In the present study, a design of a fractional order mathematical model is presented based on the schistosomiasis disease. To observe more accurate performances of the results, the use of fractional order derivatives in the mathematical model is introduce based on the schistosomiasis disease is executed. The preliminary design of the fractional order mathematical model focused on schistosomiasis disease is classified as follows: uninfected with schistosomiasis, infected with schistosomiasis, recovered from infection, susceptible snail unafflicted with schistosomiasis disease and susceptible snail afflicted with this disease. The solutions to the proposed system of the fractional order mathematical model will be presented using stochastic artificial neural network (ANN) techniques in conjunction with the Levenberg-Marquardt backpropagation (LMBP), referred to as ANN-LMBP. To illustrate the preciseness of the ANN-LMBP method, mathematical presentations of three different values focused on fractional order will be performed. These statics performances are taken in these investigations are 78% and 11% for both learning and certification. The accuracy of the ANN-LMBP method is determined by comparing the values obtained by the database Adams-Bash forth-Moulton scheme. The simulation-based error histograms (EHs), MSE, recurrence, and state transitions (STs) will be offered to achieve the capability, accuracy, steadiness, abilities, and finesse of the ANN-LMBP method.

Schistosomiasis disease (SD) is a parasitic disease produced by worms of the species Schistosoma. Schistosomiasis was treated for approximately 206 million people in 2016. However, the observed number of people expected to treat in 2016 was 89.2 million [

Schistosomiasis mathematical analysis is not new, but it is typically expressed in the form of boundary value problems. Several researchers have used a deterministic approach to develop new schistosomiasis approaches or modify existing ones [

Fractional calculus (FC) was invented in the time of Newton, but it has recently piqued the interest of many academics. Over the last thirty years, the most fascinating leaps in industry sectors have been discovered within the structure of FC. The idea of the fractional derivative has been modernized due to the complexities affiliated with the characteristic of inhomogeneity. The behavior of multifaceted media with a diffusion process can be captured using fractional differential operators. It has been a very valuable tool, but many problems can now be demonstrated more helpfully and precisely using ordinary differential equations of any order. Many scholars began to work on simplistic calculus to reveal their points of view while analyzing a wide range of complex phenomena as a result of the rapid development of computational techniques with computer software systems. Many senior researchers proposed various concepts for fractional-order and integral operators, which arranged the foundation [

Recent advances in fractional calculus (FC) theory have resulted in the creation of two notable operators, Caputo–Fabrizio and Atangana–Baleanu. The new derivatives mentioned above have non-singular kernels and do not use energy-distribution. The Atangana–Baleanu operator is mainly focused on the simple Mittag-Leffler method, which has powerful forces because its diffusional effects are collaborations with good assumptions. The Caputo–Fabrizio estimation is also based on exponential law, which can be found in a variety of natural phenomena and has a fusion effect with numerical depiction. These operators have demonstrated that they are the future for demonstrating a variety of scientific progressions such as turbulent theory, life processes, heat flux problems, operations research, and financial problems, among others [

This research is related to the development of a fractional order mathematical model focused on the evolution of the schistosomiasis disease. The use of fractional order derivatives is implemented to observe more precise performances of the mathematical equation. The novel fractional order mathematical model solutions will be described using stochastic artificial neural network (SANN) processes in conjunction with Levenberg-Marquardt backpropagation (LMBP), i.e., ANN-LMBP.

To solve the dynamic behavior of the schistosomiasis disease, the ANN system is proposed effectively using LMBP optimization processes.

The consistent overlapped results obtained by LMBP and the Adams achievements verify the proposed approach.

The performance is certified by using various statistics values to accomplish the numerical performances of the schistosomiasis disease model.

The paper is systematized as: Section 2 demonstrates the configuration of the fractional order mathematical equation, Section 3 contains the stochastic contributions, Section 4 is created on ANN-LMBP method, Section 5 is intended using simulation procedures, and the final part contains concluding remarks.

The model defines the total humanity at time

Parameters | Details |
---|---|

Susceptible human rate | |

Snail population | |

Susceptible human infected with schistosomiasis | |

Effected with infection snail | |

Infected human to recovery class rate | |

Loss immunity rate | |

Morality of snails and humans | |

ICs |

The current research aims to provide numerical simulations of a fractional order mathematical formula based on the schistosomiasis disease using artificial intelligence (AI) and ANN-LMBP. The following is the structure of the fractional order mathematical model focused on the schistosomiasis disease:

In this system,

The current section demonstrates the stochastic operator performances when solving the fractional order computational schistosomiasis disease model with ANN-LMBP. Stochastic software solvers have been examined in the literature to solve complicated, singular, and rigid systems [

A design fractional order and numerical solutions are introduced to tackle the mathematical schistosomiasis disease model.

