“Mycobacterium tuberculosis” is the bacterium that causes “tuberculosis” (TB). Although the lungs are usually affected, other organs like the spine, brain, as well as kidneys may also be affected. Droplets containing germs are released into the air and consumed by others when an infected person coughs, sneezes, or talks. The bacteria-containing particles are inhaled by others when an infected person coughs, sneezes or speaks. Once within the body, tuberculosis can spread and lead to an infection. On the other hand, not all TB patients experience active illnesses. A persistent TB infection is typically the result of the immune system suppressing the illness [1]. Congenital tuberculosis patients are asymptomatic and not contagious, but if their immune systems are weakened, the bacteria may stay dormant in their organs and later emerge active. Worldwide TB Burden: By 2020, the World Health Organization (WHO) predicts that 10 million people will get TB globally. The number of TB-related deaths: According to [2], TB claimed the lives of almost 1.5 million people in 2020. People who are actively TB-positive exhibit clinical TB symptoms and run the risk of spreading the illness to others. People who have active tuberculosis may cough up blood or sputum, have a phlegm-producing cough, chest pain, exhaustion, loss of weight, night sweats, and appetite loss. Usually, symptoms get worse with time, but they can also go away on their own and come back. One of the most important methods for characterizing and comprehending systems of various degrees that result from interactions is mathematical modelling [3]. Many mathematical models were created and examined over time to explain the dynamics of tuberculosis epidemiology in the general population. A computational framework was developed in [4] to investigate the dynamics of tuberculosis transmission that takes exogenous re-infection into account. It was demonstrated that the inclusion of exogenous recurrence in the model significantly alters the disease’s qualitative dynamics, increasing the likelihood of several chronic points of equilibrium at the crucial basic reproduction number threshold. Huo et al. [5] developed and examined an unpredictable mathematical model that took into account both hospital and home therapy for TB infectious persons. It was found that home treatment significantly impacted the dynamics of TB transmission. Few research has taken into account the fractional derivative operator while simulating the dynamics of TB epidemiology. Notably, the authors in [6] investigated the issue of optimal control for the fractional-order TB infection model combining diabetic and resistant strains using the Atangana–Baleanu–Caputo (ABC) derivative operator. A discrete fractional framework for tuberculosis in the form of the Caputo derivative was proposed by Altaf Khan et al. [7], who also looked into how the model parameters affected the system’s behavior. A TB model with continuous control measures using the Caputo fractional operator was examined by Owolabi et al. [8]. The scientists concluded that a model that included a fractional order component could provide a good control measure against the spread of tuberculosis in the population after conducting some numerical tests. Using the Laguerre polynomial, the authors in [9] applied fractional-order derivatives of Caputo type to the dynamics of TB transmission. The authors concluded that fractional-order derivatives are more capable of handling real-world scenarios than the typical classical-order models. Ullah et al. [10] analyzed a five-dimensional mathematical model that described the transmission dynamics of tuberculosis in the human population using non-integer-order derivatives of Caputo sense. In a related advancement, Farman et al. [11] used non-integer-order differential equations to create a six-dimensional system of differential equations to investigate how treatment affects the dynamics of tuberculosis transmission. In [12], the authors have done the mathematical analysis of a fractional operator-based TB model. To describe the changing patterns of TB infection in vivo, this work presents a unique model that takes host cell encounters with Mycobacterium tuberculosis into consideration. In addition to the TB virus, fractional derivatives have recently been used to study the dynamics of a recently created co-infection model of the HIV and HCV [13] as well as the dynamic behaviors of the HBV under the influence of cytokines and immune response [14].

An empirical concept called an Artificial Neural Network (ANN) aims to replicate the structure and relationships between neurons in the brain. The basic mathematical model (function) of an ANN begins with an input neuron. Three basic rules: amplification, gathering, and activation—control this model. At the time of termination of an artificial neuron, the sum of the previous weights for connections and selectivity passes through an activation function, also known as a transfer function. When ANNs are connected, their full potential and decoding capability become apparent, even though their physics and basic set of rules may not seem very remarkable. These ANNs are predicated on the idea that variation might result from a few straightforward concepts. Instead of just making the system complex and unmanageable, we often don’t link these artificial neurons carelessly to exactly benefit from the computational demands that may be met by connecting numerous ANs. Several “highly standardised” surface properties for ANNs have previously been produced by researchers. Problem solving may be facilitated by these preset traits, which have the potential to be even more successful. Various problems can be solved more successfully with different ANN topographies. Determine the type of problem that has to be solved before selecting and properly implementing the ANN’s design. It is necessary to modify the architecture and its components. The ANN is a necessary component of every ANN. Its form and function were modelled after the genuine neuron, which is thought to be the basic building component of biological NNs (operations) such as the cerebral cortex, brain stem, and surrounding glands. Recently, ANN approach has been applied to investigate the plant virus [15] and the Stuxnet virus [16].

