A novel fractionalized investigation of tuberculosis disease

ABSTRACT

In this study, we investigate novel fractional tuberculosis model with Caputo fractional derivative. The computational solution of the fractional tuberculosis disease is obtained with the help of the generalized Euler's method (GEM). This article considers two treatment strategies: protective treatment for latent populations and main treatment for infected ones. The compartmental structure of the six-dimensional model includes the susceptible, latent, infected, recovered, and treatment classes. To examine the nature of differential equations of fractional order that arise in biological illness, the findings produced utilizing the method under consideration are more accurate and straightforward. Additionally, the stability of the equilibrium point is also investigated. The fractional model has been graphically simulated using MATLAB22.

KEYWORDS:

• Tuberculosis disease
• Caputo fractional derivative
• Mittag-Leffler function
• generalized Euler's method

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 26A33
• 33E12
• 92D30
• 92-10

1. Introduction

For millennia, humans have struggled against epidemics, resulting in the deaths of countless individuals. These epidemics occasionally take the shape of diseases like smallpox, HIV, plague, avian flu, and seasonal flu. Tuberculosis (TB) is one of these epidemics that has gained widespread recognition. Annually, millions succumb to airborne infectious TB, causing mortality. Mycobacterium tuberculosis (MTB) causes tuberculosis, a chronic bacterial and infectious disease. A TB patient's sputum or Bacillus-laden droplets spat while coughing are the most common means of transmission. Although TB is a multi-organ illness, it most commonly affects the lungs and the lymph nodes in the chest cavity (mediastinum). The sickness has been in the population for a very long time, dating back to ancient Egypt, China, and India [1]. There are two main categories of tuberculosis infection:

Latent TB infection: The immune system controls germs, preventing illness. Latent TB infection is asymptomatic and cannot spread. If the immune system is impaired, the bacteria can become active and develop TB.

Active tuberculosis: This condition develops when the immune system is unable to keep the bacteria under control, enabling them to grow and lead to active illness. Individuals who have active TB experience symptoms and can transmit the germs to others.

TB is conveyed by airborne droplets laden with bacteria and is passed from one sick person to another. When a person spits, talks, coughs, or sneezes while carrying active TB germs that spread into air. Getting TB from relatives and friends is simple. When someone inhales a few TB bacterium germs, they get infected extremely quickly. Ninety percent of those with TB infection stay latent, whereas only around ten percent develop active TB infection. People who are latently infected do not spread the TB virus. People with TB infection might be identified using skin or blood testing. People who have immune-compromised illnesses (such as HIV patients and diabetics) are more susceptible to TB disease. Symptoms indicative of active TB encompass persistent cough exceeding three weeks, elevated body temperature, hemoptysis, chest unease, weariness, and nocturnal perspiration. Although TB is a disease that may be treated with medication, taking the wrong medication or administering it insufficiently might be cause a TB condition that is strictly resistant to treatment. Furthermore, the BCG vaccine is an easy way to prevent tuberculosis, but its usefulness is still up for debate, with only a 50% testified success rate [2]. Despite significant efforts in TB prevention and treatment, a latent TB infection afflicts roughly 25% of the global populace, ensuring a persistent wellspring for active TB cases.

Every year on March 24, ‘World Tuberculosis Day’ is observed to honour Robert Koch's discovered of MTB Bacillus, which put a stop to the world's TB pandemic and paved the path for identification and treatment of the illness. The danger of TB is growing globally along with the rate of HIV infection. Every day, 4,500 people die from TB, which affects 30,000 people worldwide. Since 2000, TB has been decreased by 42 percent globally and 54 million lives have been spared. The World Health Organization had put into effect both its ‘European Region Tuberculosis Action Plan 2016–2020’ and its ‘Worldwide Tuberculosis Eradication Plan’, which outline the actions to be performed after 2015. The goal in this situation is to decrease TB incidence by 90 percent worldwide by 2030.

Fractional calculus expands the realm of applied mathematics by accommodating fractional orders beyond integer calculus. This extension enriches mathematical modelling approaches, particularly in capturing complex dynamics in various fields. Its versatility offers insights into intricate phenomena, enhancing our understanding and predictive capabilities. In engineering, water resources management, machine learning, cryptography, and epidemiology, fractional calculus has recently attracted substantial interest in the scientific community [3–6]. Memory effect is present in fractional calculus, which plays a vital role in understanding various phenomena. The application of fractional derivatives has gained significant importance in epidemiological modelling. Adnan et al. [7] investigated time-varying COVID-19 mathematical models with singular kernels, Qu et al. [8] investigated a fractal-fractional mathematical model of tuberculosis, and Bhatter et al. [9] investigated calcium buffering using Hilfer fractional derivatives, among others.

Mathematical models have a crucial role in analyzing disease propagation behaviour and the formation of effective methods for the control of infectious diseases. Mathematical modelling is a significant tool for understanding and eventually managing such diseases [10–12]. The genesis, propagation, and decay of waves are studied by a wide variety of models, many of which centre on transmission dynamics. In the study of infectious illnesses, mathematical models are crucial instruments that offer essential insights into disease transmission and management. Researchers have created a plethora of models to investigate the dynamics of tuberculosis. These models may aid in the search for measures that can stop the spread of tuberculosis and lessen its influence on society at large. Mathematical models can analyze data on TB transmission and illness development to forecast TB incidence and direct public health actions. These models are crucial for comprehending the intricate dynamics of TB and creating evidence-based TB prevention strategies.

The potential of first mathematical model introduced by Waaler et al. [13] showcased through three illustrative instances that these examples elucidate its utility in forecasting the propensity of tuberculosis within a specific region, be it through abrupt and spontaneous emergence or through the implementation of specific control initiatives. These specific cases illustrate the model's utility in assessing diverse control approaches while showcasing TB's impact on its innate predisposition. Castillo-Chavet and Feng [14] demonstrated discrepancies between two TB patients with drug-resistant TB and those without. Liu and Zhang [15] investigated the influence of therapy and vaccination on the spread of TB using a newly developed deterministic mathematical model. In the research of children in 22 countries with the highest rates of tuberculosis cases, Dodd et al. [16] established a baseline for the country-level rates of tuberculosis cases and created guidelines to avoid TB cases at these rates. A model was constructed in another study and compared with data from Senegal to describe the effects of a public health education program for TB. The authors of [17] took into account two distinct therapeutic approaches. One emphasizes addressing latent tuberculosis infection in the elderly alongside existing tuberculosis control strategies, while the other involves providing anti-TB medications to individuals already infected, whereas Schulzer et al. [18] proposed a mathematical model to explore the accelerating impact of HIV infection on TB illness. It was claimed in another investigation that the impact of exogen on the qualitative dynamics of TB was excessive [19]. The probability that 10,000 clinical patients receiving various doses of moxifloxacin would achieve or surpass the level of exposure necessary to reduce their moxifloxacin resistance in TB was calculated using Monte-Carlo simulations [20]. Furthermore, it was determined that improved TB diagnostic methods had a significant influence on t-related illness and mortality rates in HIV endemic settings, and it was emphasised that as TB rates continue to rise, sophisticated diagnostic techniques should be taken into account as TB control measures [21].

Important findings over the next ten years have been obtained through analysis of the mathematical model used in research on multidrug resistant (CID) and common drug-resistant (YID) strains in South Africa, the area with the highest TB incidence globally. It shown that from 2008 to 2017, more than 7.000 instances of CID-TB might have been avoided and more than 47.000 lives may have been saved in South Africa by increasing medication sensitivity and TB awareness among adults. This results in a 47% drop in CID-TB mortality and a 17% drop in overall TB deaths [22]. According to Bowong and Tewa's [23] research, the TB system they analyzed has a single stable equilibrium and is asymptotically globally stable. They also demonstrated that, depending on the basic reproduction rate, this stable structure may exist in either disease-free or disease-affected regions. Liu and Zhang [15] examined the impact of vaccination and therapy on the spread of TB using a newly developed deterministic mathematical model. Tewa et al. [24] used the Lyapunov function to demonstrate the existence-uniqueness of linked endemic balances in quadratic forms while taking into consideration the possibility of TB-sensitive people switching from one section to another. Trauer et al. [25] the model introduced by simulated program-based responses to TB in highly endemic nations in the Asia-Pacific area and claimed that it could not be changed in accordance with the projected rate of a hit without permitting reinfection during the delay. In a different research, it was attempted to infect the fewest vulnerable individuals and to reduce the number of infected persons by using vaccination characteristics in the mathematical model [26].

