Review on Designing Multiple Neural Network-Based Intelligent Computing Procedures for Solving the Anthrax Disease Model Used in Animal

Abstract

The aim of this work is to provide the numerical representations of the anthrax disease model used in animals both in integer and fractional form by designing a soft computing platform based on multiple neural network procedures. The integer order form of the anthrax disease model used in animals will be discussed first and then the fractional order derivatives of the anthrax disease model used in animals are solve later to get more precise solutions of the model. The values of the fractional order will be taken between 0 and 1 in order to check, which results perform better either close to zero or one. The mathematical form of the anthrax disease model used in animals is categorized into four classes, named as susceptible, infected, recovered and vaccinated. The dataset is obtained through the explicit Runge-Kutta and Adams numerical technique, which is used to train the neural network procedures in order to reduce the mean square error by separating into training as 80%, testing 8% and substantiation 12%. The design of the supervised neural network procedures based on the Levenberg–Marquardt backpropagation scheme, Bayesian regularization approach and computational scale conjugate neural network solver is presented through the activation log- sigmoid function or radial basis functions. The process of supervised neural network is used with a single hidden layer structure and deep neural network construction using two or more hidden layers. The numbers of neurons will be selected by performing many tests in the hidden layer(s) for solving the anthrax disease model used in animals. The optimization is performed through the efficient and reliable Marquardt backpropagation, Bayesian regularization approach scale conjugate neural network procedures for the numerical performances of the mathematical anthrax disease system in animals. The exactness and accuracy of the designed solver is validated through the matching of the outcomes and negligible values of the absolute error that are performed around 10-05 to 10-08 for each case of the anthrax disease model used in animals. The correctness of the designed computational scale conjugate neural network has also observed through the optimal training performances, which are calculated around 10-10 to 10-12 for each case of the anthrax disease model used in animals. Furthermore, the statistical values in the form of error histogram, regression coefficient, and state transition enhance the reliability and stability of the designed computational scale conjugate neural network for presenting the numerical solutions of the anthrax disease model used in animals. The designed computational Marquardt backpropagation, Bayesian regularization and scale conjugate neural networks has never been tested before for presenting the numerical representations of the anthrax disease model used in animals.

Introduction

• Anthrax is a serious zoonotic disease caused by Bacillus anthracis, affecting both animals and humans.

• Primarily impacts herbivorous animals such as cattle, sheep, goats, and horses, but humans can be infected through direct contact.

• The bacteria form spores that can survive in soil for decades, making anthrax a persistent threat.

• Transmission occurs via:

  1. Direct contact with infected animals or carcasses.

  2. Contaminated animal products (wool, hides, bone meal).

  3. Inhalation of spores (airborne).

  4. Consumption of contaminated meat or water.

  5. Contaminated soil and water.

  6. Bioterrorism or laboratory exposure.

Types of Anthrax

• Cutaneous: Skin lesions; from direct contact.

• Gastrointestinal: From eating contaminated food.

• Inhalation/Respiratory: Highly fatal; from spores in air.

• Septicemic: Blood infection.

Symptoms include fatigue, fever, appetite loss, breathing problems, bloody discharges, and enlarged lymph nodes.

Prevention and Treatment

• Vaccination: Live spore-based vaccines for livestock.

• Antibiotics: Amoxicillin and tetracycline for infected animals.

• Control Measures: Isolation, sterilization, proper disposal of carcasses, and careful livestock management.

• Understanding outbreak areas and risk factors helps prevent zoonotic transmission and bioterrorism threats.

Anthrax Disease Modeling

• Purpose: To study disease spread, population impact, and control strategies.

• Components:

  1. Host: Susceptible animals (mainly herbivores) and humans.

  2. Pathogen: Bacillus anthracis in bacterial and spore form.

  3. Environment: Contaminated soil, water, and carcasses act as reservoirs. Environmental factors like flooding or grazing can trigger outbreaks.

• Mathematical Approaches:

  • Integer and Fractional Models: Provide numerical solutions to understand anthrax dynamics in animals.

  • Neural Networks: Used for predicting and simulating disease behavior, with techniques such as:

    • Levenberg–Marquardt backpropagation

    • Bayesian regularization

    • Radial basis and log-sigmoid activation functions

  • Models test single and deep hidden layer architectures to improve precision.

  • Fractional Calculus: Enhances modeling accuracy by capturing complex dynamics.

Literature Insights

• Anthrax is enzootic in parts of Asia and Africa, with occasional outbreaks in Europe, Australia, and America.

• Herbivores like cattle and sheep are most susceptible; dwarf pigs and certain sheep are relatively resistant.

• Chaotic and fractional-fractional systems, combined with neural network simulations, are increasingly used to study disease dynamics, stability, and numerical solutions of anthrax models.

• Various stochastic, singular, and fractional-order numerical methods have been applied across different biological and engineering systems, demonstrating the versatility of these approaches.

Key Takeaways

1. Anthrax is a persistent, high-risk zoonotic disease due to spore survival in the environment.

2. Multiple transmission routes necessitate careful control and preventive measures.

3. Mathematical and computational modeling, especially using fractional calculus and neural networks, is crucial for understanding and controlling anthrax outbreaks in animals.

4. Integrating modern computational methods with epidemiology can improve prediction, management, and prevention strategies.

Conclusion

To understand and predict anthrax outbreaks, researchers use mathematical models. These models help simulate the spread of anthrax under different conditions, evaluate control measures, and estimate infection risks [9, 10]. Some common types of models include: 1) SIR Model (Susceptible-Infected-Recovered) o This is a basic compartmental model that classifies individuals into three categories: ? Susceptible (S): Individuals who can contract the disease. ? Infected (I): Individuals currently carrying and spreading the infection. ? Recovered (R): Individuals who have survived the infection and developed immunity. o The SIR model uses differential equations to predict how an anthrax outbreak might progress over time. 2) SEIR Model (Susceptible-Exposed-Infected-Recovered) o This model is an extension of the SIR model and includes an exposed (E) category. Since anthrax has an incubation period before symptoms appear, this model accounts for individuals who have been exposed to the spores but are not yet infectious. 3) Environmental Persistence Models o These models consider how anthrax spores survive in the environment, how they resurface after environmental disturbances, and how transmission cycles between animals and humans occur over long periods. The anthrax disease mathematical model used in animals is one of the nonlinear and complicated differential system, which contains ten different classes and the numerical solutions have been presented by using the proposed stochastic solver. The fractional operator Caputo derivative is used to get more precise solutions of the anthrax disease mathematical model used in animals. An artificial neural network approach using the consistent supervised Bayesian regularization, scale conjugate gradient and Levenburg Backpropogation methods are applied effectively to solve the anthrax disease mathematical model used in animals. An activation sigmoid function along with single layer structure and multiple layers is used in the hidden layers to get the numerical results of the anthrax disease mathematical model used in animals. The reliability of the scheme is observed through the overlapping of the solutions as well as small absolute error values.

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Copyright © 2025 Vikash Panthi, Dr. Nikita Kashyap, Dr. Manoj Gupta. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.