a) Objective:

All of the above amazing uses encourage authors to explore the machine learning paradigm by developing a Bayesian regularization approach for fractional models of the TB virus (FMTBV) based on neural network (BRA-NNs). The developed determining BRA-NNs plan has the following aims in the future:

The findings obtained using the BRA-NNs approach are compared to the exact solutions for six different FMTBV variants.

b) Layout:

The study is arranged in the form that an orderly and step-by-step introduction of concepts and methods used closely follows after the results and conclusions. Section 2, first of all, a general fundamental overview is needed regarding some definitions of fractional derivatives, which are now considered to be necessary when the modelization of complex dynamic systems requires it, followed then by the introduction of TB virus fractional model, that mathematically frames the subject of the study. The discretization techniques used in the analysis are further explained in Section 3, where the Grünwald-Letnikov scheme for the discretization of fractional derivatives and BRA-NN, how to solve and analyze the model, are also presented. Section 4 states the results and the analysis of the simulation. Finally, Section 5 summarizes key findings and implications of further work that may be achieved for modeling fractional systems such as the TB virus, based on the possible impact of BRA-NN and GL methods.

In this section, firstly, we will discuss some basics regarding fractional integral and Caputo fractional order derivative. After we will discuss the methodology that is Bayesian regularization. a)

Basic Definitions:

Definition 1: For a function g: Ŗ+→Ŗ, the fractional integral of order ᾶ > 0 is defined by Іʈᾶ(g(ʈ))=1ϓ(ᾶ)∫0ʈ(ʈ−Ẋ)ᾶ−1g(Ẋ) dẊ

The Gamma function is denoted by ϒ here and elsewhere.

Definition 2: The following gives the Caputo fractional order derivative for the function g ϵ ʗn of order ᾶ: Đʈᾶc(g(ʈ))=Іn−ᾶĐng(ʈ)=1ϓ(n−ᾶ)∫0ʈgn(Ẋ)(ʈ−Ẋ)ᾶ+n−1dẊ

This is clearly defined for functions that are absolutely continuous and n−1<ᾶ<nεÑ. Note that all values of g(n), for Ẋϵ[0,ʈ], are included in the value of the Caputo fractional derivative of the function g at point t. It is obvious that cĐʈᾶ(g(ʈ)) tends to g(ʈ) as ᾶ→1.

Definition 3: The constant ẓ∗ is the system’s equilibrium point in the below-mentioned Caputo fractional dynamical system. Đʈᾶcẓ(ʈ)=f(ʈ,ẓ(ʈ)),ᾶϵ(1,0)only if and when, f(ʈ,ẓ∗)=0. b)

Mathematical Model:

We offer the proposed fractional model to explain the dynamics of TB infection (FMTBV) in this section. The whole human population is separated into five epidemiological sub-divisions to develop the model: susceptible Ș(ʈ), exposed Լ(ʈ), TB active І(ʈ), under treatment Ť(ʈ), and recovered persons after treatment Ŗ(ʈ).

The system of non-linear fractional differential equations that follows provides the transmission model for TB dynamics: cĐʈᾶȘ=ᴧ−βȘIÑ−ȗȘ  cĐʈᾶԼ=βȘIÑ−(ȗ+ε)Լ+(1−n)σŤ cĐʈᾶІ=εԼ+nσŤ−(ȗ+ϓ+σ1)І cĐʈᾶŤ=ϓІ−(ȗ+σ+σ2+ξ)Ť cĐʈᾶŖ=ξŤ−ȗŖ

The starting circumstances are Ș(0)=Ș0,Լ(0)=Լ0,І(0)=І0,Ť(0)=Ť0,andŖ(0)=Ŗ0.

The parameter ᾶ ϵ [0, 1] and the fractional derivatives used in the aforementioned model are both taken in Caputo meaning. All of the functions Ș, Լ, І, Ť, and Ŗ, together with their Caputo fractional derivativses, are considered to be continuous for all ʈ ≥ 0. Table 1 provides a detailed overview of the model parameters and their fitted or estimated numerical values.

The FMTBV mathematical paradigm is discussed in this section utilizing the recommended BRA-NNs structure. The approach consists of two elements. First, the fundamental BRA-NNs controller discoveries are given. The BRA-NNs implementation approach is also used to resolve the FMTBV mathematical framework. Fig. 1 shows the single-layer layout of neurons. The “nftool” function in MATLAB provides access to the BRA-NNs processes, which use the following data configuration: 90% for training, 5% for testing, and 5% for authorization. Fig. 2 depicts the graphical presentation of the proposed scheme, i.e., BRA-NNs for FMTBV. Table A1 illustrates the simulations of BRA-NNs for FMTBV. Reference datasets are generated using FOLotkavoltera which is based on Grunwald Letnikov method. The data set for scenario I’s case I is presented in Table A2. Complete size of all data sets for each case is 200×5 with step size 0.5.

