# Modeling the Effects of Helminth Infection on the Transmission Dynamics of Mycobacterium tuberculosis under Optimal Control Strategies

Authors: Aristide G. Lambura; Gasper G. Mwanga; Livingstone Luboobi; Dmitry Kuznetsov
Journal: Computational and Mathematical Methods in Medicine
 (2020)

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## Abstract

A deterministic mathematical model for the transmission and control of cointeraction of helminths and tuberculosis is presented, to examine the impact of helminth on tuberculosis and the effect of control strategies. The equilibrium point is established, and the effective reproduction number is computed. The disease-free equilibrium point is confirmed to be asymptotically stable whenever the effective reproduction number is less than the unit. The analysis of the effective reproduction number indicates that an increase in the helminth cases increases the tuberculosis cases, suggesting that the control of helminth infection has a positive impact on controlling the dynamics of tuberculosis. The possibility of bifurcation is investigated using the Center Manifold Theorem. Sensitivity analysis is performed to determine the effect of every parameter on the spread of the two diseases. The model is extended to incorporate control measures, and Pontryagin’s Maximum Principle is applied to derive the necessary conditions for optimal control. The optimal control problem is solved numerically by the iterative scheme by considering vaccination of infants for Mtb, treatment of individuals with active tuberculosis, mass drug administration with regular antihelminthic drugs, and sanitation control strategies. The results show that a combination of educational campaign, treatment of individuals with active tuberculosis, mass drug administration, and sanitation is the most effective strategy to control helminth-Mtb coinfection. Thus, to effectively control the helminth-Mtb coinfection, we suggest to public health stakeholders to apply intervention strategies that are aimed at controlling helminth infection and the combination of vaccination of infants and treatment of individuals with active tuberculosis.


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## Body

## 1. Introduction

Soil-transmitted helminth infections are common infections that affect poor communities of the world. The species that infect people are the roundworm (Ascaris lumbricoides), the whipworm (Trichuris trichiura), and the hookworm (Necator americanus). These parasites reside in the intestine and release eggs to contaminate the soil [1]. Approximately 1.5 billion people are infected with soil-transmitted helminths worldwide which are widely distributed in tropical and subtropical areas, with the greatest numbers occurring in sub-Saharan Africa, America, China, and East Asia [1]. On the other hand, tuberculosis (TB) is an infectious disease that is caused by the bacillus Mycobacterium tuberculosis (Mtb). Approximately 5-10% of the estimated 1.7 billion people infected with Mtb develop active TB during their lifetime [2]. The study by Watts et al. [3] suggests that infection by trichuriasis reduces the chance of latent tuberculosis (LTB) or blunts tuberculin skin tests (TST). Moreover, coinfection of helminth and Mtb has been a public health issue in developing countries [4]. The study by Dias et al. [5] suggests that intestinal helminth infection has a harmful effect to contain the Mtb infection and contributes to the development of active TB for coinfected individuals. Furthermore, coinfection with helminths increases the susceptibility to Mtb [6].The usefulness of mathematical modeling for understanding and assessing the dynamics of infectious diseases cannot be underrated and has been used over the years to investigate the optimal mechanisms for intervention strategies in controlling the spread of infectious diseases. The transmission and dynamics of Mtb with other diseases have been studied by many researchers [7–9]. Bhunu et al. [7] developed an HIV/AIDS and TB coinfection model that considers the treatment of exposed and active TB individuals and therapy for AIDS. They found that the treatment of exposed and active forms of TB results in stuck commencement of the AIDS stage of HIV infection. Fatmawati and Tasman [8] proposed an optimal control model for the transmission of TB-HIV coinfection. They found that the best and effective strategy to lower the TB-HIV infection is to apply a combination of anti-TB and ARV treatment. Okosun [9] discussed the synergy between malaria and schistosomiasis in the presence of treatment. Their study revealed that an effective control of malaria could be achieved if efficient treatment and prevention of schistosomiasis are implemented. There is a need for modeling coinfection of infectious diseases in areas with neglected tropical diseases (NTD) which include helminth infections, Mtb, and malaria to list a few [10], due to their geographical overlap. To the best of our understanding, no study has been carried out to consider the helminth-Mtb coinfection with the use of optimal control theory.In this paper, our emphasis is to propose a compartmental system for the codynamics of helminths and Mtb diseases and to explore the effect of helminths on Mtb and vice versa. The model is extended to incorporate four time-dependent controls, namely, educational campaign to sensitize the parents to take their babies to the Bacille Calmette-Guerin (BCG) vaccine clinic; treatment of individuals with active TB; treatment of individuals infected with helminth parasites through mass drug administration (MDA); and sanitation to lessen the ingestion rate of parasites from the polluted environment.