January 2005–December 2012 in mathematical model created with a focus on data related to tuberculosis in China [27] fitted the pertinent data, and using the chi-square test, they determined the model's optimal parameter values. Using these parameters, they calculated the effective number of disease reproduction each year. In light of the findings, it was proposed that additional variables, such as chronological age and/or geographical structure, which takes into account sensitivity by age, be included to the model in order to evaluate the differences in how people in rural and urban regions approach TB. A mathematical model to combine several data sources and examine the dynamics of TB was investigated in this work. This research, which details the dynamics of tuberculosis in HCMC from 1996 to 2005, was the first comprehensive study on the disease there. The authors of [17] took into account two alternative treatment modalities; one of them highlights the need of treating latent TB infection (LTBI) in the elderly in addition to current TB control measures. The second is the administration of anti-TB medications to those who are infected.

The article is arranged as follows: The essential definitions are covered in the second Section. Section 3 formulates the fractionalized model based on the variables and parameters. Section 4 investigates the solution's non-negativity and the equilibrium points. Section 5 examines the stability analysis. A numerical simulation of the model is included in Section 6. Finally, Section 7 presented the main research findings.

2. Preliminaries

This section commences with a review of essential mathematical tools employed in subsequent sections.

Definition 2.1

The usual definition of the Caputo fractional derivative (CFD) of order $0<\delta <1$ is defined [28] as follows: (1) ${}^c{\mathfrak{D}}_t^{\delta }(\varphi (t))=\frac{1}{\mathrm{\Gamma }(r-\delta )}{\int }_0^t\frac{{\varphi }^r(s)}{(t-s{)}^{\delta -r+1}}\mathrm{d}s,$(1) where $r=[\delta ]+1$.

Definition 2.2

The usual definition of Laplace transform (LT) [29] is defined as follow: (2) $\mathsf{L}[\varphi (t);\mathrm{P}]=\overline{\varphi }(\mathrm{P})={\int }_0^{\mathrm{\infty }}{e}^{-\mathrm{P}t}\varphi (t)\mathrm{d}t,t\ge 0,\mathrm{\Re }(\mathrm{P})>\alpha .$(2) The inverse LT for the function $\overline{\varphi }(\mathrm{P})$ is given as follow: (3) ${\mathsf{L}}^{\mathsf{-}\mathsf{1}}[\overline{\varphi }(\mathrm{P});t]=\varphi (t)=\frac{1}{2\mathrm{\pi i}}{\int }_{\varkappa -i\mathrm{\infty }}^{\varkappa +i\mathrm{\infty }}{e}^{\mathrm{P}t}\overline{\varphi }(\mathrm{P})\mathrm{d}\mathrm{P},$(3) where ϰ is real number.

Definition 2.3

The LT of Equation (1(1) ${}^c{\mathfrak{D}}_t^{\delta }(\varphi (t))=\frac{1}{\mathrm{\Gamma }(r-\delta )}{\int }_0^t\frac{{\varphi }^r(s)}{(t-s{)}^{\delta -r+1}}\mathrm{d}s,$(1) ) is define by (4) $\mathsf{L}[{}_0^c{\mathfrak{D}}_t^{\delta }\varphi (t)](\mathrm{P})={\mathrm{P}}^{\delta }\mathsf{L}[\varphi (t)](\mathrm{P})-\sum _{j=0}^{m-1}{\mathrm{P}}^{\delta -j-1}{\varphi }^j(0),m-1<\delta <1,m\in \mathbb{N}.$(4)

Definition 2.4

The Mittag-Leffler functions of one and two parameters are defined as follows [30, 31]: $E_{\delta }(\hslash )=\sum _{\ell =0}^{\mathrm{\infty }}\frac{{\hslash }^{\ell }}{\mathrm{\Gamma }(\mathrm{\delta \ell }+1)};(\hslash ,\delta \in \mathbb{C},\mathrm{\Re }(\delta )>0),$and $E_{\delta ,\rho }(\hslash )=\sum _{\ell =0}^{\mathrm{\infty }}\frac{{\hslash }^{\ell }}{\mathrm{\Gamma }(\mathrm{\delta \ell }+\rho )};(\hslash ,\delta ,\rho \in \mathbb{C},\mathrm{\Re }(\delta )>0,\mathrm{\Re }(\rho )>0).$

3. Mathematical model

The mathematical model for TB introduced in reference [32] delineates six distinct population compartments: susceptible ($\mathrm{S}$), latent ($\mathrm{L}$), latently infected individuals under treatment (${\mathrm{T}}_1$), infected ($\mathrm{I}$), actively infected individuals under treatment (${\mathrm{T}}_2$), and recovered ($\mathrm{R}$) so that at any time t, the total human population is define by $\mathbf{N}(t)=\mathrm{S}(t)+\mathrm{L}(t)+{\mathrm{T}}_1(t)+\mathrm{I}(t)+{\mathrm{T}}_2(t)+\mathrm{R}(t)$. The parameters in the model Equation (5(5) $\begin{array}{rl}& \frac{\mathrm{d}\mathrm{S}}{\mathrm{d}t}=\mathrm{\Theta }-(\alpha \mathrm{I}+\beta \mathrm{L}+\varsigma )\mathrm{S},\\ & \frac{\mathrm{d}\mathrm{L}}{\mathrm{d}t}=\beta \mathrm{S}\mathrm{L}-(\gamma \mathrm{I}+\varsigma +f)\mathrm{L},\\ & \frac{\mathrm{d}{\mathrm{T}}_1}{\mathrm{d}t}=f\mathrm{L}-(\varsigma +m+\varkappa ){\mathrm{T}}_1,\\ & \frac{\mathrm{d}\mathrm{I}}{\mathrm{d}t}=\gamma \mathrm{L}\mathrm{I}+\alpha \mathrm{SI}-(\theta +\varsigma +\mathfrak{z}+\varkappa )\mathrm{I},\\ & \frac{\mathrm{d}{\mathrm{T}}_2}{\mathrm{d}t}=\theta \mathrm{I}-(ϵ+\varkappa +\varsigma ){\mathrm{T}}_2,\\ & \frac{\mathrm{d}\mathrm{R}}{\mathrm{d}t}=m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2+\mathfrak{z}\mathrm{I}-\varsigma \mathrm{R},\end{array}\}$(5) ) are comprehensively detailed below.

The mathematical equations are (5) $\begin{array}{rl}& \frac{\mathrm{d}\mathrm{S}}{\mathrm{d}t}=\mathrm{\Theta }-(\alpha \mathrm{I}+\beta \mathrm{L}+\varsigma )\mathrm{S},\\ & \frac{\mathrm{d}\mathrm{L}}{\mathrm{d}t}=\beta \mathrm{S}\mathrm{L}-(\gamma \mathrm{I}+\varsigma +f)\mathrm{L},\\ & \frac{\mathrm{d}{\mathrm{T}}_1}{\mathrm{d}t}=f\mathrm{L}-(\varsigma +m+\varkappa ){\mathrm{T}}_1,\\ & \frac{\mathrm{d}\mathrm{I}}{\mathrm{d}t}=\gamma \mathrm{L}\mathrm{I}+\alpha \mathrm{SI}-(\theta +\varsigma +\mathfrak{z}+\varkappa )\mathrm{I},\\ & \frac{\mathrm{d}{\mathrm{T}}_2}{\mathrm{d}t}=\theta \mathrm{I}-(ϵ+\varkappa +\varsigma ){\mathrm{T}}_2,\\ & \frac{\mathrm{d}\mathrm{R}}{\mathrm{d}t}=m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2+\mathfrak{z}\mathrm{I}-\varsigma \mathrm{R},\end{array}\}$(5)

where

Θ	=	is recruitment rate,
α	=	is the transmission rate of susceptible individuals to infected compartment,
β	=	is the proportion of susceptible populations to enter the latent TB class,
θ	=	is the proportion of individuals treated with infected individuals,
ς	=	is normal mortality rate,
f	=	is the proportion at which latent TB populations enter the consciousness class,
γ	=	is the proportion of hidden TB individuals entering the infected class,
ϰ	=	is the disease related mortality,
m	=	is the proportion of individuals recovering with preventive treatment,
ϵ	=	is the proportion of individuals recovering by treating infected individuals,
$\mathfrak{z}$	=	is the proportion of infected people recovering without treatment,
$\mathbb{A}$	=	is saturated incidence force of latent from $\mathrm{S}$,
$\mathbb{B}$	=	is saturated incidence force of infection from $\mathrm{S}$,
$\mathbb{C}$	=	is saturated incidence force of infection from $\mathrm{L}$.

3.1. The fractional model

Here we are introducing the fractional derivative to the TB infection model given as follows (6) $\begin{array}{rl}& {}^c{\mathfrak{D}}_t^{\delta }\mathrm{S}(t)=\mathrm{\Theta }-(\alpha \mathrm{I}+\beta \mathrm{L}+\varsigma )\mathrm{S},\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{L}(t)=\beta \mathrm{SL}-(f+\gamma \mathrm{I}+\varsigma )\mathrm{L},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_1(t)=f\mathrm{L}-(\varsigma +\varkappa +m){\mathrm{T}}_1,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{I}(t)=\gamma \mathrm{LI}+\alpha \mathrm{SI}-(\theta +\mathfrak{z}+\varsigma +\varkappa )\mathrm{I},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_2(t)=\theta \mathrm{I}-(ϵ+\varkappa +\varsigma ){\mathrm{T}}_2,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{R}(t)=m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2+\mathfrak{z}\mathrm{I}-\varsigma \mathrm{R},\end{array}\}$(6) with initial conditions $\mathrm{S}(0)={\mathrm{S}}_0$, $\mathrm{L}(0)={\mathrm{L}}_0$, ${\mathrm{T}}_1(0)={\mathrm{T}}_{1_0}$, $\mathrm{I}(0)={\mathrm{I}}_0$, ${\mathrm{T}}_2(0)={\mathrm{T}}_{2_0}$, $\mathrm{R}(0)={\mathrm{R}}_0$.

