# Dynamics of Tuberculosis (TB) with Drug Resistance to First-Line Treatment and Leaky Vaccination: A Deterministic Modelling Perspective

Authors: Dominic Otoo; Shaibu Osman; Stephen Atta Poku; Elvis Kobina Donkoh
Journal: Computational and Mathematical Methods in Medicine
 (2021)

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## Abstract

A deterministic model was formulated and employed in the analysis of the dynamics of tuberculosis with a keen emphasis on vaccination and drug resistance as the first line of treatment. It was assumed that some of the susceptible population were vaccinated but with temporal immunity. This is due to the fact that vaccines do not confer permanent immunity. Moreover, part of the infected individual after treatment grows resistance to the drug. Infective immigrants were also considered to be part of the population. The basic reproductive number for the model is estimated using the next-generation matrix method. The equilibrium points of the TB model and their local and global stability were determined. It was established that if the basic reproductive number was less than unityR0<1, then the disease free equilibrium is stable and unstable if R0>1. Furthermore, we investigated the optimal prevention, treatment, and vaccination as control measures for the disease. As the objective functional was optimised, there have been a significant reduction in the number of infections and an increase in the number of recovery. The best control measure in combating tuberculosis infections is prevention and vaccination of the susceptible population.


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## Body

## 1. Introduction

Respiratory disease can be described as an infection which can be treated with time. The commonest respiratory infections include pneumonia, tuberculosis, and flu. Chronic conditions such as asthma and chronic bronchitis are persistent and sometimes long-lasting [1].Tuberculosis is among the most ancient diseases worldwide. It is very contagious. The causative organism, Mycobacterium tuberculosis, was discovered by the German microbiologist Robert Koch in 1882 [2]. The motivation behind this study is to use a deterministic model to analyse the dynamics of the infection and suggest the best optimal control measure in combating the disease.Through coughing, singing, and sneezing, pulmonary tuberculosis is spread from a sick TB patient as a droplet infection. Inhalation by an uninfected individual of these droplets may cause infection. With the frequency and duration of contact with people who have the disease, the risk of contracting TB rises.In 1993, the WHO decreed TB a global epidemic [3]. It is estimated that the risk of contracting active TB after coming into contact with an infected person is between 5% and 10%, with a greater proportion of the disease playing a crucial role which happens in the very first few years after the initial infection with the arrival of HIV [4].Biological models usually explain the transmission dynamics of infectious diseases and can determine the status of the disease in a population with time. The basic reproduction number is the threshold value that determines the persistence of a disease in a population [5–7].Optimal control theory is usually employed in biological models to determine the best optimal control strategy in combating infections in a population [8–10].


## 2. Model Description and Formulation

The model partitions the entire populace into six compartments according to their epidemiological status. We defineSt, Vt, Et, It, R1t, and Rt as the number of susceptible individuals, vaccinated individuals, exposed individuals, infectious individuals, individuals with resistance to treatment, and recovered individuals, respectively, at time t≥0.Tables1 and 2 show the variables and parameters used in the tuberculosis model. Figure 1 shows the tuberculosis (TB) model transmission dynamics.Table 1
Variable description.
VariablesDescriptionSSusceptible personsVVaccinated personsEExposed personsITB-infected personsR1Individuals resistant to treatmentRRecovered personsTable 2
Parameter description.
ParametersDescriptionΛRecruitment of susceptible individualsMImmigrants into the susceptible and infectious compartmentsαRate of inflow of immigrants into the susceptible compartmentσRate at which the cured lose their immunityμRate of natural mortalityγRate of vaccination of susceptible individualsθRate at which the vaccinated recoverβRate at which the susceptible individuals are exposed to MtbρRate at which unprotected individuals get infectedδDisease-induced death rateκRate of recovery after treatmentτRate of resistance to the treatment1−αRate of inflows of immigrants into the infected compartmentFigure 1
Tuberculosis model flow diagram.The following differential equations were obtained from the model flow diagram:(1)dSdt=Λ+αM+σR−βSI−γ+μS,(2)dVdtγS−θ+μV,(3)dEdt=βSI−ρ+μE,(4)dIdt=ρE+1−αM−τ+δ+μI,(5)dR1dt=τI−κ+μR1,(6)dRdt=κR1+θV−σ+μR.Thus, the total population is given as(7)N=S+V+E+I+R1+R,with initial conditions:
(8)S0,V0,E0,I0,R10,R0∈R+6.


## 3. Tuberculosis Model Analysis

The tuberculosis (TB) model is about human population; hence, model state variables ought to be nonnegative and limited for allt≥0. In this section, we demonstrate that the TB model is numerically and epidemiologically sensible.


### 3.1. Positivity of Solution

We prove the positivity of the variables in the model. Based on the concept of derivative of a function, the behavior of the function at a known point can be established.Theorem 1.
Let the initial set beS0, V0, E0, I0, R10, and R0 and be nonnegative; then, the solution set of St,Vt,Et,It,R1t,Rt of equation (1) is positive and bounded for all t>0, wherever they exist.Proof.
From equation (1), we can state that
(9)dSdt≥−γ+μ+βIS,dSSdSdt≥−γ+μ+βIdt,InSdSdt≥−γ+μ+βIt+c,StdSdt≥ce−γ+μ+βIt.
Att=0,S0dS/dt≥c,
(10)StdSdt≥S0e−γ+μ+βIt,since
(11)γ+μ+βI>0,StdSdt≥0.
Also,(12)dVdtdSdt≥−θ+μV,VtdSdt≥ce−θ+μt.
Att=0,V0dS/dt≥c,
(13)VtdSdt≥V0e−θ+μt,since
(14)θ+μ>0,VtdSdt≥0.
Also,(15)dEdtdSdt≥−ρ+μE,EtdSdt≥ce−ρ+μt.
Att=0,E0dS/dt≥c,
(16)EtdSdt≥E0e−ρ+μt,since
(17)ρ+μ>0,EtdSdt≥0.
Also,(18)dIdtdSdt≥−τ+μ+δI,ItdSdt≥ce−τ+μ+δt.
Att=0,I0dS/dt≥c,
(19)ItdSdt≥I0e−τ+μ+δt,since
(20)τ+μ+δ>0,ItdSdt≥0.
Also,(21)dR1dtdSdt≥−κ+μR1,R1tdSdt≥ce−κ+μt.
Att=0,R10dS/dt≥c,
(22)R1tdSdt≥R10e−κ+μt,since
(23)κ+μ>0,R1tdSdt≥0.
Also,(24)dRdtdSdt≥−σ+μR,RtdSdt≥ce−σ+μt.
Att=0,R0dS/dt≥c,
(25)RtdSdt≥R0e−σ+μt,since
(26)σ+μ>0,RtdSdt≥0.


