The current investigations are presented to solve the fractional order HBV differential infection system (FO-HBV-DIS) with the response of antibody immune using the optimization based stochastic schemes of the Levenberg-Marquardt backpropagation (LMB) neural networks (NNs), i.e., LMBNNs. The FO-HBV-DIS with the response of antibody immune is categorized into five dynamics, healthy hepatocytes (H), capsids (D), infected hepatocytes (I), free virus (V) and antibodies (W). The investigations for three different FO variants have been tested numerically to solve the nonlinear FO-HBV-DIS. The data magnitudes are implemented 75% for training, 10% for certification and 15% for testing to solve the FO-HBV-DIS with the response of antibody immune. The numerical observations are achieved using the stochastic LMBNNs procedures for soling the FO-HBV-DIS with the response of antibody immune and comparison of the results is presented through the database Adams-Bashforth-Moulton approach. To authenticate the validity, competence, consistency, capability and exactness of the LMBNNs, the numerical presentations using the mean square error (MSE), error histograms (EHs), state transitions (STs), correlation and regression are accomplished.

The hepatitis B virus (HBV) is known as a threatening disease, which directly attacks the healthy liver. Around 257 million individuals get infected per year, among them the top disease is HBV [

The fractional order (FO) derivative is a significant operator to design a real apparatus. A number of the FO derivatives have been used in the nonlinear models [_{1}, the starting to desired time (_{0} to _{1}) is reported in the literature. Therefore, the phenomena to calculate more real results of the mathematical FO derivatives has been motivated by many researchers [

A fractional order HBV differential infection system (FO-HBV-DIS) with the response of antibody immune is classified into five modules, healthy hepatocytes (

It is observed that the hepatocytes (_{1}, _{2}, _{3}, _{4} and _{5}, respectively.

The novelty of this work is to perform the numerical simulations of the nonlinear FO-HBV-DIS with the response of antibody immune using the optimization based stochastic schemes of the Levenberg-Marquardt backpropagation (LMB) neural networks (NNs), i.e., LMBNNs. The stochastic LMBNNs measures have never applied before to solve the nonlinear FO-HBV-DIS with the response of antibody immune. The data magnitudes are implemented 75% for training, 10% for certification and 15% for testing to solve the FO-HBV-DIS with the response of antibody immune. The stochastic numerical procedures have the competence and ability to solve the linear models as well as various stiff complex nonlinear systems [

A fractional order nonlinear mathematical system of differential equations is numerically handled successfully using the stochastic techniques.

The design of the artificial NNs along with the LMB method is provided first time to solve the FO-HBV-DIS with the response of the antibody immune model.

Three different FO variations have been presented for solving the biological nonlinear FO-HBV-DIS with the response of antibody immune.

The correctness of the stochastic scheme is observed using the comparison of the obtained and numerical Adams-Bashforth-Moulton approach.

The calculated absolute error (AE) validates the accuracy of the designed scheme for the biological nonlinear FO-HBV-DIS with the response of antibody immune.

The performances based on the regression, error histograms (EHs), mean square error (MSE), state transitions (STs) and correlation authorize the reliability of the designed LMBNNs procedure for the biological FO-HBV-DIS with the response of the antibody immune system.

The remaining sections are organized as: The LMBNNs procedure is described in Section 2. The numerical simulations of the FO-HBV-DIS using the LMBNNs are presented in Section 3. Concluding notes are presented in the last Section.

