The purpose of this research work is to investigate the numerical solutions of the fractional dengue transmission model (FDTM) in the presence of Wolbachia using the stochastic-based Levenberg-Marquardt neural network (LM-NN) technique. The fractional dengue transmission model (FDTM) consists of 12 compartments. The human population is divided into four compartments; susceptible humans (

Dengue fever is widespread, with an approximated 3.9 billion people at threat and 390 million new dengue infections estimated each year [

Dengue is transmitted to people through the bite of a dengue-infected female mosquito. The extrinsic incubation period (EIP) is the time it takes for the dengue virus to multiply in a mosquito’s body before it can reach the salivary glands and be transmitted to a human through a bite. Factors such as temperature, humidity, and rainfall affect the length of the EIP. A comprehensive approach that includes these factors can help to reduce the EIP of dengue virus in mosquitoes and ultimately reduce the transmission of the disease. Only mosquitoes that survive the EIP are capable of transmitting the virus to humans [

Wolbachia, a bacterium, is present in certain mosquito species and about 60% of all groups of insects. The potential use of Wolbachia-infected mosquitoes as an alternative approach to control dengue transmission. While vaccines and insecticides are effective, they have limitations such as incomplete protection and adverse effects on the environment. In contrast, Wolbachia-infected mosquitoes have shown promise in reducing dengue transmission in field trials by preventing the virus from being transmitted to humans. This method appears to have fewer environmental impacts and a lower likelihood of insecticide-resistant mosquito populations developing [

The theory of fractional derivatives has a wide range of applications in medicine, finance, engineering, physics, and many other fields [

where

An artificial neural network (ANN) is a method for simulating biological neural activity using mathematics. The essential mathematical principles of an artificial neural network employing an artificial neuron are multiplication, summation, and activation. At the start of the neuron, the input values are multiplied by the specified weights. The sum function, which adds all weighted inputs and biases, is located in the middle layer of the artificial neuron. The summation of previously weighted inputs and bias is sent via an activation function, also known as a transfer function, at the end of an artificial neuron [

In ANN, an activation function is utilized to provide nonlinear features. In a neural network, the yn are input variables, wn are the weights, and fn shows the output in

In literature, many fractional-order mathematical models along with artificial neural network approach models for real-world problems have been proposed utilizing the ANNs approach. Pornsawad et al. [

The aim of this work was to develop and numerically solve a fractional dengue transmission model (FDTM) with Wolbachia using neural network. In the present model, we incorporated Wolbachia-infected mosquitoes along with Wolbachia-free mosquitoes and humans. The impact of vertical transmission probability on the persistence of dengue transmission in humans and mosquitoes has been analyzed using the Levenberg-Marquardt neural network (LM-NN) approach. The convergence and accuracy of the proposed LM-NN technique have been checked through mean square error, regression analysis plots, and state transition results. The results obtained from the proposed LM-NN approach for FDTM have been compared with the reference solutions obtained using the Adams-Bashforth-Moulton (ABM) method and they were found to be in good agreement.

A fractional dengue transmission model in the presence of Wolbachia is developed to observe the dynamics of dengue between humans and mosquitoes. The model consists of 12 compartments, namely, susceptible human

The reproduction rate of Wolbachia-carrying and Wolbachia-free mosquitoes are denoted by

The model flowchart is given in

All the parameters that appear in model (1) are defined in

Parameters | Description | Unit | Min | Used values | Max |
---|---|---|---|---|---|

Transmission probability of dengue from Wolbachia-free mosquitoes | N/A | 0 | 0.2614 | 1 | |

Transmission probability from Wolbachia-infected mosquitoes to humans | N/A | 0 | 0.1307 | 1 | |

The biting rate of Wolbachia-free mosquitoes | 0 | 0.63 | 1 | ||

Adult mosquito death rate of non-Wolbachia mosquitoes | 1/30 | 1/14 | 1/10 | ||

Death rate of non-Wolbachia mosquitoes | 1/30 | 1/14 | 1/10 | ||

rate of infected humans | 1/14 | 1/5 | 1/3 | ||

Maturation rate of mosquitoes that carrying Wolbachia | 1/12 | 1/10 | 1/8 | ||

Reproduction rate of mosquitoes that are Wolbachia-free | 1 | 1.25 | 2.5 | ||

Reproduction rate of Wolbachia-carrying mosquitoes | 0.7 | 1.1875 | 2.5 | ||

Vertical transmission probability | N/A | 0.6 | 0.8 | 1 | |

The rate at which exposed become infectious | 1/12 | 1/10 | 1/8 | ||

Maturation rate of mosquitoes that are Wolbachia-free | 1/17 | 1/10 | 1/6 | ||

The death rate of the aquatic (egg, larvae, and pupae) stage of Wolbachia-infected mosquitoes | 1/20 | 1/14 | 1/7 | ||

Progression rate from exposed to infectious human | 1/7 | 1/5.5 | 1/4 | ||

Death rate of aquatic mosquitoes that are Wolbachia-free | 1/20 | 1/14 | 1/7 | ||

Progression rate from exposed to infectious non-Wolbachia | 1/12 | 1/10 | 1/8 | ||