Figures relating to performance plot, i.e., mean squared error (MSE), transition state (TS), fitness curve (FC), error histogram (EH), and regression analysis (RA) are included in Figs. 3 and 4, which illustrate the BRA-NNs numerical simulation for FMTBV for scenarios I-Case III and II-Case III, respectively. In neural networks, the Mean Squared Error (MSE) loss function is frequently used to calculate the discrepancy between expected and actual output values. It measures the total discrepancy between the predicted values of the network and the exact target values. The neural network’s training function is referred to as “trainFcn”. The network’s training performance and speed can be considerably impacted by the training function that is used. Regularization is a method for keeping neural networks from overfitting. It aids in striking a balance between the network’s capacity (complexity) and adaptability to new inputs. Typically, a regularization parameter with a value between 0 and 1 is indicated. Overfitting may be avoided by using a larger regularization value, prioritizing minimizing squared biases and weights above reducing errors. A preprocessing method called normalization is used to scale data into a particular range. Normalization is used in the context of neural networks to affect the input data and target values. Normalization choices include “no,” “standard,” and “percent.” For BRA-NNs, the transition state consists of five different components: the gradient, Mu, Num parameters, sum squared parameter, and validation check. The vector of a function’s partial derivatives about its inputs is referred to as the gradient. The method of updating the model’s parameters during training is employed in neural network optimization. As a hyper parameter used in optimization methods like gradient descent, Mu often refers to the learning rate. The step size used for parameter updates is determined by the learning rate. The number of learnable weights and biases is represented by the num-parameter in the context of neural networks. In machine learning, the term “sum squared parameter” is not commonly used. Without more information, it’s difficult to give a precise explanation, however, it might be a reference to a certain formula or technique. The process of assessing a trained model’s performance using a different dataset known as the validation set is known as the validation check.

Figs. 5 and 6 show a graphic representation of the FMTBV numerical results and absolute error (AE) for all situations, each with five cases. Despite the behavior of the TB virus model, we see Figs. 5 and 6a,c,e,g,i for α=0.80,0.85,0.90,0.95,1, respectively. We can see from Fig. 5a that the number of sensitive individuals grows as α rises for scenario I, but in Fig. 6a the results fluctuated over time in scenario II. Fig. 5c,e,g,i shows that this tendency alters when α increases, leading to a rapid increase in the population in the latent, infected, treated, and recovered classes. Similarly, in Fig. 6c,e,g, the results have a chaotic behavior whereas Fig. 6i by the increase of fractional order the recovery rate got increased. The simulation’s outputs using BRA-NNs and the FOLotkaVolterra framework are essentially in line with the related AE. As can be observed, the AE for all cases is in the range of 103→10−5. A visual representation of the AEs between actual (ground truth) values and anticipated values in a dataset is called an AE graph. It is frequently employed to assess the effectiveness of a regression model whose objective is to forecast continuous values. We can learn more about the model’s performance by looking at the absolute error graph. It is preferable to have lower absolute errors and a more condensed distribution of vertical lines near 0, which show that the model’s predictions are nearer to the actual values. On the other hand, a wide range of vertical lines with big absolute errors indicates that the model might need to be tweaked or improved.

The TB-virus fractional model is discussed in this article. The purpose of this investigation is to supply a fractional order assessment using a mathematical structure with an emphasis on epidemic phenomena to deliver more reliable system efficiency. BRA-NNs are used to solve the FMTBV five-chamber design computationally. While the best under distinct conditions corresponded to different fractional orders, the orders of 0.80 and 0.95 were always the best in a comprehensive context. Such particular orders brought model precision together with computation efficiency, thus describing the subtle dynamics of the TB virus more effectively than other fractional values. The performance at 0.80 and 0.95 indicates that those orders are very well suited to the representation of the complexity inherent in the system, offering strong and reliable predictions across a variety of scenarios for simulations. The elements that follow are the main components of the mathematically calculated FMTBV findings:

Using the suggested randomized computing paradigm offered by BRA-NNs, the FMTBV simulated solution has been effectively discovered. Modifications to the settings significantly alter how FMTBV functions.

Limitations: The study’s reliance on specific fractional orders (0.80 and 0.95) limits its generalizability to other epidemic models. While BRA-NNs improve computational efficiency, the lack of comparison with other numerical methods leaves uncertainty about its optimality. Additionally, chaotic behavior in Scenario II suggests sensitivity to parameter variations, requiring further stability analysis. The absence of validation with real-world TB data and exploration of alternative AI techniques also restricts the model’s practical applicability.

In the future, the authors intend to implement AI solver based on transfer learning, physics-informing, and deep learning for epidemic systems.