The purpose of these investigations is to find the numerical outcomes of the fractional kind of biological system based on Leptospirosis by exploiting the strength of artificial neural networks aided by scale conjugate gradient, called ANNs-SCG. The fractional derivatives have been applied to get more reliable performances of the system. The mathematical form of the biological Leptospirosis system is divided into five categories, and the numerical performances of each model class will be provided by using the ANNs-SCG. The exactness of the ANNs-SCG is performed using the comparison of the reference and obtained results. The reference solutions have been obtained by using the Adams numerical scheme. For these investigations, the data selection is performed at 82% for training, while the statics for both testing and authentication is selected as 9%. The procedures based on the recurrence, mean square error, error histograms, regression, state transitions, and correlation will be accomplished to validate the fitness, accuracy, and reliability of the ANNs-SCG scheme.

The spreading virus ratio creates several diseases in the population with high or low rates [

The mathematical modeling provides the basic operators using the good policies of minimizing or eliminating Leptospirosis and predicting its future occurrences [

Newton proposed fractional calculus many years ago, but it obtained enormous fame and significance in recent years. Over the last thirty years, fractional calculus has been widely used in industrial areas. The idea of the fractional order derivative has been modernized through the complexities connected to the typical inhomogeneity. The multilayered media construction based on the diffusion procedure is reported using the operators based on the fractional calculus. A significant apparatus presents the precise performance using the ordinary differential form of the systems. There is a wider range of complicated phenomena that have been proposed by many researchers based on the software models. There are various fractional/integer kinds of schemes reported in [

These investigations aim to authenticate the numerical outcomes based on the fractional kind of biological system based on Leptospirosis by exploiting the strength of artificial neural networks (ANNs) aided by scale conjugate gradient (SCG), called ANNs-SCG. The derivatives based on the fractional order have been used to achieve precise mathematical performances. Moreover, the stochastic numerical computing performances based on the ANNs-SCG procedure for the mathematical form based on the biological Leptospirosis model have not been presented before in terms of the fractional form. The solution of this novel fraction order model will be presented first by using the stochastic ANNs-SCG scheme. The stochastic numerical performances have been applied to solve the corneal shape of model [

The remaining structure of the paper is given as follows: Section 2 presents the formulation of the fractional Leptospirosis system, Section shows the methodology of the stochastic ANNs-SCG scheme, Section 4 describes the simulations together with some important findings, while the last Section 5 presents the concluding remarks.

The mathematical kind of the biological Leptospirosis system is divided into five categories (Human and vector). The Susceptible-Infected-Recovered (SIR) population dynamics based on the vector (rat) and human population are shown as [

The integer kind of differential Leptospirosis system is presented in

Parameter | Details |
---|---|

Recovered human | |

Susceptible human | |

Infected human | |

Susceptible rat | |

Infected rat | |

Individual rate of immune to becoming again susceptible | |

The recruitment rate of the human population | |

The transmission rate of Leptospirosis from infected rats to susceptible human | |

The natural rate of death in the human population | |

Infected human die through disease | |

Recovered human rate | |

The recruitment rate of the rat population | |

Rate of infection that rat dies | |

Natural death of the population of rat | |

Transmission of Leptospirosis from infected rat to susceptible rat | |

Time | |

Initial conditions |

The current work provides the numerical representations of the mathematical model based on Leptospirosis using artificial intelligence (AI) together through the artificial neural networks (ANNs) along with the scale conjugate gradient (SCG), i.e., ANNs-SCG. As a result, the mathematical model based on Leptospirosis is presented as follows:

The fractional derivatives provide more reliable performances of the mathematical Leptospirosis system.

The design of the stochastic ANNs-SCG is provided for the numerical investigations of the fractional kind of mathematical model based on Leptospirosis.

Three different variants of the fraction order have been used to perform the numerical simulations of the fractional kind of mathematical model based on Leptospirosis.

The correctness of the stochastic ANNs-SCG scheme is performed to compare the obtained results based on the ANNs-SCG and the reference solutions.

The excellence of the stochastic ANNs-SCG scheme is approved through the obtained negligible absolute error magnitude based fractional kind of mathematical Leptospirosis system.

The recurrence, mean square error, error histograms, regression, state transitions, and correlation supports the reliability of the ANNs-SCG scheme using the fractional kind of mathematical model based on Leptospirosis.