### 3.2. Boundedness of the System

The region in which solutions of the tuberculosis (TB) model system are uniformly bounded is the proper subset, and it is given by(27)Γ=S,V,E,I,R1,R∈R+6:N≤Λ+M−δIμ,μ≠0.Proof.
(28)N=S+V+E+I+R1+R,dNdt=Λ−μN+M−δIdNΛ−μN+M−δI=dt−1μ∫−μΛ−μN+M−δIdN=∫dt−1μInΛ−μN+M−δI=t+c,InΛ−μN+M−dI=−μt+c1,Λ−μN+M−δI=c2e−μt,Nt=Λ+M−δI−c2e−μtμ,Att=0,N0=N0,I0=I0,N0=Λ+M−δI0−c2μ,c2=Λ+M−δI0−μN0,Nt=Λ+M−δI−Λ+M−δI0−μN0e−μtμ.
So ast⟶∞, N⟶Λ+M−δI/μ∈R+.
Therefore,Γ is a positive invariant.


### 3.3. Existence of Disease-Free Equilibrium Point

The disease-free equilibrium of the dynamical system (1) is obtained by setting dS/dt=dV/dt=dE/dt=dI/dt=dR1/dt=dR/dt=0, and since there is no disease E=I=R1=R=0,
(29)Λ+αM−γ+μS=0⇒S=Λ+αMγ+μ.Therefore, the disease-free equilibrium of the dynamical system (1) is
(30)C0=S0,V0,E0,I0,R10,R0=Λ+αMγ+μ,γΛ+αMθ+μγ+μ,0,0,0,0.


### 3.4. Basic Reproductive Number

The basic reproductive number can be computed utilizing the cutting edge matrix approach. The basic reproduction number determines the state of a disease with time in a dynamical system [11, 12]. It is utilized to predict the stability of the disease equilibrium. The basic reproductive number is characterized as the quantity of secondary infections that one tainted person can create in a completely susceptible population [13, 14]. According to [13, 15], the next-generation matrix is defined as K=FG−1 and R0=ρFG−1, where ρFG−1 denotes the spectral radius of FG−1.Using the next-generation matrix, we consider only the infectious compartments in the system of differential equation in (1).
(31)dEdt=βSI−ρ+μE,dIdt=ρE+1−αM−τ+δ+μI,dR1dt=τI−κ+μR1.Letf be the count of emerging infection moving into the system and g be the count of infections exiting the system.
(32)f=βSI,0,0,g=ρ+μE,−ρE−1−αM+τ+μ+δI,−γI+k+μR1.The Jacobian matrix off and g are obtained by
(33)F=0βS0000000,G=ρ+μ00−ρτ+μ+δ00−γk+μ.ButR0=ρFG−1.From the relationFG−1, the inverse of G can be calculated:
(34)G−1=γ+μ+δρ+μτ+μ+δ00ρρ+μτ+μ+δ1τ+μ+δ0γρρ+μτ+μ+δk+μγτ+μ+δk+μγ+μ+δτ+μ+δk+μ.Computing the product ofFG−1,
(35)FG−1=0βS0000000γ+μ+δρ+μτ+μ+δ00ρρ+μτ+μ+δ1τ+μ+δ0γρρ+μτ+μ+δk+μγτ+μ+δk+μγ+μ+δτ+μ+δk+μ,FG−1=βρSρ+μτ+μ+δβSτ+μ+δ0000000.By selecting the dominant eigenvalue ofFG−1, the basic reproductive number is
(36)R0=βρSρ+μτ+μ+δ.At the disease-free equilibrium, we substituteS=Λ+αM/γ+μ into the basic reproductive number, R0.This therefore implies that(37)R0=βρΛ+αMγ+μρ+μτ+μ+δ.


### 3.5. Local Stability of the Disease-Free Equilibrium

Theorem 2.
The disease-free equilibrium pointC0 of the dynamical system (1) is locally asymptotically stable if R0<1 and unstable R0>1.Proof.
The Jacobian matrix of the dynamical system (1) at the DFE point C0=Λ+αM/γ+μ,γΛ+αM/θ+μγ+μ,0,0,0,0
is given by(38)JC0=−γ+μ00−βΛ+αMγ+μ0σγ−θ+μ000000−ρ+μβΛ+αMγ+μ0000ρ−τ+μ+δ00000τ−κ+μ00θ00κ−σ+μ.
The corresponding characteristic equation for the eigenvaluesλ is λI−JC0=0.
(39)λ+γ+μ00βΛ+αMγ+μ0−σ−γλ+θ+μ000000λ+ρ+μ−βΛ+αMγ+μ0000−ρλ+τ+μ+δ00000−τλ+κ+μ00−θ00−κλ+σ+μ=0.λ+κ+μλ+γ+μλ+θ+μλ+σ+μ−γθσλ+τ+μ+δλ+ρ+μ−ρβΛ+αMγ+μ=0,λ+κ+μ=0⇒λ1=−κ−μ,λ+γ+μλ+θ+μλ+σ+μ−σγθ=0,λ3+γ+θ+σ+3μλ2+θ+μσ+μγ+μθ+σ+2μλ+μ3+γ+θ+σμ2+γθ+σγ+σθμ=0.
This is(40)λ3+Qλ2+Rλ+T=0.
According to the Routh-Hurwitz criterion, sinceQ>0, R>0, and T>0, λ2, λ3, and λ4 will have negative real part as roots.
Also,(41)ρ+μτ+μ+δ−ρβΛ+αMγ+μ>0,λ2+ρ+τ+δ+2μλ+ρ+μτ+μ+δ−ρβΛ+αMγ+μ=0.
The roots,λ5 and λ6, of this characteristic polynomial will have negative real part if and only if
(42)ρ+μτ+μ+δ−ρβΛ+αMγ+μ>0,1−ρβΛ+αMγ+μρ+μτ+μ+δ>0,1−R0>0,R0<1.
Therefore,C0 is asymptotically stable since R0<1 and unstable if R0>1.


