Mathematical Analysis of the TB Model with Treatment via Caputo-Type Fractional Derivative

College of Science, Hunan City University, Yiyang 413000, China Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Khyber Pakhtunkhwa, Pakistan Institute for Ground Water Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein, South Africa Department of Clinical Laboratory Sciences, College of Applied Medical Sciences, King Khalid University, P.O. Box 61413, Abha 9088, Saudi Arabia Department of Mathematics, College of Sciences, King Khalid University, Abha 61413, Saudi Arabia


Introduction
Tuberculosis (TB) usually caused by a bacterium called Mycobacterium tuberculosis bacterium (MTB) is a contagious infectious disease. is life-threatening disease is still imposing an alarming situation and health challenge across the globe, specifically for developing countries. TB is listed in the topmost death-causing diseases due to the high rate of mortality. Tuberculosis spread from MTB infects healthy people through the air, when they sneeze, spit, speak, or cough. Common symptoms of this disease are high fever with chills, chronic cough, night sweats, nail clubbing, weight loss, and fatigue [1]. e total number of people who died from TB in the year 2019 is 1.4 million, among which 208000 were reported HIV-positive. According to the WHO, around 10 million TB cases were estimated worldwide, in which over 95% has occurred in developing countries [2].
Several mathematical models for the transmission dynamics and on how to curtail the disease were developed. In the study of infectious diseases, the mathematical models play a key role and provide helpful information on how to control the spread and disease. ere are a lot of models developed by researchers for TB dynamics. In 1962, a TB model was proposed by Waaler et al. [3], and Revelle et al.
formulated a TB disease compartmental model [4]. Liu and Zhang provided a TB model and discussed the effect of vaccination and treatment [5]. Liu et al. formulated another model and used statistical data of TB to check the seasonal effects [6]. e rabies transmission and control analysis were proposed in [7]. Wallis explored a TB model with reactivation, while Kim et al. presented a model to reduce the spread of TB with optimal control strategies in the Philippines [8,9]. Recently, a TB model with slow and fast exposed classes using real data of Pakistan was proposed by Khan et al. [10].
All the abovementioned models are formulated via integer-order derivatives. ese classical models do not provide information about the learning mechanism and memory effect and thus have some limitations, while fractional models provide more compatible and realistic results. Many fractional-order derivatives were proposed in [11,12] and have a wide range of applications in the fields of epidemiology, physics, engineering, fluid dynamics, and many others [13][14][15][16]. e fractional TB mathematical model for children and adults is investigated in [17]. In recent years, fractional-order models gained much attention due to vast applications. In biological systems, fractional models are used for better understanding of the dynamics of various diseases. Recently, in the current situation of the pandemic, many fractional-order COVID-19 models and dengue transmission models were developed in [18][19][20]. e fractional calculus using the series approach of the type (p, q)-Mathieu-type series has been suggested in [21].
Tuberculosis disease is a main cause of mortality and morbidity and so is a massive health challenge in Pakistan. Pakistan is at the fifth position in the list of high-burden TB countries [22]. In Pakistan, the current incidence rate is more than 0.5 million and more than 50,000 die annually [23,24]. TB is considered as a massive burden in the province Khyber Pakhtunkhwa (KP), Pakistan. A report issued by the National TB Control Programme showed that an estimated total of 462920 new cases were reported and treated in KP, Pakistan, from 2002 to 2017.
Motivated by the abovementioned work and the previous literature in view, we study the dynamical TB model with standard incidence rate explored in [10] by considering Caputo fractional derivatives for more insights into the disease. We also used the real data from year 2002 to year 2017 of Khyber Pakhtunkhwa (KP) for the parametrization of the model [25]. e remaining work is organized as follows: Section 2 contains preliminaries, and model description is given in Section 3. e analysis of the model and estimation of parameters are given in Section 4, and Section 5 contains the numerical simulation. Finally, the study is concluded in Section 6.

Fractional Basics Concepts
e basic definitions are presented related to fractional calculus.

Definition 1.
e derivative for the function w ∈ C p having order α in the Caputo operator is defined as [26] (1) Definition 2. For the given function w: R + ⟶ R having fractional order α > 0, the fractional integral is where Γ(.) denotes the gamma function.