The current section provides the stochastic artificial neural networks (ANNs) along with the scale conjugate gradient (SCG), i.e., ANNs-SCG scheme, along with the numerical results procedures to achieve the fractional kind of mathematical model based on the Leptospirosis as shown in

The computational ANNs-SCG scheme is provided in two steps: the system dataset is generated using the generalized Adams scheme and the significant operators to execute the estimated results of the fractional kind of mathematical model based on the Leptospirosis using the ANNs-SCG. The essential information based on the generalized Adam method can be observed in the reported investigation [

The approximate numerical performances of the mathematical model of Leptospirosis with the artificial intelligence abilities based on the computing framework using the supervised neural networks trained with the scaled conjugate gradient in terms of best negotiation between the measures, including the complexity, premature convergence, underfitting, and overfitting situations.

Moreover, all the network parameters are adjusted after comprehensive simulation investigations, knowledge, care, experience, and minor dissimilarities in these situations to degrade the network’s performance.

This section provides three cases-based fraction Leptospirosis systems through the artificial neural networks (ANNs) along with the scale conjugate gradient (SCG), i.e., ANNs-SCG. The mathematical representation of each case (C) is provided as:

The simulations of the fractional kind of mathematical model based on Leptospirosis using the computational ANNs-SCG approach have been provided by taking 14 numbers of neurons.

The fractional kind of mathematical Leptospirosis system simulations using the computational ANNs-SCG approach has been provided by taking 14 numbers of neurons together with data selection as 82% for training, and the statics for both testing and authentication is selected as 9%.

^{−08}, 1.85270 × 10^{−09}, and 6.90443 × 10^{−10}. The gradient representations are performed as 7.32 × 10^{−07}, 9.90 × 10^{−08}, and 9.92 × 10^{−08} for the respective cases of the fractional order model based on Leptospirosis. ^{−04}, 9.33 × 10^{−06}, and 3.09 × 10^{−06}.

C | MSE | Performance | Gradient | Mu | Iteration | Time | ||
---|---|---|---|---|---|---|---|---|

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

I | 9.46 × 10^{−08} |
2.88 × 10^{−09} |
3.23 × 10^{−09} |
7.32 × 10^{−07} |
8.89 × 10^{−08} |
1 × 10^{−08} |
125 | 3 |

2 | 1.85 × 10^{−09} |
2.37 × 10^{−10} |
5.34 × 10^{−10} |
9.90 × 10^{−08} |
1.85 × 10^{−09} |
1 × 10^{−09} |
509 | 5 |

3 | 6.90 × 10^{−10} |
3.18 × 10^{−10} |
4.33 × 10^{−10} |
9.90 × 10^{−08} |
6.90 × 10^{−10} |
1 × 10^{−09} |
122 | 3 |

The result comparisons and the values of the AE are performed in ^{−03}–10^{−05}, 10^{−04}–10^{−05} and 10^{−05}–10^{−06}. For ^{−04}–10^{−06}, 10^{−05}–10^{−06} and 10^{−05}–10^{−07}. For ^{−04}–10^{−05}, 10^{−05}–10^{−06} and 10^{−06}–10^{−08} for C (1–3). For the fractional kind of system based on Leptospirosis, the AE for ^{−03}–10^{−05}, 10^{−04}–10^{−06} and 10^{−05}–10^{−06}. For C (1–3), the AE for ^{−04}–10^{−06}, 10^{−05}–10^{−06} and 10^{−05}–10^{−07}. These best measures calculated through the AE perform the exactness of the stochastic ANNs-SCG procedure for the fractional kind of model based on Leptospirosis.

In this work, the solutions of the fractional kind of biological model based on Leptospirosis have been provided by exploiting the strength of artificial neural networks aided by the scaled conjugate gradient. The reliable and accurate results of the model have been achieved by using fractional kinds of derivatives. The biological Leptospirosis mathematical form is categorized into five classes. The numerical performances of each category of the biological Leptospirosis mathematical model have been presented through the stochastic artificial neural networks (ANNs) along with the scale conjugate gradient (SCG), i.e., the ANNs-SCG procedure. The correctness of the computing ANNs-SCG procedure is performed by comparing the achieved and reference solutions (Adam numerical method). The simulations of the fractional kind of mathematical model based on Leptospirosis using the computational ANNs-SCG approach have been provided by taking 14 numbers of neurons together with the selection of the data as 82% for training. At the same time the statics for both testing and authentication is selected as 9%. The AE has been calculated for the respective cases of the fractional kind of system are found as 10^{−06} to 10^{−08}. The numerical procedures based on the recurrence, error histograms, regression, mean square error, state transitions, and correlation have been accomplished to validate the fitness, capability, accuracy, steadiness and reliability of the proposed ANNs-SCG method.

In upcoming work, the proposed ANNs-SCG numerical procedure can be implemented to solve different applications of extreme interest to researchers [

_{2}gas with an efficient method