## 2. Materials and Methods

### 2.1. Model Formulation

The helminth-Mtb infection model is formulated under the following model assumptions:(i)
The susceptible individuals cannot simultaneously get infected by both diseases but rather get infected with one disease and later by the other disease(ii)
The population is homogeneously mixed(iii)
The incidence rate for helminth infection follows the logistic model while Mtb is assumed to be frequency-dependent(iv)
An individual gets helminth infection after sufficient contact with the polluted environment(v)
Due to the fact that Mtb infection is more severe than helminth infection, we assume that coinfected individuals seek TB treatment rather than helminth infection(vi)
The Bacille Calmette-Guerin (BCG) vaccine does not confer a total immunityThe model is formulated by considering two populations, namely, the human population and the parasite populationM that are present in the contaminated environment. The human population is subdivided into nine classes, namely, susceptible S, the vaccinated against Mtb VT, those infected with Mtb but do not show clinical symptoms of TB ET, those with active TB IT, TB-recovered individuals RT, those infected by helminths IH, those who have recovered from helminths RH, and those who are dually infected by helminths and Mtb IHT. Thus, the total human population is given by N=S+VT+ET+IT+IH+IHT+RT+RH+RHT. The susceptible population increases by recruitment through birth at the rate 1−ηbN and by those who lose their temporary immunity against the helminths and Mtb at rates γ1 and γ2, respectively. The susceptible individuals get infected with Mtb upon sufficient contact with an infectious individual with the force of infection λT=βTIT+σIHT/N, where βT is the transmission rate and σ>1 is the modification parameter for coinfected individuals that models infectivity for susceptible individuals. The susceptible individuals also get infected by helminths after ingestion of eggs/larvae or skin penetration by infective larvae with the force of infection λH=βHM/K+M, where βH is the ingestion rate of parasitic worms and K is the density of parasites in the soil. The helminth-infected individuals increase due to susceptible individuals being infected with helminths, Mtb-vaccinated individuals infected with helminths, and coinfected individuals who recover from Mtb at the rate τ2. Furthermore, due to the fact that the BCG vaccine does not grant total protection [11], then individuals exposed to Mtb increase due to Mtb-vaccinated individuals losing their protection against Mtb and getting infected at the rate ρλT, with ρ∈0,1 such that 1−ρ is the efficacy of the vaccine.The chronic infection by helminths alters the chance of getting infected and the progression of coinfecting parasites (virus or bacteria) [12]; as a result, individuals infested with helminths are at higher risk of developing tuberculosis [13]. Therefore, individuals coinfected with helminth and Mtb are increased by helminth-infected individuals being infected with Mtb at the force of infection gλT, where g>1 is the modification parameter that models increased susceptibility to Mtb, and Mtb-exposed individuals getting infected by helminths at a force of infection λH. In addition, the number of individuals with active TB increases as individuals progressed from the exposed class at the ratekand coinfected individuals recovered from helminth infection at the rateτ1and decreases due to natural recovery at the rateτ2. The parasite populations in the environment increase as they are released by the helminth-infected individuals at the rate ε1 and coinfected individuals at the rate ε2 and are cleared from the environment at a rate μM.The description above is illustrated in Figure 1, and the governing system of nonlinear differential equations is shown in model (1). The model parameters are described in Table 1.
(1)dSdt=1−ηbN+γ1RH+γ2RT−μ+λT+λHS,dVTdt=ηbN−μ+ρλT+λHVT,dETdt=λTS+ρλTVT+λTRH−μ+k+λHET,dITdt=kET+τ1IHT−μ+dT+τ2+λHIT,dIHdt=λHS+λHVT+λHRT+τ2IHT−gλT+μ+dH+τ1+ε1IH,dIHTdt=gλTIH+λHET+IT−μ+dHT+τ1+τ2+ε2IHT,dRTdt=τ2IT−λH+μ+γ2RT,RHdt=τ1IH−λT+μ+γ1RH,dMdt=ε1IH+ε2IHT−μMM,where
(2)λT=βTIT+σIHTN,λH=βHMM+K.Figure 1
Compartmental diagram for the helminth and Mycobacterium tuberculosis coinfection. The dotted lines indicate the interaction of individuals with the polluted environment.Table 1
Description of the model parameters.
ParametersDefinitionEstimated valueReferencebNatural birth rate1/18250day−1[14]βTTransmission rate for Mtb0.42day−1[15]μNatural death rate1/60×365day−1[16]γ1Loss of temporary immunity for helminth-recovered individuals1/70day−1AssumedβHIngestion rate of parasitic worms1parasiteday−1AssumedρVaccine wane rate0.7 dimensionless[17]τ1Natural recovery rate for helminth-infected individuals1/28day−1[18]dHHelminth disease-induced death rate35/1000day−1[14]γ2Loss of temporary immunity for Mtb-recovered individuals0.3day−1[19]τ2Natural recovery rate for Mtb-infected individuals0.2/365day−1[20]dTMtb disease-induced death rate0.08day−1[17]dHTHelminth-Mtb disease-induced death rate0.08day−1AssumedkProgression to active TB0.00013/365day−1[21]μMClearance rate of parasitic worms13/37500day−1[14]ε1Shading rate for helminth-infected individuals0.09day−1[18]KNumber of parasites in the environment105 parasites[18]ε2Shading rate for coinfected individuals0.1day−1AssumedgModification parameter1.12 dimensionlessAssumedσModification parameter1.5 dimensionlessAssumedWe perform a qualitative analysis of model (1) using a dimensionless version such that dimensionless variables obtained by setting s=S/N, vT=VT/N, eT=ET/N, iT=IT/N, iH=IH/N, iHT=IHT/N, rT=RT/N, rH=RH/N, rHT=RHT/N, and m=M/K replace the original variables S, VT, ET, IT, IT, IHT, RT, RH, RHT, and M, respectively, in which the rate of change of the population at time t is given by
(3)dNdt=b−μN+dTIT−dH+ε1IH−dHT+ε2IHT.Upon differentiating dimensionless variables with respect to time and using equation (3), system (1) becomes
(4)dsdt=1−ηb+γ1rH+γ2rT−b+λT∗+λH∗−dTiT−dH+ε1iH−dHT+ε2iHTs,dvTdt=ηb−b+ρλT∗+λH∗−dTiT−dH+ε1iH−dHT+ε2iHTvT,deTdt=λT∗s+ρvT+rH−b+k+λH∗−dTiT−dH+ε1iH−dHT+ε2iHTeT,diTdt=keT+τ1iHT−b+dT+λH∗+τ2−dTiT−dH+ε1iH−dHT+ε2iHTiT,diHdt=λH∗s+vT+rT+τ2iHT−b+dH+τ1+ε1+gλT∗−dTiT−dH+ε1iH−dHT+ε2iHTiH,diHTdt=gλT∗iH+λH∗eT+iT−b+τ1+τ2+ε2+dHT−dTiT−dH+ε1iH−dHT+ε2iHTiHT,drTdt=τ2iT−b+γ2+λH∗−dTiT−dH+ε1iH−dHT+ε2iHTrT,drHdt=τ1iH−b+γ1+λT∗−dTiT−dH+ε1iH−dHT+ε2iHTrH,dmdt=ε1iH+ε2iHT−μMm,subject to condition s+vT+eT+iT+iH+iHT+rT+rH=1.


