# Swarming Computational Procedures for the Coronavirus-Based Mathematical SEIR-NDC Model

Authors: Suthep Suantai; Zulqurnain Sabir; Muhammad Asif Zahoor Raja; Watcharaporn Cholamjiak
Journal: Journal of Mathematics
 (2022)

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## Abstract

The motive of the current work is related to solving the coronavirus-based mathematical system of susceptible (S), exposed (E), infected (I), recovered (R), overall population (N), civic observation (D), and cumulative performance (C), called as SEIR-NDC. The numerical solutions of the SEIR-NDC model are presented by using the computational framework of artificial neural networks (ANNs) together with the swarming optimization procedures aided with the sequential quadratic programming. The swarming procedure based on the particle swarm optimization (PSO) works as a global search, while the sequential quadratic programming (SQP) is used as a local search algorithm. A merit function is constructed by using the nonlinear dynamics of the SEIR-NDC mathematical system based on its 7 classes, and the optimization of the merit function is performed through the PSOSQP. The numerical expressions of system are accessible with the ANNs using the PSOSQP optimization with 30 variables. The correctness of the stochastic computing scheme performances is verified by using the comparison of the obtained performances of the mathematical SEIR-NDC system and the reference Runge–Kutta scheme. Furthermore, the graphical illustrations of the performance indices, absolute error, and convergence curves are derived to validate the robustness of the proposed ANN-PSOSQP approach for the mathematical SEIR-NDC system.


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## Body

## 1. Introduction

The focus of this study is to highlight the health issues along with various diseases that are caused by infections with viruses or bacteria. In addition to disrupting human life, infectious diseases are also seen as a significant threat to economies, businesses, education, and other facets of daily life. One of these viruses, the coronavirus, spread fast throughout the entire world. The transmitting ratio of the coronavirus was very high, and it rapidly spread in the humans from one to another. Millions of people were infected by the coronavirus, which did not discriminate between developed, developing, and underdeveloped nations. After 2.5 years, there are still high number of positive cases reported daily throughout the world. However, the recovery ratio of this virus was so high in its start and now the recovery rate is much higher due to the process of vaccination [1–3]. The symptom of this disease alters with its new shapes, like runny noses, fever, coughs, sore throats, headaches, and respiratory indications (shortness of breath, high fever, bleeding, phlegm, cough, and chest pain) [4–6].In recent years, one of the main focuses of the researchers is to analyze the outcomes of the coronavirus dynamics by using different effects. Wang [7] presented a mathematical system formulation based on the coronavirus using its applications and capacities. Based on the coronavirus as well as the pregnancy effects, Donders et al. [8] provided the Worldwide Society of Infectious Illness in Gynecology and Obstetrics. Rhodes et al. [9] introduced the coronavirus mathematical model to control the public problems. Khrapov et al. [10] performed the comparative investigations using the mathematical systems of the coronavirus dynamics based on the data of different regions. Jewell et al. [11] presented the potential disruptive impacts using the HIV packages in sub-Saharan Africa based on coronavirus. A coronavirus dynamical system is constructed by Sánchez et al. [12]. Thompson [13] proposed an epidemiological system to consider the significant operators using the coronavirus supervisory interferences. Elsonbaty et al. [14] designed the discrete fractional-order dynamical coronavirus system. Umer et al. [15,16] studied the numerical performances using the swarming as well as heuristic schemes to present the solutions of the coronavirus system. Shikongo et al. [17] presented the fractional-order operator using the principle of quarantine and isolation based on the coronavirus.Mathematical models appear in a wide range of disciplines, including biology, chemistry, economics, civil and mechanical engineering, and health. According to Side et al. [18] mathematical systems can be used to assess the disease’s evolution. Owolabi et al. [19] designed an efficient scheme for the biological stoichiometry model based on tumour dynamics. In another study, they presented the fitted numerical scheme using the HIV model based on the dynamics of cancer-immune system [20]. Some more relevant studies have been presented in [21]. Moreover, few mathematical systems can be accessible analytically, but it is better to perform numerical simulations of the abovementioned systems. Therefore, the computational framework of artificial neural networks (ANNs) together with the swarming optimization procedures aided with the sequential quadratic programming is provided the solve the coronavirus-based mathematical SEIR-NDC system. The swarming procedure based on the particle swarm optimization (PSO) works as a global search, while the sequential quadratic programming (SQP) is used as a local search algorithm. The ANN-based swarming procedures together with PSOSQP have never been applied before to solve the coronavirus-based mathematical SEIR-NDC system. Few famous submissions of the computing stochastic solvers are the prey-predator models [22], delay singular differential models [23, 24], multifractional systems [25–27], periodic singular systems [28,29], nonlinear design of the optics [30], HIV infection system [31], eye surgery models [32, 33], functional singular model [34], mosquito spreading model [35], Thomas–Fermi system [36], smoking model [37], and food chain nonlinear models [38].The mathematical SEIR-NDC system depends upon seven dynamics, susceptible (S(y)), exposed (E (y)), infected (I(y)), recovered (R(y)), overall population (N (y)), civic observation (D(y)), and growing performance (C(y)), shown as [4](1)dSydy=−ψSy−α0SyEyN−βIySyN,S0=q1,dEydy=α0EySyN+βSyIyN−ρ+ψEy,E0=q2,dIydy=ρEy−ϑ+ψIy,I0=q3,dRydy=−ψRy+ϑIy,R0=q4,dNydy=−ψNy,N0=q5,dDydy=−ϕDy+δϑIy,D0=q6,dCydy=ρEy,C0=q7,where ψ, ρ, α0, β, ϑ, ϕ, and δ are the emigration ratio, rate of initial spread, transmission ratio, infected rate, public response, and the severe case. The initial conditions of the model are q1, q2, q3, q4, q5, q6, and q7. The biological aspects of these initial conditions are well defined in a reported study [19] along with the setting of the parameters such as existence, uniqueness, bounded, and nonnegative of the solution.The novel topographies of the ANN-PSOSQP are provided as follows:(i)
A neuro swarming computing scheme is presented to solve the coronavirus-based mathematical SEIR-NDC system.(ii)
The swarming procedure based on the PSO is applied as a global search, while the SQP is used as a local search algorithm to solve the coronavirus-based mathematical SEIR-NDC system.(iii)
The stochastic computing scheme is presented efficiently to solve the coronavirus-based mathematical model.(iv)
The comparison of the obtained and reference solutions demonstrates the stochastic approach's correctness.(v)
The absolute error (AE) values are in good measures for the coronavirus-based mathematical SEIR-NDC system, which signifies the exactness of the swarming computing scheme.(vi)
The robustness of the stochastic swarming computing scheme together with PSOSQP is provided by using the statistical performances for multiple trials.The rest of the paper is organized as follows. Section2 shows the designed procedures. Section 3 is designed based on the numerical solutions. The concluding comments are presented in the last section.


## 2. Methodology

The current section provides the stochastic ANN procedure in two steps using the optimization of PSOSQP procedures for solving the mathematical SEIR-NDC system.(i)
A cost function based on the differential SEIR-NDS system is provided.(ii)
Optimization performances of the PSOSQP are provided.