### 3.6. Global Stability of the Disease-Free Equilibrium

Theorem 3.
The disease-free equilibrium pointC0 of the dynamical system (1) is globally asymptotically stable in Λ if R0<1 and unstable R0>1.Proof.
Using the Perron eigenvector to prove the global stability of the disease free equilibrium as in [16–18], we apply the matrix-theoretic method. In the dynamical system, the disease compartment is x=EIR1T∈R3 and the nondisease compartment is y∈R6.
Taking the same path as [16, 18], let us set
(43)fx,y≔F−Gx−Fx,y+Gx,y.
Then, the equation of the disease compartment can be written as(44)x1=F−Gx−fx,y.Theorem 4.
LetRo be defined as in equation (11). Then, the threshold property holds for system (1).Proof.
Using the condition outlined in Theorem3, we set the Lyapunov function for the disease-free equilibrium (DFE).
We first findwT (the left eigenvector of the nonnegative matrix G−1F):
(45)G−1F=1ρ+μ00ρρ+μγ+μ+δ1γ+μ+δ0γρρ+μγ+μ+δκ+μγγ+μ+δκ+μ1κ+μ0βS0000000=0βSρ+μ00ρβSρ+μγ+μ+δ00ρβγSρ+μγ+μ+δκ+μ0=0R0γ+μ+δρ00R000R0γκ+μ0⇒wT=0R0γ+μ+δρ00R000R0γκ+μ0n1n2n3=n1n2n3R0⇒R0γ+μ+δρn2=R0n1,R0n2=R0n2andR0γκ+μn2=R0n3.
∴wT=γ+μ+δ/ρ1γ/κ+μ, and any multiple of this becomes our eigenvector.
From equation (40),
(46)x1=F−Gx−fx,y.
That is,fx,y=F−Gx−x1,
(47)fx,y=0βS0000000−ρ+μ00−ρτ+μ+δ00−γκ+μx−βSI00+ρ+μE−ρE+1−αM+τ+μ+δI−γI+κ+μR1=−ρ+μβS0ρ−τ+μ+δ00γ−κ+μEIR1+−βSI+ρ+μE−ρE+1−αM+τ+μ+δI−γI+κ+μR1=βSI−ρ+μEρE−τ+μ+δIγI−κ+μR1+−βSI+ρ+μE−ρE+1−αM+τ+μ+δI−γI+κ+μR1=0,where α=1 at the disease-free equilibrium.
Therefore,fx,y=0, and this satisfies the demand of Theorem 4. The Lyapunov function D is
(48)D=wTG−1x,D=γ+μ+δρ1γκ+μ1ρ+μ00ρρ+μγ+μ+δ1γ+μ+δ0ργρ+μγ+μ+δκ+μγγ+μ+δκ+μ1κ+μEIR1=γ+μ+δρ1γκ+μ1ρ+μEρρ+μγ+μ+δE+1γ+μ+δIργρ+μγ+μ+δκ+μE+γγ+μ+δκ+μI+1κ+μR1=γ+μ+δρρ+μ+ρρ+μγ+μ+δ+ργ2ρ+μγ+μ+δκ+μ2E+1γ+μ+δ+γ2γ+μ+δκ+μ2I+γκ+μ2R1=γ+μ+δ2κ+μ2+ρ2ρ+μκ+μ+ρ2γ2ρρ+μγ+μ+δκ+μ2E+κ+μ2+γ2γ+μ+δκ+μ2I+γκ+μ2R1.
But(49)D1=wTV−1x1=wTV−1F−Vx−wTV−1fx,y,D1=R0−1wTx−wTV−1fx,y.
SincewT>0, V−1, and fx,y=0,
(50)⇒D1<0ifR0<1.
From the derivative of the Lyapunov function,D1<0 when R0<1, which satisfies the condition that the disease-free equilibrium is asymptotically stable and unstable when R0>1.


### 3.7. Existence of the Endemic Equilibrium

The endemic equilibrium point is acquired by mounting the right-hand side of the dynamical system (1) equal to zero and solving them simultaneously [19, 20]. The endemic equilibrium point is C∗=S∗,V∗,E∗,I∗,R1∗,R∗, where
(51)S∗=θ+μΛ+αMσ+μκ+μ+κστI∗κ+μσ+μθ+μγ+μ+βI∗−σθγ,V∗=γΛ+αMσ+μκ+μ+κστI∗κ+μσ+μθ+μγ+μ+βI∗−σθγ,E∗=βθ+μΛ+αMσ+μκ+μ+κστI∗I∗ρ+μκ+μσ+μθ+μγ+μ+βI∗−σθγ,R1∗=τI∗κ+μ,R∗=1σ+μκ+μκτI∗+θγΛ+αMσ+μκ+μ+κστI∗σ+μθ+μγ+μ+βI∗−σθγ.I∗ is the positive root of AI∗2+BI∗+C=0, that is, I∗=−B+B2−4AC/2A>0.We have three possibilities of getting the value ofI∗:
(1)
IfB2−4AC<0, then there is no endemic equilibrium state(2)
IfB2−4AC=0, then again, the endemic equilibrium point does not exist(3)
IfB2−4AC>0, then the endemic equilibrium point exists when AC<0where A=κστρβθ+μ,
(52)B=βσ+μκ+μρθ+μΛ+αM+ρ+μτ+μ+δθ+μ+ρ+μ1−αM,C=ρ+μκ+μθγσ−σ+μθ+μγ+μτ+μ+δ−1−αM.


### 3.8. Local Stability of the Endemic Equilibrium

Theorem 5.
The positive endemic equilibrium pointC∗ of the system (1) is locally asymptotically stable if Ro>1.Proof.
The Jacobian matrix of the system of equation (1) at the endemic point is
(53)JC∗=M1100M140σγM220000M310M33M340000ρM4400000τM5500θ00κM66,where M11=−γ+μ+βI∗, M14=−βS∗, M22=−θ+μ, M31=−βI∗, M33=−ρ+μ, M34=−βS∗, M44=−τ+μ+δ, M55=−κ+μ, and M66=−σ+μ.
The corresponding characteristic equation isJC∗ denoted by λI−JC∗=0 and is given as
(54)λ−M1100M140σγλ−M220000M310λ−M33M340000ρλ−M4400000τλ−M5500θ00κλ−M66=0.
The matrixJC∗ is a strictly column diagonally dominant matrix. Again, all the diagonal entries are negative. Hence, all eigenvalues of JC∗ have negative real part. Now applying the Gershgorin circle theorem [21], C∗ is locally asymptotically stable if M11>M14+σ, M22>γ, M33>M31+M34, M44>ρ, M55>τ, and M66>θ+κ.


### 3.9. Global Stability of the Endemic Equilibrium

Theorem 6.
The dynamical system (1) is said to have an endemic equilibrium if Ro>1, and it is globally asymptotically stable.Proof.
Consider the Lyapunov function defined by(55)QC∗=S−S∗−S∗InS∗S+V−V∗−V∗InV∗V+E−E∗−E∗InE∗E+I−I∗−I∗InI∗I+R1−R1∗−R1∗InR1∗R1+R−R∗−R∗InR∗R.
Computing the derivative ofQ along the solution of the dynamical system in (1) directly,
(56)dQdt=S−S∗SdSdt+V−V∗VdVdt+E−E∗EdEdt+I−I∗IdIdt+R1−R1∗R1dR1dt+R−R∗RdRdt,dQC∗dt=S−S∗SΛ+αM+σR−γ+μS−βSI+V−V∗VγS−θ+μS+E−E∗EβSI−ρ+μE+I−I∗IρE+1−αM−τ+δ+μI+R1−R1∗R1τI−κ+μR1+R−R∗RκR1+θV−σ+μR,dQdt=Λ+M+μN∗+γS∗+σV∗+ρE∗+τ+δI∗+κR1∗+σR∗+βS∗I−μN+Λ+αM+σRS∗S+γSV∗V+βSIE∗E+δI+ρE+1−αMI∗I+τIR1∗R1+κR1+θVR∗R⇒dQdt=Z−Y,where Z=Λ+M+μN∗+γS∗+σV∗+ρE∗+τ+δI∗+κR1∗+σR∗+βS∗I and Y=μN+Λ+αM+σRS∗/S+γSV∗/V+βSIE∗/E+δI+ρE+1−αMI∗/I+τIR1∗/R1+κR1+θVR∗/R.
Imposing the condition thatZ<Y, the derivative of the Lyapunov function with respect to time is less than or equal to zero.
IfZ<Y, then dQ/dt≤0.
ButdQ/dt=0 if and only if S=S∗, V=V∗, E=E∗, I=I∗, R1=R1∗, and R=R∗.
Therefore, the endemic equilibrium pointC∗ is globally asymptotically stable in Γ if Z<Y.
The largest invariant set inC∗=S∗,V∗,E∗,I∗,R1∗,R∗∈Γ:dQ/dt=0 is a singleton, where C∗ is the endemic equilibrium point.


