The Chinese government declared the agent virus on 1^st January 2020 [1]. The critical source of spreaders of this dangerous disease is due to Bats, as discussed in [2,3]. The effect of the COVID-19 epidemic on human health is that it changes daily life routines [4]. Due to the significant problems of COVID-19, the governments of different countries are forced to think about this disease and make strict public policies to face created challenges to save their populations. The crown infection of 2019 is also known as crown infection of Wuhan because its presence is found first in Wuhan city of China, the capital of Hubei territory. It is observed that this is a transmitted infection and is considered a respiratory disease rapidly transmitted from one human being to another. The epidemic ofCOVID-19 spread in the whole world in a few months, and millions of people were hunted by this disease. According to the news agencies and World Health Organization (WHO), data collected up to 27^th may 2021 in three countries, Pakistan, India, and Italy are given as, in Pakistan the total confirmed infected cases are 927000, recovered 85300, number of deaths is 21022, and fully vaccinated people are 1513144 which are 0.7% of the total population. In India, the total confirmed infected cases are 28.4 M, recovered 26.4 M, the number of deaths is 338 K, and fully vaccinated people are 42419560, 3.1% of the population. While in Italy, the total confirmed infected cases are 4.22 M, recovered 3.89 M, the number of deaths is 126 K, and fully vaccinated people are 11245499, 19.1% of the population. The past medical history of viruses tells us that the spreaders of different viruses are always due to some logical reasoning, and medical companies of the world can prepare medications in time for better treatment. The rate of spread of coronavirus is much high as compared to other viruses in human history. Furthermore, the source of transmission of this disease is humans to humans only [5]. First time in history, all industries and trade centers have been closed for a long time. The developed and developing countries’ economy is badly affected. In the first phase of the COVID-19 epidemic, the much-effected countries are China, the USA, Italy, Brazil, etc. But in the second and third phases, Asian countries like India, Iran, Pakistan, and many Arabian countries like Saudi Arabia, UAE, etc., are highly affected. The vaccination process has started worldwide and is expected to be reduced after some weeks. No doubt, this disease affects a large population of the world. However, many health organizations play a vital role in decreasing this disease and overcoming the difficulties of human beings. Meanwhile, the research community also takes an active part in helping world organizations by designing COVID-19 models to create awareness of the current situation about death rate, infected people, recovered people, and the ratio of spreading coronavirus in different areas of the world. These research techniques are necessary for solving future problems and planning coming issues. The mathematical models used by the research community for linear and nonlinear virus disease variations are studied in [6–10]. The well-known models on the strategy of the epidemic are studied in [11–32]. The model considered for the study is presented in [33]. Many countries like Germany, the USA, Spain, and Italy are affected due to this disease [34], and their governments try to overthrow the virus by imposing social distancing. Infectious diseases have significant effects the health as well as finance [35]. Learning the dynamic forces of disease transmission is very important with the help of mathematical tools [36], the prospect of guessing many real-world time sequences in various fields like economics [37], business, and the environment. The mathematical model is also helpful in predicting the spread of diseases and thus supports the governments being organized and making the necessary decisions [38]. The paper’s strategy is as follows: Section 2 goes to innovation work. Section 3 goes to mathematical modeling of coronavirus. Section 4 goes to numerical results via MATLAB and graphical illustration of each compartment; the Last section goes to a conclusion.

The innovative contributions of the study introduced here for Bayesian-regularization backpropagation artificial neural networks applied on COVID-19 models are emphasized below.

A revolutionary scheme on the base of the two-layers design of the Bayesian-regularization backpropagation artificial neural networks (BRANNs) is described to investigate the dynamics of the COVID-19 model that are expressed with a system of eight ordinary differential equations corresponding to the initial value problems.

In our investigation, we use the proposed model, which consists of eight classes the susceptible (S), the exposed (G), the symptomatic and infectious (M), the super spreaders (P), the infection but asymptomatic (L), the hospitalized (H), the recovery (R) and fatality (F) classes, for numerical investigations. The graphical description of the model is in Fig. 1.

The general mathematical description of the model that explains the relations of all classes with the included parameters is defined as a set of the system (1–2). The physical relevance of the model is presented in Table 1.