4. Qualitative analysis

This segment discusses the existence, uniqueness, positivity, and boundness of the system's solution (6(6) $\begin{array}{rl}& {}^c{\mathfrak{D}}_t^{\delta }\mathrm{S}(t)=\mathrm{\Theta }-(\alpha \mathrm{I}+\beta \mathrm{L}+\varsigma )\mathrm{S},\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{L}(t)=\beta \mathrm{SL}-(f+\gamma \mathrm{I}+\varsigma )\mathrm{L},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_1(t)=f\mathrm{L}-(\varsigma +\varkappa +m){\mathrm{T}}_1,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{I}(t)=\gamma \mathrm{LI}+\alpha \mathrm{SI}-(\theta +\mathfrak{z}+\varsigma +\varkappa )\mathrm{I},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_2(t)=\theta \mathrm{I}-(ϵ+\varkappa +\varsigma ){\mathrm{T}}_2,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{R}(t)=m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2+\mathfrak{z}\mathrm{I}-\varsigma \mathrm{R},\end{array}\}$(6) ).

4.1. Existence and uniqueness

The symbol ${\mathbb{R}}_{+}$ indicates the set of all non-negative real numbers and presumes that $\begin{array}{rl}\mathrm{\Lambda }& =\{(\mathrm{S},\mathrm{L},{\mathrm{T}}_1,\mathrm{I},{\mathrm{T}}_2,\mathrm{R})\\ & \in {\mathbb{R}}_{+}:\mathrm{S},\mathrm{L},{\mathrm{T}}_1,\mathrm{I},{\mathrm{T}}_2,\mathrm{R}\ge 0;\max (|\mathrm{S}|,|\mathrm{L}|,|\mathrm{T}|,|{\mathrm{T}}_1|,|{\mathrm{T}}_2|,|\mathrm{R}|)\le \mathbf{N}\}.\end{array}$In the region $\mathrm{\Lambda }\times (0,T]$, it is demonstrated [33] that the fractional order model (6(6) $\begin{array}{rl}& {}^c{\mathfrak{D}}_t^{\delta }\mathrm{S}(t)=\mathrm{\Theta }-(\alpha \mathrm{I}+\beta \mathrm{L}+\varsigma )\mathrm{S},\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{L}(t)=\beta \mathrm{SL}-(f+\gamma \mathrm{I}+\varsigma )\mathrm{L},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_1(t)=f\mathrm{L}-(\varsigma +\varkappa +m){\mathrm{T}}_1,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{I}(t)=\gamma \mathrm{LI}+\alpha \mathrm{SI}-(\theta +\mathfrak{z}+\varsigma +\varkappa )\mathrm{I},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_2(t)=\theta \mathrm{I}-(ϵ+\varkappa +\varsigma ){\mathrm{T}}_2,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{R}(t)=m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2+\mathfrak{z}\mathrm{I}-\varsigma \mathrm{R},\end{array}\}$(6) ) solution exists and is unique.

Theorem 4.1

If ${\mathrm{ð}}_0=(\mathrm{S}(0),\mathrm{L}(0),{\mathrm{T}}_1(0),\mathrm{I}(0),{\mathrm{T}}_2(0),\mathrm{R}(0))\in \mathrm{\Lambda }$ is a starting condition, then the fractional order model (6(6) $\begin{array}{rl}& {}^c{\mathfrak{D}}_t^{\delta }\mathrm{S}(t)=\mathrm{\Theta }-(\alpha \mathrm{I}+\beta \mathrm{L}+\varsigma )\mathrm{S},\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{L}(t)=\beta \mathrm{SL}-(f+\gamma \mathrm{I}+\varsigma )\mathrm{L},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_1(t)=f\mathrm{L}-(\varsigma +\varkappa +m){\mathrm{T}}_1,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{I}(t)=\gamma \mathrm{LI}+\alpha \mathrm{SI}-(\theta +\mathfrak{z}+\varsigma +\varkappa )\mathrm{I},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_2(t)=\theta \mathrm{I}-(ϵ+\varkappa +\varsigma ){\mathrm{T}}_2,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{R}(t)=m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2+\mathfrak{z}\mathrm{I}-\varsigma \mathrm{R},\end{array}\}$(6) ) solution $\mathrm{ð}=(\mathrm{S},\mathrm{L},{\mathrm{T}}_1,\mathrm{I},{\mathrm{T}}_2,\mathrm{R})\in \mathrm{\Lambda }$ is unique for $t\le 0$.

Proof.

$\begin{array}{rl}& {\mathcal{G}}_1(\mathrm{ð})=\mathrm{\Theta }-(\mathbb{A}+\mathbb{B}+\varsigma )\mathrm{S},\\ & {\mathcal{G}}_2(\mathrm{ð})=\mathbb{A}\mathrm{S}-(f+\mathbb{C}+\varsigma )\mathrm{L},\\ & {\mathcal{G}}_3(\mathrm{ð})=f\mathrm{L}-(\varsigma +m+\varkappa ){\mathrm{T}}_1,\\ & {\mathcal{G}}_4(\mathrm{ð})=\mathbb{C}\mathrm{L}+\mathbb{B}\mathrm{S}-(\mathfrak{z}+\theta +\varsigma +\varkappa )\mathrm{I},\\ & {\mathcal{G}}_5(\mathrm{ð})=\theta \mathrm{I}-(\varsigma +ϵ+\varkappa ){\mathrm{T}}_2,\\ & {\mathcal{G}}_6(\mathrm{ð})=\mathfrak{z}\mathrm{I}+m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2-\varsigma \mathrm{R}.\end{array}$Where, $\mathbb{A}=\beta \mathrm{L}$, $\mathbb{B}=\alpha \mathrm{I}$ and $\mathbb{C}=\gamma \mathrm{I}$.

Here, for $\mathrm{ð},\overline{\mathrm{ð}}\in \mathrm{\Lambda }$, one obtains $\begin{array}{rl}& \Vert \mathcal{G}(\overline{\mathrm{ð}})-\mathcal{G}(\mathrm{ð})\Vert =\Vert {\mathcal{G}}_1(\overline{\mathrm{ð}})-{\mathcal{G}}_1(\mathrm{ð})\Vert +\Vert {\mathcal{G}}_2(\overline{\mathrm{ð}})-{\mathcal{G}}_2(\mathrm{ð})\Vert +\Vert {\mathcal{G}}_3(\overline{\mathrm{ð}})-{\mathcal{G}}_3(\mathrm{ð})\Vert \\ & +\Vert {\mathcal{G}}_4(\overline{\mathrm{ð}})-{\mathcal{G}}_4(\mathrm{ð})\Vert +\Vert {\mathcal{G}}_5(\overline{\mathrm{ð}})-{\mathcal{G}}_5(\mathrm{ð})\Vert +\Vert {\mathcal{G}}_6(\overline{\mathrm{ð}})-{\mathcal{G}}_6(\mathrm{ð})\Vert ,\\ & \le (2\mathbb{A}+2\mathbb{B}+\varsigma )|\overline{\mathrm{S}}-\mathrm{S}|+(2\mathbb{C}+2f+\varsigma )|\overline{\mathrm{L}}-\mathrm{L}|+(2m+\varsigma +\varkappa )|{\overline{\mathrm{T}}}_1-{\mathrm{T}}_1|\\ & +(2\theta +2\mathfrak{z}+\varsigma +\varkappa )|\overline{\mathrm{I}}-\mathrm{I}|+(2ϵ+\varkappa +\varsigma )|{\overline{\mathrm{T}}}_2-{\mathrm{T}}_2|+\varsigma |\overline{\mathrm{R}}-\mathrm{R}|\\ & \le \mathcal{M}\Vert \overline{\mathrm{ð}}-\mathrm{ð}\Vert ,\end{array}$where $\mathcal{M}=\max \{(2\mathbb{A}+2\mathbb{B}+\varsigma ),(2\mathbb{C}+2f+\varsigma ),(2m+\varsigma +\varkappa ),(2\theta +2\mathfrak{z}+\varkappa +\varsigma ),(2ϵ+\varkappa +\varsigma ),\varsigma \}$.

