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

Authors: Xiao-Hong Zhang; Aatif Ali; Muhammad Altaf Khan; Mohammad Y. Alshahrani; Taseer Muhammad; Saeed Islam
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

In this study, we formulate a noninteger-order mathematical model via the Caputo operator for the transmission dynamics of the bacterial disease tuberculosis (TB) in Khyber Pakhtunkhwa (KP), Pakistan. The number of confirmed cases from 2002 to 2017 is considered as incidence data for the estimation of parameters or to parameterize the model and analysis. The positivity and boundedness of the model solution are derived. For the dynamics of the tuberculosis model, we find the equilibrium points and the basic reproduction number. The proposed model is locally and globally stable at disease-free equilibrium, if the reproduction numberℛ0<1. Furthermore, to examine the behavior of the various parameters and different values of fractional-order derivative graphically, the most common iterative scheme based on fundamental theorem and Lagrange interpolation polynomial is implemented. From the numerical result, it is observed that the contact rate and treatment rate have a great impact on curtailing the tuberculosis disease. Furthermore, proper treatment is a key factor in reducing the TB transmission and prevalence. Also, the results are more precise for lower fractional order. The results from various numerical plots show that the fractional model gives more insights into the disease dynamics and on how to curtail the disease spread.


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## Body

## 1. Introduction

Tuberculosis (TB) usually caused by a bacterium calledMycobacterium tuberculosis bacterium (MTB) is a contagious infectious disease. This 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]. The 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. There 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]. The 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. These 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 [13–16]. The 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 [18–20]. The 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]. The remaining work is organized as follows: Section 2 contains preliminaries, and model description is given in Section 3. The 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.


## 2. Fractional Basics Concepts

The basic definitions are presented related to fractional calculus.Definition 1.
The derivative for the functionw∈Cp having order α in the Caputo operator is defined as [26](1)DCtαwt=1Γp−α∫0twpεt−εp−α−1dε.Definition 2.
For the given functionw:R+⟶R having fractional order α>0, the fractional integral is(2)Itαwt=1Γα∫0tt−εα−1wεdε,where Γ. denotes the gamma function.Definition 3.
For the Caputo fractional dynamical system, letp∗ be the equilibrium point. Then,(3)DCtαpt=wt,pt,α∈0,1,iff wt,p∗=0.


## 3. 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. The model is divided into six compartments: the susceptible compartment St, slow and fast exposed compartments E1t and E2t, and It, Tt, and Rt 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. Then, the sum of all compartments is ℕt, that is,(4)ℕt=St+E1t+E2t+It+Tt+Rt.The nonlinear fractional differential system governed by these assumptions is described as follows [10]:(5)DCtαS=Δ−BSIN−νS,DCtαE1=ωBSIN−ν+ς1E1+1−ρδT,DCtαE2=1−ωBSIN+ς1E1−ν+ς2E2,DCtαI=ς2E2+ρδT−ν+γ+σ1I,DCtαT=γI−ν+δ+σ2+ϕT,DCtαR=ϕT−νR.In model (5), after interaction between susceptible and infected persons, a ratio ω0<ω<1 of susceptible individuals St joins E1t and a fraction 1−ω enters into fast exposed class E2 directly. Δ and ν denote the birth and death rates, respectively, while β represents the successful transmission coefficient. The compartments I- and T-induced rates due to disease are given by σ1 and σ2, respectively; the rate of progression from E1 to E2 is ς1 and from E2 to I is ς2. For the infected individuals, the per capita treatment rate is represented by γ and δ is the rate, where individuals quit the Tt compartment due to incomplete treatment. A fraction of δ that reenters into the infected class I is denoted by ρδT and the remaining δT1−ρ rejoins the slow exposed class depending on treatment state of the individuals. The evolution rate is ϕ, where treated T becomes recovered R. The parameter ρ0<ρ<0 in 1−ρδ represents the ratio of drug defiance people in compartment T; let(6)m1=ν+ς1,m2=ν+ς2,m3=ν+γ+σ1,m4=ν+δ+σ2+ϕ.Model (5) can be written as(7)DCtαS=Δ−BSIN−νS,DCtαE1=ωBSIN−m1E1+1−ρδT,DCtαE2=1−ωBSIN+ς1E1−m2E2,DCtαI=ς2E2+ρδT−m3I,DCtαT=γI−m4T,DCtαR=ϕT−νR,subjected to appropriate nonnegative conditions(8)S0≥0,E10≥0,E20≥0,I0≥0,T0≥0,R0≥0.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.Table 1
Descriptions of parameters and their numerical values.
ParameterDescriptionValue (years−1)ReferenceΔBirth rateν×ℕ0EstimatedνNatural death rate1/67.7[23]βRate of contact0.6001FittedγI treatment rate0.1500FittedρFailure of treatment0.2959Fittedσ1Disease-induced death rate inI0.2738Fittedσ2Disease-induced death rate during treatment0.1000Fittedς1Rate of moving fromE1 to E20.2738Fittedς2Transfer rate fromE2 to I0.1000FittedδLeaving rate ofT reentering to I or E0.0649FittedωFraction ofS becoming I0.5259FittedϕRecovery rate0.0100Fitted