### 2.1. Modeling: ANNs

The mathematical representations of the SEIR-NDC system are stated using the feed-forward ANNs based on the solutions of 1st derivative as(2)S^y,E^y,I^y,R^y,N^y,D^y,C^y=∑n=1mbS,nTwS,ny+uS,n,∑n=1mbE,nTwE,ny+uE,n,∑n=1mbI,nTwI,ny+uI,n,∑n=1mbR,nTwR,ny+uR,n,∑n=1mbN,nTwN,ny+uN,n,∑n=1mbD,nTwD,ny+uD,n,∑n=1mbC,nTwC,ny+uC,n,(3)ddyS^y,ddyE^y,ddyI^y,ddyR^y,ddyN^y,ddyD^y,ddyC^y=∑n=1mbS,nddyTwS,ny+uS,n,∑n=1mbE,nddyTwE,ny+uE,n,∑n=1mbI,nddyTwI,ky+uI,k,∑n=1mbR,nddyTwR,ny+uR,n,∑n=1mbN,nddyTwN,ky+uN,k,∑n=1mbD,nddyTwD,ny+uD,n,∑n=1mbC,nddyTwC,ky+uC,k,where m shows the number of neurons, T is the activation function, and the first derivative is used due to the nature of the 1st order SEIR-NDC system. The unidentified weights are represented by W and written as W=WSWEWIWRWNWDWCt, where WS=bSwSuSt, WE=bEwEuEt, WI=bIwIuIt, WR=bRwRuRt, WN=bNwNuNt, WD=bDwDuDt, and WC=bCwCuCt.bS=bS,1,bS,2,…,bS,m, bE=bE,1,bE,2,…,bE,m, bI=bI,1,bI,2,…,bI,m, bR=bR,1,bR,2,…,bR,m, bN=bN,1,bN,2,…,bN,m, bD=bD,1,bD,2,…,bD,m, bC=bC,1,bC,2,…,bC,m, wS=wS,1,wS,2,…,wS,m, wE=wE,1,wE,2,…,wE,m, wI=wI,1,wI,2,…,wI,m, wR=wR,1,wR,2,…,wR,m, wN=wN,1,wN,2,…,wN,m, wD=wD,1,wD,2,…,wD,m, wC=wC,1,wC,2,…,wC,m, uS=uS,1,uS,2,…,uS,m, uE=uE,1,uE,2,…,uE,m, uI=uI,1,uI,2,…,uI,m, uR=uR,1,uR,2,…,uR,m, uN=uN,1,uN,2,…,uN,m, uD=uD,1,uD,2,…,uD,m, uC=uC,1,uC,2,…,uC,m.A log-sigmoid activation function (LSAF) is used in this work, and the mathematical formulation of LSAF is provided asTy=1/1+e−y. Put the values of LSAF in equation (2), and then first-order derivative has been performed, which is shown in the following equation:(4)S^y,E^y,I^y,R^y,N^y,D^y,C^y=∑n=1mbS,n1+e−wS,ny+uS,n,∑n=1mbE,n1+e−wE,ny+uE,n,∑n=1mbI,n1+e−wI,ny+uI,n∑n=1mbR,n1+e−wR,ny+uR,n,∑n=1mbN,n1+e−wN,ny+uN,n,∑n=1mbD,n1+e−wD,ny+uD,n,∑n=1mbC,n1+e−wC,ny+uC,n,ddyS^y,ddyE^y,ddyI^y,ddyR^y,ddyN^y,ddyD^y,ddyC^y=∑n=1mwS,nbS,ne−wS,ny+uS,n1+e−wS,kℵ+pS,k2,∑n=1mwE,nbE,ne−wE,ny+uE,n1+e−wE,ny+uE,n2,∑n=1mwI,nbI,ne−wI,ny+uI,n1+e−wI,ny+uI,n2,∑n=1mwR,nbR,ne−wR,ny+uR,n1+e−wR,ny+uR,n2,∑n=1mwN,nbN,ne−wN,ny+uN,n1+e−wN,ny+uN,n2,∑n=1mwD,nbD,ne−wD,ny+uD,m1+e−wD,ny+uD,m2,∑n=1mwC,nbC,ne−wC,ny+uC,n1+e−wC,ny+uC,n2.The cost function is provided as(5)EC=EC−1+EE−2+EE−3+EE−4+EE−5+EE−6+EF−7+EF−8,(6)EC−1=1N∑n=1NdS^dyn+α0S^nE^nN+βI^nS^nN+ψS^k2,(7)EC−2=1N∑n=1NdE^dyn−α0S^nE^nN−βI^nS^nN+ψE^n+ρE^n2,(8)EC−3=1N∑n=1NdI^dyn+ϑI^n+ψI^n−ρE^n2,(9)EC−4=1N∑n=1NdR^dyn−ϑI^n+ψR^n2,(10)EC−5=1N∑n=1NdN^dyn+ψN^n2,(11)EC−6=1N∑n=1NdD^dyn+ϕD^n−δϑI^n2,(12)EC−7=1N∑n=1NdC^dyn−ρE^n2,(13)EC−8=17S^0−q12+E^0−q22+I^0−q32+R^0−q42+N^0−q52+D^0−q62+C^0−q72,where Nh=1, E^n=E^yn, D^n=D^yn,S^n=S^yn,N^n=N^yn,I^n=I^yn,R^n=R^yn,C^n=C^yn, and yn=nh. The proposed results are S^n=S^yn,I^n=I^yn,E^n=E^yn,R^n=R^yn,N^n=N^yn,D^n=D^yn, and C^n=C^yn. Moreover, EC−1, EC−2, EC−3, EC−4, EC−5, EC−6, EC−7, and EC−8 are the cost functions using SEIR-NDC system and its initial conditions, respectively.


### 2.2. Optimization: PSOSQP

The optimization performances based on the swarming procedures based on the PSO along with the SQP procedures are presented using the mathematical SEIR-NDC system.The global search swarming procedure PSO is used as an alteration of the genetic algorithm. PSO was discovered in the 7th decade of the 19th century. PSO provides best solution performance to solve the stiff and nonstiff natured problems. Recently, PSO has been applied in widespread applications, such as mixed-variable optimization systems [39], solar energy systems [40], plant diseases diagnosis [41], feature selection in cataloging [42], organizing the single, double, and three diode photovoltaic systems [43], particle filter noise reduction in mechanical fault diagnosis [44], big data excavation of hot subjects about recycled water based on the microblog [45], 2nd order functional singular differential system [34], and green coal production problem [46].The efficient and rapid performances of the results have been obtained through the hybridization of the swarming optimization schemes with the local search method. Hence, SQP is used as a local search scheme with the hybridization of PSO. The local search PSO approach is used by taking the initial PSO input to perform the quick results. In recent decades, PSO has been functional in the variety of applications, like optimal power flow problem [47], dynamic economic dispatch [48], constrained nonlinear control allocation with singularity avoidance [49], four-level integrated supply chain with the aim of determining the optimum stockpile and period length [50], and multivariate regression based on the fuel cell using the electric vehicle [51]. The present investigations are related to indicate the numerical performances based on the mathematical system using the PSOSQP. The stochastic PSOSQP procedure for the mathematical SEIR-NDC system is provided in Figure 1. The descriptions of the PSO and SQP are tabulated in Table 1.Figure 1
Stochastic PSOSQP scheme for the mathematical SEIR-NDC system.Table 1
Optimization PSOSQP scheme for the mathematical SEIR-NDC system.
PSO procedure[Inputs]: The chromosomes are presented asW = [bw, u].[Population]: Chromosome sets are presented asW=WSWEWIWRWNWDWCt, where WS=bSwSuSt, WE=bEwEuEt, WI=bIwIuIt, WR=bRwRuRt, WN=bNwNuNt, WD=bDwDuDt and WC=bCwCuCt.[Output]: Best PSO weights are denoted as WBPSO.Initialization: The collection of chromosomes represents the WBPSO values.[Fitness Valuation]: Adjust the fitness (EC) using the population [P] through systems 4–12.(i) [Terminating Standards]: Terminate the process, when the below criteria meets, [TolCon = 10–19], [Generations = 120], [EC  = 10–19], [TolFun = 10–21], [StallLimit = 110], and [Size of population = 210].Move to [storage].[Ranking]: WBPSO shows the population rank forEC.[Storage]: [WBPSO],EC, [iterations], [time], and [Counts of function] for PSO.[PSO] process ends.[SQP] process starts.[Inputs]: WBPSO.[Output]: WPSOSQP represents the best values of the weights.[Initialize]: WPSOSQP, Iterations and Assignments.[Stopping rules]: [Iterations = 170], [EC  = 10–20], [Max-Fun-Evals = 170000] and [TolFun = 10–19], [Tol-X = 10–22].[FIT Calculation]: Compute [WBPSO] andEC through equations (5)–(13).[Amendments]: Regulate [fmincon] for SQP,EC to perform the ‘W’ for equations (5)–(13).[Store]: Transform WBPSO,nt.[SQP] process ends.