As a result, $\mathcal{G}(\mathrm{ð})$ fulfills the Lipschitz condition, implying that the solution to model (6(6) $\begin{array}{rl}& {}^c{\mathfrak{D}}_t^{\delta }\mathrm{S}(t)=\mathrm{\Theta }-(\alpha \mathrm{I}+\beta \mathrm{L}+\varsigma )\mathrm{S},\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{L}(t)=\beta \mathrm{SL}-(f+\gamma \mathrm{I}+\varsigma )\mathrm{L},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_1(t)=f\mathrm{L}-(\varsigma +\varkappa +m){\mathrm{T}}_1,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{I}(t)=\gamma \mathrm{LI}+\alpha \mathrm{SI}-(\theta +\mathfrak{z}+\varsigma +\varkappa )\mathrm{I},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_2(t)=\theta \mathrm{I}-(ϵ+\varkappa +\varsigma ){\mathrm{T}}_2,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{R}(t)=m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2+\mathfrak{z}\mathrm{I}-\varsigma \mathrm{R},\end{array}\}$(6) ) exists and is unique.

4.2. Non-negativity and boundness of solution

Theorem 4.2

Suppose the primary solution of our model is $\{\mathrm{S}(0),\mathrm{L}(0),{\mathrm{T}}_1(0),\mathrm{I}(0),$ ${\mathrm{T}}_2(0),\mathrm{R}(0)\ge 0\}\in \mathrm{\Lambda }$. It follows that the solution set {$\mathrm{S}(t),\mathrm{L}(t),{\mathrm{T}}_1(t),\mathrm{I}(t),{\mathrm{T}}_2(t),\mathrm{R}(t)$} for the system (6(6) $\begin{array}{rl}& {}^c{\mathfrak{D}}_t^{\delta }\mathrm{S}(t)=\mathrm{\Theta }-(\alpha \mathrm{I}+\beta \mathrm{L}+\varsigma )\mathrm{S},\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{L}(t)=\beta \mathrm{SL}-(f+\gamma \mathrm{I}+\varsigma )\mathrm{L},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_1(t)=f\mathrm{L}-(\varsigma +\varkappa +m){\mathrm{T}}_1,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{I}(t)=\gamma \mathrm{LI}+\alpha \mathrm{SI}-(\theta +\mathfrak{z}+\varsigma +\varkappa )\mathrm{I},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_2(t)=\theta \mathrm{I}-(ϵ+\varkappa +\varsigma ){\mathrm{T}}_2,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{R}(t)=m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2+\mathfrak{z}\mathrm{I}-\varsigma \mathrm{R},\end{array}\}$(6) ) remains non-negative for all $t\ge 0$.

Proof.

The first equation of (6(6) $\begin{array}{rl}& {}^c{\mathfrak{D}}_t^{\delta }\mathrm{S}(t)=\mathrm{\Theta }-(\alpha \mathrm{I}+\beta \mathrm{L}+\varsigma )\mathrm{S},\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{L}(t)=\beta \mathrm{SL}-(f+\gamma \mathrm{I}+\varsigma )\mathrm{L},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_1(t)=f\mathrm{L}-(\varsigma +\varkappa +m){\mathrm{T}}_1,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{I}(t)=\gamma \mathrm{LI}+\alpha \mathrm{SI}-(\theta +\mathfrak{z}+\varsigma +\varkappa )\mathrm{I},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_2(t)=\theta \mathrm{I}-(ϵ+\varkappa +\varsigma ){\mathrm{T}}_2,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{R}(t)=m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2+\mathfrak{z}\mathrm{I}-\varsigma \mathrm{R},\end{array}\}$(6) ) yields $\begin{array}{rl}& {}^c{\mathfrak{D}}_t^{\delta }\mathrm{S}(t)=\mathrm{\Theta }-(\mathbb{A}+\mathbb{B}+\varsigma )\mathrm{S}(t),\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{S}(t)\ge -(\mathbb{A}+\mathbb{B}+\varsigma )\mathrm{S}(t).\end{array}$Applying Laplace transformation, $\begin{array}{rl}& {\mathrm{P}}^{\delta }\mathrm{S}(\mathrm{P})-\sum _{k=0}^{n-1}{\mathrm{P}}^{\delta -k-1}{\mathrm{S}}^k(0)\ge -(\mathbb{A}+\mathbb{B}+\varsigma )\mathrm{S}(\mathrm{P}),\\ & \mathrm{S}(\mathrm{P})\ge \sum _{k=0}^{n-1}\frac{{\mathrm{P}}^{\delta -k-1}}{{\mathrm{P}}^{\delta }+(\mathbb{A}+\mathbb{B}+\varsigma )}{\mathrm{S}}^k(0),\\ & \mathrm{S}(\mathrm{P})\ge \sum _{k=0}^{n-1}\mathsf{L}[t^k{E}_{\delta ,k+1}(-(\mathbb{A}+\mathbb{B}+\varsigma )t^{\delta })]{\mathrm{S}}^k(0),\mathrm{for}n=1\\ & \mathrm{S}(\mathrm{P})\ge \mathsf{L}[E_{\delta ,1}(-(\mathbb{A}+\mathbb{B}+\varsigma )t^{\delta })]\mathrm{S}(0).\end{array}$Taking inverse LT $\begin{array}{rl}& \mathrm{S}(t)\ge E_{\delta ,1}(-(\mathbb{A}+\mathbb{B}+\varsigma )t^{\delta })\mathrm{S}(0),\delta \to 1\\ & \mathrm{S}(t)\ge \exp (-(\mathbb{A}+\mathbb{B}+\varsigma )t)\mathrm{S}(0)\ge 0,\\ & \mathrm{S}(t)\ge 0.\end{array}$Similarly $\mathrm{L}(t)\ge 0,{\mathrm{T}}_1(t)\ge 0,\mathrm{I}(t)\ge 0,{\mathrm{T}}_2(t)\ge 0,\mathrm{R}(t)\ge 0.$

As a result, for any $t\ge 0$, the solution set $\{\mathrm{S}(t),\mathrm{L}(t),{\mathrm{T}}_1(t),\mathrm{I}(t),{\mathrm{T}}_2(t),\mathrm{R}(t)\}$ of system (6(6) $\begin{array}{rl}& {}^c{\mathfrak{D}}_t^{\delta }\mathrm{S}(t)=\mathrm{\Theta }-(\alpha \mathrm{I}+\beta \mathrm{L}+\varsigma )\mathrm{S},\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{L}(t)=\beta \mathrm{SL}-(f+\gamma \mathrm{I}+\varsigma )\mathrm{L},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_1(t)=f\mathrm{L}-(\varsigma +\varkappa +m){\mathrm{T}}_1,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{I}(t)=\gamma \mathrm{LI}+\alpha \mathrm{SI}-(\theta +\mathfrak{z}+\varsigma +\varkappa )\mathrm{I},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_2(t)=\theta \mathrm{I}-(ϵ+\varkappa +\varsigma ){\mathrm{T}}_2,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{R}(t)=m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2+\mathfrak{z}\mathrm{I}-\varsigma \mathrm{R},\end{array}\}$(6) ) is positive.

Theorem 4.3

The model (6(6) $\begin{array}{rl}& {}^c{\mathfrak{D}}_t^{\delta }\mathrm{S}(t)=\mathrm{\Theta }-(\alpha \mathrm{I}+\beta \mathrm{L}+\varsigma )\mathrm{S},\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{L}(t)=\beta \mathrm{SL}-(f+\gamma \mathrm{I}+\varsigma )\mathrm{L},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_1(t)=f\mathrm{L}-(\varsigma +\varkappa +m){\mathrm{T}}_1,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{I}(t)=\gamma \mathrm{LI}+\alpha \mathrm{SI}-(\theta +\mathfrak{z}+\varsigma +\varkappa )\mathrm{I},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_2(t)=\theta \mathrm{I}-(ϵ+\varkappa +\varsigma ){\mathrm{T}}_2,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{R}(t)=m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2+\mathfrak{z}\mathrm{I}-\varsigma \mathrm{R},\end{array}\}$(6) ) has uniformly bounded solution begins in ${\mathrm{\Lambda }}^{+}=\{(\mathrm{S},\mathrm{L},{\mathrm{T}}_1,\mathrm{I},{\mathrm{T}}_2,\mathrm{R})\in \mathrm{\Lambda };0\le \mathrm{S}+\mathrm{L}+{\mathrm{T}}_1+\mathrm{I}+{\mathrm{T}}_2+\mathrm{R}\le \frac{\mathrm{\Theta }}{\varsigma }\}.$

Proof.

In equation $\mathbf{N}(t)$ symbolizes the overall population at time t,

$\mathbf{N}(t)=\mathrm{S}(t)+\mathrm{L}(t)+{\mathrm{T}}_1(t)+\mathrm{I}(t)+{\mathrm{T}}_2(t)+\mathrm{R}(t)$.