The designed procedures for the biological nonlinear FO-HBV-DIS with the response of the antibody immune system are presented in this section. The classification of the LMBNNs procedure is provided in two steps. First, the essential executions based on the LMBNNs are provided, while the implementation process is provided in second phase. An optimization method is drawn in

In this section, the numerical presentations of the nonlinear FO-HBV-DIS are provided by using the LMBNNs. The literature values to solve the nonlinear FO-HBV-DIS are

The graphical plots have been derived using the LMBNNs for solving the FO-HBV-DIS in ^{−06}, 7.7456 × 10^{−07} and 1.102 × 10^{−08}, respectively. The gradient measures based on the proposed LMBNNs to solve the nonlinear FO-HBV-DIS are calculated 8.0165 × 10^{−03}, 3.1268 × 10^{−05} and 7.3161 × 10^{−06}, respectively. The achieved values in these figures represent the precision, convergence and accuracy of the proposed LMBNNs for the FO-HBV-DIS. The values of the fitting curve have been drawn in ^{−03}, 6.68 × 10^{−04} and 1.21 × 10^{−04} for 1^{st}, 2^{nd} and 3^{rd} variation. The performances based on the regression are derived in

Case | M.S.E | Performance | Mu | Gradient | Iterations | Time | ||
---|---|---|---|---|---|---|---|---|

Training | Testing | Validation | ||||||

1 | 8.91 × 10^{−06} |
9.11 × 10^{−06} |
2.06 × 10^{−06} |
5.63 × 10^{−06} |
1.00 × 10^{−07} |
8.02 × 10^{−03} |
95 | 3 |

2 | 1.32 × 10^{−07} |
1.98 × 10^{−08} |
7.74 × 10^{−08} |
1.19 × 10^{−06} |
1.00 × 10^{−07} |
3.13 × 10^{−05} |
173 | 4 |

3 | 7.81 × 10^{−08} |
2.08 × 10^{−08} |
1.10 × 10^{−08} |
7.72 × 10^{−08} |
1.00 × 10^{−07} |
7.32 × 10^{−06} |
371 | 5 |

The AE plots AE have been derived using the comparisons based on the nonlinear FO-HBV-DIS in

The AE measures for each category of the nonlinear FO-HBV-DIS with the response of antibody immune are demonstrated in ^{−03} to 10^{−04} for variation 1, 10^{−03} to 10^{−06} for variation 2 and 10^{−04} to 10^{−06} for variation 3 for solving the nonlinear FO-HBV-DIS. The AE values for the infected hepatocytes ^{−03} to 10^{−05} for variation 1, 10^{−04} to 10^{−05} for variation 2 and 10^{−04} to 10^{−06} for variation 3 for solving the nonlinear FO-HBV-DIS.

The AE values for the capsids ^{−02} to 10^{−04}, 10^{−03} to 10^{−04} and 10^{−03} to 10^{−05} for case 1 to 3 for solving the nonlinear FO-HBV-DIS. The AE values for the free virus ^{−03} to 10^{−05} for case 1 and 2, while for case 3 the AE is found around 10^{−04} to 10^{−06} for solving the nonlinear FO-HBV-DIS. The AE values for the antibodies class ^{−03} to 10^{−07}, 10^{−04} to 10^{−06} and 10^{−04} to 10^{−07} cases 1 to 3 of the nonlinear FO-HBV-DIS. The AE plots enhance the correctness and exactness of the stochastic scheme for the nonlinear biological model.

The aim of this study is to present the numerical simulations of the fractional order HBV differential infection system with the response of antibody immune using the stochastic procedures of the LMBNNs. The fractional order HBV differential infection system is implemented to solve three different deviations using different values of the fractional order. The data magnitudes are implemented 75% for training, 10% for certification and 15% for testing to solve the FO-HBV-DIS with the response of antibody immune. Twelve numbers of neurons have been implemented to solve the biological differential system. The solutions of the FO-HBV-DIS with the response of antibody immune have been presented by using the LMBNNs, however the comparative performances have been accessible through the reference results. The numerical performances of FO biological system have been simulated using the LMBNNs to lessen the MSE. To authenticate the reliability, aptitude and capability of the LMBNNs, the numerical outcomes have been plotted using the STs, regression, Ehs, MSE and correlation. The matching performances indicate the precision of the designed stochastic method. The AE in good assortments shows the correctness of the nonlinear biological fractional order system. The other performance plots signify the consistency and dependability of the proposed method.

In future work, the stochastic LMBNNs procedures have been used to achieve the numerical measures/treatments of the many potential nonlinear fractional/integer order systems [