### 2.3. Statistical Performance

The current section presents the statistical representations of the mean square error (MSE), Theil’s inequality coefficient (TIC), and semi-interquartile range (SIR) for mathematical SEIR-NDC system. The statistical performances are used to check the reliability of the stochastic scheme. The mathematical form of these operators is given as(14)MSES,MSEE,MSEI,MSER,MSEN,MSED,MSEC=∑j=1nSj−S^j2,∑j=1nEj−E^j2,∑j=1nIj−I^j2,∑j=1nRj−R^j2,∑j=1nNj−N^j2,∑j=1nDj−D^j2,∑j=1nCj−C^j2A,TICS,TICE,TICI,TICR,TICN,TICD,TICC=1/n∑j=1nSj−S^j21/n∑j=1nSj2+1/n∑j=1nS^j2,1/n∑j=1nEj−E^j21/n∑j=1nEj2+1/n∑j=1nE^j2,1/n∑j=1nIj−I^j21/n∑j=1nIj2+1/n∑j=1nI^j2,1/n∑j=1nRj−R^j21/n∑j=1nRj2+1/n∑j=1nR^k2,1/n∑j=1nNj−N^j21/n∑j=1nNj2+1/n∑j=1nN^j2,1/n∑j=1nDj−D^j21/n∑j=1nD^j2+1/n∑j=1nD^j2,1/n∑j=1nCk−C^k21/n∑j=1nC^j2+1/n∑j=1nC^j2,SIR=−0.51stQuartile−3rdQuartile,where S(y), E (y), I(y), R(y), N(y), D(y), and C(y) indicate the reference solutions, while the hat terms present the proposed solutions that have been obtained by using the proposed scheme.


## 2.1. Modeling: ANNs

The mathematical representations of the SEIR-NDC system are stated using the feed-forward ANNs based on the solutions of 1st derivative as(2)S^y,E^y,I^y,R^y,N^y,D^y,C^y=∑n=1mbS,nTwS,ny+uS,n,∑n=1mbE,nTwE,ny+uE,n,∑n=1mbI,nTwI,ny+uI,n,∑n=1mbR,nTwR,ny+uR,n,∑n=1mbN,nTwN,ny+uN,n,∑n=1mbD,nTwD,ny+uD,n,∑n=1mbC,nTwC,ny+uC,n,(3)ddyS^y,ddyE^y,ddyI^y,ddyR^y,ddyN^y,ddyD^y,ddyC^y=∑n=1mbS,nddyTwS,ny+uS,n,∑n=1mbE,nddyTwE,ny+uE,n,∑n=1mbI,nddyTwI,ky+uI,k,∑n=1mbR,nddyTwR,ny+uR,n,∑n=1mbN,nddyTwN,ky+uN,k,∑n=1mbD,nddyTwD,ny+uD,n,∑n=1mbC,nddyTwC,ky+uC,k,where m shows the number of neurons, T is the activation function, and the first derivative is used due to the nature of the 1st order SEIR-NDC system. The unidentified weights are represented by W and written as W=WSWEWIWRWNWDWCt, where WS=bSwSuSt, WE=bEwEuEt, WI=bIwIuIt, WR=bRwRuRt, WN=bNwNuNt, WD=bDwDuDt, and WC=bCwCuCt.bS=bS,1,bS,2,…,bS,m, bE=bE,1,bE,2,…,bE,m, bI=bI,1,bI,2,…,bI,m, bR=bR,1,bR,2,…,bR,m, bN=bN,1,bN,2,…,bN,m, bD=bD,1,bD,2,…,bD,m, bC=bC,1,bC,2,…,bC,m, wS=wS,1,wS,2,…,wS,m, wE=wE,1,wE,2,…,wE,m, wI=wI,1,wI,2,…,wI,m, wR=wR,1,wR,2,…,wR,m, wN=wN,1,wN,2,…,wN,m, wD=wD,1,wD,2,…,wD,m, wC=wC,1,wC,2,…,wC,m, uS=uS,1,uS,2,…,uS,m, uE=uE,1,uE,2,…,uE,m, uI=uI,1,uI,2,…,uI,m, uR=uR,1,uR,2,…,uR,m, uN=uN,1,uN,2,…,uN,m, uD=uD,1,uD,2,…,uD,m, uC=uC,1,uC,2,…,uC,m.A log-sigmoid activation function (LSAF) is used in this work, and the mathematical formulation of LSAF is provided asTy=1/1+e−y. Put the values of LSAF in equation (2), and then first-order derivative has been performed, which is shown in the following equation:(4)S^y,E^y,I^y,R^y,N^y,D^y,C^y=∑n=1mbS,n1+e−wS,ny+uS,n,∑n=1mbE,n1+e−wE,ny+uE,n,∑n=1mbI,n1+e−wI,ny+uI,n∑n=1mbR,n1+e−wR,ny+uR,n,∑n=1mbN,n1+e−wN,ny+uN,n,∑n=1mbD,n1+e−wD,ny+uD,n,∑n=1mbC,n1+e−wC,ny+uC,n,ddyS^y,ddyE^y,ddyI^y,ddyR^y,ddyN^y,ddyD^y,ddyC^y=∑n=1mwS,nbS,ne−wS,ny+uS,n1+e−wS,kℵ+pS,k2,∑n=1mwE,nbE,ne−wE,ny+uE,n1+e−wE,ny+uE,n2,∑n=1mwI,nbI,ne−wI,ny+uI,n1+e−wI,ny+uI,n2,∑n=1mwR,nbR,ne−wR,ny+uR,n1+e−wR,ny+uR,n2,∑n=1mwN,nbN,ne−wN,ny+uN,n1+e−wN,ny+uN,n2,∑n=1mwD,nbD,ne−wD,ny+uD,m1+e−wD,ny+uD,m2,∑n=1mwC,nbC,ne−wC,ny+uC,n1+e−wC,ny+uC,n2.The cost function is provided as(5)EC=EC−1+EE−2+EE−3+EE−4+EE−5+EE−6+EF−7+EF−8,(6)EC−1=1N∑n=1NdS^dyn+α0S^nE^nN+βI^nS^nN+ψS^k2,(7)EC−2=1N∑n=1NdE^dyn−α0S^nE^nN−βI^nS^nN+ψE^n+ρE^n2,(8)EC−3=1N∑n=1NdI^dyn+ϑI^n+ψI^n−ρE^n2,(9)EC−4=1N∑n=1NdR^dyn−ϑI^n+ψR^n2,(10)EC−5=1N∑n=1NdN^dyn+ψN^n2,(11)EC−6=1N∑n=1NdD^dyn+ϕD^n−δϑI^n2,(12)EC−7=1N∑n=1NdC^dyn−ρE^n2,(13)EC−8=17S^0−q12+E^0−q22+I^0−q32+R^0−q42+N^0−q52+D^0−q62+C^0−q72,where Nh=1, E^n=E^yn, D^n=D^yn,S^n=S^yn,N^n=N^yn,I^n=I^yn,R^n=R^yn,C^n=C^yn, and yn=nh. The proposed results are S^n=S^yn,I^n=I^yn,E^n=E^yn,R^n=R^yn,N^n=N^yn,D^n=D^yn, and C^n=C^yn. Moreover, EC−1, EC−2, EC−3, EC−4, EC−5, EC−6, EC−7, and EC−8 are the cost functions using SEIR-NDC system and its initial conditions, respectively.