Description of the Model
To analyze the transmission dynamics of TB disease, we consider the tuberculosis model proposed in [10] for more insight into the disease dynamics studied via the Caputo fractional operator. e model is divided into six compartments: the susceptible compartment S(t), slow and fast exposed compartments E 1 (t) and E 2 (t), and I(t), T(t), and R(t) representing the infected, treated, and recovered compartments, respectively. When the susceptible person after getting infection by interacting with infected individual remains in the incubation period, then it is due to the nature of infection that the individual goes to fast or slow exposed class. We assumed that the slow exposed individuals before entering into the infected compartment must join the fast exposed individuals. en, the sum of all compartments is N(t), that is, e nonlinear fractional differential system governed by these assumptions is described as follows [10]: In model (5), after interaction between susceptible and infected persons, a ratio ω(0 < ω < 1) of susceptible individuals S(t) joins E 1 (t) and a fraction (1 − ω) enters into fast exposed class E 2 directly. Δ and ] denote the birth and death 2 Discrete Dynamics in Nature and Society rates, respectively, while β represents the successful transmission coefficient. e compartments Iand T-induced rates due to disease are given by σ 1 and σ 2 , respectively; the rate of progression from E 1 to E 2 is ς 1 and from E 2 to I is ς 2 . For the infected individuals, the per capita treatment rate is represented by c and δ is the rate, where individuals quit the T(t) compartment due to incomplete treatment. A fraction of δ that reenters into the infected class I is denoted by ρδT and the remaining δT(1 − ρ) rejoins the slow exposed class depending on treatment state of the individuals. e evolution rate is ϕ, where treated T becomes recovered R. e parameter ρ(0 < ρ < 0) in (1 − ρ)δ represents the ratio of drug defiance people in compartment T; let Model (5) can be written as subjected to appropriate nonnegative conditions In model (7), the fractional derivatives α ∈ (0, 1] considered as in Caputo sense and with biological parameter values, both estimated and fitted, are displayed in Table 1. (7) is explored in a feasible region Ξ ⊂ R 6 + , such that

Invariant Region and Attractivity. Model
Lemma 1. Ξ ⊂ R 6 + is a positively invariant region with nonnegative initial conditions for model (7) Proof. e net population becomes and then, we have Now by Laplace transform, we obtained us, the model solution with nonnegative conditions in Ξ remains in Ξ. So, Ξ is positive invariant and hence attracts all solutions in R 6 + .

□
Now, for the model solution positivity, Corollary 1 (see [27]). We assumed that Discrete Dynamics in Nature and Society

Proposition 1. Model (7) solution is nonnegative and bounded by
Proof. Using the result given in [28], we show the nonnegativity of the proposed model R 6 So, Corollary 1 gives the required result, and we can say that the solution is in e sum of all terms is positive; thus, the solution of model (7) is bounded.  Figure 1 illustrates the model's best fitted curve, and the values of parameter with description are tabulated in Table 1.
e value of control basic reproductive number is R 0 ≈ 1.38, estimated via fitted parameters. e curve fitting is summarized in a few steps for model (7) as follows: Let Ξ denote the unknown parameters and w be the vector-dependent variables. e objective function is taken for better possible fit and is given as where w t k and w t k are considered as the model solution and actual data points at time t k . To obtain model parameters by minimizing the objective function and for better agreement, we follow the optimization algorithm as well [29].

Model Equilibria and Reproduction Number.
To get the equilibria for fractional-order TB model (5), we have Model (7) has two equilibrium points: (1) e risk-free or DFE E 0 is By using the next generation technique given in [30], we have us, the reproductive number R 0 is given by (2) Endemic equilibrium (EE) E 1 is given by It is clear from above that E 1 exists only if R 0 > 1.

Stability of the DFE
Proof.
e Jacobian measure of system (7) evaluated at E 0 is given by We have − ] (twice), while for the others, the following is shown: where It can be observed that the coefficients shown above are positive, i.e., c i > 0, for i � 1, . . . , 4; furthermore, the Routh-Hurwitz criteria can be satisfied easily, So, the TB model is locally asymptotic at the DFE when R 0 < 1. us, |arg(λ)| > απ/2 for all α ∈ [0, 1), i.e., local asymptotic stability. e fractional tuberculosis model is GAS at riskfree equilibrium (E 0 ) if R 0 < 1.

