A computational stochastic procedure for solving the epidemic breathing transmission system

This work provides numerical simulations of the nonlinear breathing transmission epidemic system using the proposed stochastic scale conjugate gradient neural networks (SCGGNNs) procedure. The mathematical model categorizes the breathing transmission epidemic model into four dynamics based on a nonlinear stiff ordinary differential system: susceptible, exposed, infected, and recovered. Three different cases of the model are taken and numerically presented by applying the stochastic SCGGNNs. An activation function ‘log-sigmoid’ uses twenty neurons in the hidden layers. The precision of SCGGNNs is obtained by comparing the proposed and database solutions. While the negligible absolute error is performed around 10–06 to 10–07, it enhances the accuracy of the scheme. The obtained results of the breathing transmission epidemic system have been provided using the training, verification, and testing procedures to reduce the mean square error. Moreover, the exactness and capability of the stochastic SCGGNNs are approved through error histograms, regression values, correlation tests, and state transitions.

The classification of this model is introduced into four dynamics: susceptible, exposed, infected, and recovered, which is mathematically given as 21 : where b shows the rate of recruitment into the susceptible population, µ N represents the natural death rate, β is the disease transmission probability, Ar shows the actual risk population, k indicates the recovered rate that returns to the susceptible population based on the immunity loss, υ signifies the seroconversion rate, α denotes the recovered rate and µ D expresses the death rate of disease induced.Figure 1 presents the graphical illustrations based on the nonlinear breathing transmission epidemic model presented in system (1).
The nonlinear breathing transmission epidemic model has never been solved using the proposed stochastic operators.Some more novel features of the current work are highlighted as: • The proposed stochastic procedure successfully solves the nonlinear breathing transmission epidemic system.
• Three different values of the disease transmission probability ( β ) based on the breathing transmission epi- demic system are employed to validate the scheme's performance.• The SCGGNNs' accuracy is performed by matching the reference and calculated solutions.
• The comparative evaluation using the error histograms (EHs) and correlation/regression are used to evaluate metrics to demonstrate the competence of SCGGNNs.
The organization of the remaining parts of the paper is presented as Methodology in Section II, numerical solutions in Section III, and the conclusions are shown in the last Section.

Methodology
This section presents the proposed stochastic procedure based on SCGGNNs for the breathing transmission epidemic model.The graphical procedures and the execution performances of the stochastic solvers are also presented.

Neural network
This Section provides the structure of a neural network by taking 20 numbers of neurons in the hidden layers.An activation function based on the log-sigmoid (LS) is applied in the hidden layers, mathematically given as: (1) where ω shows the weights in hidden layers, b is the bias, while S(τ ) , E(τ ) , I(τ ) and R(τ ) are the outputs, and L represents the LS activation function, mathematically given as [20][21][22][23][24] : where r indicates the neurons.Figure 2 represents the different steps to solve the mathematical model.Construction of the neural network by taking 20 neurons is provided in 1st part of Fig. 2, the mathematical formulations are shown in 2nd half of Fig. 2, third portion of Fig. 2 represents the obtained performances of the results.
Figure 3 represents the structure of different layers for solving the nonlinear dynamics based on the breathing transmission epidemic system.The layered structure has been used in numerous areas, including design, communication, and technology.The process of layers effectively involves breaking down complicated processes into interconnected, manageable, and distinct components or levels.The construction of the layers-based output/ input and hidden layers using twenty neurons is also presented.
Figure 4 describes the neural network procedure using the LS activation function in the hidden layers for solving the breathing transmission epidemic system using the stochastic performances of SCGGNNs.The mathematical LS function serves as an activation function in neural networks and machine learning.It can compress large inputs to small positive values that are close to zero, while it keeps large negative to larger negative values approaching negative infinity.This is particularly useful in certain computations where numerical uncertainty is a concern, especially when dealing with extreme values.