## 3.1. Positivity of Solution

We prove the positivity of the variables in the model. Based on the concept of derivative of a function, the behavior of the function at a known point can be established.Theorem 1.
Let the initial set beS0, V0, E0, I0, R10, and R0 and be nonnegative; then, the solution set of St,Vt,Et,It,R1t,Rt of equation (1) is positive and bounded for all t>0, wherever they exist.Proof.
From equation (1), we can state that
(9)dSdt≥−γ+μ+βIS,dSSdSdt≥−γ+μ+βIdt,InSdSdt≥−γ+μ+βIt+c,StdSdt≥ce−γ+μ+βIt.
Att=0,S0dS/dt≥c,
(10)StdSdt≥S0e−γ+μ+βIt,since
(11)γ+μ+βI>0,StdSdt≥0.
Also,(12)dVdtdSdt≥−θ+μV,VtdSdt≥ce−θ+μt.
Att=0,V0dS/dt≥c,
(13)VtdSdt≥V0e−θ+μt,since
(14)θ+μ>0,VtdSdt≥0.
Also,(15)dEdtdSdt≥−ρ+μE,EtdSdt≥ce−ρ+μt.
Att=0,E0dS/dt≥c,
(16)EtdSdt≥E0e−ρ+μt,since
(17)ρ+μ>0,EtdSdt≥0.
Also,(18)dIdtdSdt≥−τ+μ+δI,ItdSdt≥ce−τ+μ+δt.
Att=0,I0dS/dt≥c,
(19)ItdSdt≥I0e−τ+μ+δt,since
(20)τ+μ+δ>0,ItdSdt≥0.
Also,(21)dR1dtdSdt≥−κ+μR1,R1tdSdt≥ce−κ+μt.
Att=0,R10dS/dt≥c,
(22)R1tdSdt≥R10e−κ+μt,since
(23)κ+μ>0,R1tdSdt≥0.
Also,(24)dRdtdSdt≥−σ+μR,RtdSdt≥ce−σ+μt.
Att=0,R0dS/dt≥c,
(25)RtdSdt≥R0e−σ+μt,since
(26)σ+μ>0,RtdSdt≥0.


## 3.2. Boundedness of the System

The region in which solutions of the tuberculosis (TB) model system are uniformly bounded is the proper subset, and it is given by(27)Γ=S,V,E,I,R1,R∈R+6:N≤Λ+M−δIμ,μ≠0.Proof.
(28)N=S+V+E+I+R1+R,dNdt=Λ−μN+M−δIdNΛ−μN+M−δI=dt−1μ∫−μΛ−μN+M−δIdN=∫dt−1μInΛ−μN+M−δI=t+c,InΛ−μN+M−dI=−μt+c1,Λ−μN+M−δI=c2e−μt,Nt=Λ+M−δI−c2e−μtμ,Att=0,N0=N0,I0=I0,N0=Λ+M−δI0−c2μ,c2=Λ+M−δI0−μN0,Nt=Λ+M−δI−Λ+M−δI0−μN0e−μtμ.
So ast⟶∞, N⟶Λ+M−δI/μ∈R+.
Therefore,Γ is a positive invariant.


## 3.3. Existence of Disease-Free Equilibrium Point

The disease-free equilibrium of the dynamical system (1) is obtained by setting dS/dt=dV/dt=dE/dt=dI/dt=dR1/dt=dR/dt=0, and since there is no disease E=I=R1=R=0,
(29)Λ+αM−γ+μS=0⇒S=Λ+αMγ+μ.Therefore, the disease-free equilibrium of the dynamical system (1) is
(30)C0=S0,V0,E0,I0,R10,R0=Λ+αMγ+μ,γΛ+αMθ+μγ+μ,0,0,0,0.


## 3.4. Basic Reproductive Number

The basic reproductive number can be computed utilizing the cutting edge matrix approach. The basic reproduction number determines the state of a disease with time in a dynamical system [11, 12]. It is utilized to predict the stability of the disease equilibrium. The basic reproductive number is characterized as the quantity of secondary infections that one tainted person can create in a completely susceptible population [13, 14]. According to [13, 15], the next-generation matrix is defined as K=FG−1 and R0=ρFG−1, where ρFG−1 denotes the spectral radius of FG−1.Using the next-generation matrix, we consider only the infectious compartments in the system of differential equation in (1).
(31)dEdt=βSI−ρ+μE,dIdt=ρE+1−αM−τ+δ+μI,dR1dt=τI−κ+μR1.Letf be the count of emerging infection moving into the system and g be the count of infections exiting the system.
(32)f=βSI,0,0,g=ρ+μE,−ρE−1−αM+τ+μ+δI,−γI+k+μR1.The Jacobian matrix off and g are obtained by
(33)F=0βS0000000,G=ρ+μ00−ρτ+μ+δ00−γk+μ.ButR0=ρFG−1.From the relationFG−1, the inverse of G can be calculated:
(34)G−1=γ+μ+δρ+μτ+μ+δ00ρρ+μτ+μ+δ1τ+μ+δ0γρρ+μτ+μ+δk+μγτ+μ+δk+μγ+μ+δτ+μ+δk+μ.Computing the product ofFG−1,
(35)FG−1=0βS0000000γ+μ+δρ+μτ+μ+δ00ρρ+μτ+μ+δ1τ+μ+δ0γρρ+μτ+μ+δk+μγτ+μ+δk+μγ+μ+δτ+μ+δk+μ,FG−1=βρSρ+μτ+μ+δβSτ+μ+δ0000000.By selecting the dominant eigenvalue ofFG−1, the basic reproductive number is
(36)R0=βρSρ+μτ+μ+δ.At the disease-free equilibrium, we substituteS=Λ+αM/γ+μ into the basic reproductive number, R0.This therefore implies that(37)R0=βρΛ+αMγ+μρ+μτ+μ+δ.