Using the stochastic ANN-LMBP methods, the numerical performance evaluation of the built mathematical schistosomiasis model is provided.

The proposed ANN-LMBP method is validated by computing the mathematical results of three multiple varieties depending on fractional order derivatives.

The accuracy and precision of the computing ANN-LMBP method are validated by comparing obtained and citation (Adams-Bashforth-Moulton) discussions.

The correctness of the ANN-LMBP method is obtained through absolute error (AE), which is skillful in good procedures for performing mathematical schistosomiasis disease model.

The stagnation, STs, correlation, MSE, and EHs measures validate the reliability and serviceability of the constructed ANN-LMBP method for mathematical schistosomiasis model solution.

This part of the study describes the proposed ANN-LMBP method for presenting the numerical schistosomiasis disease model solutions. The proposed ANN-LMBP scheme is introduced in two stages: the substantial performances of the ANN-LMBP method and the operational plans for solving the mathematical schistosomiasis model using the ANN-LMBP method. The significant operator performances-based L-MBNNs technique is provided.

The designed approach for solving the nonlinear fractional differential model is displayed in

In this portion, three different variability of fractional order differential equations of the mathematical schistosomiasis disease model using ANN-LMBP are presented. Each type mathematical representation is as follows:

The numerical representations using the outcomes of the fractional order mathematical schistosomiasis disease model are discussed utilizing ANN-LMBP method with 13 numbers of neurons and data selection as 78%, 11%, and 11% for training, certification, and testing, respectively.

^{−10}, 4.9824 × 10^{−10} and 8.6225 ×10^{−10}. For the cases 1, 2 and 3, the gradient performances are 9.75 × 10^{−08}, 9.2105 × 10^{−08}, and 8.4198 × 10^{−08}. The graphical methodologies demonstrate the convergence of the ANN-LMBP method for solving the fractional order mathematical (SD) model. The profitability of the results and EHs for the fractional order mathematical schistosomiasis disease (SD) model using the ANN-LMBP technique is shown in ^{−05}, −3.6 × 10^{−06}, and 3.16 × 10^{−07} for cases 1, 2, and 3. The correlation schemes for the fractional order mathematical (SDM) using the ANN-LMBP method are shown in

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

[Training] | [Verification] | [Testing] | ||||||

I | 3.38 × 10^{−09} |
8.62 × 10^{−10} |
3.94 × 10^{−09} |
3.38 × 10^{−09} |
8.42 × 10^{−08} |
1 × 10^{−10} |
46 | 2 |

2 | 1.17 × 10^{−09} |
1.75 × 10^{−10} |
7.11 × 10^{−10} |
1.18 × 10^{−09} |
9.75 × 10^{−08} |
1 × 10^{−10} |
38 | 1 |

3 | 3.98 × 10^{−11} |
1.38 × 10^{−11} |
7.82 × 10^{−11} |
3.99 × 10^{−11} |
9.21 × 10^{−08} |
1 × 10^{−11} |
41 | 2 |

The correctness of the proposed ANN-LMBP method for the fractional order mathematical (SD) model is observed in

The AE is provided based on the not infected with schistosomiasis ^{−04} to 10^{−06}, 10^{−05} to 10^{−07} and 10^{−05} to 10^{−07} for the case 1, 2 and 3. The AE for the people infected with schistosomiasis ^{−04} to 10^{−04}, 10^{−05} to 10^{−04} and 10^{−05} to 10^{−06} for the case 1, 2 and 3. The AE for the people recovered from infection ^{−04} to 10^{−04}, 10^{−05} to 10^{−04} and 10^{−05} to 10^{−06} for the case 1, 2 and 3. Similarly, the AE for the behaviors ^{−04} to 10^{−05}, 10^{−05} to 10^{−05} and 10^{−06} to 10^{−07} for the case 1, 2 and 3. These best AE values represent the exactness of proposed ANN-LMBP method for the fractional order mathematical (SD) model.

In this presents study the numerical solutions of the fractional order mathematical model are presented based on the schistosomiasis disease. The goal of finding fractional order numerical solutions is to improve the accuracy of the mathematical model performance. The nonlinear mathematical system is divided into five dynamics, susceptible snail not infected with (SD)

In future given methodology can be implemented to many fractional and integer order systems of utmost significance [