### 2.2. Disease-Free Equilibrium (DFE)

The DFE point of system (4) is obtained by setting the right-hand sides of the equations to zero with eT=0, iT=0, iH=0, iHT=0, rT=0, rH=0, rHT=0, and m=0 to obtain the DFE E0=1−η,η,0,0,0,0,0,0,0,0.


### 2.3. Effective Reproduction Number

The principles of the next-generation matrix [22] are used to obtain the effective reproduction number.
(5)RT=βTk1+ηρ−1b+kb+dT+τ2,(6)RH=βHε1μMb+dH+ε1+τ1,corresponding to the reproduction numbers for the Mtb and helminth infection transmission models obtained from the positive eigenvalues of the next-generation matrix, respectively.


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

Theorem 1.
The disease-free equilibrium point is locally asymptotically stable ifRHT<1 and unstable if either RT>1 or RH>1.Proof.
We first obtain the Jacobian matrix at the disease-free equilibriumE0 in order to prove this theorem. The Jacobian matrix is given by
(7)JE0=−b00l0−l1−l2γ2γ1−βHs00−b0ρβT−dTvT0−l3−l400−βHvT000−b+kβTs0+ρvT00βTσs0+ρvT000000k−l50τ10000000−l6τ200βH00000−l7000000τ200−l8000000τ100−l900000ε1ε200−μM,where
(8)l0=dT−βTs0,l1=dH+ε1s0,l2=σβT−dHT+ε2s0,l3=dH+ε1vT0,l4=ρσβT−dHT+ε2vT0,l5=b+dT+τ2,l6=b+dH+τ1,l7=b+τ1+τ2+ε2+dHT,l8=b+γ2,l9=b+γ1.
Now, the characteristic polynomial of the Jacobian matrixJE0 is given by
(9)pλ=b+λ2b+γ1+λb+γ2+λb+dHT+τ1+τ2+λ+ε2λ2+2b+k+dT+τ2λ+b+kb+dT+τ21−RTλ2+b+dH+ε1+μM+τ1λ+μMb+dH+ε1+τ11−RH.
Clearly, from equation (9), the eigenvalues of JE0 are as follows: λ1,2=−b<0, λ2=−b+γ1<0, λ3=−b+γ2<0, λ4=−b+dHT+τ1+τ2+ε2<0, and
(10)λ2+2b+k+dT+τ2λ+b+kb+dT+τ21−RTλ2+b+dH+ε1+μM+τ1λ+μMb+dH+ε1+τ11−RH=0.
Applying Routh-Hurwitz conditions [23] on equation (10), it has strictly negative real roots if RT<1 and RH<1. Thus, the DFE point E0 is locally asymptotically stable whenever RHT<1 and unstable if either RT>1 or RH>1.