By applying a direct sum to the equations in TB model (6(6) $\begin{array}{rl}& {}^c{\mathfrak{D}}_t^{\delta }\mathrm{S}(t)=\mathrm{\Theta }-(\alpha \mathrm{I}+\beta \mathrm{L}+\varsigma )\mathrm{S},\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{L}(t)=\beta \mathrm{SL}-(f+\gamma \mathrm{I}+\varsigma )\mathrm{L},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_1(t)=f\mathrm{L}-(\varsigma +\varkappa +m){\mathrm{T}}_1,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{I}(t)=\gamma \mathrm{LI}+\alpha \mathrm{SI}-(\theta +\mathfrak{z}+\varsigma +\varkappa )\mathrm{I},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_2(t)=\theta \mathrm{I}-(ϵ+\varkappa +\varsigma ){\mathrm{T}}_2,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{R}(t)=m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2+\mathfrak{z}\mathrm{I}-\varsigma \mathrm{R},\end{array}\}$(6) ) of fractional order, we obtain $\begin{array}{rl}{}^c{\mathfrak{D}}^{\delta }\mathbf{N}(t)& =\mathrm{\Theta }-(\varsigma \mathrm{S}(t)+\varsigma \mathrm{L}(t)+(\varkappa +\varsigma ){\mathrm{T}}_1(t)\\ & +(\varkappa +\varsigma )\mathrm{I}(t)+(\varsigma +\varkappa ){\mathrm{T}}_2(t)+\varsigma \mathrm{R}(t))\\ & =\mathrm{\Theta }-\varsigma \mathbf{N}(t)-\varkappa (\mathrm{I}(t)+{\mathrm{T}}_1(t)+{\mathrm{T}}_2(t))\\ & \le \mathrm{\Theta }-\varsigma \mathbf{N}(t).\end{array}$Thus ${}^c{\mathfrak{D}}^{\delta }\mathbf{N}(t)+\varsigma \mathbf{N}(t)\le \mathrm{\Theta }.$Following to [34], we get $0\le \mathbf{N}(t)\le \mathbf{N}(0)E_{\delta }(-\varsigma t^{\varsigma })+\frac{\mathrm{\Theta }}{\varsigma }(1-E_{\delta ,1}(-\varsigma t^{\delta })).$In accordance with [34], one gets $0\le \mathbf{N}(t)\le \frac{\mathrm{\Theta }}{\varsigma },t\to \mathrm{\infty },$As a result, the solution to the TB model (6(6) $\begin{array}{rl}& {}^c{\mathfrak{D}}_t^{\delta }\mathrm{S}(t)=\mathrm{\Theta }-(\alpha \mathrm{I}+\beta \mathrm{L}+\varsigma )\mathrm{S},\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{L}(t)=\beta \mathrm{SL}-(f+\gamma \mathrm{I}+\varsigma )\mathrm{L},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_1(t)=f\mathrm{L}-(\varsigma +\varkappa +m){\mathrm{T}}_1,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{I}(t)=\gamma \mathrm{LI}+\alpha \mathrm{SI}-(\theta +\mathfrak{z}+\varsigma +\varkappa )\mathrm{I},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_2(t)=\theta \mathrm{I}-(ϵ+\varkappa +\varsigma ){\mathrm{T}}_2,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{R}(t)=m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2+\mathfrak{z}\mathrm{I}-\varsigma \mathrm{R},\end{array}\}$(6) ) remains uniformly bounded in the region ${\mathrm{\Lambda }}^{+}$, starting in ${\mathrm{\Lambda }}^{+}$.

5. Stability analysis

The fundamental reproductive number ${\mathcal{R}}_0$ has been determined in this section. The infectious-free equilibrium point's, local asymptotic stability is also investigated.

5.1. The infectious-free-equilibrium (IFE) point

To determine the fractionalized TB model (6(6) $\begin{array}{rl}& {}^c{\mathfrak{D}}_t^{\delta }\mathrm{S}(t)=\mathrm{\Theta }-(\alpha \mathrm{I}+\beta \mathrm{L}+\varsigma )\mathrm{S},\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{L}(t)=\beta \mathrm{SL}-(f+\gamma \mathrm{I}+\varsigma )\mathrm{L},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_1(t)=f\mathrm{L}-(\varsigma +\varkappa +m){\mathrm{T}}_1,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{I}(t)=\gamma \mathrm{LI}+\alpha \mathrm{SI}-(\theta +\mathfrak{z}+\varsigma +\varkappa )\mathrm{I},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_2(t)=\theta \mathrm{I}-(ϵ+\varkappa +\varsigma ){\mathrm{T}}_2,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{R}(t)=m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2+\mathfrak{z}\mathrm{I}-\varsigma \mathrm{R},\end{array}\}$(6) ) equilibrium points, we choose to: (7) $\begin{array}{rl}& \mathrm{\Theta }-(\alpha \mathrm{I}+\varsigma +\beta \mathrm{L})\mathrm{S}=0,\\ & \beta \mathrm{SL}-(f+\gamma \mathrm{I}+\varsigma )\mathrm{L}=0,\\ & f\mathrm{L}-(\varsigma +m+\varkappa ){\mathrm{T}}_1=0,\\ & \gamma \mathrm{LI}+\alpha \mathrm{SI}-(\varsigma +\mathfrak{z}+\theta +\varkappa )\mathrm{I}=0,\\ & \theta \mathrm{I}-(ϵ+\varsigma +\varkappa ){\mathrm{T}}_2=0,\\ & m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2+\mathfrak{z}\mathrm{I}-\varsigma \mathrm{R}=0.\end{array}\}$(7) Model (6(6) $\begin{array}{rl}& {}^c{\mathfrak{D}}_t^{\delta }\mathrm{S}(t)=\mathrm{\Theta }-(\alpha \mathrm{I}+\beta \mathrm{L}+\varsigma )\mathrm{S},\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{L}(t)=\beta \mathrm{SL}-(f+\gamma \mathrm{I}+\varsigma )\mathrm{L},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_1(t)=f\mathrm{L}-(\varsigma +\varkappa +m){\mathrm{T}}_1,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{I}(t)=\gamma \mathrm{LI}+\alpha \mathrm{SI}-(\theta +\mathfrak{z}+\varsigma +\varkappa )\mathrm{I},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_2(t)=\theta \mathrm{I}-(ϵ+\varkappa +\varsigma ){\mathrm{T}}_2,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{R}(t)=m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2+\mathfrak{z}\mathrm{I}-\varsigma \mathrm{R},\end{array}\}$(6) ) is stable when no TB exists in the population under investigation. That is when ${\mathrm{L}}^0={\mathrm{I}}^0=0$.

Hence, the IFE denoted by ${\beth }^0$, of the model (6(6) $\begin{array}{rl}& {}^c{\mathfrak{D}}_t^{\delta }\mathrm{S}(t)=\mathrm{\Theta }-(\alpha \mathrm{I}+\beta \mathrm{L}+\varsigma )\mathrm{S},\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{L}(t)=\beta \mathrm{SL}-(f+\gamma \mathrm{I}+\varsigma )\mathrm{L},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_1(t)=f\mathrm{L}-(\varsigma +\varkappa +m){\mathrm{T}}_1,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{I}(t)=\gamma \mathrm{LI}+\alpha \mathrm{SI}-(\theta +\mathfrak{z}+\varsigma +\varkappa )\mathrm{I},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_2(t)=\theta \mathrm{I}-(ϵ+\varkappa +\varsigma ){\mathrm{T}}_2,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{R}(t)=m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2+\mathfrak{z}\mathrm{I}-\varsigma \mathrm{R},\end{array}\}$(6) ) is produced by replacing ${\mathrm{L}}^0={\mathrm{I}}^0=0$ in above system. $\mathrm{\Theta }-\varsigma {\mathrm{S}}^0=0.$Therefore, ${\mathrm{S}}^0=\frac{\mathrm{\Theta }}{\varsigma }.$We can easily obtain the IFE ${\beth }^0=(\frac{\mathrm{\Theta }}{\varsigma },0,0,0,0,0)$.

5.2. The basic reproductive number

In the conventional meaning, the fundamental reproductive number is the anticipated number of secondary infected transmitted by one infectious individual in a completely susceptible population throughout the individual duration of infectiousness. This is a state quantity acquired from the conventional ‘next-generation matrix approach’ [35].