## 2.2. Optimization: PSOSQP

The optimization performances based on the swarming procedures based on the PSO along with the SQP procedures are presented using the mathematical SEIR-NDC system.The global search swarming procedure PSO is used as an alteration of the genetic algorithm. PSO was discovered in the 7th decade of the 19th century. PSO provides best solution performance to solve the stiff and nonstiff natured problems. Recently, PSO has been applied in widespread applications, such as mixed-variable optimization systems [39], solar energy systems [40], plant diseases diagnosis [41], feature selection in cataloging [42], organizing the single, double, and three diode photovoltaic systems [43], particle filter noise reduction in mechanical fault diagnosis [44], big data excavation of hot subjects about recycled water based on the microblog [45], 2nd order functional singular differential system [34], and green coal production problem [46].The efficient and rapid performances of the results have been obtained through the hybridization of the swarming optimization schemes with the local search method. Hence, SQP is used as a local search scheme with the hybridization of PSO. The local search PSO approach is used by taking the initial PSO input to perform the quick results. In recent decades, PSO has been functional in the variety of applications, like optimal power flow problem [47], dynamic economic dispatch [48], constrained nonlinear control allocation with singularity avoidance [49], four-level integrated supply chain with the aim of determining the optimum stockpile and period length [50], and multivariate regression based on the fuel cell using the electric vehicle [51]. The present investigations are related to indicate the numerical performances based on the mathematical system using the PSOSQP. The stochastic PSOSQP procedure for the mathematical SEIR-NDC system is provided in Figure 1. The descriptions of the PSO and SQP are tabulated in Table 1.Figure 1
Stochastic PSOSQP scheme for the mathematical SEIR-NDC system.Table 1
Optimization PSOSQP scheme for the mathematical SEIR-NDC system.
PSO procedure[Inputs]: The chromosomes are presented asW = [bw, u].[Population]: Chromosome sets are presented asW=WSWEWIWRWNWDWCt, where WS=bSwSuSt, WE=bEwEuEt, WI=bIwIuIt, WR=bRwRuRt, WN=bNwNuNt, WD=bDwDuDt and WC=bCwCuCt.[Output]: Best PSO weights are denoted as WBPSO.Initialization: The collection of chromosomes represents the WBPSO values.[Fitness Valuation]: Adjust the fitness (EC) using the population [P] through systems 4–12.(i) [Terminating Standards]: Terminate the process, when the below criteria meets, [TolCon = 10–19], [Generations = 120], [EC  = 10–19], [TolFun = 10–21], [StallLimit = 110], and [Size of population = 210].Move to [storage].[Ranking]: WBPSO shows the population rank forEC.[Storage]: [WBPSO],EC, [iterations], [time], and [Counts of function] for PSO.[PSO] process ends.[SQP] process starts.[Inputs]: WBPSO.[Output]: WPSOSQP represents the best values of the weights.[Initialize]: WPSOSQP, Iterations and Assignments.[Stopping rules]: [Iterations = 170], [EC  = 10–20], [Max-Fun-Evals = 170000] and [TolFun = 10–19], [Tol-X = 10–22].[FIT Calculation]: Compute [WBPSO] andEC through equations (5)–(13).[Amendments]: Regulate [fmincon] for SQP,EC to perform the ‘W’ for equations (5)–(13).[Store]: Transform WBPSO,nt.[SQP] process ends.


## 2.3. Statistical Performance

The current section presents the statistical representations of the mean square error (MSE), Theil’s inequality coefficient (TIC), and semi-interquartile range (SIR) for mathematical SEIR-NDC system. The statistical performances are used to check the reliability of the stochastic scheme. The mathematical form of these operators is given as(14)MSES,MSEE,MSEI,MSER,MSEN,MSED,MSEC=∑j=1nSj−S^j2,∑j=1nEj−E^j2,∑j=1nIj−I^j2,∑j=1nRj−R^j2,∑j=1nNj−N^j2,∑j=1nDj−D^j2,∑j=1nCj−C^j2A,TICS,TICE,TICI,TICR,TICN,TICD,TICC=1/n∑j=1nSj−S^j21/n∑j=1nSj2+1/n∑j=1nS^j2,1/n∑j=1nEj−E^j21/n∑j=1nEj2+1/n∑j=1nE^j2,1/n∑j=1nIj−I^j21/n∑j=1nIj2+1/n∑j=1nI^j2,1/n∑j=1nRj−R^j21/n∑j=1nRj2+1/n∑j=1nR^k2,1/n∑j=1nNj−N^j21/n∑j=1nNj2+1/n∑j=1nN^j2,1/n∑j=1nDj−D^j21/n∑j=1nD^j2+1/n∑j=1nD^j2,1/n∑j=1nCk−C^k21/n∑j=1nC^j2+1/n∑j=1nC^j2,SIR=−0.51stQuartile−3rdQuartile,where S(y), E (y), I(y), R(y), N(y), D(y), and C(y) indicate the reference solutions, while the hat terms present the proposed solutions that have been obtained by using the proposed scheme.


## 3. Result Performances

The current section shows the numerical performances of the results for the mathematical SEIR-NDC system by applying the proposed ANN-PSOSQP procedure.