## 4. Fractional Model Analysis

### 4.1. Invariant Region and Attractivity

Model (7) is explored in a feasible region Ξ⊂ℝ+6, such that(9)Ξ=St,Et,E1t,Tt,Rt∈ℝ+6:ℕt≤Δν.Lemma 1.
Ξ⊂ℝ+6 is a positively invariant region with nonnegative initial conditions for model (7) in ℝ+6.Proof.
The net population becomes(10)DCtαℕt=DCtαSt+DCtαE1t+DCtαIt+DCtαE2t+DCtαTt+DCtαRt,and then, we have(11)DCtαℕt+νℕt≤Δ.
Now by Laplace transform, we obtained(12)ℕt≤ℕ0Eα,1νtα+ΔtαEα,α+1νtα,ℕt≤Δν.
Thus, the model solution with nonnegative conditions inΞ remains in Ξ. So, Ξ is positive invariant and hence attracts all solutions in ℝ+6.Now, for the model solution positivity,(13)R+6=u∈R6|z≥0andut=St,Et,E1t,E2t,Tt,RtT.Corollary 1 (see [27]).
We assumed thatjt∈Cc,d and DCtαjt∈c,d, where α∈0,1. IfDCtαjt≥0,∀u∈c,d,thenjtis nondecreasingDCtαjt≤0,∀u∈c,d,thenjtis nonincreasing


### 4.2. Positivity and Boundedness

Proposition 1.
Model (7) solution is nonnegative and bounded by ∀S0,E10,E20,I0,T0,R0∈ℝ+6, for t>0.Proof.
Using the result given in [28], we show the nonnegativity of the proposed model R+6. System (7) gives(14)DCtαS|S=0=Δ>0,DCtαE1|E1=0=ωβSIS+E2+I+T+R+1−ρδT≥0,δ>ρδ,DCtαE2|E2=0=1−ωβSIS+E1+I+T+R+ς1E1≥0,DCtαI|I=0=ς2E2+ρδT≥0,DCtαT|T=0=γI≥0,DCtαR|R=0=ϕT≥0.
So, Corollary1 gives the required result, and we can say that the solution is in ℝ+6.(15)Ξ=S,E1,E2,I,T,R∈R+6:S,E1,E2,I,T,R≥0.
The sum of all terms is positive; thus, the solution of model (7) is bounded.


### 4.3. Parameter Estimation

In this section, the method of least-square curve fitting is used to estimate the parameters from the confirmed cases of tuberculosis in Khyber Pakhtunkhwa, Pakistan, since2002−2017. The birth rate Δ and natural death rate ν are estimated from the literature. The other biological parameters are fitted from incidence data. Figure 1 illustrates the model’s best fitted curve, and the values of parameter with description are tabulated in Table 1. The value of control basic reproductive number is ℛ0≈1.38, estimated via fitted parameters. The curve fitting is summarized in a few steps for model (7) as follows:(16)dwdt=Ϝt,w,Ξ,wt0=w0.Figure 1
Model fitting (solid blue line) to the TB reported cases (red) from 2002 to 2017.LetΞ denote the unknown parameters and w be the vector-dependent variables. The objective function is taken for better possible fit and is given as(17)Ξ^=∑k=1nwtk−w˜tk2,where wtk and w˜tk are considered as the model solution and actual data points at time tk. To obtain model parameters by minimizing the objective function and for better agreement, we follow the optimization algorithm as well [29].