## 3.5. Local Stability of the Disease-Free Equilibrium

Theorem 2.
The disease-free equilibrium pointC0 of the dynamical system (1) is locally asymptotically stable if R0<1 and unstable R0>1.Proof.
The Jacobian matrix of the dynamical system (1) at the DFE point C0=Λ+αM/γ+μ,γΛ+αM/θ+μγ+μ,0,0,0,0
is given by(38)JC0=−γ+μ00−βΛ+αMγ+μ0σγ−θ+μ000000−ρ+μβΛ+αMγ+μ0000ρ−τ+μ+δ00000τ−κ+μ00θ00κ−σ+μ.
The corresponding characteristic equation for the eigenvaluesλ is λI−JC0=0.
(39)λ+γ+μ00βΛ+αMγ+μ0−σ−γλ+θ+μ000000λ+ρ+μ−βΛ+αMγ+μ0000−ρλ+τ+μ+δ00000−τλ+κ+μ00−θ00−κλ+σ+μ=0.λ+κ+μλ+γ+μλ+θ+μλ+σ+μ−γθσλ+τ+μ+δλ+ρ+μ−ρβΛ+αMγ+μ=0,λ+κ+μ=0⇒λ1=−κ−μ,λ+γ+μλ+θ+μλ+σ+μ−σγθ=0,λ3+γ+θ+σ+3μλ2+θ+μσ+μγ+μθ+σ+2μλ+μ3+γ+θ+σμ2+γθ+σγ+σθμ=0.
This is(40)λ3+Qλ2+Rλ+T=0.
According to the Routh-Hurwitz criterion, sinceQ>0, R>0, and T>0, λ2, λ3, and λ4 will have negative real part as roots.
Also,(41)ρ+μτ+μ+δ−ρβΛ+αMγ+μ>0,λ2+ρ+τ+δ+2μλ+ρ+μτ+μ+δ−ρβΛ+αMγ+μ=0.
The roots,λ5 and λ6, of this characteristic polynomial will have negative real part if and only if
(42)ρ+μτ+μ+δ−ρβΛ+αMγ+μ>0,1−ρβΛ+αMγ+μρ+μτ+μ+δ>0,1−R0>0,R0<1.
Therefore,C0 is asymptotically stable since R0<1 and unstable if R0>1.


## 3.6. Global Stability of the Disease-Free Equilibrium

Theorem 3.
The disease-free equilibrium pointC0 of the dynamical system (1) is globally asymptotically stable in Λ if R0<1 and unstable R0>1.Proof.
Using the Perron eigenvector to prove the global stability of the disease free equilibrium as in [16–18], we apply the matrix-theoretic method. In the dynamical system, the disease compartment is x=EIR1T∈R3 and the nondisease compartment is y∈R6.
Taking the same path as [16, 18], let us set
(43)fx,y≔F−Gx−Fx,y+Gx,y.
Then, the equation of the disease compartment can be written as(44)x1=F−Gx−fx,y.Theorem 4.
LetRo be defined as in equation (11). Then, the threshold property holds for system (1).Proof.
Using the condition outlined in Theorem3, we set the Lyapunov function for the disease-free equilibrium (DFE).
We first findwT (the left eigenvector of the nonnegative matrix G−1F):
(45)G−1F=1ρ+μ00ρρ+μγ+μ+δ1γ+μ+δ0γρρ+μγ+μ+δκ+μγγ+μ+δκ+μ1κ+μ0βS0000000=0βSρ+μ00ρβSρ+μγ+μ+δ00ρβγSρ+μγ+μ+δκ+μ0=0R0γ+μ+δρ00R000R0γκ+μ0⇒wT=0R0γ+μ+δρ00R000R0γκ+μ0n1n2n3=n1n2n3R0⇒R0γ+μ+δρn2=R0n1,R0n2=R0n2andR0γκ+μn2=R0n3.
∴wT=γ+μ+δ/ρ1γ/κ+μ, and any multiple of this becomes our eigenvector.
From equation (40),
(46)x1=F−Gx−fx,y.
That is,fx,y=F−Gx−x1,
(47)fx,y=0βS0000000−ρ+μ00−ρτ+μ+δ00−γκ+μx−βSI00+ρ+μE−ρE+1−αM+τ+μ+δI−γI+κ+μR1=−ρ+μβS0ρ−τ+μ+δ00γ−κ+μEIR1+−βSI+ρ+μE−ρE+1−αM+τ+μ+δI−γI+κ+μR1=βSI−ρ+μEρE−τ+μ+δIγI−κ+μR1+−βSI+ρ+μE−ρE+1−αM+τ+μ+δI−γI+κ+μR1=0,where α=1 at the disease-free equilibrium.
Therefore,fx,y=0, and this satisfies the demand of Theorem 4. The Lyapunov function D is
(48)D=wTG−1x,D=γ+μ+δρ1γκ+μ1ρ+μ00ρρ+μγ+μ+δ1γ+μ+δ0ργρ+μγ+μ+δκ+μγγ+μ+δκ+μ1κ+μEIR1=γ+μ+δρ1γκ+μ1ρ+μEρρ+μγ+μ+δE+1γ+μ+δIργρ+μγ+μ+δκ+μE+γγ+μ+δκ+μI+1κ+μR1=γ+μ+δρρ+μ+ρρ+μγ+μ+δ+ργ2ρ+μγ+μ+δκ+μ2E+1γ+μ+δ+γ2γ+μ+δκ+μ2I+γκ+μ2R1=γ+μ+δ2κ+μ2+ρ2ρ+μκ+μ+ρ2γ2ρρ+μγ+μ+δκ+μ2E+κ+μ2+γ2γ+μ+δκ+μ2I+γκ+μ2R1.
But(49)D1=wTV−1x1=wTV−1F−Vx−wTV−1fx,y,D1=R0−1wTx−wTV−1fx,y.
SincewT>0, V−1, and fx,y=0,
(50)⇒D1<0ifR0<1.
From the derivative of the Lyapunov function,D1<0 when R0<1, which satisfies the condition that the disease-free equilibrium is asymptotically stable and unstable when R0>1.


## 3.7. Existence of the Endemic Equilibrium

The endemic equilibrium point is acquired by mounting the right-hand side of the dynamical system (1) equal to zero and solving them simultaneously [19, 20]. The endemic equilibrium point is C∗=S∗,V∗,E∗,I∗,R1∗,R∗, where
(51)S∗=θ+μΛ+αMσ+μκ+μ+κστI∗κ+μσ+μθ+μγ+μ+βI∗−σθγ,V∗=γΛ+αMσ+μκ+μ+κστI∗κ+μσ+μθ+μγ+μ+βI∗−σθγ,E∗=βθ+μΛ+αMσ+μκ+μ+κστI∗I∗ρ+μκ+μσ+μθ+μγ+μ+βI∗−σθγ,R1∗=τI∗κ+μ,R∗=1σ+μκ+μκτI∗+θγΛ+αMσ+μκ+μ+κστI∗σ+μθ+μγ+μ+βI∗−σθγ.I∗ is the positive root of AI∗2+BI∗+C=0, that is, I∗=−B+B2−4AC/2A>0.We have three possibilities of getting the value ofI∗:
(1)
IfB2−4AC<0, then there is no endemic equilibrium state(2)
IfB2−4AC=0, then again, the endemic equilibrium point does not exist(3)
IfB2−4AC>0, then the endemic equilibrium point exists when AC<0where A=κστρβθ+μ,
(52)B=βσ+μκ+μρθ+μΛ+αM+ρ+μτ+μ+δθ+μ+ρ+μ1−αM,C=ρ+μκ+μθγσ−σ+μθ+μγ+μτ+μ+δ−1−αM.