Numerical results
The current part of this study shows the mathematical structure of the nonlinear breathing epidemic model.Three cases based on the disease transmission probability ( β ) are taken as 1.1, 4.  5 (iv-vi).The best validation values for solving the nonlinear breathing transmission epidemic model are calculated at epochs 44, 49, and 67, which are given as 3.4398 × 10 -10 , 5.5815 × 10 -10 , and 6.6384 × 10 -10 , respectively.In the process of model training and machine learning, the purpose is to design a model that can be used to generalize the hidden data.The performance of validation can also be measured using metrics like MSE, which indicates the performance of the model on unseen and new data.The perception of best validation performances in the perspective of MSE is associated with the hyperparameter tuning as well as the selection of the model.Gradient, Mu, and validation checks are considered significant concepts to train and optimize the models based on the neural networks.The gradient presents a vector and gives the information of each parameter that should be amended to degrade the loss or error of the model.Mu is an improvement to conventional gradient descent optimization schemes.It is utilized for faster optimization convergence and to avoid local minima.Validation checks are the fundamental training part based on the models of machine learning to confirm that the model is simplifying better to unseen or new data.In the process of training, a model is adjusted iteratively using the data of training.However, to avoid the case of overfitting which means the performance of the model is well based on the data of training and poorly performed on the new data, we need to consider its recital on the data of validation.The validation check is typically accomplished on each epoch.The gradient values of the operator have been presented as 7.3698 × 10 -08 , 9.7808 × 10 -08 , and 4.0822 × 10 -08 for the breathing transmission epidemic model.These values represent the convergence performances of SCGGNNs for solving the breathing transmission epidemic system.The fitness values are performed in Fig. 6i to iii using the training outputs/targets, error performances, test outputs, validation targets/outputs, fitness, and test targets.The EHs based on zero error, authentication, test, and training for cases 1 to 3 of the breathing transmission epidemic model have been presented in Fig. 6iv-vi using the proposed SCCGNNs.The EHs are performed as 1.15 × 10 -06 for case 1.15 × 10 -06 and 5.38 × 10 -06 for cases 1, 2, and 3 of the breathing transmission epidemic model.The regression analysis based on the training/testing/ validation has been illustrated in Fig. 7, and it is noticed that regression values are performed as 1 for each case (perfect model).Table 1 shows the MSE convergence using Epochs, Mu, gradient, and complexity for solving the breathing transmission epidemic model.
Figures 8 and 9 illustrate the comparison plots and AE for solving the breathing transmission epidemic mathematical model.The preciseness of the scheme is observed by overlapping the proposed and reference solutions.Moreover, the negligible AE enhances the accurateness of the procedure for solving the breathing transmission epidemic system. (3)

Conclusions
The numerical results of the nonlinear breathing transmission epidemic system by applying the proposed stochastic scale conjugate gradient neural network process have been presented in this study.The nonlinear form of the breathing transmission epidemic is divided into four dynamics, susceptible, exposed, infected, and recovered.The conclusions of this study are as follows: • The nonlinear breathing transmission epidemic system has been presented through the stochastic SCGGNNs using the LS activation function along with twenty numbers of hidden neurons.In the future, the breathing mathematical model can be implemented in real-life scenarios with applications across different areas of healthcare, nonlinear systems, and a variety of other differential models [25][26][27][28][29][30][31][32][33][34][35] .www.nature.com/scientificreports/

Figure 1 .
Figure 1.Graphical depictions of the nonlinear breathing transmission epidemic systems.
1 and 7.1, while other values are b = 0.061, α = 7.222 , υ = 0.004107 , Ar = 0.2 , κ = 0.95, µ N = 0.000024, and µ D = 0.00000088 including initial conditions 0.1.0.2, 0.3 and 0.4.Figures 5, 6 and 7 represent the performances of MSE, STs, function fitness, EHs and regression tests for three different variations of the breathing transmission epidemic system.The stochastic SCGGNNs procedures have been presented together with the LS activation function and twenty numbers of hidden neurons.Figure 5i to iii indicates the best mean square error (MSE) based on the best validation values along with Epochs.Whereas the values of Mu, gradient, and authentication checks in terms of STs are illustrated in Fig.

Figure 2 . 2 •
Figure 2. Mathematical model, layer constructions, neural structure, and result performances of the breathing transmission epidemic system.

Figure 3 .
Figure 3.A layers structure to solve the breathing transmission epidemic system.

Figure 4 .
Figure 4.A design of layer structure for solving the breathing transmission epidemic system.

Figure 5 .Figure 6 .
Figure 5. MSE tests and STs for breathing transmission system.

Figure 7 .
Figure 7. Regression tests for breathing epidemic system.

Figure 8 .
Figure 8.The comparison of the results performances for the breathing transmission epidemic system.

Table 1 .
Stochastic solvers for the breathing transmission epidemic system.