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

We investigate the global asymptotic stability of the disease-free equilibrium of model (9) by using the theory applied by Chavez et al. [24]. System (9) is written as
(11)dXdt=FX,Y,dYdt=GX,Y,GX,0=0,where X=s,vT,rT,rH∈ℝ+4 represents the number of uninfected individuals and Y=eT,iT,iH,iHT,m∈ℝ+5 represents the number of infected individuals including the latent and infectious individuals.The DFE is given byE0=X0∗,0, where X0∗=1−η,η. The conditions in equation (12) must be satisfied to guarantee local asymptotic stability:
(12)dXdt=FX∗,0,X∗isgloballyasmptoticallystableg.a.s,GX,Y=BY−G^X,Y,whereG^X,Y≥0forX,Y∈Σ,and B=DYG^X∗,0 is M-matrix (that is, the off-diagonal elements of matrix B are nonnegative); and Σ is the region where the model makes biological sense. If the equations in (4) satisfy the conditions in (12), then the following theorem holds.Theorem 2.
The fixed pointE0 is a globally asymptotically stable equilibrium of system (4) given that RHT<1 and that the conditions in (12) are fulfilled.Proof.
System (4) as written in equation (12) can be rewritten in a reduced form:
(13)dXdtY=0=1−ηb−bsηb−bvT0.
Solving equation one in (13) gives st=1−η+s0−1−ηe−bt implying that st⟶1−η as t⟶∞. Similarly, solving equation two in (13) gives vTt=η+vT0−ηe−bt implying that vTt⟶η as t⟶0. Therefore, the first condition in (12) is satisfied.
Let(14)G^1X,YG^2X,YG^3X,YG^4X,YG^5X,Y=χ1λT∗1−s+ρvTχ1−λT∗rH+λH∗eT−dTiTeT−χ2iHeT−χ3iHTeTλH∗iT−dTiT2−χ2iHiT−χ3iHTiTβHm1−s+vT+rT1+m+gλT∗iH−dTiTiH−χ2iH2−χ3iHTiH−gλT∗iH−λH∗eT+iT−dTiTiHT−χ2iHiHT−χ3iHT20,B=−b+kβTs0+ρvT00βTσs0+ρvT00k−b+dT+τ20τ1000−b+dH+τ1+ε1τ2βH000−Γ000ε1ε2−μM,where χ1=1+ηρ−1, χ2=dH+ε1, χ3=dHT+ε3, and Γ=b+τ1+τ2+ε2+dHT. From equation (14), we observe that G^4X,Y<0 implying that the second condition in (12) is not fulfilled, so E0 may not be globally asymptotically stable. This suggests the existence of multiple equilibria.


### 2.6. The Effect of Helminth Infection on Mtb and Vice Versa

We analyze the effect of helminth infection on Mtb infection and vice versa, by first expressingRH in terms of RT. We solve for b from the expression of RT in equation (5) to get
(15)b=−θ1RT+θ2RT2+θ3RT2RT,where θ1=k+dT+τ2, θ2=k+dT+τ22−4kdT+τ2, θ3=4βTk1+ηρ−1.Then, substitutingb into the expression for RH, we get
(16)RH=βHε1μM−θ1RT+θ2RT2+θ3RT/2RT+dH+ε1+τ1.Differentiating equation (16) partially with respect to RT gives
(17)∂RH∂RT=θ3βHε1RTμMRTθ2RT+θ32RTdH+ε1+τ1−θ1RT+RTθ2RT+θ32.Now, whenever the right-hand side of equation (17) is strictly positive, an increase in the Mtb incidence in the society results in an increase in the incidence of helminth infection in the society.If RHS of equation (17) is equal to zero, it means that Mtb incidences have no influence on the spread of helminth infection.Also, expressingb in terms of RH from the expression of RH in equation (6), we get
(18)b=βHε1−θ6RHμMRH,where θ6=μMdH+τ1. Then, expressing RT in terms of RH, we have
(19)RT=βTk1+ηρ−1μM2RH2βHε1+μMk−θ6RHβHε1+μMdT+τ2−θ6RH,DifferentiatingRT with respect to RH, we get
(20)∂RT∂RH=kε1RHμM2βHβT1+ηρ−1RHμMdT+k+τ2−2θ6+2βHε1RHkμM−θ6+βHε12RHμMdT+τ2−θ6+βHε12.WheneverμMdT+k+τ2≥2θ6, then ∂RT/∂RH is strictly positive. This indicates that helminth infection incidence results in an increase in Mtb infection incidence in the society. Thus, helminth infection enhances Mtb infection in the society.