## 3.8. Local Stability of the Endemic Equilibrium

Theorem 5.
The positive endemic equilibrium pointC∗ of the system (1) is locally asymptotically stable if Ro>1.Proof.
The Jacobian matrix of the system of equation (1) at the endemic point is
(53)JC∗=M1100M140σγM220000M310M33M340000ρM4400000τM5500θ00κM66,where M11=−γ+μ+βI∗, M14=−βS∗, M22=−θ+μ, M31=−βI∗, M33=−ρ+μ, M34=−βS∗, M44=−τ+μ+δ, M55=−κ+μ, and M66=−σ+μ.
The corresponding characteristic equation isJC∗ denoted by λI−JC∗=0 and is given as
(54)λ−M1100M140σγλ−M220000M310λ−M33M340000ρλ−M4400000τλ−M5500θ00κλ−M66=0.
The matrixJC∗ is a strictly column diagonally dominant matrix. Again, all the diagonal entries are negative. Hence, all eigenvalues of JC∗ have negative real part. Now applying the Gershgorin circle theorem [21], C∗ is locally asymptotically stable if M11>M14+σ, M22>γ, M33>M31+M34, M44>ρ, M55>τ, and M66>θ+κ.


## 3.9. Global Stability of the Endemic Equilibrium

Theorem 6.
The dynamical system (1) is said to have an endemic equilibrium if Ro>1, and it is globally asymptotically stable.Proof.
Consider the Lyapunov function defined by(55)QC∗=S−S∗−S∗InS∗S+V−V∗−V∗InV∗V+E−E∗−E∗InE∗E+I−I∗−I∗InI∗I+R1−R1∗−R1∗InR1∗R1+R−R∗−R∗InR∗R.
Computing the derivative ofQ along the solution of the dynamical system in (1) directly,
(56)dQdt=S−S∗SdSdt+V−V∗VdVdt+E−E∗EdEdt+I−I∗IdIdt+R1−R1∗R1dR1dt+R−R∗RdRdt,dQC∗dt=S−S∗SΛ+αM+σR−γ+μS−βSI+V−V∗VγS−θ+μS+E−E∗EβSI−ρ+μE+I−I∗IρE+1−αM−τ+δ+μI+R1−R1∗R1τI−κ+μR1+R−R∗RκR1+θV−σ+μR,dQdt=Λ+M+μN∗+γS∗+σV∗+ρE∗+τ+δI∗+κR1∗+σR∗+βS∗I−μN+Λ+αM+σRS∗S+γSV∗V+βSIE∗E+δI+ρE+1−αMI∗I+τIR1∗R1+κR1+θVR∗R⇒dQdt=Z−Y,where Z=Λ+M+μN∗+γS∗+σV∗+ρE∗+τ+δI∗+κR1∗+σR∗+βS∗I and Y=μN+Λ+αM+σRS∗/S+γSV∗/V+βSIE∗/E+δI+ρE+1−αMI∗/I+τIR1∗/R1+κR1+θVR∗/R.
Imposing the condition thatZ<Y, the derivative of the Lyapunov function with respect to time is less than or equal to zero.
IfZ<Y, then dQ/dt≤0.
ButdQ/dt=0 if and only if S=S∗, V=V∗, E=E∗, I=I∗, R1=R1∗, and R=R∗.
Therefore, the endemic equilibrium pointC∗ is globally asymptotically stable in Γ if Z<Y.
The largest invariant set inC∗=S∗,V∗,E∗,I∗,R1∗,R∗∈Γ:dQ/dt=0 is a singleton, where C∗ is the endemic equilibrium point.


## 4. TB Model Extension to Optimal Control

An analysis of the optimal controls to ascertain its effects on the model is been conducted. The optimal control problem is obtained by integrating the undermentioned control functions into the tuberculosis model (1) and introducing an objective functional that desires to minimize the controls u1,u2,u3, where u1 is the vaccination of the susceptible population S as a control measure, u2 is the treatment of the infected individuals I as a control measure, and u3 is the education/sensitization of the exposed population E as a control measure.By inserting the various controls, the system with the optimal controls becomes(57)dSdt=Λ+αM+σR−u1γS−μS−βSI,(58)dVdt=u1γS−θ+μV,(59)dEdt=βSI−1−u3ρE−μE,(60)dIdt=1−u3ρE+1−αM−u2τI−δ+μI,(61)dR1dtu2τI−1−u2κR1−μR1,(62)dRdt1−u2κR1−1−u1θV−σ+μR.Let the optimal levels of the control set beu, which is Lebesgue measurable and defined as
(63)U=u1t,u2t,u3t:0≤u1<1,0≤u2<1,0≤u3<1,0≤t≤tf.The quadratic nature of the control efforts as a result of the assumption that costs is generally nonlinear in nature. Our objective is to minimise the number of infections and reduce the cost of treatment.The problem is to find a controlut and its associated state variables St, Vt, Et, It, R1t, and Rt to minimize the objective functional J given by
(64)J=minu1,u2,u3∫0tfa1I+a2R1+∑i=13wiui2dt.That is,J=minu1,u2,u3∫0tfa1I+a2R1+w1u12+w2u22+w3u32dt subject to the differential equation system (57), where a1, a2, w1, w2, and w3 are the weight constants to balance the terms in the integrals to abstain the ascendance of one over the others.Also,a1, I, and a2R1 are the cost associated with the infected individuals and the individuals with resistance to treatment, respectively, while w1u12, w2u22, and w3u32 are the cost associated with vaccination, treatment, and sensitization as preventive measures. tf is the period of the intervention.The purpose of inserting the controls is to minimize the number of infections and at the same time reduce the cost of treatment.Our task at this point is to find the optimal functions:u1∗t, u2∗t, and u3∗t such that Ju1∗t,u1∗t,u1∗t=minu1,u2,u3∈∪Ju1,u2,u3, where U=ui:0≤uit≤1,t∈0,tf,i=1,2,3 is referred to as the control set.