### 4.4. Model Equilibria and Reproduction Number

To get the equilibria for fractional-order TB model (5), we have(18)DCtαS=DCtαE1=DCtαE2=DCtαI=DCtαT=DCtαR=0.Model (7) has two equilibrium points:(1)
The risk-free or DFEE0 is(19)E0=Δν,0,0,0,0,0.By using the next generation technique given in [30], we have(20)F=00ωβ0001−ωβ000000000,V=m100−δ1−ρ−ς1m2000−ς2m3−ρδ00−γm4.Thus, the reproductive numberℛ0 is given by(21)ℛ0=ρFV−1=βς1ς2ωm1m2m3+βς21−ωm2m3+γδς1ς21−ρm1m2m3m4+γδρm3m4.(2)
Endemic equilibrium (EE)E1 is given by E1=S∗,E1∗,E2∗,I∗,T∗,R∗, where(22)S∗=ℕ∗ℛ0,E1∗=1m1m4m4ωβℛ0+1−ργδI∗,E2∗=1ς2m4m4m3−ργδI∗.T∗=γm4I∗,R∗=ϕγνm4I∗,I∗=m1m4νς2ℛ0−1ℕ∗m4νς2ωβ+νm1m3m4−ργδ+γς2m1+1−ρδ+m1ς2ϕγ+m4νℛ0.It is clear from above thatE1 exists only if ℛ0>1.


### 4.5. Stability of the DFE

Theorem 1.
The DFEE0 is LAS if the eigenvalues λj of model (7) satisfy argλj>απ/2.Proof.
The Jacobian measure of system (7) evaluated at E0 is given by(23)JE0=−ν00−β000−m10βωδ1−ρ00ς1−m21−ωβ0000ς2−m3ρδ0000γ−m400000ϕ−ν.
We have−ν (twice), while for the others, the following is shown:(24)Pλ=λ4+c1λ3+c2λ2+c3λ+c4,where(25)c1=m1+m2+m3+m4,c2=m2m4+m1m2+m3+m4+m2m31−ℛ2+m3m41−ℛ4,c3=m1+m2m3m41−ℛ4+m1m2m31−ℛ1+ℛ2+m4m1m2−βς21−ω,c4=m1m2m3m41−ℛ0.
It can be observed that the coefficients shown above are positive, i.e.,ci>0, for i=1,…,4; furthermore, the Routh–Hurwitz criteria can be satisfied easily, c1c2c3−c12c−4−c32>0. So, the TB model is locally asymptotic at the DFE when ℛ0<1. Thus, argλ>απ/2 for all α∈0,1, i.e., local asymptotic stability.


### 4.6. Global Stability of DFE

Theorem 2.
The fractional tuberculosis model is GAS at risk-free equilibriumE0 if ℛ0<1.Proof.
The appropriate Lyapunov functionℒt for global stability of model (7) is defined as(26)ℒt=l1E1+l2E2+l3I+l4T,where ln>0, with n=1,…,4, are positive constants and the time fractional derivative of ℒt is(27)DCtαℒt=l1CDtαE1+l2CDtαE2+l3CDtαI+l4CDtαT.
Considering fractional system (7), we obtain(28)DCtαℒE1,E2,I,T=l1ωβSIN−m1E1+1−ρδT+l21−ωβSIN+ς1E1−m2E2+l3ς2E2+ρδT−m3I+l4γI−m4T,≤l1ωβI−m1E1+1−ρδT+l21−ωβI+ς1E1−m2E2+l3ς2E2+ρδT−m3I+l4γI−m4T,S≤N.=l1ωβ+l21−ωβ+l4γ−l3m3I+l2ς1−l1m1E1+l3ς2−l2m2E2+l11−ρδ+l3ρδ−l4m4T,=l3m3l1ωβ+l21−ωβ+l4γl3m3−1I+l2ς1−l1m1E1+l3ς2−l2m2E2+l11−ρδ+l3ρδ−l4m4T.
Now, we choose(29)l1=ς1,l2=m1,l3=m1m2ς2,l4=ς1ς21−ρδ+m1m2ρδς2m4,DCtαℒt≤l3m3ℛ0−1I.
Thus,DCtαℒt≤0 if ℛ0≤1. Therefore, the variables and parameters are nonnegative with DCtαℒt≤0 iff E1=E2=I=T=0. Hence, E1,E2,I,T⟶0,0,0,0 as t⟶∞. We get S⟶Δ/ν and R⟶0 as t⟶∞ from system (7). Thus, the solution of model (7) with nonnegative initial conditions as t⟶∞ approaches E0 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.Lemma 2.
Model (7) is unstable at DFE if ℛ0>1.