### 3.1. Mathematical SEIR-NDC System

The updated form of the mathematical SEIR-NDC system is provided by adjusting the appropriate parameter performances, written as(15)dSydy=−0.5SyEyN−0.6ISyN−0.0205Sy,S0=0.9,dEydy=0.5ESyN+0.6ISyN−0.3505Ey,E0=0.1,dIydy=13Ey−0.225Iy,I0=0,dRydy=−0.025Ry+15Iy,R0=0,dNydy=−0.0205Ny,N0=14,dDydy=−111.2Dy+0.04Iy,D0=0,dCydy=13Ey,C0=0.The cost function is designed in terms of mean square error, which is based on the differential SEIR-NDC system given in equation (1) by keeping the methodology of equation (5) shown as(16)EC=1N∑n=1NdS^dyn+0.6S^nI^nN+0.5S^nE^nN+0.0205S^k2+dE^dyn−0.6S^nI^nN+0.3505E^n−0.5E^nS^nN2+dI^dyn−0.333E^n+0.225I^n2+dR^dyn−0.2I^n+0.025R^n2dN^dyn+0.0205N^n2+dD^dyn+111.2D^n−0.04I^n2+dC^dyn−0.33E^n2+17S^0−0.92+E^0−0.12+I^02+R^02+N^0−142+D^02+C^02.The solutions of the mathematical model are obtained through the optimization-based PSOSQP for 30 runs together with 10 neurons and 30 variables. The best values of the weights using the ANNs along with the optimization procedures of PSOSQP are illustrated in Figure2. These weight vectors are capable to achieve the proposed outcomes based on the mathematical SEIR-NDC system. The obtained solutions are presented as(17)S^y=−0.26421+e−−0.3453y−2.7523−1.11081+e−−0.0416y−3.3393+1.15751+e−−0.0138y−1.4023−2.70901+e−0.0464y−4.5616−1.67021+e−−0.3570y−4.4423−2.66031+e−−0.0283y−4.2419+1.75471+e−−0.029y−0.3310+0.00911+e−−2.652y−3.6896−1.55761+e−−0.3541y−3.7727+0.74271+e−0.3299y−1.7304,(18)E^y=−0.65391+e−0.4382y−2.7725+0.72971+e−0.2924y−0.8677+0.26591+e−0.0822y−2.7119−0.18121+e−0.1846y−2.6274−0.60851+e−−0.4121y−3.4072−0.14001+e−−1.906y−4.1492−0.37121+e−0.3892y−2.6735+0.69671+e−−0.186y−1.9214+0.15591+e−−2.6372y−4.3152−0.20111+e−0.9194y+0.5422,(19)I^y=−0.05741+e−1.2464y−0.8872+0.21871+e−−1.3408y+0.2756+0.06171+e−0.4227y−1.7866−0.29801+e−−1.2379y+0.2365−0.01331+e−−1.605y+1.2206−0.46631+e−−0.4652y+0.2808+0.73941+e−−0.400y−0.5017+0.64931+e−0.4898y+1.4072−0.72981+e−−0.0279y+0.0603−0.24521+e−0.3425y−0.3986,(20)R^y=0.15591+e−−0.0877y+0.1975+1.28321+e−0.0576y−0.8472+1.03991+e−−0.065y−0.8056−1.28871+e−−0.574y+0.1095−1.28871+e−−0.5745y+0.1095+2.06751+e−0.0278y−0.8404+0.63451+e−0.2984y−1.3477−0.89351+e−0.4460y+0.1236+0.15271+e−0.7777y+0.88888−1.03341+e−0.6031y−0.1048,(21)N^y=−0.38841+e−0.6558y−1.2335+3.51901+e−−0.6532y+1.0132−1.68261+e−1.4236y−1.3726+1.79111+e−−1.3963y−1.700+4.05161+e−−0.1159y+2.0362+5.36511+e−0.5579y+2.6055+0.87741+e−1.6682y−1.5831−0.17551+e−0.3419y−0.5896+4.72501+e−0.4341y+0.3914+0.75091+e−1.1986y−2.0756,(22)D^y=0.03461+e−0.4044y+0.1385+0.03691+e−−0.6492y−1.3941+0.02521+e−0.6736y−3.6858−0.19541+e−−0.0098y−0.552−0.46201+e−−0.3519y−1.7583−0.00311+e−2.5703y+1.1318−0.20871+e−0.4861y+0.7123+0.44751+e−0.1428y+1.0332−0.35251+e−0.1370y−1.0306+0.04851+e−0.0037y−0.5402,(23)C^y=0.40091+e−−0.9719y−1.4703−0.01801+e−1.8583y+1.2249−0.04801+e−−1.3170y−1.1695−0.23471+e−0.5107y−0.4262−0.28111+e−0.3620y−0.3718−0.20711+e−−0.1348y+0.1319+1.232801+e−0.2702y+0.8495−1.25491+e−−0.0358y−0.3046−0.55491+e−−0.4663y−1.5261+0.10581+e−−0.7565y−0.8043.Figure 2
Weights for solving the mathematical SEIR-NDC system. (a) Weights:S^y. (b) Weights: E^y. (c) Weights: I^y. (d) Weights: R^y. (e) Weights: N^y. (f) Weights: D^y. (g) Weights: C^y.
(a)(b)(c)(d)(e)(f)(g)The above equations are given to evaluate the outputs of the SEIR-NDC model by operating the ANNs and PSOSQP. The achieved outcomes are illustrated in Figures2–4 using 10 neurons and 30 variables. Figures 2(a)–2(g) represent the optimal weight vectors for the model. These optimal weights have been obtained by using equations (17)–(23). The solutions have been obtained by using these weight vectors. Figures 3(a)–3(g) show the proposed, mean, and best result comparison for the mathematical SEIR-NDC system. The proposed results have been obtained by using the designed ANN procedure along with the computing swarming scheme, and the best solutions indicate the ideal solutions, while the mean solutions have been achieved by using the mean performances. The matching of the mean, obtained, and best results indicates the exactness of the designed ANN procedure along with the computing swarming scheme. Figure 4 presents the AE performances based on the mean and the best solutions for the mathematical SEIR-NDC system. It is shown that the best outcomes of the AE to solve the respective categories of the model are 10−05–10−06, 10−05–10−07, 10−03–10−05, 10−04–10−07, 10−07–10−08, 10−05–10−08, and 10−06–10−09 for each category of the model. The performances based on the mean for the respective categories of the model are presented as 10−04–10−05, 10−03–10−05, 10−03–10−04, 10−04–10−05, 10−05–10−06, 10−04–10−06, and 10−05–10−06 for each dynamics of the model. These observed AE values show the accurateness of ANN-PSOSQP procedure for SEIR-NDC. It is shown that the proposed stochastic ANN-PSOSQP procedure is precisely based on the AE for the SEIR-NDC.Figure 3
Comparison of the mean, exact, and best results for the mathematical SEIR-NDC system. (a) Result comparison ofS^y. (b) Result comparison of E^y. (c) Result comparison of I^y. (d) Result comparison of R^y. (e) Result comparison of N^y. (f) Result comparison of D^y. (g) Result comparison of C^y.
(a)(b)(c)(d)(e)(f)(g)Figure 4
AE using the mean and best results for the nonlinear system. (a) AE performances forS^y. (b) AE performances for E^y. (c) AE performances for I^y. (d) AE performances for R^y. (e) AE performances for N^y. (f) AE performances for D^y. (g) AE performances for C^y.
(a)(b)(c)(d)(e)(f)(g)Figure5 presents the statistical performances using the MSE and TIC operators to solve the mathematical model. The MSE measures for the SEIR-NDC model are found as 10−08–10−10, 10−09–10−10, 10−07–10−08, 10−08–10−09, 10−11–10−12, 10−09–10−10, and 10−10–10−12 for each dynamics of the model. The TIC performance for the SEIR-NDC model are found around 10−06–10−08, 10−07–10−08, 10−03–10−04, 10−04–10−05, 10−12–10−13, 10−04–10−05, and 10−08–10−10 for respective dynamics of the system. These optimal small performances enhance the value and correctness of proposed stochastic ANN-PSOSQP procedure.Figure 5
Statistical performances for the mathematical SEIR-NDC system.Figure6 shows the convergence study based on the MSE and TIC operators. It was noted that most of the runs resulted in a high level of fitness. Based on these analyses, it is shown that most of the executions produced higher fitness levels that proves the procedure is reliable and stable.Figure 6
MSE and TIC convergence plots for the mathematical SEIR-NDC system.Tables2–8 provide the correctness, precision, and accurateness of the designed ANN-PSOSQP approach based on the statistical representations of the classes S^y, I^yR^y, N^y, D^y, and C^y of the SEIR-NDC mathematical system. The statistical illustrations are presented for the maximum (Max), semi-interquartile range (SIR), minimum (Min), standard deviation (SD), mean, and median (MD) operators. The Max and Min operators specify the worst and best runs. The Max values based on the worst runs lie as 10−04 to 10−06 for the classes. The statistical Min operator values performed were found as 10−06–10−13, 10−07–10−10, 10−05–10−12, 10−05–10−10, 10−07–10−11, 10−06–10−11, and 10−07–10−11 for each dynamics of the system. The SIR, MD, mean, and SD statistical values lie around 10−06–10−09 of each dynamics of the model. These statistical performances present the consistency and trustworthiness of the designed ANN-PSOSQP.Table 2
Statistical performances for the classSy of the model.
ySyMaxSIRMinSDMeanMD04.14847E-072.45733E-074.77285E-139.72224E-075.65438E-071.61416E-070.13.43018E-051.35393E-064.75690E-087.41284E-063.45398E-061.47969E-060.24.35970E-051.56468E-066.72517E-078.38919E-065.53205E-063.15588E-060.35.48928E-051.75403E-066.36321E-071.06886E-058.22091E-065.54958E-060.46.04801E-052.10799E-061.52531E-061.27852E-051.12576E-056.80979E-060.56.10558E-052.00471E-064.91716E-061.32654E-051.37476E-058.76556E-060.65.71444E-051.72297E-065.65694E-061.25290E-051.54038E-051.17188E-050.75.73368E-052.59359E-067.15833E-061.11478E-051.62769E-051.29955E-050.85.28638E-052.15910E-066.92380E-069.13535E-061.60719E-051.30170E-050.94.29129E-051.66618E-067.26516E-067.99805E-061.51653E-051.26148E-0514.59409E-058.96879E-073.19198E-067.32537E-061.30260E-051.16572E-05Table 3
Statistical performances for the classEy to solve the mathematical SEIR-NDC system.
yEyMaxSIRMinSDMeanMD04.75086E-062.96939E-079.83620E-109.46658E-074.96867E-079.25922E-080.11.28771E-055.14479E-072.09799E-082.87660E-061.62623E-066.62741E-070.23.46180E-051.59348E-061.58369E-078.31965E-065.00211E-061.88446E-060.35.05547E-051.76810E-065.24284E-071.10046E-057.01565E-063.36419E-060.45.27680E-051.93202E-064.44707E-071.03637E-057.89142E-065.38967E-060.53.68427E-052.42961E-065.06646E-077.47919E-068.75240E-067.61206E-060.63.24343E-051.28777E-062.93991E-074.97252E-069.60879E-069.24771E-060.73.30130E-051.08012E-063.30355E-064.76950E-061.10794E-051.04494E-050.83.35403E-051.07412E-062.26763E-066.10328E-061.23142E-051.12846E-050.93.35538E-051.54209E-065.45111E-076.27136E-061.19936E-051.06522E-0512.19173E-059.50185E-072.27305E-083.62661E-069.06845E-069.08091E-06Table 4
Statistical performances for the classIy to solve the mathematical SEIR-NDC system.
yIyMaxSIRMinSDMeanMD05.41754E-062.75761E-073.72458E-121.17847E-065.61374E-071.36275E-070.12.75677E-041.28020E-062.53537E-044.11470E-062.59946E-042.59379E-040.24.55826E-041.39173E-064.31680E-044.66773E-064.40432E-044.40370E-040.35.58989E-042.08120E-065.34904E-045.08382E-065.47084E-045.47243E-040.45.94001E-041.23786E-065.67049E-046.05079E-065.84125E-045.84972E-040.55.68161E-042.76336E-065.36930E-047.14524E-065.55266E-045.56218E-040.64.84700E-042.47091E-064.44792E-048.47217E-064.63991E-044.64920E-040.73.39830E-041.73645E-062.94297E-049.89722E-063.13563E-043.13775E-040.81.34826E-041.87554E-068.57211E-051.07004E-051.06965E-041.06549E-040.91.74484E-041.13857E-061.28803E-049.96170E-061.53142E-041.53830E-0414.90050E-041.01315E-064.51017E-047.32199E-064.64416E-044.64311E-04Table 5
Statistical performances for the classRy to solve the mathematical SEIR-NDC system.
yRyMaxSIRMinSDMeanMD03.41437E-052.37597E-072.34978E-109.85762E-075.45699E-071.29018E-070.11.18197E-041.13076E-064.57222E-072.72914E-063.90519E-063.00401E-060.21.94783E-041.31799E-061.68986E-064.15290E-069.76711E-069.79606E-060.33.11757E-042.95319E-061.55537E-075.99365E-061.90733E-051.95954E-050.44.12688E-043.13432E-061.72611E-055.37846E-063.11529E-053.13887E-050.56.15166E-043.31077E-063.30247E-056.07381E-064.31572E-054.27264E-050.68.09984E-051.96736E-064.47792E-057.79507E-065.33325E-055.19418E-050.79.38333E-051.84372E-064.38602E-059.07666E-066.03332E-055.89380E-050.89.77846E-057.95787E-074.05124E-059.75804E-066.31963E-056.27128E-050.99.12796E-052.02403E-063.79806E-059.65600E-066.12207E-056.10542E-0517.33207E-059.80744E-073.80090E-056.72537E-065.38149E-055.40220E-05Table 6
Statistical performances for the classNy to solve the mathematical SEIR-NDC system.
yNyMaxSIRMinSDMeanMD09.65096E-053.16692E-072.32632E-111.93612E-069.21061E-072.73802E-070.12.18383E-051.02040E-062.76323E-084.15033E-062.61757E-061.70645E-060.22.76927E-053.08333E-067.57758E-086.90415E-066.25072E-063.82658E-060.33.96641E-053.99790E-067.15828E-099.78686E-068.47392E-064.29770E-060.44.95862E-055.51404E-061.25805E-071.20188E-058.60042E-063.68434E-060.55.11064E-056.05062E-065.79059E-091.24962E-058.84438E-062.22204E-060.65.30185E-054.70764E-067.41051E-091.31610E-058.51820E-064.00156E-060.77.28155E-052.57856E-061.63822E-071.46255E-057.24511E-062.27404E-060.88.19951E-051.53702E-064.43392E-081.54233E-056.19774E-062.04275E-060.97.54090E-052.01301E-061.02477E-071.40382E-056.21569E-062.59263E-0614.79690E-051.98560E-064.90144E-081.02712E-055.75728E-062.06130E-06Table 7
Statistical performances for the classDy to solve the mathematical SEIR-NDC system.
yDyMaxSIRMinSDMeanMD02.55036E-052.42996E-071.20788E-114.62219E-061.15625E-061.05220E-070.11.84948E-059.71835E-074.59191E-084.10983E-062.36323E-069.52892E-070.21.77093E-051.26636E-061.68545E-074.13578E-063.82845E-062.47213E-060.31.88607E-051.70762E-064.08993E-074.28106E-065.61650E-064.59504E-060.42.64931E-051.55731E-061.28199E-065.09365E-067.41478E-066.54103E-060.53.03275E-052.01054E-065.69120E-075.35037E-068.93916E-068.43291E-060.62.81061E-051.74536E-061.29764E-064.75049E-061.06058E-051.02894E-050.73.14862E-051.60627E-063.06673E-064.88070E-061.22384E-051.17988E-050.84.09111E-057.70645E-079.81581E-076.41026E-061.35938E-051.26136E-050.94.21191E-051.73597E-065.17936E-067.19353E-061.42641E-051.25719E-0513.50801E-051.30583E-062.03979E-066.07739E-061.26104E-051.12803E-05Table 8
Statistical performances for the classCy to solve the mathematical SEIR-NDC system.
yCyMaxSIRMinSDMeanMD07.76923E-061.99620E-073.87874E-111.76960E-067.18293E-071.16157E-070.11.43901E-056.81331E-073.83744E-083.66051E-062.17221E-068.58069E-070.21.75513E-051.10384E-066.95938E-084.42693E-062.73246E-061.12893E-060.31.54017E-051.17901E-062.12959E-073.28945E-062.87969E-061.63906E-060.41.28635E-052.11604E-068.01746E-083.48020E-063.21899E-061.66457E-060.52.70540E-052.87815E-061.90426E-085.80604E-064.39566E-061.79130E-060.64.36450E-052.43142E-066.15750E-099.11304E-065.35181E-062.75264E-060.75.57509E-051.44891E-063.58756E-071.23557E-055.72802E-062.31043E-060.86.04390E-051.19585E-064.22483E-081.41987E-055.91669E-061.74317E-060.95.51061E-051.26357E-066.25928E-081.29102E-056.43255E-062.76458E-0613.74735E-051.27784E-061.29893E-077.96444E-064.65701E-062.25638E-06