### 2.7. Existence of Backward Bifurcation

To determine the local asymptotic stability of the endemic equilibrium, we apply the Center Manifold Theorem developed by [25]. To use the theory, we make the subsequent change of variables: let x1=s, x2=vT, x3=eT, x4=iT, x5=iH, x6=iHT, x7=rT, x8=rH, and x9=m. Then, model system (4) is often written within the following form:
(21)dXdt=Fx=f1x,f2x,f3x,f4x,f5x,f6x,f7x,f8x,f9xwhere
(22)dx1dt=f1x=1−ηb+γ1x8+γ2x7−b+βTx4+σx6−dTx4−dH+ε1x5−dHT+ε2x6x1,dx2dt=f2x=ηb−b+ρβTx4+σx6+βHx91+x9−dTx4−dH+ε1x5−dHT+ε2x6x2,dx3dt=f3x=βTx4+σx6x1+ρx2+x8−b+k+βHx91+x9−dTx4−dH+ε1x5−dHT+ε2x6x3,dx4dt=f4x=kx3+τ1x6−b+dT+τ2+βHx91+x9−dTx4−dH+ε1x5−dHT+ε2x6x4,dx5dt=f5x=βHx91+x9x1+x2+x7+τ2x6−b+dH+τ1+ε1+gβTx4+σx6−dTx4−dH+ε1x5−dHT+ε2x6x5,dx6dt=f6x=gβTx4+σx6x5+βHx91+x9x3+x4−b+τ1+τ2+ε2+dHT−dTx4−dH+ε1x5−dHT+ε2x6x6,dx7dt=f7x=τ2x4−b+γ2+βHx91+x9−dTx4−dH+ε1x5−dHT+ε2x6x7,dx8dt=f8x=τ1x5−b+γ1+βTx4+σx6−dTx4−dH+ε1x5−dHT+ε2x6x8,dx10dt=f9x=ε1x5+ε2x6−μMx10.So the Jacobian matrix of system (22) at the disease-free equilibrium (E0) is given by
(23)JE0=−b00J0−J1−J2γ2γ1−βHs00−b0ρβT−dTvT0−J3−J400−βHvT000−b+kβT∗s0+ρvT00βT∗σs0+ρvT000000k−J50τ10000000−J6τ200βH00000−J7000000τ200−J8000000τ100−J900000ε1ε200−μM,where J0=dT−βTs0, J1=dH+ε1s0, J2=σβT−dHT+ε2s0, J3=dH+ε1vT0, J4=ρσβT∗−dHT+ε2vT0, J5=b+dT+τ2, J6=b+dH+τ1, J7=b+τ1+τ2+ε2+dHT, J8=b+γ2, and J9=b+γ1.Recall thatRHT=maxRH,RT. If we consider RHT=1 (that is, RH<RT=1), let βT=βT∗ be a bifurcation parameter. Then, RT=1 gives
(24)βT=βT∗=b+kb+dT+τ2k1+ηρ−1.The JacobianJE0 has a simple zero eigenvalue whose related right eigenvectors are denoted by w=w1,w2,w3,w4,w5,w6,w7,w8,w9T.where w1=J8dT−βT∗s0+γ2τ2/bJ8w4+μMγ1τ1−J1J9−βT∗ε1s0J9/bμMJ9w5, w2=ρβT∗−dTvT0/bw4−μMdH+ε1+βHε1/bμMw5, w3=βT∗s0+ρvT0w4/b+k, w4=w4>0, w5=w5>0, w6=0, w7=τ2w4/J8, w8=τ1w5/J9, and w9=ε1w5/μM.The left eigenvectors ofJE0 associated with the simple eigenvalues are denoted by v=v1,v2,v3,v4,v5,v6,v7,v8,v9T, where v1=v2=v7=v8=0, v3=kv4/b+k, v4=v4>0, v5=v5>0, v6=kβT∗σs0+ρvT0+τ1b+k/J7b+kv4+τ2μM+ε2βH/J7μMv5, and v9=βHv5/μM.The coefficientsa and b that decide the local dynamics of the endemic equilibrium point are defined in equations (25) and (26):