Proof.
e appropriate Lyapunov function (L(t)) for global stability of model (7) is defined as where l n > 0, with n � 1, . . . , 4, are positive constants and the time fractional derivative of L(t) is Considering fractional system (7), we obtain Now, we choose us, C D α t L(t) ≤ 0 if R 0 ≤ 1. erefore, the variables and parameters are nonnegative with C D α t L(t) ≤ 0 iff E 1 � E 2 � I � T � 0. Hence, (E 1 , E 2 , I, T) ⟶ (0, 0, 0, 0) as t ⟶ ∞. We get S ⟶ Δ/] and R ⟶ 0 as t ⟶ ∞ from system (7). us, the solution of model (7) with nonnegative initial conditions as t ⟶ ∞ approaches E 0 according to the fractional case developed in [31], in the feasible region. Hence, it complies that the disease-free equilibrium of system (7) is GAS.

Numerical Scheme and Simulation
e numerical iterative method [32] is used for the numerical solution of model (7). A differential system is given as By applying the fractional integral operator, we obtained By using the trapezoidal quadrature formula, e right-hand side of (32) yields      Figure 4: Behavior of the TB model with an impact of contact rate β at α � 1.  10 Discrete Dynamics in Nature and Society

S (t)
When nodes t i � t 0 + i h are equispaced, then (34) reduces to

12
Discrete Dynamics in Nature and Society where now We have a simplified expression as follows: us, the predictor w P n + 1 is determined by We explore the dynamics of fractional TB model (7) numerically by the generalized predictor-corrector of Adams-Bashforth-Moulton method mentioned above. To control the disease up to some extent and for memory effect, we analyze the effect of different parameters by varying some sensitive parameters and fractional order α. e infection must be reduced with the help of control strategies and treatment. To illustrate the effects of sensitive parameters, the population in 2017 of Khyber Pakhtunkhwa was 30, 523, 371 [33], which is to be considered as N(0). e state variables assumed as E 1 (0), E 2 (0), and I(0) are 83000, 50000, and 8010, respectively. Moreover, no treated or recovered cases are considered initially, that is, en, the susceptible individuals can be obtained easily as S(0) � 30.
e behaviors of cumulative infected people for R 0 � 1.38 when β � 0.6001 and R 0 � 0.9300 when β � 0.4001 are shown in Figure 2. e dynamical behavior of model (7) is depicted in Figure 3 for fractional orders α � 1, 0.97, 0.94, 0.91, 0.88, 0.85. Figure 3 shows that only susceptible individuals increased, while the infected individuals decreased significantly by decreasing α. Figures 4 and 5 illustrate the impact of contact rate β on model (7) for α � 1 and α � 0.85, respectively. e decrease is observed in infected compartments with a decrease in β, and for smaller order, the impact of α is more pronounced. Also, the behaviors of cumulative infected people with an impact of treatment rate and infectious people with variation of failure of treatment ρ are plotted in Figure 6 for various fractional orders α. So, by increasing treatment rate c, a decay in cumulative infected individuals is observed and vice versa for infective compartments in case of ρ. Finally, the behavior of total infected people with and without treatment is depicted in Figure 7, and results become more precise for smaller values of fractional order α. From numerical results, we can say that the tuberculosis infection can be controlled with proper treatment.

Conclusion
Tuberculosis has resulted in a lot of infected cases and deaths in Pakistan. e government of Pakistan and particularly the province KP have put many efforts for its minimization by treating the infected cases and also the relapse cases. To understand the TB infections in the KP province, we studied a mathematical model with slow and fast exposed cases and its effect on the model dynamics. We studied the essential mathematics involved in the modeling of the fractional-order model. en, we investigated the stability of the model and proved the TB model to be locally as well as globally asymptotically stable. e stability results were obtained in the context of Caputo operator. Some discussion on the estimation procedures for the investigations of the model parameters is considered. We utilized the realistic parameters and obtained results graphically. We provided an efficient numerical scheme based on the Adams-Bashforth-Moulton method and obtained the graphical results. Numerical results were achieved by considering different parameters and fractional order values, and we discussed its impact on disease eradications. e effect of treatment rate and its related results was explored. Some results with treatment and without treatment are discussed and shown graphically, which show the disease can be eliminated by treating the infected people. While considering the effect of fractional-order parameter values, there were observed decreases in the infective compartments. From the analysis, the work shown suggests that the disease can be minimized more efficiently if the government takes serious actions by educating the people, making awareness, etc., and provides better treatment at doorsteps. Regarding the future of modeling of TB in Pakistan, we will explore and extend the results by considering the cases across the country with vaccinations and relapse model.

Data Availability
Data are available on the reasonable request to the corresponding author.

Conflicts of Interest
e authors declare that no conflicts of interest.