## 4.1. Invariant Region and Attractivity

Model (7) is explored in a feasible region Ξ⊂ℝ+6, such that(9)Ξ=St,Et,E1t,Tt,Rt∈ℝ+6:ℕt≤Δν.Lemma 1.
Ξ⊂ℝ+6 is a positively invariant region with nonnegative initial conditions for model (7) in ℝ+6.Proof.
The net population becomes(10)DCtαℕt=DCtαSt+DCtαE1t+DCtαIt+DCtαE2t+DCtαTt+DCtαRt,and then, we have(11)DCtαℕt+νℕt≤Δ.
Now by Laplace transform, we obtained(12)ℕt≤ℕ0Eα,1νtα+ΔtαEα,α+1νtα,ℕt≤Δν.
Thus, the model solution with nonnegative conditions inΞ remains in Ξ. So, Ξ is positive invariant and hence attracts all solutions in ℝ+6.Now, for the model solution positivity,(13)R+6=u∈R6|z≥0andut=St,Et,E1t,E2t,Tt,RtT.Corollary 1 (see [27]).
We assumed thatjt∈Cc,d and DCtαjt∈c,d, where α∈0,1. IfDCtαjt≥0,∀u∈c,d,thenjtis nondecreasingDCtαjt≤0,∀u∈c,d,thenjtis nonincreasing


## 4.2. Positivity and Boundedness

Proposition 1.
Model (7) solution is nonnegative and bounded by ∀S0,E10,E20,I0,T0,R0∈ℝ+6, for t>0.Proof.
Using the result given in [28], we show the nonnegativity of the proposed model R+6. System (7) gives(14)DCtαS|S=0=Δ>0,DCtαE1|E1=0=ωβSIS+E2+I+T+R+1−ρδT≥0,δ>ρδ,DCtαE2|E2=0=1−ωβSIS+E1+I+T+R+ς1E1≥0,DCtαI|I=0=ς2E2+ρδT≥0,DCtαT|T=0=γI≥0,DCtαR|R=0=ϕT≥0.
So, Corollary1 gives the required result, and we can say that the solution is in ℝ+6.(15)Ξ=S,E1,E2,I,T,R∈R+6:S,E1,E2,I,T,R≥0.
The sum of all terms is positive; thus, the solution of model (7) is bounded.


## 4.3. Parameter Estimation

In this section, the method of least-square curve fitting is used to estimate the parameters from the confirmed cases of tuberculosis in Khyber Pakhtunkhwa, Pakistan, since2002−2017. The birth rate Δ and natural death rate ν are estimated from the literature. The other biological parameters are fitted from incidence data. Figure 1 illustrates the model’s best fitted curve, and the values of parameter with description are tabulated in Table 1. The value of control basic reproductive number is ℛ0≈1.38, estimated via fitted parameters. The curve fitting is summarized in a few steps for model (7) as follows:(16)dwdt=Ϝt,w,Ξ,wt0=w0.Figure 1
Model fitting (solid blue line) to the TB reported cases (red) from 2002 to 2017.LetΞ denote the unknown parameters and w be the vector-dependent variables. The objective function is taken for better possible fit and is given as(17)Ξ^=∑k=1nwtk−w˜tk2,where wtk and w˜tk are considered as the model solution and actual data points at time tk. To obtain model parameters by minimizing the objective function and for better agreement, we follow the optimization algorithm as well [29].