(25)a=∑k,j,i=1nvkwiwj∂2fk∂xi∂xjE0,βT∗,(26)b=∑k,i=1nvkwi∂2fk∂xi∂βT∗E0,βT∗.From system (22), the nonzero partial derivatives of F related to a at DFE are given by
(27)∂2f1∂x4∂x1=dT−βT∗,∂2f1∂x5∂x1=dH+ε1,∂2f1∂x6∂x1=dH+ε2−σβT∗,∂2f2∂x4∂x2=dT−ρβT∗,∂2f2∂x5∂x2=dH+ε1,∂2f2∂x6∂x2=dHT+ε2−σρβT∗,∂2f2∂x9∂x2=−βH,∂2f3∂x4∂x3=dT,∂2f3∂x5∂x3=dH+ε1,∂2f3∂x6∂x3=dHT+ε2,∂2f3∂x9∂x3=−βH,∂2f3∂x4∂x1=βT,∂2f3∂x6∂x1=σβT,∂2f3∂x4∂x2=ρβT,∂2f3∂x6∂x2=ρσβT,∂2f3∂x4∂x8=βT,∂2f3∂x6∂x8=σβT,∂2f4∂x42=dT,∂2f4∂x5∂x4=dH+ε1,∂2f4∂x6∂x4=dHT+ε2,∂2f4∂x9∂x4=−βH,∂2f5∂x4∂x5=dT−gβT∗,∂2f5∂x52=dH+ε1,∂2f5∂x6∂x5=dHT+ε2−gσβT∗,∂2f5∂x9∂x1=βH,∂2f5∂x9∂x2=βH,∂2f5∂x9∂x7=βH,∂2f6∂x4∂x6=dT,∂2f6∂x5∂x6=dH+ε1+gσβT∗,∂2f6∂x62=dHT+ε2,∂2f6∂x4∂x5=gβT∗,∂2f6∂x9∂x3=βH,∂2f6∂x9∂x4=βH,∂2f7∂x9∂x7=−βH,∂2f7∂x4∂x7=dT,∂2f7∂x5∂x7=dH+ε1,∂2f7∂x6∂x7=dHT+ε2,∂2f8∂x4∂x8=dT−βT∗,∂2f8∂x5∂x8=dH+ε1,∂2f8∂x6∂x8=dHT+ε2−σβT∗.Therefore, equation (25) becomes
(28)a=a0w42bb+kJ8+a1w4w5bJμMJ8J9b+kv3+dTw42+μMdH+ε1−βHε1μMw4w5v4+a2w4w5bJ8μM2+a3w52bJ9μM2v5+a4w4w5μMb+kv6,where
(29)a0=bdTβT∗J8s0+ρvT0+J8ρβT∗vT0b+kρβT∗−dT+J8dT−βT∗s0+γ2τ2σβT∗,a1=βT∗s0+ρvT0bJ8J9μMdH+ε1−βHε1+βT∗μMb+kσJ9μMγ1τ1−J1J8−βT∗ε1s0J9−bτ1J8−ρβT∗J8J9bb+kμMdH+ε1+βHε1,a2=dT−gβT∗bJ8μM2+J8dT−βT∗s0+γ2τ2ε1βHμM+bμMε1βHτ2,a3=bJ9μM2dH+ε1+βHε1μMγ1τ1−J1J9−βT∗ε1s0J9,a4=μMβT∗b+k+βT∗s0+ρvT0ε1βH+βHε1b+k.To determine the sign ofb, we find the subsequent nonvanishing partial derivatives of F.
(30)∂2f1∂x4∂βT∗=−s0,∂2f1∂x6∂βT∗=−σs0,∂2f3∂x4∂βT∗=s0+ρvT0,∂2f3∂x6∂βT∗=σs0+ρvT0.Therefore,b=βT∗ks0+ρvT02/b+k2w4v4.Now, according to [25], the signs of a and b dictate the local dynamics; thus, we have the following lemma.Lemma 3.
Suppose thatb>0. Then, we have the following:
(i)
System (4) undergoes backward bifurcation if the coefficient a is positive(ii)
System (4) will exhibit transcritical bifurcation if the coefficient a is negative