$(\mathrm{S}(t),\mathrm{L}(t),{\mathrm{T}}_1(t),\mathrm{I}(t),{\mathrm{T}}_2(t),\mathrm{R}(t))$ remains in the biologically significant region Λ, which is positively invariant for model (6(6) $\begin{array}{rl}& {}^c{\mathfrak{D}}_t^{\delta }\mathrm{S}(t)=\mathrm{\Theta }-(\alpha \mathrm{I}+\beta \mathrm{L}+\varsigma )\mathrm{S},\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{L}(t)=\beta \mathrm{SL}-(f+\gamma \mathrm{I}+\varsigma )\mathrm{L},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_1(t)=f\mathrm{L}-(\varsigma +\varkappa +m){\mathrm{T}}_1,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{I}(t)=\gamma \mathrm{LI}+\alpha \mathrm{SI}-(\theta +\mathfrak{z}+\varsigma +\varkappa )\mathrm{I},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_2(t)=\theta \mathrm{I}-(ϵ+\varkappa +\varsigma ){\mathrm{T}}_2,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{R}(t)=m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2+\mathfrak{z}\mathrm{I}-\varsigma \mathrm{R},\end{array}\}$(6) ). The model equations are rewritten, beginning with potentially infective classes. $\begin{array}{rl}& {}^c{\mathfrak{D}}_t^{\delta }\mathrm{L}(t)=\beta \mathrm{S}(t)\mathrm{L}(t)-(\varsigma +\gamma \mathrm{I}(t)+f)\mathrm{L}(t),\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{I}(t)=\gamma \mathrm{L}(t)\mathrm{I}(t)+\alpha \mathrm{S}(t)\mathrm{I}(t)-(\theta +\varkappa +\mathfrak{z}+\varsigma )\mathrm{I}(t).\end{array}$Then, using the next-generation matrix concept, we can gain for the model (6(6) $\begin{array}{rl}& {}^c{\mathfrak{D}}_t^{\delta }\mathrm{S}(t)=\mathrm{\Theta }-(\alpha \mathrm{I}+\beta \mathrm{L}+\varsigma )\mathrm{S},\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{L}(t)=\beta \mathrm{SL}-(f+\gamma \mathrm{I}+\varsigma )\mathrm{L},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_1(t)=f\mathrm{L}-(\varsigma +\varkappa +m){\mathrm{T}}_1,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{I}(t)=\gamma \mathrm{LI}+\alpha \mathrm{SI}-(\theta +\mathfrak{z}+\varsigma +\varkappa )\mathrm{I},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_2(t)=\theta \mathrm{I}-(ϵ+\varkappa +\varsigma ){\mathrm{T}}_2,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{R}(t)=m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2+\mathfrak{z}\mathrm{I}-\varsigma \mathrm{R},\end{array}\}$(6) ), the disease component is $x=(\mathrm{L},\mathrm{I}{)}^T$ and the non-disease component $y=(\mathrm{S},{\mathrm{T}}_1,{\mathrm{T}}_2,\mathrm{R}{)}^T$, then we have ${}^c{\mathfrak{D}}_t^{\delta }x=\mathbf{F}(x)-\mathbf{V}(x),$where $\mathbf{F}(x,y)=[\beta \mathrm{S}\mathrm{L}\alpha \mathrm{S}\mathrm{I}{]}^T,\mathbf{V}(x,y)=[(\gamma \mathrm{I}+f+\varsigma )\mathrm{L}(\theta +\varsigma +\varkappa +\mathfrak{z})\mathrm{I}-\gamma \mathrm{L}\mathrm{I}{]}^T.$The Jacobian matrix of $\mathbf{F}(x)$ and $\mathbf{V}(x)$ with regard to $\mathrm{L}$ and $\mathrm{I}$ at the IFE ${\beth }^0=(\frac{\mathrm{\Theta }}{\varsigma },0,0,0,0,0)$ is obtained next. We assigned $\mathbf{F}(x)$ and $\mathbf{V}(x)$ in the following manner in order to streamline our work: $\mathbf{F}(x,y)=[{\mathbf{F}}_1{\mathbf{F}}_2{]}^T\mathbf{V}(x,y)=[{\mathbf{V}}_1{\mathbf{V}}_2{]}^T,$where ${\mathbf{F}}_1=\beta \mathrm{S}\mathrm{L},{\mathbf{F}}_2=\alpha \mathrm{S}\mathrm{I},{\mathbf{V}}_1=(\gamma \mathrm{I}+f+\varsigma )\mathrm{L},{\mathbf{V}}_2=(\theta +\mathfrak{z}+\varsigma +\varkappa )\mathrm{I}-\gamma \mathrm{L}\mathrm{I}.$At the IFE point ${\beth }^0$, $\mathbf{F}$, and $\mathbf{V}$ respectively receive the Jacobian matrix of $\mathbf{F}(x)$ and $\mathbf{V}(x)$; $\begin{array}{c}\mathbf{F}=\left[\begin{array}{cc}{\displaystyle \frac{\beta \mathrm{\Theta }}{\varsigma }}& 0\\ 0& {\displaystyle \frac{\alpha \mathrm{\Theta }}{\varsigma }}\end{array}\right],\mathrm{and}\mathbf{V}=\left[\begin{array}{cc}\varsigma +f& 0\\ 0& (\theta +\varsigma +\mathfrak{z}+\varkappa )\end{array}\right].\end{array}$Where the matrix $\mathbf{F}$ is nonsingular and the matrix $\mathbf{V}$ is non-negative, the next-generation matrix $\mathbf{F}{\mathbf{V}}^{-1}$ is defined as using the next-generation matrix approach. This matrix's $\mathbf{F}{\mathbf{V}}^{-1}$ spectral radius is evaluated at the equilibrium point ${\beth }^0$ to compute the basic reproductive number of the disease within the system (6(6) $\begin{array}{rl}& {}^c{\mathfrak{D}}_t^{\delta }\mathrm{S}(t)=\mathrm{\Theta }-(\alpha \mathrm{I}+\beta \mathrm{L}+\varsigma )\mathrm{S},\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{L}(t)=\beta \mathrm{SL}-(f+\gamma \mathrm{I}+\varsigma )\mathrm{L},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_1(t)=f\mathrm{L}-(\varsigma +\varkappa +m){\mathrm{T}}_1,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{I}(t)=\gamma \mathrm{LI}+\alpha \mathrm{SI}-(\theta +\mathfrak{z}+\varsigma +\varkappa )\mathrm{I},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_2(t)=\theta \mathrm{I}-(ϵ+\varkappa +\varsigma ){\mathrm{T}}_2,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{R}(t)=m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2+\mathfrak{z}\mathrm{I}-\varsigma \mathrm{R},\end{array}\}$(6) ), explained by two distinct cases as ${\mathcal{R}}_{0_1}$ and ${\mathcal{R}}_{0{}_2}$. ${\mathcal{R}}_{0_1}=\frac{\beta \mathrm{\Theta }}{\varsigma (\varsigma +f)},{\mathcal{R}}_{0_2}=\frac{\alpha \mathrm{\Theta }}{\varsigma (\theta +\varsigma +\mathfrak{z}+\varkappa )}$in which ${\mathcal{R}}_0=max[{\mathcal{R}}_{0_1},{\mathcal{R}}_{0_2}]$.

5.3. Local stability of the IF steady state

Theorem 5.1

The IFE ${\beth }^0=(\frac{\mathrm{\Theta }}{\varsigma },0,0,0,0,0)$ is locally asymptotically stable if ${\mathcal{R}}_0<1$ and unstable if ${\mathcal{R}}_0>1$.

Proof.

For the model (6(6) $\begin{array}{rl}& {}^c{\mathfrak{D}}_t^{\delta }\mathrm{S}(t)=\mathrm{\Theta }-(\alpha \mathrm{I}+\beta \mathrm{L}+\varsigma )\mathrm{S},\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{L}(t)=\beta \mathrm{SL}-(f+\gamma \mathrm{I}+\varsigma )\mathrm{L},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_1(t)=f\mathrm{L}-(\varsigma +\varkappa +m){\mathrm{T}}_1,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{I}(t)=\gamma \mathrm{LI}+\alpha \mathrm{SI}-(\theta +\mathfrak{z}+\varsigma +\varkappa )\mathrm{I},\\ & {}^c{\mathfrak{D}}_t^{\delta }{\mathrm{T}}_2(t)=\theta \mathrm{I}-(ϵ+\varkappa +\varsigma ){\mathrm{T}}_2,\\ & {}^c{\mathfrak{D}}_t^{\delta }\mathrm{R}(t)=m{\mathrm{T}}_1+ϵ{\mathrm{T}}_2+\mathfrak{z}\mathrm{I}-\varsigma \mathrm{R},\end{array}\}$(6) ) at ${\beth }^0=(\frac{\mathrm{\Theta }}{\varsigma },0,0,0,0,0)$, the Jacobian matrix ${\mathbb{J}}_{{\beth }^0}$ is provided by $\begin{array}{rl}{\mathbb{J}}_{{\beth }^0}& =[\begin{array}{ccc}-\varsigma & -{\displaystyle \frac{\beta \mathrm{\Theta }}{\varsigma }}& 0\\ 0& {\displaystyle \frac{\beta \mathrm{\Theta }}{\varsigma }}-(\varsigma +f)& 0\\ 0& f& -(\varsigma +\varkappa +m)\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& m\end{array}\\ & \begin{array}{ccc}-{\displaystyle \frac{\alpha \mathrm{\Theta }}{\varsigma }}& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ {\displaystyle \frac{\alpha \mathrm{\Theta }}{\varsigma }}-(\mathfrak{z}+\theta +\varsigma +\varkappa )& 0& 0\\ \theta & -(ϵ+\varsigma +\varkappa )& 0\\ \mathfrak{z}& ϵ& -\varsigma \end{array}]\end{array}$The eigen values of Jacobian matrix ${\mathbb{J}}_{{\beth }^0}$ are ${\lambda }_1=-\varsigma$, ${\lambda }_2=\frac{\beta \mathrm{\Theta }}{\varsigma }-(\varsigma +f)$, ${\lambda }_3=-(\varsigma +m+\varkappa )$, ${\lambda }_4=\frac{\alpha \mathrm{\Theta }}{\varsigma }-(\theta +\mathfrak{z}+\varsigma +\varkappa )$, ${\lambda }_5=-(ϵ+\varkappa +\varsigma )$, ${\lambda }_6=-\varsigma$.