## 3.1. Mathematical SEIR-NDC System

The updated form of the mathematical SEIR-NDC system is provided by adjusting the appropriate parameter performances, written as(15)dSydy=−0.5SyEyN−0.6ISyN−0.0205Sy,S0=0.9,dEydy=0.5ESyN+0.6ISyN−0.3505Ey,E0=0.1,dIydy=13Ey−0.225Iy,I0=0,dRydy=−0.025Ry+15Iy,R0=0,dNydy=−0.0205Ny,N0=14,dDydy=−111.2Dy+0.04Iy,D0=0,dCydy=13Ey,C0=0.The cost function is designed in terms of mean square error, which is based on the differential SEIR-NDC system given in equation (1) by keeping the methodology of equation (5) shown as(16)EC=1N∑n=1NdS^dyn+0.6S^nI^nN+0.5S^nE^nN+0.0205S^k2+dE^dyn−0.6S^nI^nN+0.3505E^n−0.5E^nS^nN2+dI^dyn−0.333E^n+0.225I^n2+dR^dyn−0.2I^n+0.025R^n2dN^dyn+0.0205N^n2+dD^dyn+111.2D^n−0.04I^n2+dC^dyn−0.33E^n2+17S^0−0.92+E^0−0.12+I^02+R^02+N^0−142+D^02+C^02.The solutions of the mathematical model are obtained through the optimization-based PSOSQP for 30 runs together with 10 neurons and 30 variables. The best values of the weights using the ANNs along with the optimization procedures of PSOSQP are illustrated in Figure2. These weight vectors are capable to achieve the proposed outcomes based on the mathematical SEIR-NDC system. The obtained solutions are presented as(17)S^y=−0.26421+e−−0.3453y−2.7523−1.11081+e−−0.0416y−3.3393+1.15751+e−−0.0138y−1.4023−2.70901+e−0.0464y−4.5616−1.67021+e−−0.3570y−4.4423−2.66031+e−−0.0283y−4.2419+1.75471+e−−0.029y−0.3310+0.00911+e−−2.652y−3.6896−1.55761+e−−0.3541y−3.7727+0.74271+e−0.3299y−1.7304,(18)E^y=−0.65391+e−0.4382y−2.7725+0.72971+e−0.2924y−0.8677+0.26591+e−0.0822y−2.7119−0.18121+e−0.1846y−2.6274−0.60851+e−−0.4121y−3.4072−0.14001+e−−1.906y−4.1492−0.37121+e−0.3892y−2.6735+0.69671+e−−0.186y−1.9214+0.15591+e−−2.6372y−4.3152−0.20111+e−0.9194y+0.5422,(19)I^y=−0.05741+e−1.2464y−0.8872+0.21871+e−−1.3408y+0.2756+0.06171+e−0.4227y−1.7866−0.29801+e−−1.2379y+0.2365−0.01331+e−−1.605y+1.2206−0.46631+e−−0.4652y+0.2808+0.73941+e−−0.400y−0.5017+0.64931+e−0.4898y+1.4072−0.72981+e−−0.0279y+0.0603−0.24521+e−0.3425y−0.3986,(20)R^y=0.15591+e−−0.0877y+0.1975+1.28321+e−0.0576y−0.8472+1.03991+e−−0.065y−0.8056−1.28871+e−−0.574y+0.1095−1.28871+e−−0.5745y+0.1095+2.06751+e−0.0278y−0.8404+0.63451+e−0.2984y−1.3477−0.89351+e−0.4460y+0.1236+0.15271+e−0.7777y+0.88888−1.03341+e−0.6031y−0.1048,(21)N^y=−0.38841+e−0.6558y−1.2335+3.51901+e−−0.6532y+1.0132−1.68261+e−1.4236y−1.3726+1.79111+e−−1.3963y−1.700+4.05161+e−−0.1159y+2.0362+5.36511+e−0.5579y+2.6055+0.87741+e−1.6682y−1.5831−0.17551+e−0.3419y−0.5896+4.72501+e−0.4341y+0.3914+0.75091+e−1.1986y−2.0756,(22)D^y=0.03461+e−0.4044y+0.1385+0.03691+e−−0.6492y−1.3941+0.02521+e−0.6736y−3.6858−0.19541+e−−0.0098y−0.552−0.46201+e−−0.3519y−1.7583−0.00311+e−2.5703y+1.1318−0.20871+e−0.4861y+0.7123+0.44751+e−0.1428y+1.0332−0.35251+e−0.1370y−1.0306+0.04851+e−0.0037y−0.5402,(23)C^y=0.40091+e−−0.9719y−1.4703−0.01801+e−1.8583y+1.2249−0.04801+e−−1.3170y−1.1695−0.23471+e−0.5107y−0.4262−0.28111+e−0.3620y−0.3718−0.20711+e−−0.1348y+0.1319+1.232801+e−0.2702y+0.8495−1.25491+e−−0.0358y−0.3046−0.55491+e−−0.4663y−1.5261+0.10581+e−−0.7565y−0.8043.Figure 2
Weights for solving the mathematical SEIR-NDC system. (a) Weights:S^y. (b) Weights: E^y. (c) Weights: I^y. (d) Weights: R^y. (e) Weights: N^y. (f) Weights: D^y. (g) Weights: C^y.
(a)(b)(c)(d)(e)(f)(g)The above equations are given to evaluate the outputs of the SEIR-NDC model by operating the ANNs and PSOSQP. The achieved outcomes are illustrated in Figures2–4 using 10 neurons and 30 variables. Figures 2(a)–2(g) represent the optimal weight vectors for the model. These optimal weights have been obtained by using equations (17)–(23). The solutions have been obtained by using these weight vectors. Figures 3(a)–3(g) show the proposed, mean, and best result comparison for the mathematical SEIR-NDC system. The proposed results have been obtained by using the designed ANN procedure along with the computing swarming scheme, and the best solutions indicate the ideal solutions, while the mean solutions have been achieved by using the mean performances. The matching of the mean, obtained, and best results indicates the exactness of the designed ANN procedure along with the computing swarming scheme. Figure 4 presents the AE performances based on the mean and the best solutions for the mathematical SEIR-NDC system. It is shown that the best outcomes of the AE to solve the respective categories of the model are 10−05–10−06, 10−05–10−07, 10−03–10−05, 10−04–10−07, 10−07–10−08, 10−05–10−08, and 10−06–10−09 for each category of the model. The performances based on the mean for the respective categories of the model are presented as 10−04–10−05, 10−03–10−05, 10−03–10−04, 10−04–10−05, 10−05–10−06, 10−04–10−06, and 10−05–10−06 for each dynamics of the model. These observed AE values show the accurateness of ANN-PSOSQP procedure for SEIR-NDC. It is shown that the proposed stochastic ANN-PSOSQP procedure is precisely based on the AE for the SEIR-NDC.Figure 3
Comparison of the mean, exact, and best results for the mathematical SEIR-NDC system. (a) Result comparison ofS^y. (b) Result comparison of E^y. (c) Result comparison of I^y. (d) Result comparison of R^y. (e) Result comparison of N^y. (f) Result comparison of D^y. (g) Result comparison of C^y.
(a)(b)(c)(d)(e)(f)(g)Figure 4
AE using the mean and best results for the nonlinear system. (a) AE performances forS^y. (b) AE performances for E^y. (c) AE performances for I^y. (d) AE performances for R^y. (e) AE performances for N^y. (f) AE performances for D^y. (g) AE performances for C^y.
(a)(b)(c)(d)(e)(f)(g)Figure5 presents the statistical performances using the MSE and TIC operators to solve the mathematical model. The MSE measures for the SEIR-NDC model are found as 10−08–10−10, 10−09–10−10, 10−07–10−08, 10−08–10−09, 10−11–10−12, 10−09–10−10, and 10−10–10−12 for each dynamics of the model. The TIC performance for the SEIR-NDC model are found around 10−06–10−08, 10−07–10−08, 10−03–10−04, 10−04–10−05, 10−12–10−13, 10−04–10−05, and 10−08–10−10 for respective dynamics of the system. These optimal small performances enhance the value and correctness of proposed stochastic ANN-PSOSQP procedure.Figure 5
Statistical performances for the mathematical SEIR-NDC system.Figure6 shows the convergence study based on the MSE and TIC operators. It was noted that most of the runs resulted in a high level of fitness. Based on these analyses, it is shown that most of the executions produced higher fitness levels that proves the procedure is reliable and stable.Figure 6
