Artificial intelligence is demonstrated by machines, unlike the natural intelligence displayed by animals, including humans. Artificial intelligence research has been defined as the field of study of intelligent agents, which refers to any system that perceives its environment and takes actions that maximize its chance of achieving its goals. The techniques of intelligent computing solve many applications of mathematical modeling. The research work was designed via a particular method of artificial neural networks to solve the mathematical model of coronavirus. The representation of the mathematical model is made via systems of nonlinear ordinary differential equations. These differential equations are established by collecting the susceptible, the exposed, the symptomatic, super spreaders, infection with asymptomatic, hospitalized, recovery, and fatality classes. The generation of the coronavirus model’s dataset is exploited by the strength of the explicit Runge Kutta method for different countries like India, Pakistan, Italy, and many more. The generated dataset is approximately used for training, validation, and testing processes for each cyclic update in Bayesian Regularization Backpropagation for the numerical treatment of the dynamics of the desired model. The performance and effectiveness of the designed methodology are checked through mean squared error, error histograms, numerical solutions, absolute error, and regression analysis.

The Chinese government declared the agent virus on 1^{st} January 2020 [^{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 [

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.

The analysis of the results obtained computationally is performed effectively by the mean square error (MSE) index to structure a merit function for the design of Bayesian-regularization backpropagation by considering the reference solutions of eight classes established on the model for the COVID-19 pandemic by using artificial neural network techniques.

Bayesian-regularization backpropagation is utilized to execute validation, testing processes, and training to get the decision variables tuned for artificial neural network techniques for incrementing epoch index each. To satisfy the target, we have considered a model of eight complex classes that is established on the Susceptible (S_{cov}), the Exposed (G_{cov}), the Symptomatic and Infectious (M_{cov}), the Super Propagation (P_{cov}), the Infection but the Asymptomatic (L_{cov}), the Hospitalized (H_{cov}), the Recovery (R_{cov}) and the Fatality (F_{cov}) classes.

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

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

Nomenclature | |||
---|---|---|---|

Susceptible group | Rate of transmission coefficient (infection) | ||

Exposed group | _{1} |
Rate of transmission coefficient (super spreaders) | |

Infectious group of COVID-19 Epidemic | Rate of exposed infectious | ||

Super Propagation Group | _{1} |
Exposed infected rate | |

Infectious but asymptomatic group | _{2} |
Rate of exposure to superspreaders | |

Hospitalized group | _{a} |
hospitalized rate | |

Recovery Group | _{i} |
Rate of recovery without hospitalized | |

Fatality group | _{r} |
Rate of recovery of hospitalized infectors | |

_{h} |
Hospitalized Death rate | _{i} |
Rate of death because infected people |

Transmissibility of Hospitalized Relative | _{p} |
Rate of death because of superspreaders |

The system of differential equations is as follows:

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.

Let us suppose the dynamical system for differential equations describing the COVID-19 model for Pakistan.

Let us suppose the dynamical system for the differential equations describing the COVID-19 model for Italy

The expression for predictor step-2 formula for the case for 1^{st} equation of dataset is expressed by

When the step-2 correlation formula for the case for the 1^{st} equation of the dataset is expressed in following

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

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:

Here ^{th} input, respectively, when the ^{2} 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

Parameter | Value | Units | Parameter | value | Units |
---|---|---|---|---|---|

^{’} |
7.65 | _{a} |
0.94 | ||

2.55 | _{r} |
0.5 | |||

0.25 | _{i} |
3.5 | |||

_{1} |
0.580 | dimensionless | _{h} |
0.3 | |

_{2} |
0.001 | dimensionless | _{p} |
1 | |

_{i} |
0.27 | 1.56 | Dimensionless |

The plot of the training performance is described in

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

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).

ODEs showing the fast spread of COVID-19 are tackled by employing a governing system, Bayesian-regularization backpropagation artificial neural networks (BRANNs).

A comparison of the projected outcomes with the reference of the numerical solution obtained displays the consistency and accuracy of the suggested Bayesian-regularization backpropagation artificial neural networks (BRANNs).

Moreover, the suggested method’s scenario is justified by the numerical and graphical expression based on the error histograms, convergence plots, i.e., regression dynamics, and square errors.

Differences in the parameters of interest significantly affect the dynamics of the model.

The demonstration of the process of computation is good in case of the complexity found in connection with time series, histogram, regression, mean square error, etc.

In the future, anyone will be able to solve their systems and differential equations with different sigmoidal, radial base, and violet initiation purposes to solve neutral, single, multi, fuzzy, and partial variations.

Thanks to our families and colleagues who supported us morally. The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Large Groups (Project under Grant Number (RGP.2/116/43)).