It is obivious that ${\lambda }_1,{\lambda }_3,{\lambda }_5,{\lambda }_6$ are negative and remaining eigen values, $\begin{array}{rl}& {\lambda }_2=\frac{\beta \mathrm{\Theta }}{\varsigma }-(f+\varsigma )=({\mathcal{R}}_{0_1}-1)(f+\varsigma )\\ & \mathrm{if}{\mathcal{R}}_{0_1}<1\mathrm{then}{\lambda }_2\mathrm{is}\mathrm{negative},\\ \mathrm{and}& {\lambda }_4=\frac{\alpha \mathrm{\Theta }}{\varsigma }-(\theta +\varsigma +\mathfrak{z}+\varkappa )=({\mathcal{R}}_{0_2}-1)(\theta +\varsigma +\varkappa +\mathfrak{z})\\ & \mathrm{if}{\mathcal{R}}_{0_2}<1\mathrm{then}{\lambda }_4\mathrm{is}\mathrm{negative}.\end{array}$It can be observed that eigen values ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5,{\lambda }_6$ are negative, which means the IFE point is locally asymtotically stable if ${\mathcal{R}}_0=max[{\mathcal{R}}_{0_1},{\mathcal{R}}_{0_2}]<1$.

6. Application of GEM

We analyze the following problem in the impression of Odibat and Momani [36], which is a generalization of Euler's classical technique. (8) ${}^c{\mathfrak{D}}_t^{\delta }\mathtt{Z}=f(t,\mathtt{Z}),t\in [0,T],0<\delta \le 1.$(8) The interval $[0,\mathsf{b}]$ over which we wish to locate the solution to the problem (8(8) ${}^c{\mathfrak{D}}_t^{\delta }\mathtt{Z}=f(t,\mathtt{Z}),t\in [0,T],0<\delta \le 1.$(8) ) can be subdivided into k subintervals $[t_i,t_{i+1}]$ of similar width $h=\frac{\mathsf{b}}{k}$, by utilizing the nodes $t_i=\mathrm{ih}$ for $i=0,1,2,\dots ,k$. Expand Z pertaining to $t=t_0=0$ and utilizing the generalized Taylor's formula [37]. Assume that $\mathtt{Z},{}^c{\mathfrak{D}}^{\delta }\mathtt{Z},{}^c{\mathfrak{D}}^{2\delta }\mathtt{Z}$ are continuous on $[0,\mathsf{b}]$. As a consequence, there is an ${\mathsf{a}}_1$ value for every t value, meaning that (9) $\mathtt{Z}(t)=\mathtt{Z}(t_0)+({}^c{\mathfrak{D}}_t^{\delta }\mathtt{Z}(t))(t_0)\frac{t^{\delta }}{\mathrm{\Gamma }(\delta +1)}+({}^c{\mathfrak{D}}_t^{2\delta }\mathtt{Z}(t))({\mathsf{a}}_1)\frac{t^{2\delta }}{\mathrm{\Gamma }(2\delta +1)}.$(9) When $({}^c{\mathfrak{D}}_t^{\delta }\mathtt{Y}(t))(t_0)=f(t_0,\mathtt{Z}(t_0))$ and $h=t_1$ are subsituted into Equation (9(9) $\mathtt{Z}(t)=\mathtt{Z}(t_0)+({}^c{\mathfrak{D}}_t^{\delta }\mathtt{Z}(t))(t_0)\frac{t^{\delta }}{\mathrm{\Gamma }(\delta +1)}+({}^c{\mathfrak{D}}_t^{2\delta }\mathtt{Z}(t))({\mathsf{a}}_1)\frac{t^{2\delta }}{\mathrm{\Gamma }(2\delta +1)}.$(9) ), the outcome is an expression for $\mathtt{Y}(t_1):$ $\mathtt{Z}(t_1)=\mathtt{Z}(t_0)+f(t_0,\mathtt{Z}(t_0))\frac{t^{\delta }}{\mathrm{\Gamma }(\delta +1)}+({}^c{\mathfrak{D}}_t^{2\delta }\mathtt{Y}(t))({\mathsf{a}}_1)\frac{t^{2\delta }}{\mathrm{\Gamma }(2\delta +1)}.$The solution can be achieved by neglecting the second-order term (involving $h^{2\delta }$) when the step size h is sufficiently small. $\mathtt{Z}(t_1)=\mathtt{Z}(t_0)+f(t_0,\mathtt{Z}(t_0))\frac{t^{\delta }}{\mathrm{\Gamma }(\delta +1)}.$Repeat the procedure until the solution $\mathtt{Z}(t)$ is approximated by a series of points. When $t_{i+1}=t_i+h$, the formula for (10) $\mathtt{Z}(t_{i+1})=\mathtt{Z}(t_i)+\frac{h^{\delta }}{\mathrm{\Gamma }(\delta +1)}f(t_i,\mathtt{Z}(t_i)),$(10) for $i=0,1,2,\dots ,k-1.$ It is evident that if $\delta =1$, then the GEM (10(10) $\mathtt{Z}(t_{i+1})=\mathtt{Z}(t_i)+\frac{h^{\delta }}{\mathrm{\Gamma }(\delta +1)}f(t_i,\mathtt{Z}(t_i)),$(10) ) simplifies to the typical Euler's technique. Hence $\begin{array}{rl}& \mathrm{S}(t_{i+1})=\mathrm{S}(t_i)+\frac{h^{\delta }}{\mathrm{\Gamma }(\delta +1)}[\mathrm{\Theta }-(\beta \mathrm{L}(t_i)+\alpha \mathrm{I}(t_i)+\varsigma )\mathrm{S}(t_i)],\\ & \mathrm{L}(t_{i+1})=\mathrm{L}(t_i)+\frac{h^{\delta }}{\mathrm{\Gamma }(\delta +1)}[\beta \mathrm{S}(t_i)\mathrm{L}(t_i)-(\gamma \mathrm{I}(t_i)+f+\varsigma )\mathrm{L}(t_i)],\\ & {\mathrm{T}}_1(t_{i+1})={\mathrm{T}}_1(t_i)+\frac{h^{\delta }}{\mathrm{\Gamma }(\delta +1)}[f\mathrm{L}(t_i)-(m+\varsigma +\varkappa ){\mathrm{T}}_1(t_i)],\\ & \mathrm{I}(t_{i+1})=\mathrm{I}(t_i)+\frac{h^{\delta }}{\mathrm{\Gamma }(\delta +1)}[\gamma \mathrm{L}(t_i)\mathrm{I}(t_i)+\alpha \mathrm{S}(t_i)\mathrm{I}(t_i)-(\theta +\mathfrak{z}+\varsigma +\varkappa )\mathrm{I}(t_i)],\\ & {\mathrm{T}}_2(t_{i+1})={\mathrm{T}}_2(t_i)+\frac{h^{\delta }}{\mathrm{\Gamma }(\delta +1)}[\theta \mathrm{I}(t_i)-(ϵ+\varsigma +\varkappa ){\mathrm{T}}_2(t_i)],\\ & \mathrm{R}(t_{i+1})=\mathrm{R}(t_i)+\frac{h^{\delta }}{\mathrm{\Gamma }(\delta +1)}[m{\mathrm{T}}_1(t_i)+ϵ{\mathrm{T}}_2(t_i)+\mathfrak{z}\mathrm{I}(t_i)-\varsigma \mathrm{R}(t_i)].\end{array}$In doing so, one obtains (11) $\begin{array}{rl}& {\mathrm{S}}_{n+1}={\mathrm{S}}_n+\mathrm{\wp }[\mathrm{\Theta }-(\beta {\mathrm{L}}_n+\alpha {\mathrm{I}}_n+\varsigma ){\mathrm{S}}_n],\\ & {\mathrm{L}}_{n+1}={\mathrm{L}}_n+\mathrm{\wp }[\beta {\mathrm{S}}_n{\mathrm{L}}_n-(\gamma {\mathrm{I}}_n+f+\varsigma ){\mathrm{L}}_n],\\ & {\mathrm{T}}_{1_{n+1}}={\mathrm{T}}_{1_n}+\mathrm{\wp }[f{\mathrm{L}}_n-(m+\varsigma +\varkappa ){\mathrm{T}}_{1_n}],\\ & {\mathrm{I}}_{n+1}={\mathrm{I}}_n+\mathrm{\wp }[\gamma {\mathrm{L}}_n{\mathrm{I}}_n+\alpha {\mathrm{S}}_n{\mathrm{I}}_n-(\theta +\mathfrak{z}+\varsigma +\varkappa ){\mathrm{I}}_n],\\ & {\mathrm{T}}_{2_{n+1}}={\mathrm{T}}_{2_n}+\mathrm{\wp }[\theta {\mathrm{I}}_n-(ϵ+\varsigma +\varkappa ){\mathrm{T}}_{2_n}],\\ & {\mathrm{R}}_{n+1}={\mathrm{R}}_n+\mathrm{\wp }[m{\mathrm{T}}_{1_n}+ϵ{\mathrm{T}}_{2_n}+\mathfrak{z}{\mathrm{I}}_n-\varsigma {\mathrm{R}}_n],\end{array}$(11) where $0<\mathrm{\wp }=\frac{h^{\delta }}{\mathrm{\Gamma }(\delta +1)}<1$ is the step size and is dependent on the initial circumstances, ${\mathrm{S}}_0$, ${\mathrm{L}}_0$, ${\mathrm{T}}_{1_0}$, ${\mathrm{I}}_0$, ${\mathrm{T}}_{2_0}$, ${\mathrm{R}}_0$. We have ${\mathrm{S}}_{n+1}={\mathrm{S}}_n=\mathrm{S},{\mathrm{L}}_{n+1}={\mathrm{L}}_n=\mathrm{L},{\mathrm{T}}_{1_{n+1}}={\mathrm{T}}_{1_n}={\mathrm{T}}_1,{\mathrm{I}}_{n+1}={\mathrm{I}}_n=\mathrm{I},{\mathrm{T}}_{2_{n+1}}={\mathrm{T}}_{2_n}={\mathrm{T}}_2,{\mathrm{R}}_{n+1}={\mathrm{R}}_n=\mathrm{R}$ at a fixed point.