## 2.1. Model Formulation

The helminth-Mtb infection model is formulated under the following model assumptions:(i)
The susceptible individuals cannot simultaneously get infected by both diseases but rather get infected with one disease and later by the other disease(ii)
The population is homogeneously mixed(iii)
The incidence rate for helminth infection follows the logistic model while Mtb is assumed to be frequency-dependent(iv)
An individual gets helminth infection after sufficient contact with the polluted environment(v)
Due to the fact that Mtb infection is more severe than helminth infection, we assume that coinfected individuals seek TB treatment rather than helminth infection(vi)
The Bacille Calmette-Guerin (BCG) vaccine does not confer a total immunityThe model is formulated by considering two populations, namely, the human population and the parasite populationM that are present in the contaminated environment. The human population is subdivided into nine classes, namely, susceptible S, the vaccinated against Mtb VT, those infected with Mtb but do not show clinical symptoms of TB ET, those with active TB IT, TB-recovered individuals RT, those infected by helminths IH, those who have recovered from helminths RH, and those who are dually infected by helminths and Mtb IHT. Thus, the total human population is given by N=S+VT+ET+IT+IH+IHT+RT+RH+RHT. The susceptible population increases by recruitment through birth at the rate 1−ηbN and by those who lose their temporary immunity against the helminths and Mtb at rates γ1 and γ2, respectively. The susceptible individuals get infected with Mtb upon sufficient contact with an infectious individual with the force of infection λT=βTIT+σIHT/N, where βT is the transmission rate and σ>1 is the modification parameter for coinfected individuals that models infectivity for susceptible individuals. The susceptible individuals also get infected by helminths after ingestion of eggs/larvae or skin penetration by infective larvae with the force of infection λH=βHM/K+M, where βH is the ingestion rate of parasitic worms and K is the density of parasites in the soil. The helminth-infected individuals increase due to susceptible individuals being infected with helminths, Mtb-vaccinated individuals infected with helminths, and coinfected individuals who recover from Mtb at the rate τ2. Furthermore, due to the fact that the BCG vaccine does not grant total protection [11], then individuals exposed to Mtb increase due to Mtb-vaccinated individuals losing their protection against Mtb and getting infected at the rate ρλT, with ρ∈0,1 such that 1−ρ is the efficacy of the vaccine.The chronic infection by helminths alters the chance of getting infected and the progression of coinfecting parasites (virus or bacteria) [12]; as a result, individuals infested with helminths are at higher risk of developing tuberculosis [13]. Therefore, individuals coinfected with helminth and Mtb are increased by helminth-infected individuals being infected with Mtb at the force of infection gλT, where g>1 is the modification parameter that models increased susceptibility to Mtb, and Mtb-exposed individuals getting infected by helminths at a force of infection λH. In addition, the number of individuals with active TB increases as individuals progressed from the exposed class at the ratekand coinfected individuals recovered from helminth infection at the rateτ1and decreases due to natural recovery at the rateτ2. The parasite populations in the environment increase as they are released by the helminth-infected individuals at the rate ε1 and coinfected individuals at the rate ε2 and are cleared from the environment at a rate μM.The description above is illustrated in Figure 1, and the governing system of nonlinear differential equations is shown in model (1). The model parameters are described in Table 1.
(1)dSdt=1−ηbN+γ1RH+γ2RT−μ+λT+λHS,dVTdt=ηbN−μ+ρλT+λHVT,dETdt=λTS+ρλTVT+λTRH−μ+k+λHET,dITdt=kET+τ1IHT−μ+dT+τ2+λHIT,dIHdt=λHS+λHVT+λHRT+τ2IHT−gλT+μ+dH+τ1+ε1IH,dIHTdt=gλTIH+λHET+IT−μ+dHT+τ1+τ2+ε2IHT,dRTdt=τ2IT−λH+μ+γ2RT,RHdt=τ1IH−λT+μ+γ1RH,dMdt=ε1IH+ε2IHT−μMM,where
(2)λT=βTIT+σIHTN,λH=βHMM+K.Figure 1
Compartmental diagram for the helminth and Mycobacterium tuberculosis coinfection. The dotted lines indicate the interaction of individuals with the polluted environment.Table 1
Description of the model parameters.
ParametersDefinitionEstimated valueReferencebNatural birth rate1/18250day−1[14]βTTransmission rate for Mtb0.42day−1[15]μNatural death rate1/60×365day−1[16]γ1Loss of temporary immunity for helminth-recovered individuals1/70day−1AssumedβHIngestion rate of parasitic worms1parasiteday−1AssumedρVaccine wane rate0.7 dimensionless[17]τ1Natural recovery rate for helminth-infected individuals1/28day−1[18]dHHelminth disease-induced death rate35/1000day−1[14]γ2Loss of temporary immunity for Mtb-recovered individuals0.3day−1[19]τ2Natural recovery rate for Mtb-infected individuals0.2/365day−1[20]dTMtb disease-induced death rate0.08day−1[17]dHTHelminth-Mtb disease-induced death rate0.08day−1AssumedkProgression to active TB0.00013/365day−1[21]μMClearance rate of parasitic worms13/37500day−1[14]ε1Shading rate for helminth-infected individuals0.09day−1[18]KNumber of parasites in the environment105 parasites[18]ε2Shading rate for coinfected individuals0.1day−1AssumedgModification parameter1.12 dimensionlessAssumedσModification parameter1.5 dimensionlessAssumedWe perform a qualitative analysis of model (1) using a dimensionless version such that dimensionless variables obtained by setting s=S/N, vT=VT/N, eT=ET/N, iT=IT/N, iH=IH/N, iHT=IHT/N, rT=RT/N, rH=RH/N, rHT=RHT/N, and m=M/K replace the original variables S, VT, ET, IT, IT, IHT, RT, RH, RHT, and M, respectively, in which the rate of change of the population at time t is given by
(3)dNdt=b−μN+dTIT−dH+ε1IH−dHT+ε2IHT.Upon differentiating dimensionless variables with respect to time and using equation (3), system (1) becomes
(4)dsdt=1−ηb+γ1rH+γ2rT−b+λT∗+λH∗−dTiT−dH+ε1iH−dHT+ε2iHTs,dvTdt=ηb−b+ρλT∗+λH∗−dTiT−dH+ε1iH−dHT+ε2iHTvT,deTdt=λT∗s+ρvT+rH−b+k+λH∗−dTiT−dH+ε1iH−dHT+ε2iHTeT,diTdt=keT+τ1iHT−b+dT+λH∗+τ2−dTiT−dH+ε1iH−dHT+ε2iHTiT,diHdt=λH∗s+vT+rT+τ2iHT−b+dH+τ1+ε1+gλT∗−dTiT−dH+ε1iH−dHT+ε2iHTiH,diHTdt=gλT∗iH+λH∗eT+iT−b+τ1+τ2+ε2+dHT−dTiT−dH+ε1iH−dHT+ε2iHTiHT,drTdt=τ2iT−b+γ2+λH∗−dTiT−dH+ε1iH−dHT+ε2iHTrT,drHdt=τ1iH−b+γ1+λT∗−dTiT−dH+ε1iH−dHT+ε2iHTrH,dmdt=ε1iH+ε2iHT−μMm,subject to condition s+vT+eT+iT+iH+iHT+rT+rH=1.


## 2.2. Disease-Free Equilibrium (DFE)

The DFE point of system (4) is obtained by setting the right-hand sides of the equations to zero with eT=0, iT=0, iH=0, iHT=0, rT=0, rH=0, rHT=0, and m=0 to obtain the DFE E0=1−η,η,0,0,0,0,0,0,0,0.


## 2.3. Effective Reproduction Number

The principles of the next-generation matrix [22] are used to obtain the effective reproduction number.
(5)RT=βTk1+ηρ−1b+kb+dT+τ2,(6)RH=βHε1μMb+dH+ε1+τ1,corresponding to the reproduction numbers for the Mtb and helminth infection transmission models obtained from the positive eigenvalues of the next-generation matrix, respectively.


