# Computational Intelligent Paradigms to Solve the Nonlinear SIR System for Spreading Infection and Treatment Using Levenberg–Marquardt Backpropagation

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## Abstract

**:**

## 1. Introduction

- A novel integrated design of an intelligent computing scheme is introduced via modeling competency of Levenberg–Marquardt backpropagation neural network applied to scrutinize the dynamics of both the categories of SIR systems represented with set of nonlinear ordinary differential equations.
- The designed LMB neural networks operate effectively on a dataset generated from numerical Adam method for different variants of the nonlinear modeling and treatment-based SIR systems.
- The performance via comparative investigations from reference results of Adam method on correlation, error histograms, regression and mean square error (MSE) metrics establish the worth of designed Levenberg–Marquardt backpropagation neural networks.
- Advantage of the proposed LMB neural network methodology is the smooth implementation, simplicity of the concept, stability and exhaustive applicability.

## 2. Methodology

- Essential descriptions are given to make or formulate the LMB neural networks dataset by the used of standard numerical methods—i.e., Runge–Kutta or Adam numerical solvers.
- Implementation procedure approved for LMB neural networks is introduced to find the approximate solution of both modified SIR (M.SIR) and SIR treatment (T.SIR) models presented in set of Equations (1) and (2).

## 3. Numerical Measures with Analysis

^{−12}, 10

^{−10}, 10

^{−12}, 10

^{−12}, 10

^{−10}and 10

^{−10}, respectively. The gradient-based values using the step size (Mu) of the designed LMB process given for the M.SIR and T.SIR systems are [9.48 × 10

^{−8}, 9.75 × 10

^{−08}, 9.83 × 10

^{−08}, 9.97 × 10

^{−08}, 9.83 × 10

^{−08}and 9.94 × 10

^{−08}] and [10

^{−13}, 10

^{−11}, 10

^{−11}, 10

^{−12}, 10

^{−10}, 10

^{−09}] and are shown in Figure 5. These plots designate the specificity as well as the convergence of the LMB neural network for each case of the M.SIR and T.SIR systems.

^{−06}to 10

^{−07}for each case of the M.SIR and T.SIR. The values of the error histograms (EHs) are plotted in Figure 12.

^{−05}, 10

^{−07}], while for $R(\tau )$ these values lie around [10

^{−06}, 10

^{−08}]. In the T.SIR system, the AE values for the all the parameters lie around [10

^{−05}, 10

^{−07}]. These outcomes indicate the exactness of the LMB designed neural network scheme.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Workflow diagram of the Levenberg–Marquardt backpropagation (LMB) proposed neural network for the nonlinear susceptible (S), infected (I) and recovered (R) (SIR) system.

**Figure 4.**Performance curves based on the mean square error (MSE) using the LMB designed neural networks for both of the SIR systems.

**Figure 5.**State transition for the LMB designed neural networks for each case of modified SIR (M.SIR) and SIR treatment (T.SIR) systems.

**Figure 12.**Plots of the error histograms (EHs) for LMB designed neural networks for all cases of the M.SIR and T.SIR systems.

**Figure 14.**Comparison of the obtained results through LMB neural network for each case of the M.SIR.

**Figure 15.**Comparison of the obtained results through LMB neural network for each case of the T.SIR.

**Figure 16.**Absolute error (AE) based on the obtained results and Adam results via LMB neural network for each case of the M.SIR.

**Figure 17.**AE based on the obtained results and Adam results via LMB neural network for each case of the T.SIR.

Case | MSE | Performance | Gradient | Mu | Epoch | Time | ||
---|---|---|---|---|---|---|---|---|

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

1 | 1.90 × 10^{−12} | 6.21 × 10^{−12} | 2.71 × 10^{−12} | 1.90 × 10^{−12} | 9.49 × 10^{−08} | 1.00 × 10^{−13} | 25 | 1 |

2 | 9.04× 10^{−12} | 3.16 × 10^{−10} | 1.50 × 10^{−11} | 9.04 × 10^{−12} | 9.76 × 10^{−08} | 1.00 × 10^{−11} | 36 | 1 |

3 | 6.80 × 10^{−13} | 2.61 × 10^{−12} | 7.86 × 10^{−11} | 6.80 × 10^{−08} | 9.83 × 10^{−08} | 1.00 × 10^{−11} | 89 | 1 |

Case | MSE | Performance | Gradient | Mu | Epoch | Time | ||
---|---|---|---|---|---|---|---|---|

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

1 | 5.26 × 10^{−12} | 8.82× 10^{−12} | 4.36 × 10^{−12} | 5.26 × 10^{−12} | 9.98 × 10^{−08} | 1.00 × 10^{−12} | 32 | 1 |

2 | 8.05× 10^{−11} | 1.83 × 10^{−10} | 1.00 × 10^{−10} | 8.50 × 10^{−12} | 9.83 × 10^{−08} | 1.00 × 10^{−10} | 551 | 3 |

3 | 6.89× 10^{−11} | 4.24 × 10^{−10} | 6.81 × 10^{−10} | 6.90 × 10^{−11} | 9.94 × 10^{−08} | 1.00 × 10^{−09} | 265 | 4 |

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**MDPI and ACS Style**

Umar, M.; Sabir, Z.; Zahoor Raja, M.A.; Gupta, M.; Le, D.-N.; Aly, A.A.; Guerrero-Sánchez, Y.
Computational Intelligent Paradigms to Solve the Nonlinear SIR System for Spreading Infection and Treatment Using Levenberg–Marquardt Backpropagation. *Symmetry* **2021**, *13*, 618.
https://doi.org/10.3390/sym13040618

**AMA Style**

Umar M, Sabir Z, Zahoor Raja MA, Gupta M, Le D-N, Aly AA, Guerrero-Sánchez Y.
Computational Intelligent Paradigms to Solve the Nonlinear SIR System for Spreading Infection and Treatment Using Levenberg–Marquardt Backpropagation. *Symmetry*. 2021; 13(4):618.
https://doi.org/10.3390/sym13040618

**Chicago/Turabian Style**

Umar, Muhammad, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Manoj Gupta, Dac-Nhuong Le, Ayman A. Aly, and Yolanda Guerrero-Sánchez.
2021. "Computational Intelligent Paradigms to Solve the Nonlinear SIR System for Spreading Infection and Treatment Using Levenberg–Marquardt Backpropagation" *Symmetry* 13, no. 4: 618.
https://doi.org/10.3390/sym13040618