Initial exposed human | N/A | 1 | 2 | 5 | |

Ratio of |
N/A | 3 | |||

The biting rate of Wolbachia-infected mosquitoes | 0 | 0.5985 | 1 | ||

maximum carrying capacity | 3 |
||||

Total human population | 150, 000 | ||||

The death rate of Wolbachia-infected adult mosquitoes | 1/30 | 1/14 | 1/10 | ||

Death rate the Wolbachia-infected mosquito | 1/30 | 1/14 | 1/10 | ||

Recruitment/death rate of the human compartments |

According to the model, a susceptible human becomes exposed to a bite from an infected mosquito. The biting rates of Wolbachia-infected and Wolbachia-free mosquitoes are

The exposed Wolbachia-free and Wolbachia-carrying mosquitoes move to the infected class after biting the dengue-infected human with rate

In this section, we explained the proposed Levenberg-Marquardt neural network (LM-NN) method that is used to solve the FDTM model (1). The Pseudocode of the proposed LM-NN technique is also given in

Levenberg-Marquardt neural network | |
---|---|

(1) Training data-80% | |

(2) Testing data-10% | |

(3) Validation data-10% | |

(4) Hidden neurons-10 | |

(1) Maximum iteration has been achieved | |

(2) Mu achieve its maximum value | |

(3) Value of performance should be minimum | |

(4) The value of the gradient becomes smaller than the minimum gradient | |

If the required results are achieved, save the outputs otherwise retrain the network. | |

The novel properties of the proposed research work is as follows:

Levenberg-Marquardt neural network (LM-NN) is based on a backpropagation process with a novel methodology or design is created for solving the fractional-order model (1) dealing with Wolbachia carrying and Wolbachia-free mosquitoes with humans.

For the fractional dengue transmission model (FDTM), a stochastic-based back propagated numerical Levenberg-Marquardt neural network (LM-NN) procedure is utilized to obtain the best-approximated solution.

Adams-Bashforth-Moulton (ABM) numerical method has been employed to generate the reference dataset.

Numerical results obtained from the proposed LM-NN technique show good agreement with the reference solutions achieved using the ABM method.

The reliability and consistency of the proposed LM-NN approach are investigated through statistical analysis.

The best performance of the proposed LM-NN technique is analyzed using regression analysis (RA) plots, mean square error (MSE) graphs, fitness curves, and state transitions (STs) for different classes.

Error analysis performed for different classes of FDTM is used to check the accuracy of the proposed LM-NN technique.

This section describes the numerical simulations based on the outcomes of three scenarios of the nonlinear fractional dengue transmission model (FDTM) performed with the suggested LM-NN approach. The simulations were conducted on scenarios with only Wolbachia-free mosquitoes present, both Wolbachia-free and Wolbachia-carrying mosquitoes present, and only Wolbachia-carrying mosquitoes present. The mathematical structure for each case is given as:

For this case, we consider the following values of the parameters

For this case, the parameter values being examined are

Similarly, after putting the values of parameters in the model (1)

The numerical results of the FDTM model is obtained through LM-NN technique which is implemented in MATLAB using the built-in package ‘nftool’ together with 10 neurons. To train the model using the Levenberg method, we used 80% of the data for training purposes, 10% of the data is used for validation and the remaining 10% of the data is used for testing.

Features | Case I | Case II | Case III |
---|---|---|---|

Training | |||

Validation | |||

Testing | |||

Gradient | |||

Performance | |||

Mu | |||

Epoches | 176 | 426 | 146 |

The mean square error results obtained from the proposed neural network approach for each scenario of the FDTM are presented in

Regression is a graphical representation of the precision of the LM-NN solution to the reference values for training, validation, and testing points separately. A straight line represents an available reference solution in these plots, while dots or small circles represent LM-NN results. The numerical value of regression can also be used to assess the computation’s accuracy and precision.

Error plots were generated for three different scenarios involving the classes

The purpose of this research study was to develop the fractional dengue transmission model (FDTM) in the presence of Wolbachia-carrying mosquitoes. The model incorporated the vertical transmission of Wolbachia in mosquitoes. In this study, we proposed the numerical solution of the fractional dengue transmission model (FDTM) using the stochastic-based Levenberg-Marquardt neural networks (LM-NN) technique. The FDTM model was solved using the Adams-Bashforth-Moulton (ABM) technique to create a reference dataset. The created dataset was used for training (80%), testing (10%), and validation (10%) of the proposed LM-NN technique with 10 hidden layers of neurons. The graphs of mean square error were presented for three different cases, when Wolbachia-free mosquitoes persist only

Levenberg-Marquardt

Fractional dengue transmission model

Artificial neural network

Mean square error

Adams-Bashforth-Moulton

The authors are thankful to the anonymous reviewers for improving this article.

The authors received no specific funding for this study.

The authors confirm their contribution to the paper as follows: study conception and design: SJ, IA, and DB; methodology: ZF; analysis and interpretation of results: SJ, IA, ZF, and DB; draft manuscript preparation: SJ, IA, ZF, and DB; validation: SJ and DB; review and editing: IF and ZF. All authors reviewed the results and approved the final version of the manuscript.

Not applicable.

The authors declare that they have no conflicts of interest to report regarding the present study.