MSE and TIC convergence plots for the mathematical SEIR-NDC system.Tables2–8 provide the correctness, precision, and accurateness of the designed ANN-PSOSQP approach based on the statistical representations of the classes S^y, I^yR^y, N^y, D^y, and C^y of the SEIR-NDC mathematical system. The statistical illustrations are presented for the maximum (Max), semi-interquartile range (SIR), minimum (Min), standard deviation (SD), mean, and median (MD) operators. The Max and Min operators specify the worst and best runs. The Max values based on the worst runs lie as 10−04 to 10−06 for the classes. The statistical Min operator values performed were found as 10−06–10−13, 10−07–10−10, 10−05–10−12, 10−05–10−10, 10−07–10−11, 10−06–10−11, and 10−07–10−11 for each dynamics of the system. The SIR, MD, mean, and SD statistical values lie around 10−06–10−09 of each dynamics of the model. These statistical performances present the consistency and trustworthiness of the designed ANN-PSOSQP.Table 2
Statistical performances for the classSy of the model.
ySyMaxSIRMinSDMeanMD04.14847E-072.45733E-074.77285E-139.72224E-075.65438E-071.61416E-070.13.43018E-051.35393E-064.75690E-087.41284E-063.45398E-061.47969E-060.24.35970E-051.56468E-066.72517E-078.38919E-065.53205E-063.15588E-060.35.48928E-051.75403E-066.36321E-071.06886E-058.22091E-065.54958E-060.46.04801E-052.10799E-061.52531E-061.27852E-051.12576E-056.80979E-060.56.10558E-052.00471E-064.91716E-061.32654E-051.37476E-058.76556E-060.65.71444E-051.72297E-065.65694E-061.25290E-051.54038E-051.17188E-050.75.73368E-052.59359E-067.15833E-061.11478E-051.62769E-051.29955E-050.85.28638E-052.15910E-066.92380E-069.13535E-061.60719E-051.30170E-050.94.29129E-051.66618E-067.26516E-067.99805E-061.51653E-051.26148E-0514.59409E-058.96879E-073.19198E-067.32537E-061.30260E-051.16572E-05Table 3
Statistical performances for the classEy to solve the mathematical SEIR-NDC system.
yEyMaxSIRMinSDMeanMD04.75086E-062.96939E-079.83620E-109.46658E-074.96867E-079.25922E-080.11.28771E-055.14479E-072.09799E-082.87660E-061.62623E-066.62741E-070.23.46180E-051.59348E-061.58369E-078.31965E-065.00211E-061.88446E-060.35.05547E-051.76810E-065.24284E-071.10046E-057.01565E-063.36419E-060.45.27680E-051.93202E-064.44707E-071.03637E-057.89142E-065.38967E-060.53.68427E-052.42961E-065.06646E-077.47919E-068.75240E-067.61206E-060.63.24343E-051.28777E-062.93991E-074.97252E-069.60879E-069.24771E-060.73.30130E-051.08012E-063.30355E-064.76950E-061.10794E-051.04494E-050.83.35403E-051.07412E-062.26763E-066.10328E-061.23142E-051.12846E-050.93.35538E-051.54209E-065.45111E-076.27136E-061.19936E-051.06522E-0512.19173E-059.50185E-072.27305E-083.62661E-069.06845E-069.08091E-06Table 4
Statistical performances for the classIy to solve the mathematical SEIR-NDC system.
yIyMaxSIRMinSDMeanMD05.41754E-062.75761E-073.72458E-121.17847E-065.61374E-071.36275E-070.12.75677E-041.28020E-062.53537E-044.11470E-062.59946E-042.59379E-040.24.55826E-041.39173E-064.31680E-044.66773E-064.40432E-044.40370E-040.35.58989E-042.08120E-065.34904E-045.08382E-065.47084E-045.47243E-040.45.94001E-041.23786E-065.67049E-046.05079E-065.84125E-045.84972E-040.55.68161E-042.76336E-065.36930E-047.14524E-065.55266E-045.56218E-040.64.84700E-042.47091E-064.44792E-048.47217E-064.63991E-044.64920E-040.73.39830E-041.73645E-062.94297E-049.89722E-063.13563E-043.13775E-040.81.34826E-041.87554E-068.57211E-051.07004E-051.06965E-041.06549E-040.91.74484E-041.13857E-061.28803E-049.96170E-061.53142E-041.53830E-0414.90050E-041.01315E-064.51017E-047.32199E-064.64416E-044.64311E-04Table 5
Statistical performances for the classRy to solve the mathematical SEIR-NDC system.
yRyMaxSIRMinSDMeanMD03.41437E-052.37597E-072.34978E-109.85762E-075.45699E-071.29018E-070.11.18197E-041.13076E-064.57222E-072.72914E-063.90519E-063.00401E-060.21.94783E-041.31799E-061.68986E-064.15290E-069.76711E-069.79606E-060.33.11757E-042.95319E-061.55537E-075.99365E-061.90733E-051.95954E-050.44.12688E-043.13432E-061.72611E-055.37846E-063.11529E-053.13887E-050.56.15166E-043.31077E-063.30247E-056.07381E-064.31572E-054.27264E-050.68.09984E-051.96736E-064.47792E-057.79507E-065.33325E-055.19418E-050.79.38333E-051.84372E-064.38602E-059.07666E-066.03332E-055.89380E-050.89.77846E-057.95787E-074.05124E-059.75804E-066.31963E-056.27128E-050.99.12796E-052.02403E-063.79806E-059.65600E-066.12207E-056.10542E-0517.33207E-059.80744E-073.80090E-056.72537E-065.38149E-055.40220E-05Table 6
Statistical performances for the classNy to solve the mathematical SEIR-NDC system.
yNyMaxSIRMinSDMeanMD09.65096E-053.16692E-072.32632E-111.93612E-069.21061E-072.73802E-070.12.18383E-051.02040E-062.76323E-084.15033E-062.61757E-061.70645E-060.22.76927E-053.08333E-067.57758E-086.90415E-066.25072E-063.82658E-060.33.96641E-053.99790E-067.15828E-099.78686E-068.47392E-064.29770E-060.44.95862E-055.51404E-061.25805E-071.20188E-058.60042E-063.68434E-060.55.11064E-056.05062E-065.79059E-091.24962E-058.84438E-062.22204E-060.65.30185E-054.70764E-067.41051E-091.31610E-058.51820E-064.00156E-060.77.28155E-052.57856E-061.63822E-071.46255E-057.24511E-062.27404E-060.88.19951E-051.53702E-064.43392E-081.54233E-056.19774E-062.04275E-060.97.54090E-052.01301E-061.02477E-071.40382E-056.21569E-062.59263E-0614.79690E-051.98560E-064.90144E-081.02712E-055.75728E-062.06130E-06Table 7
Statistical performances for the classDy to solve the mathematical SEIR-NDC system.
yDyMaxSIRMinSDMeanMD02.55036E-052.42996E-071.20788E-114.62219E-061.15625E-061.05220E-070.11.84948E-059.71835E-074.59191E-084.10983E-062.36323E-069.52892E-070.21.77093E-051.26636E-061.68545E-074.13578E-063.82845E-062.47213E-060.31.88607E-051.70762E-064.08993E-074.28106E-065.61650E-064.59504E-060.42.64931E-051.55731E-061.28199E-065.09365E-067.41478E-066.54103E-060.53.03275E-052.01054E-065.69120E-075.35037E-068.93916E-068.43291E-060.62.81061E-051.74536E-061.29764E-064.75049E-061.06058E-051.02894E-050.73.14862E-051.60627E-063.06673E-064.88070E-061.22384E-051.17988E-050.84.09111E-057.70645E-079.81581E-076.41026E-061.35938E-051.26136E-050.94.21191E-051.73597E-065.17936E-067.19353E-061.42641E-051.25719E-0513.50801E-051.30583E-062.03979E-066.07739E-061.26104E-051.12803E-05Table 8
Statistical performances for the classCy to solve the mathematical SEIR-NDC system.
yCyMaxSIRMinSDMeanMD07.76923E-061.99620E-073.87874E-111.76960E-067.18293E-071.16157E-070.11.43901E-056.81331E-073.83744E-083.66051E-062.17221E-068.58069E-070.21.75513E-051.10384E-066.95938E-084.42693E-062.73246E-061.12893E-060.31.54017E-051.17901E-062.12959E-073.28945E-062.87969E-061.63906E-060.41.28635E-052.11604E-068.01746E-083.48020E-063.21899E-061.66457E-060.52.70540E-052.87815E-061.90426E-085.80604E-064.39566E-061.79130E-060.64.36450E-052.43142E-066.15750E-099.11304E-065.35181E-062.75264E-060.75.57509E-051.44891E-063.58756E-071.23557E-055.72802E-062.31043E-060.86.04390E-051.19585E-064.22483E-081.41987E-055.91669E-061.74317E-060.95.51061E-051.26357E-066.25928E-081.29102E-056.43255E-062.76458E-0613.74735E-051.27784E-061.29893E-077.96444E-064.65701E-062.25638E-06