The system of differential equations is as follows: (1)dScovdt=−βMcovNScov−lβHcovNScov−β′PcovNScov (2)dGcovdt=−βMcovNScov+lβHcovNScov+β′PcovNScov−KGcov (3)dMcovdt=Kρ1Gcov−(γa+γi)Mcov−δiMcov (4)dPcovdt=Kρ2Gcov−(γa+γi)Pcov−δPPcov (5)dLcovdt=K(1−ρ1−ρ2)Gcov (6)dHcovdt=γa(Mcov+Pcov)−γiHcov (7)dRcovdt=γi(Mcov+Pcov)−γrHcov (8)dFcovdt=δiMcov+δPPcov+δhHcov (9)Scov=N−6,Gcov=0,Mcov=1,Pcov=5,Lcov=0,Hcov=0,Rcov=0,Fcov=0

This section describes the necessary information about the proposed modeling mathematically with the help of a performance matrix—modeling of applied mathematics dependent upon the three steps. In step one, the COVID-19 model is established for three-word countries geographically located in Europe and Asia; such type of investigation is known as the input reference dataset. On the other hand, the step neural network layer structure and the training we take into account models. Bayesian-regularization backpropagation artificial neural networks (BRANNs) are performed in step third with a Bayesian-regularization solver.

Let us suppose the dynamical system for differential equations describing the COVID-19 model for India. (10)dScovdt=−4.5751×10−6Mcov(t)Scov(t)−7.1372×10−6Hcov(t)Scov(t)−1.37253×10−6Pcov(t)Scov(t) (11)dGcovdt=4.5751.8375×10−6Mcov(t)Scov(t)+7.1372×10−6Hcov(t)Scov(t)+1.37253×10−6Pcov(t)Scov(t)−0.25Gcov(t) (12)dMcovdt=0.145Gcov(t)−1.21Mcov(t)−3.5Mcov(t) (13)dPcovdt=2.5×10−4Gcov(t)−2.21Pcov(t) (14)dLcovdt=0.1047Gcov(t) (15)dHcovdt=0.94(Mcov(t)+Pcov(t))−Hcov(t)−0.3Hcov(t) (16)dRcovdt=3.5(Mcov(t)+Pcov(t))+0.5Hcov(t) (17)dFcovdt=3.5Mcov(t)+Pcov(t)+0.3Hcov(t) (18)S(0)=5633,G(0)=0,M(0)=1,P(0)=5,L(0)=0,H(0)=0,R(0)=0,F(0)=0,

Let us suppose the dynamical system for differential equations describing the COVID-19 model for Pakistan. (19)dScovdt=−1.05579×10−5Mcov(t)Scov(t)−1.64703×10−5Hcov(t)Scov(t)−3.16736×10−5Pcov(t)Scov(t) (20)dGcovdt=1.05579×10−5Mcov(t)Scov(t)+1.64703×10−5Hcov(t)Scov(t)+3.16736×10−5Pcov(t)Scov(t)−0.25Gcov(t) (21)dMcovdt=0.145Gcov(t)−1.21Mcov(t)−3.5Mcov(t) (22)dPcovdt=2.5×10−4Gcov(t)−2.21Pcov(t) (23)dLcovdt=0.1047Gcov(t) (24)dHcovdt=0.94(Mcov(t)+Pcov(t))−Hcov(t)−0.3Hcov(t) (25)dRcovdt=3.5(Mcov(t)+Pcov(t))+0.5Hcov(t) (26)dFcovdt=3.5Mcov(t)+Pcov(t)+0.3Hcov(t) (27)S(0)=5633,G(0)=0,M(0)=1,P(0)=5,L(0)=0,H(0)=0,R(0)=0,F(0)=0

Let us suppose the dynamical system for the differential equations describing the COVID-19 model for Italy (28)dScovdt=−2.8375×10−6Mcov(t)Scov(t)−4.4265×10−6Hcov(t)Scov(t)−8.5125×10−6Pcov(t)Scov(t) (29)dGcovdt=2.8375×10−6Mcov(t)Scov(t)+4.4265×10−6Hcov(t)Scov(t)+8.5125×10−6Pcov(t)Scov(t)−0.25Gcov(t) (30)dMcovdt=0.145Gcov(t)−1.21Mcov(t)−3.5Mcov(t) (31)dPcovdt=2.5×10−4Gcov(t)−2.21Pcov(t) (32)dPcovdt=2.5×10−4Gcov(t)−2.21Pcov(t) (33)dHcovdt=0.94(Mcov(t)+Pcov(t))−Hcov(t)−0.3Hcov(t) (34)dRcovdt=3.5(Mcov(t)+Pcov(t))+0.5Hcov(t) (35)dFcovdt=3.5Mcov(t)+Pcov(t)+0.3Hcov(t) (36)S(0)=5633,G(0)=0,M(0)=1,P(0)=5,L(0)=0,H(0)=0,R(0)=0,F(0)=0

The expression for predictor step-2 formula for the case for 1^st equation of dataset is expressed by (37)Sl+1=Sl+1.5hf(tl,Sl)−0.5hf(tl−1,Sl−1).