### 4.1. Pontryagin’s Maximum Principle

Consider the Lagrangian function:(65)LI,R1,u1,u2,u3,t=a1I+a2R1+w1u12+w2u22+w3u32.The Pontryagin maximum principle provides the essential condition that the optimal must satisfy. This changes the system of the differential equation into minimization problem pointwise HamiltonianH with respect to u1,u2,u3.Hence, the HamiltonianH becomes
(66)HS,V,E,I,R1,R,t=LI,R1,u1,u2,u3,t+λ1dSdt+λ2dVdt+λ3dEdt+λ4dIdt+λ5dR1dt+λ6dRdt,where λ1, λ2, λ3, λ4, λ5, and λ6 are disjoint variables.
(67)H=a1I+a2R1+w1u12+w2u22+w3u32+λ1Λ+αM+σR−u1γS−μS−βSI+λ2u1γS−θ+μV+λ3βSI−1−u3ρE−μE+λ41−u3ρE+1−αM−u2τI−δ+μI+λ5u2τI−1−u2κR1−μR1+λ61−u2κR1+1−u1θV−σ+μR,considering the relation
(68)dλidt=−∂H∂x•t.By taking partial derivatives of the Hamiltonian function with respect toS,V,E,I,R1,R and negating each of them, the following costate variables are the solutions of the adjoint systems.
(69)dλ1dt=−∂H∂S=λ1−λ2u1γ+λ1−λ3βI+μλ1,dλ2dt=−∂H∂V=λ2−λ6θ+u2λ2+u1θλ6,dλ3dt−∂H∂E=1−u3λ3−λ4ρ+μλ3,dλ4dt−∂H∂I=λ1−λ3βS+λ4−λ5u2τ+μ+δλ4,dλ5dt−∂H∂R1=1−u2λ5−λ6κ+μλ5,dλ6dt−∂H∂R=λ6−λ1σ+μλ6.The above satisfy the transversality condition:(70)λ1tf=λ2tf=λ3tf=λ4tf=λ5tf=λ6tf=0.Moreover, the characterization of the optimal control is obtained by solving(71)∂H∂ui=0,where ui=ui∗,i=1,2,3.
(72)∂H∂u1=2w1u1+λ2−λ1γS−λ6θV⇒2w1u1+λ2−λ1γS−λ6θV=0∴u1∗=λ1−λ2γS∗+λ6θV∗2w1,∂H∂u2=2w2u2−λ4−λ5τI∗−λ6−λ5κR1∗⇒2w2u2−λ4−λ5τI∗−λ6−λ5κR1∗=0∴u2∗=λ4−λ5τI∗+λ6−λ5κR1∗2w2,∂H∂u3=2w3u3−λ4−λ3ρE∗⇒2w3u3−λ4−λ3ρE∗=0∴u3∗=λ4−λ3ρE∗2w3,u1∗=λ2−λ1γS∗+λ6θV∗2w1,u2∗=λ4−λ5τI∗+λ6−λ5κR1∗2w2,u1∗=λ4−λ3ρE∗2w3.Theorem 7.
Given the optimal control vectoru1∗t,u2∗t,u3∗t and the solutions S∗,V∗,E∗,I∗,R1∗,R∗ of the corresponding state system (34) and (35) that minimise the objective functional J over ∪, then there exist adjoint variables λ1, λ2, λ3, λ4, λ5, and λ6, where
(73)u1∗t=max0,min1,λ2−λ1γS∗+λ6θV∗2w1,u2∗t=max0,min1,λ4−λ5τI∗+λ6−λ5κR1∗2w2,u3∗t=max0,min1,λ4−λ3ρE∗2w3,where λ1, λ2, λ3, λ4, λ5, and λ6 are the solutions of equations (28) and (30).Proof.
The presence of optimal control is as an aftereffect of the convexity of the integral ofJ regarding u1, u2, and u3, the Lipschitz property of the state system concerning the state factors from the earlier boundedness of the state arrangements [22, 23].
The differential conditions administering the adjoint factors are acquired by separation of the Hamiltonian work, assessed at the ideal control. By standard control contentions including the limits on the control, we conclude(74)u1∗=0,ifη1∗≤0η1∗,if0<η1∗<11,ifη1∗≥1u2∗=0,ifη2∗≤0η2∗,if0<η2∗<11,ifη2∗≥1u3∗=0,ifη3∗≤0η3∗,if0<η3∗<11,ifη3∗≥1,where η1∗=λ1−λ2γS∗+λ6θV∗/2w1, η2∗=λ4−λ5τI∗+λ6−λ5κR1∗/2w2, and η3∗=λ4−λ3ρE∗/2w3.


## 4.1. Pontryagin’s Maximum Principle

Consider the Lagrangian function:(65)LI,R1,u1,u2,u3,t=a1I+a2R1+w1u12+w2u22+w3u32.The Pontryagin maximum principle provides the essential condition that the optimal must satisfy. This changes the system of the differential equation into minimization problem pointwise HamiltonianH with respect to u1,u2,u3.Hence, the HamiltonianH becomes
(66)HS,V,E,I,R1,R,t=LI,R1,u1,u2,u3,t+λ1dSdt+λ2dVdt+λ3dEdt+λ4dIdt+λ5dR1dt+λ6dRdt,where λ1, λ2, λ3, λ4, λ5, and λ6 are disjoint variables.
(67)H=a1I+a2R1+w1u12+w2u22+w3u32+λ1Λ+αM+σR−u1γS−μS−βSI+λ2u1γS−θ+μV+λ3βSI−1−u3ρE−μE+λ41−u3ρE+1−αM−u2τI−δ+μI+λ5u2τI−1−u2κR1−μR1+λ61−u2κR1+1−u1θV−σ+μR,considering the relation
(68)dλidt=−∂H∂x•t.By taking partial derivatives of the Hamiltonian function with respect toS,V,E,I,R1,R and negating each of them, the following costate variables are the solutions of the adjoint systems.
(69)dλ1dt=−∂H∂S=λ1−λ2u1γ+λ1−λ3βI+μλ1,dλ2dt=−∂H∂V=λ2−λ6θ+u2λ2+u1θλ6,dλ3dt−∂H∂E=1−u3λ3−λ4ρ+μλ3,dλ4dt−∂H∂I=λ1−λ3βS+λ4−λ5u2τ+μ+δλ4,dλ5dt−∂H∂R1=1−u2λ5−λ6κ+μλ5,dλ6dt−∂H∂R=λ6−λ1σ+μλ6.The above satisfy the transversality condition:(70)λ1tf=λ2tf=λ3tf=λ4tf=λ5tf=λ6tf=0.Moreover, the characterization of the optimal control is obtained by solving(71)∂H∂ui=0,where ui=ui∗,i=1,2,3.
(72)∂H∂u1=2w1u1+λ2−λ1γS−λ6θV⇒2w1u1+λ2−λ1γS−λ6θV=0∴u1∗=λ1−λ2γS∗+λ6θV∗2w1,∂H∂u2=2w2u2−λ4−λ5τI∗−λ6−λ5κR1∗⇒2w2u2−λ4−λ5τI∗−λ6−λ5κR1∗=0∴u2∗=λ4−λ5τI∗+λ6−λ5κR1∗2w2,∂H∂u3=2w3u3−λ4−λ3ρE∗⇒2w3u3−λ4−λ3ρE∗=0∴u3∗=λ4−λ3ρE∗2w3,u1∗=λ2−λ1γS∗+λ6θV∗2w1,u2∗=λ4−λ5τI∗+λ6−λ5κR1∗2w2,u1∗=λ4−λ3ρE∗2w3.Theorem 7.
Given the optimal control vectoru1∗t,u2∗t,u3∗t and the solutions S∗,V∗,E∗,I∗,R1∗,R∗ of the corresponding state system (34) and (35) that minimise the objective functional J over ∪, then there exist adjoint variables λ1, λ2, λ3, λ4, λ5, and λ6, where
(73)u1∗t=max0,min1,λ2−λ1γS∗+λ6θV∗2w1,u2∗t=max0,min1,λ4−λ5τI∗+λ6−λ5κR1∗2w2,u3∗t=max0,min1,λ4−λ3ρE∗2w3,where λ1, λ2, λ3, λ4, λ5, and λ6 are the solutions of equations (28) and (30).Proof.
The presence of optimal control is as an aftereffect of the convexity of the integral ofJ regarding u1, u2, and u3, the Lipschitz property of the state system concerning the state factors from the earlier boundedness of the state arrangements [22, 23].
The differential conditions administering the adjoint factors are acquired by separation of the Hamiltonian work, assessed at the ideal control. By standard control contentions including the limits on the control, we conclude(74)u1∗=0,ifη1∗≤0η1∗,if0<η1∗<11,ifη1∗≥1u2∗=0,ifη2∗≤0η2∗,if0<η2∗<11,ifη2∗≥1u3∗=0,ifη3∗≤0η3∗,if0<η3∗<11,ifη3∗≥1,where η1∗=λ1−λ2γS∗+λ6θV∗/2w1, η2∗=λ4−λ5τI∗+λ6−λ5κR1∗/2w2, and η3∗=λ4−λ3ρE∗/2w3.