## 4. Concluding Remarks

The current work obtains the numerical solutions of the coronavirus mathematical system using the computational structure of artificial neural networks, swarming optimization procedures, and the sequential quadratic programming. The particle swarm optimization works as a global swarming procedure, whereas the sequential quadratic programming is implemented as a local search algorithm approach. The optimization of the cost function which is constructed using the mathematical SEIR-NDC system is performed through the stochastic swarming procedures. The correctness of the stochastic ANN-PSOSQP computing scheme has been verified through the comparison of the obtained performances of the mathematical SEIR-NDC system and the Runge–Kutta scheme. The statistical MSE and TIC operators have been provided in good ranges for 10 neurons or 30 variables for the mathematical SEIR-NDC system. The negligible statistical Max, Min, SIR, SD, mean, and MD operator values show the accuracy. Moreover, the graphical illustrations of the performance indices, absolute error, and convergence curves have been derived to validate the robustness of the proposed ANN-PSOSQP approach for the mathematical SEIR-NDC system. These statistical operator performances authenticate that the proposed ANN-PSOSQP approach is stable, reliable, accurate, and robust for the numerical solutions of the mathematical SEIR-NDC system.The proposed stochastic paradigms can be explored in the future to present the numerical solutions of the nonlinear mathematical systems [52–61].



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*Source: 5755885-2022-10-29.xml*