When the step-2 correlation formula for the case for the 1^st equation of the dataset is expressed in following (38)Sl+1=Sl+0.5h(f(tl+1,Sl+1)+f(tl,Sl)).

Consequently, the formulas for the predictive and accurate technique of the remaining equations are formulated in a set. The two-layer structure for Bayesian-regularization backpropagation artificial neural networks (BRANNs) models with ten hidden neurons layer. The Bayesian-regularization backpropagation artificial neural networks (BRANNs) built-in architecture is presented in Fig. 2.

An error-based merit function is explained through the training of Bayesian-regularization backpropagation artificial neural networks (BRANNs) with the Levenberg-Marquardt method (LMM) and Bayesian Regulation (BR). The objective function is extracted from the metric of the objective function, and mean square error (MSE) is optimized with Bayesian-regularization backpropagation artificial neural networks (BRANNs) for all the cases inclusively corresponding to both Wuhan and Karachi cities. The mathematical performance matrix by merit data, absolute error (AE) that is, mean square error (MSE), and regression coefficient are listed underneath: (39){AE=|Scovj−S⌢covj|,j=1,2,3,4,……..k,MSE=1k∑j=1k(Scovj−S⌢covj)2R2=1−∑j=1k(Scovj−Scovj¯)2∑j=1k(S⌢covj−Scovj¯)2}

Here Scovj, S⌢covj, and Scovj¯ represent, for reference, the mean and approximate solution of i^th input, respectively, when the k represents the input values as a whole. The importance of R, i.e., square root R² is the desired parameter for the complete modeling, whereas the AE and the MSE become zero for the complete modeling.

The synthetic study of numbers with the required specifications is presented here for the first-order nonlinear system in which the model of COVID-19 is solved by the proposed Bayesian-regularization backpropagation artificial neural networks (BRANNs) method. The pandemic model is represented. The parametric values of the model corresponding to the three countries are discussed in Table 2.

The plot of the training performance is described in Fig. 3. Network-trained functions provide 19 training values, and validation and test performances are presented graphically. As we can see, the train curves, test, and confirmation are almost identical in Fig. 3. This means it’s about to be the most delusional time of the year. If the test curves increase faster than the verification curves, then overfitting is likely. The value of Mu and gradient corresponding to the distance determines the slow or fast convergence rate, as shown in Fig. 4. When the validation check touches its maximum value at epoch 1000 with decent gradient value = 0.00412, 0.00258, and 0.00552, respectively, Mu = 500 for each value training stops. The convergence depends on the value of Mu and gradient. The error is a difference between target and output values in neural networks, represented as a histogram in Fig. 5.

The vertical bars in the error histogram represent the number of boxes that are 20. The y-axis shows the number of samples taken from Mathematica’s dataset; how many models are in each box. We can observe that the error in the middle part of the error histogram is equal to 0.001411, in which the blue part shows dataset training at an altitude of 4000, but the upper green and red part shows confirmation and test from 4500. Up to 5000, The X-axis represents the zero-error line in Fig. 6. When the network completes training, testing, and validation, the regression analysis is performed in Bayesian-regularization backpropagation artificial neural networks (BRANNs). Estimating the relationship between target values and output values is statistically called regression. The plot regression function uses the values of two parameters (targets and outputs). The plot regression function graphically represents the relationship between targets and output values in Fig. 7. The logarithmic scale is used in the image for small changes because it enables us to understand tiny changes. The proposed Bayesian-regularization backpropagation artificial neural networks (BRANNs) solver is used in the neural network toolbox in the environment through the ‘nets tool’ (neural network fitting tool). In contrast, Bayesian Regularization (BR) is used for training the weights. The designed Bayesian-regularization backpropagation artificial neural networks (BRANNs) have been conducted for two countries based on the accurate data set of India, Pakistan, and Italy. Furthermore, this table explains numerically the time taken during the running of the proposed solver, testing, training, gradient, and performance values for each country and describes the accurate performance.

Integrated computing intelligent platform based on artificial intelligence introduced by artificial neural networks with backpropagation of Bayesian regulation to mathematics for COVID-19 manifesting coronavirus spread through multifaceted classes in different countries India, Pakistan, and Italy model to be solved based on the accurate data. The dataset for the model for COVID-19 has been developed using a solution of colored quotas for different classes. 70% of the reference data is employed for validation, testing, and training of the Bayesian-regularization backpropagation artificial neural networks (BRANNs).