7. Discussion of results

This section discusses the pivotal role of numerical data for the considered TB model with the inclusion of the consciousness component. We employed the parameters from Table 1 for the simulations in our current paper, treating them as the ‘year’ parameter. These parameter values are utilized as ‘years’ in all our simulations and graphs. The fractional model has been graphically simulated using MATLAB22 software [38].

Table 1. Parameter values.

Parameters	Values	Parameters	Values
$\mathrm{S}(0)$	30201	$\mathrm{L}(0)$	830
${\mathrm{T}}_1(0)$	0	$\mathrm{I}(0)$	801
${\mathrm{T}}_2(0)$	0	$\mathrm{R}(0)$	0
Θ	1562	α	0.0006
β	0.0005	ϵ	0.06
$\mathrm{\varkappa V}$	0.004	f	0.6
γ	0.00027	θ	0.35
m	0.5	ς	0.03
$\mathfrak{z}$	0.4

Figure 1 demonstrates for integral value of δ, the number of susceptible class decreases with time. Meanwhile, for fractional orders, it can be observed that the growth of the susceptible population is slower than the integral value one. In Figure 2 one can observe that the latent class increases rapidly at different fractional operators in beginning then decline fastly as compared to integer order but after some years become stable. In Figure 3 the same behaviour occurs for the treatment of latently infected individuals as latent class at different fractional operators.

Figure 1. Dynamical behaviour of susceptible population $\mathrm{S}(t)$ for δ.

Figure 2. Dynamical behaviour of latent population $\mathrm{L}(t)$ for δ.

Figure 3. Dynamical behaviour of latently infected treatment population ${\mathrm{T}}_1(t)$ for δ.

In Figure 4 it is evident that, in the absence of therapy and preventative measures, TB infection rises first at different fractional orders. However, with treatment and care, the infection decreased, demonstrating the stability and convergent behaviour of various fractional orders to integer orders. Figure 5 suggests that as the illness progresses, the treatment must also be increased in early phases at varied arbitrary sequences. Applying care and treatment will help decline the disease from society, as the number of afflicted people declines, so will the need for therapy. In Figure 6 shows that as the infected population decreases at different fractional order the recovered cases increases at various fractional order.

Figure 4. Dynamical behaviour of infected population $\mathrm{I}(t)$ for δ.

Figure 5. Dynamical behaviour of actively infected treatment population ${\mathrm{T}}_2(t)$ for δ.

Figure 6. Dynamical behaviour of recovered population $\mathrm{R}(t)$ for δ.

The effectiveness of parameter β is demonstrated in the Figure 7, which shows the rate of sensitive individuals enter the latent class. According to Figure 7, it is clear that the latent population grows at a fractional order relative to the integral value one as the parameter increases from 0.0005 to 0.0020. In Figure 8, we observe the influence of the parameter α, which illustrates the rate of transition of susceptible population to the infected class. If we look at Figure 8, we can deduce that when parameter increases from 0.0006 to 0.0014, then the infected population declines at fractional order as compared to integeral value. According to the Figure 9, we show that as parameter f grows from 0.55 to 0.75, in compared to an integer value, the number of TB-infected people declines more rapidly at fractional order. This finding is pivotal for discerning a parameter that helps in reducing TB incidence. Figure 10 illustrates the population density of the $\mathrm{R}$ class concerning parameter $\mathfrak{z}$. The graphic representation unmistakably demonstrates a rise in the $\mathrm{R}$ class density as the parameter values grow from 0.3 to 0.7. Figure 11 illustrates the population density of treatment for hidden infected individuals $({\mathrm{T}}_1)$ and actively infected individuals $({\mathrm{T}}_2)$ at varied fractional orders. Figure 12 shows the graphical relation of latent class and infected class at $\delta =(1.00,0.90,0.80)$. The comparison of integer value and fractional value of densities and behaviours of the model's population is illustrated in Figure 13 as population dynamics.

Figure 8. The infected class population density, $\mathrm{I}(t)$, for various values of $\alpha =0.0006,0.0008,0.00012,0.0014$ for $\delta =0.90,1.00$.

Figure 9. The infected class population density $\mathrm{I}(t)$, for various values of consciousness parameter for $\delta =0.90,1.00$.

Figure 10. The population density of recovered class $\mathrm{R}(t)$ for various values of $\mathfrak{z}$ for $\delta =0.90,1.00$.

Figure 11. The densities of ${\mathrm{T}}_1(t)$ and ${\mathrm{T}}_2(t)$ for $\delta =0.80,0.90,1.00$.

Figure 12. The proportions of infected $\mathrm{I}(t)$ and latent $\mathrm{L}(t)$ individuals for $\delta =0.80,0.90,1.00$.

Figure 13. Population densities in the TB model vary for δ values of 0.80 and 1.00.

Figure 7. The hidden TB population density $\mathrm{L}(t)$, for various values of $\beta =0.0005,0.0010,0.00015,0.0020$ for $\delta =0.90,1.00$.

8. Conclusion

In this article, we studied at a TB model ($\mathrm{SL}{T}_{1}I{T}_{2}R$) under CFD that has a useful consciousness method for the tuberculosis pandemic disease. We have emphasized the value and varied elements of consciousness with the aid of the aforementioned model. Numerous theoretical properties of the model, including its existence and uniqueness as well as its positivity, equilibria, and reproduction ratio, have been researched. For a numerical analysis utilizing the GEM, the fractionalized TB model is examined. Infection rates among the community dropped as a result of the treatment. Additionally, lowering the incidence of harmful contacts and taking preventative measures would aid in the eradication of the TB illness. The ability of the Caputo fractional derivative to accurately model key biological phenomena inherent to tuberculosis dynamics, such as non-locality, memory effects, and anomalous diffusion, which are crucial for capturing the complex dynamics of the disease spread within populations. As can be seen from the depicted figures, in comparison to the comparable integer order,the presented fractional order model in terms of the Caputo operator yields more flexible results. As a result, it is clear that the CFD provides more biologically plausible behaviour about the dynamics of the tuberculosis disease. Thus, we can conclude that CFD is a reliable technique for simulating physical occurrences. Overall, we can state that the fractional order differential equation has produced comparably better predictions than the classical derivatives. It should be highlighted that this model can be further investigated in the future by performing a comparative study using actual data and making use of various forms of fractional derivatives.

Author Contribution

Each author made an equal contribution to this paper. All authors have read and approved the final manuscript.

Acknowledgments

The authors express their sincere thanks to the editor and reviewers for their valuable comments and suggestions that improved the quality of the manuscript.

Data Availability and Access

All data generated or analyzed during this study are included in this article. Further, we declare that no human data or participants have been involved in the study.

Disclosure statement

No potential conflict of interest was reported by the author(s).