## 4.4. Model Equilibria and Reproduction Number

To get the equilibria for fractional-order TB model (5), we have(18)DCtαS=DCtαE1=DCtαE2=DCtαI=DCtαT=DCtαR=0.Model (7) has two equilibrium points:(1)
The risk-free or DFEE0 is(19)E0=Δν,0,0,0,0,0.By using the next generation technique given in [30], we have(20)F=00ωβ0001−ωβ000000000,V=m100−δ1−ρ−ς1m2000−ς2m3−ρδ00−γm4.Thus, the reproductive numberℛ0 is given by(21)ℛ0=ρFV−1=βς1ς2ωm1m2m3+βς21−ωm2m3+γδς1ς21−ρm1m2m3m4+γδρm3m4.(2)
Endemic equilibrium (EE)E1 is given by E1=S∗,E1∗,E2∗,I∗,T∗,R∗, where(22)S∗=ℕ∗ℛ0,E1∗=1m1m4m4ωβℛ0+1−ργδI∗,E2∗=1ς2m4m4m3−ργδI∗.T∗=γm4I∗,R∗=ϕγνm4I∗,I∗=m1m4νς2ℛ0−1ℕ∗m4νς2ωβ+νm1m3m4−ργδ+γς2m1+1−ρδ+m1ς2ϕγ+m4νℛ0.It is clear from above thatE1 exists only if ℛ0>1.


## 4.5. Stability of the DFE

Theorem 1.
The DFEE0 is LAS if the eigenvalues λj of model (7) satisfy argλj>απ/2.Proof.
The Jacobian measure of system (7) evaluated at E0 is given by(23)JE0=−ν00−β000−m10βωδ1−ρ00ς1−m21−ωβ0000ς2−m3ρδ0000γ−m400000ϕ−ν.
We have−ν (twice), while for the others, the following is shown:(24)Pλ=λ4+c1λ3+c2λ2+c3λ+c4,where(25)c1=m1+m2+m3+m4,c2=m2m4+m1m2+m3+m4+m2m31−ℛ2+m3m41−ℛ4,c3=m1+m2m3m41−ℛ4+m1m2m31−ℛ1+ℛ2+m4m1m2−βς21−ω,c4=m1m2m3m41−ℛ0.
It can be observed that the coefficients shown above are positive, i.e.,ci>0, for i=1,…,4; furthermore, the Routh–Hurwitz criteria can be satisfied easily, c1c2c3−c12c−4−c32>0. So, the TB model is locally asymptotic at the DFE when ℛ0<1. Thus, argλ>απ/2 for all α∈0,1, i.e., local asymptotic stability.


## 4.6. Global Stability of DFE

Theorem 2.
The fractional tuberculosis model is GAS at risk-free equilibriumE0 if ℛ0<1.Proof.
The appropriate Lyapunov functionℒt for global stability of model (7) is defined as(26)ℒt=l1E1+l2E2+l3I+l4T,where ln>0, with n=1,…,4, are positive constants and the time fractional derivative of ℒt is(27)DCtαℒt=l1CDtαE1+l2CDtαE2+l3CDtαI+l4CDtαT.
Considering fractional system (7), we obtain(28)DCtαℒE1,E2,I,T=l1ωβSIN−m1E1+1−ρδT+l21−ωβSIN+ς1E1−m2E2+l3ς2E2+ρδT−m3I+l4γI−m4T,≤l1ωβI−m1E1+1−ρδT+l21−ωβI+ς1E1−m2E2+l3ς2E2+ρδT−m3I+l4γI−m4T,S≤N.=l1ωβ+l21−ωβ+l4γ−l3m3I+l2ς1−l1m1E1+l3ς2−l2m2E2+l11−ρδ+l3ρδ−l4m4T,=l3m3l1ωβ+l21−ωβ+l4γl3m3−1I+l2ς1−l1m1E1+l3ς2−l2m2E2+l11−ρδ+l3ρδ−l4m4T.
Now, we choose(29)l1=ς1,l2=m1,l3=m1m2ς2,l4=ς1ς21−ρδ+m1m2ρδς2m4,DCtαℒt≤l3m3ℛ0−1I.
Thus,DCtαℒt≤0 if ℛ0≤1. Therefore, the variables and parameters are nonnegative with DCtαℒt≤0 iff E1=E2=I=T=0. Hence, E1,E2,I,T⟶0,0,0,0 as t⟶∞. We get S⟶Δ/ν and R⟶0 as t⟶∞ from system (7). Thus, the solution of model (7) with nonnegative initial conditions as t⟶∞ approaches E0 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.Lemma 2.
Model (7) is unstable at DFE if ℛ0>1.


