median	S.I.R	STD
0	6.69144E-11	6.76242E-06	2.49468E-08	4.13129E-07	6.56059E-08	1.49924E-06
0.1	1.56388E-07	1.70055E-05	2.79619E-06	3.22756E-06	6.25032E-07	3.39717E-06
0.2	5.54984E-06	2.23443E-05	1.01927E-05	1.07093E-05	1.58972E-06	3.59795E-06
0.3	1.72329E-05	2.63359E-05	1.96740E-05	2.06821E-05	2.08094E-06	3.04426E-06
0.4	7.94102E-06	3.91228E-05	3.14578E-05	3.13013E-05	1.69664E-06	6.24737E-06
0.5	1.87792E-06	5.06177E-05	4.30155E-05	4.13918E-05	1.44107E-06	9.68471E-06
0.6	8.99977E-06	5.96879E-05	5.19618E-05	5.04315E-05	9.06125E-07	1.01120E-05
0.7	1.10951E-05	6.94103E-05	5.85033E-05	5.68530E-05	1.25266E-06	1.11715E-05
0.8	6.33613E-06	7.38015E-05	6.27220E-05	5.99176E-05	1.53876E-06	1.30255E-05
0.9	6.73912E-06	7.11628E-05	6.14471E-05	5.91862E-05	1.72105E-06	1.28110E-05
1	2.93008E-05	5.98895E-05	5.49934E-05	5.36986E-05	1.68790E-06	6.21892E-06

Table 6

Statistics results of the nonlinear SEIR-NDC model for.

ℵ	N(ℵ)A_h(χ)
	Min	Max	Mean	Median	S.I.R	STD
0	4.22933E-11	1.59967E-05	7.05358E-08	1.12599E-06	2.03306E-07	3.55582E-06
0.1	1.36108E-07	1.81456E-05	9.01977E-07	2.85176E-06	8.03353E-07	4.71349E-06
0.2	1.33456E-07	1.79176E-05	2.20318E-06	3.73526E-06	2.11311E-06	4.70232E-06
0.3	1.42898E-07	1.28182E-05	2.77591E-06	4.21140E-06	2.55727E-06	3.62734E-06
0.4	1.78081E-07	2.02060E-05	2.66575E-06	3.93302E-06	2.04317E-06	4.68974E-06
0.5	3.67748E-08	2.17867E-05	1.56158E-06	3.51696E-06	1.83709E-06	5.41211E-06
0.6	5.99113E-08	1.70662E-05	1.01354E-06	3.06191E-06	9.14093E-07	5.10185E-06
0.7	1.90916E-08	1.80211E-05	1.02757E-06	2.99907E-06	2.13178E-06	4.44307E-06
0.8	1.21499E-07	1.73111E-05	1.13196E-06	3.04855E-06	9.62549E-07	4.85588E-06
0.9	3.65011E-07	1.55190E-05	2.52975E-06	4.13153E-06	1.97983E-06	4.32362E-06
1	6.93436E-08	1.03581E-05	1.87436E-06	2.79862E-06	2.04956E-06	2.96112E-06

Table 7

Statistics results of the nonlinear SEIR-NDC model for.

ℵ	D(ℵ)A_h(χ)
	Min	Max	Mean	Median	S.I.R	STD
0	1.10686E-10	1.10597E-05	2.10310E-08	8.65604E-07	8.70769E-08	2.57431E-06
0.1	1.02404E-07	1.58579E-05	8.49577E-07	1.96336E-06	4.55001E-07	3.59481E-06
0.2	1.57822E-07	1.30637E-05	2.91483E-06	3.70358E-06	1.59103E-06	2.93113E-06
0.3	3.43656E-07	1.53426E-05	4.70454E-06	5.29591E-06	1.12594E-06	3.24657E-06
0.4	1.07689E-07	2.45192E-05	6.07023E-06	6.99999E-06	1.63718E-06	5.25789E-06
0.5	6.78080E-07	3.10306E-05	8.45347E-06	9.74805E-06	1.95175E-06	6.23115E-06
0.6	2.38683E-06	3.35138E-05	1.04739E-05	1.22721E-05	2.12751E-06	6.60005E-06
0.7	2.58242E-06	3.20802E-05	1.20257E-05	1.37990E-05	1.48098E-06	6.65682E-06
0.8	1.24887E-06	3.38603E-05	1.29028E-05	1.41286E-05	8.45757E-07	6.56672E-06
0.9	1.21663E-06	3.18514E-05	1.28942E-05	1.34596E-05	1.37818E-06	6.24252E-06
1	3.16966E-06	2.54620E-05	1.15257E-05	1.21126E-05	1.35198E-06	5.21726E-06

Table 8

Statistics results of the nonlinear SEIR-NDC model for.

ℵ	C(ℵ)A_h(χ)
	Min	Max	Mean	Median	S.I.R	STD
0	3.90810E-11	4.23918E-06	1.35893E-08	3.31828E-07	7.62130E-08	9.55850E-07
0.1	5.62845E-08	5.39600E-04	4.44063E-07	2.79071E-05	3.90929E-07	1.20452E-04
0.2	1.51668E-07	5.00345E-04	1.23355E-06	2.67422E-05	1.26634E-06	1.11486E-04
0.3	3.87496E-08	5.00317E-04	3.18914E-06	2.86553E-05	2.10489E-06	1.11072E-04
0.4	2.15256E-07	4.97742E-04	3.22013E-06	2.98108E-05	2.43583E-06	1.10297E-04
0.5	1.36397E-07	5.00520E-04	3.44456E-06	2.97193E-05	2.61805E-06	1.11135E-04
0.6	2.20479E-07	5.03994E-04	1.54931E-06	2.94355E-05	1.51904E-06	1.12172E-04
0.7	9.23625E-08	5.05010E-04	1.53333E-06	2.95272E-05	1.16646E-06	1.12428E-04
0.8	4.11928E-07	5.03711E-04	1.32369E-06	2.88498E-05	9.23804E-07	1.12209E-04
0.9	1.65679E-07	5.01960E-04	2.08305E-06	2.92883E-05	1.22298E-06	1.11559E-04
1	3.10505E-07	5.02207E-04	1.98610E-06	2.90106E-05	1.39515E-06	1.11553E-04

Concluding remarks

This work aims to solve the SEIR-NDC nonlinear model using the strength of artificial neural networks and the optimization procedures of genetic algorithms and active-set techniques. The SEIR-NDC-based COVID-19 nonlinear mathematical model has never been solved using the stochastic approaches based on the GA-ASA. The validation of the proposed ANNs through the optimization of GA-ASA is observed by using the comparison of the reference and obtained results. It is noticed that the absolute error is found in good measures that are calculated as 10-07 to 10-10 for each dynamic of the SEIR-NDC system. The absolute error values are also authenticated in good measures for solving the nonlinear SEIR-NDC mathematical model.

Furthermore, the operator's TIC, EVAF, and MAD performances have been examined in good procedures for 30 numbers of variables to solve the SEIR-NDC nonlinear model. Moreover, the statistical-based operator's Min, Max, S.I.R. Mean, Median, and STD are in good ranges for solving the nonlinear SEIR-NDC nonlinear mathematical model. It is proved through these witnesses that the designed scheme is reliable, stable, robust, and accurate for solving the nonlinear SIER-NDC mathematical model.

Future research directions

the proposed stochastic computing framework can be applied to solve the nonlinear mathematical models arising in fluid dynamics [[54], [55], [56], [57]], thermal explosion models [58,59], singular studies [60,61], and food chain systems [62,63].

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.