## 5. Numerical Results

The state systems, adjoint equations, and the transversality terms are solved simultaneously to get the optimal strategies. The optimal problem is a two-point boundary value problem with two abstracted boundary conditions at initial timest=0 and t=tf, where tf=3 months. This represents the period at which preventive strategies and treatment are expected to be stopped. The numerical simulation was conducted by solving the state equations, the adjoint equations, and the transversality conditions using the Runge-Kutta fourth-order scheme by guessing the controls over a simulated time. We then use the current iteration of the state equation, the adjoint equations, and the transversality conditions by a backward method. Further iterations are done until values of the unknown variables at the previous iteration are very close to those at the present iteration [18, 24, 25].Table3 shows the various parameter values used in the TB model simulations.Table 3
Numerical values.
ParameterValueReferenceΛ10Assumedα0.9[26]β0.05Assumedγ0.2Assumedσ0.4Assumedμ0.01874[27]θ0.1Assumedρ0.00114[27]δ0.1577[27]κ1.00Assumedτ0.4[28]


### 5.1. Strategy 1: Treatment, Prevention, and Vaccination of the Susceptible Population

Objective functional was optimised by using treatment, prevention, and vaccination as control measures. As a result of these control measures, there have been significant reduction of infections and an increase in the number of recovered populations as shown in Figures2 and 3.Figure 2
Optimal treatment of the infected population.Figure 3
Optimal prevention and vaccination of the susceptible population.


### 5.2. Strategy 2: Prevention and Treatment of the Infected Population

Objective functional was optimised by using prevention, vaccination, and treatment as control measures. The outcome of these control measures indicates a reduction of the population infected and increased recoveries, indicating that these variables have greatly impacted in the combat of the spread of infections as shown in Figures4 and 5.Figure 4
Optimal prevention and treatment of the infected population.Figure 5
Optimal prevention and vaccination of the susceptible population.


### 5.3. Strategy 3: Vaccination and Treatment of the Infected Population

Objective functional was optimised by using treatment, vaccination, and prevention of the susceptible population as control measures. Figures6 and 7 show the effects of treatment and vaccination, respectively: an increase in the recovery population, a decrease in the infectious population, and a decrease in the number of population susceptible.Figure 6
Optimal treatment of the population infected.Figure 7
Optimal vaccination of the susceptible population.


## 5.1. Strategy 1: Treatment, Prevention, and Vaccination of the Susceptible Population

Objective functional was optimised by using treatment, prevention, and vaccination as control measures. As a result of these control measures, there have been significant reduction of infections and an increase in the number of recovered populations as shown in Figures2 and 3.Figure 2
Optimal treatment of the infected population.Figure 3
Optimal prevention and vaccination of the susceptible population.


## 5.2. Strategy 2: Prevention and Treatment of the Infected Population

Objective functional was optimised by using prevention, vaccination, and treatment as control measures. The outcome of these control measures indicates a reduction of the population infected and increased recoveries, indicating that these variables have greatly impacted in the combat of the spread of infections as shown in Figures4 and 5.Figure 4
Optimal prevention and treatment of the infected population.Figure 5
Optimal prevention and vaccination of the susceptible population.


## 5.3. Strategy 3: Vaccination and Treatment of the Infected Population

Objective functional was optimised by using treatment, vaccination, and prevention of the susceptible population as control measures. Figures6 and 7 show the effects of treatment and vaccination, respectively: an increase in the recovery population, a decrease in the infectious population, and a decrease in the number of population susceptible.Figure 6
Optimal treatment of the population infected.Figure 7
Optimal vaccination of the susceptible population.


## 6. Conclusion

A deterministic model for tuberculosis was formulated and analysed. The basic reproductive number for the TB model is estimated using the next-generation matrix method.The equilibrium points of the TB model and their local and global stability were determined. It was established that if the basic reproductive number was less than unityR0<1, then the disease-free equilibrium is stable and unstable if R0>1. Furthermore, we investigated the optimal prevention, treatment, and vaccination as control measures for the disease.Objective functional was optimised by using treatment, prevention, and vaccination as control measures. As a result of these control measures, there have been significant reduction of infections and an increase in the number of recovered populations as shown in Figures2 and 3.Objective functional was optimised by using prevention, vaccination, and treatment as control measures. The outcome of these control measures indicates a reduction of the population infected and increased recoveries as shown in Figures4 and 5, indicating that these variables have greatly impacted in the combat of the spread of infections.Objective functional was optimised by using treatment, vaccination, and prevention of the susceptible population as control measures. An increase in the recovery population, a decrease in the infectious population, and a decrease in the number of population susceptible are shown in Figures6 and 7.It was established that the best control measure in combating tuberculosis infections is prevention and vaccination of the susceptible population.



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*Source: 5593864-2021-07-28.xml*