## 5. Numerical Scheme and Simulation

The numerical iterative method [32] is used for the numerical solution of model (7). A differential system is given as(30)Dt0αw−w0t=Ft,wt,wt0=w0.By applying the fractional integral operator, we obtained(31)wt−wt0=1Γα∫t0tt−ℏα−1Fℏ,wℏdℏ.By using the trapezoidal quadrature formula,(32)∫t0tn+1tn+1−ℏα−1jℏdℏ≈∫t0tn+1tn+1−ℏα−1jn+1ℏdℏ.The right-hand side of (32) yields(33)∫t0tn+1tn+1−ℏα−1jn+1ℏdℏ=∑i=0n+1ai,n+1jti,where(34)ai,n+1=∫t0tn+1tn+1−ℏα−1ϕj,n+1ℏdℏ(35)χi,n+1ℏ=ℏ−ti−1ti−ti−1,ifti−1<ℏ<ti,ti+1−ℏti+1−ti,ifti<ℏ<ti+1,0,otherwise.When nodesti=t0+ih are equispaced, then (34) reduces to(36)ai,n+1ℏ=gααα+1nα+1−n−αn+1α,ifi=0,gααα+1,ifi=n+1,ai,n+1=gααα+1n−i+2α+1−2n−i+1α+1+n−iα+1.The one-step fractional Adams–Moulton equation is(37)wn+1=w0+1Γα∑i=0nai,n+1Fti,wi+an+1,n+1Ftn+1,wn+1P.(31) reduces to(38)∫t0tn+1tn+1−ℏα−1jℏdℏ≈∑i=0nbj,n+1jti,where now(39)bi,n+1=∫titi+1tn+1−ℏα−1dℏ=1αtn+1−tiα−tn+1−ti+1α.We have a simplified expression as follows:(40)bi,n+1=gααn+1−iα−n−iα.Thus, the predictorwnP+1 is determined by(41)wn+1P=w0+1Γα∑i=0nbi,n+1Fti,wti,wi.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. The 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 ℕ0. The state variables assumed as E10,E20, and I0 are 83000,50000, and 8010, respectively. Moreover, no treated or recovered cases are considered initially, that is, T0=R0=0. Then, the susceptible individuals can be obtained easily as S0=30. The behaviors of cumulative infected people for ℛ0=1.38 when β=0.6001 and ℛ0=0.9300 when β=0.4001 are shown in Figure 2. The 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. The 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 γ, 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.Figure 2
Numerical visualization of cumulative infected compartments of the TB model whenℛ0>1 (a) and ℛ0<1 (b).
(a)(b)Figure 3
Numerical results of model (7) for different fractional orders α.
(a)(b)(c)(d)(e)(f)Figure 4
Behavior of the TB model with an impact of contact rateβ at α = 1.
(a)(b)(c)(d)(e)(f)Figure 5
Behavior of the TB model with an impact of contact rateβ at α=0.85.
(a)(b)(c)(d)(e)(f)Figure 6
The treatment rateγ impact on cumulative infected persons (a–c) and variation of ρ on infectious individuals (d–f) for α=1,α=0.95, and α=0.90.
(a)(b)(c)(d)(e)(f)Figure 7
Behavior with and without treatment on total infected individuals forα=1,0.95,0.90,0.85.
(a)(b)(c)(d)


## 6. Conclusion

Tuberculosis has resulted in a lot of infected cases and deaths in Pakistan. The 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. Then, we investigated the stability of the model and proved the TB model to be locally as well as globally asymptotically stable. The 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. The 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.



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*Source: 9512371-2021-11-16.xml*