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

Theorem 1.
The disease-free equilibrium point is locally asymptotically stable ifRHT<1 and unstable if either RT>1 or RH>1.Proof.
We first obtain the Jacobian matrix at the disease-free equilibriumE0 in order to prove this theorem. The Jacobian matrix is given by
(7)JE0=−b00l0−l1−l2γ2γ1−βHs00−b0ρβT−dTvT0−l3−l400−βHvT000−b+kβTs0+ρvT00βTσs0+ρvT000000k−l50τ10000000−l6τ200βH00000−l7000000τ200−l8000000τ100−l900000ε1ε200−μM,where
(8)l0=dT−βTs0,l1=dH+ε1s0,l2=σβT−dHT+ε2s0,l3=dH+ε1vT0,l4=ρσβT−dHT+ε2vT0,l5=b+dT+τ2,l6=b+dH+τ1,l7=b+τ1+τ2+ε2+dHT,l8=b+γ2,l9=b+γ1.
Now, the characteristic polynomial of the Jacobian matrixJE0 is given by
(9)pλ=b+λ2b+γ1+λb+γ2+λb+dHT+τ1+τ2+λ+ε2λ2+2b+k+dT+τ2λ+b+kb+dT+τ21−RTλ2+b+dH+ε1+μM+τ1λ+μMb+dH+ε1+τ11−RH.
Clearly, from equation (9), the eigenvalues of JE0 are as follows: λ1,2=−b<0, λ2=−b+γ1<0, λ3=−b+γ2<0, λ4=−b+dHT+τ1+τ2+ε2<0, and
(10)λ2+2b+k+dT+τ2λ+b+kb+dT+τ21−RTλ2+b+dH+ε1+μM+τ1λ+μMb+dH+ε1+τ11−RH=0.
Applying Routh-Hurwitz conditions [23] on equation (10), it has strictly negative real roots if RT<1 and RH<1. Thus, the DFE point E0 is locally asymptotically stable whenever RHT<1 and unstable if either RT>1 or RH>1.


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

We investigate the global asymptotic stability of the disease-free equilibrium of model (9) by using the theory applied by Chavez et al. [24]. System (9) is written as
(11)dXdt=FX,Y,dYdt=GX,Y,GX,0=0,where X=s,vT,rT,rH∈ℝ+4 represents the number of uninfected individuals and Y=eT,iT,iH,iHT,m∈ℝ+5 represents the number of infected individuals including the latent and infectious individuals.The DFE is given byE0=X0∗,0, where X0∗=1−η,η. The conditions in equation (12) must be satisfied to guarantee local asymptotic stability:
(12)dXdt=FX∗,0,X∗isgloballyasmptoticallystableg.a.s,GX,Y=BY−G^X,Y,whereG^X,Y≥0forX,Y∈Σ,and B=DYG^X∗,0 is M-matrix (that is, the off-diagonal elements of matrix B are nonnegative); and Σ is the region where the model makes biological sense. If the equations in (4) satisfy the conditions in (12), then the following theorem holds.Theorem 2.
The fixed pointE0 is a globally asymptotically stable equilibrium of system (4) given that RHT<1 and that the conditions in (12) are fulfilled.Proof.
System (4) as written in equation (12) can be rewritten in a reduced form:
(13)dXdtY=0=1−ηb−bsηb−bvT0.
Solving equation one in (13) gives st=1−η+s0−1−ηe−bt implying that st⟶1−η as t⟶∞. Similarly, solving equation two in (13) gives vTt=η+vT0−ηe−bt implying that vTt⟶η as t⟶0. Therefore, the first condition in (12) is satisfied.
Let(14)G^1X,YG^2X,YG^3X,YG^4X,YG^5X,Y=χ1λT∗1−s+ρvTχ1−λT∗rH+λH∗eT−dTiTeT−χ2iHeT−χ3iHTeTλH∗iT−dTiT2−χ2iHiT−χ3iHTiTβHm1−s+vT+rT1+m+gλT∗iH−dTiTiH−χ2iH2−χ3iHTiH−gλT∗iH−λH∗eT+iT−dTiTiHT−χ2iHiHT−χ3iHT20,B=−b+kβTs0+ρvT00βTσs0+ρvT00k−b+dT+τ20τ1000−b+dH+τ1+ε1τ2βH000−Γ000ε1ε2−μM,where χ1=1+ηρ−1, χ2=dH+ε1, χ3=dHT+ε3, and Γ=b+τ1+τ2+ε2+dHT. From equation (14), we observe that G^4X,Y<0 implying that the second condition in (12) is not fulfilled, so E0 may not be globally asymptotically stable. This suggests the existence of multiple equilibria.


## 2.6. The Effect of Helminth Infection on Mtb and Vice Versa

We analyze the effect of helminth infection on Mtb infection and vice versa, by first expressingRH in terms of RT. We solve for b from the expression of RT in equation (5) to get
(15)b=−θ1RT+θ2RT2+θ3RT2RT,where θ1=k+dT+τ2, θ2=k+dT+τ22−4kdT+τ2, θ3=4βTk1+ηρ−1.Then, substitutingb into the expression for RH, we get
(16)RH=βHε1μM−θ1RT+θ2RT2+θ3RT/2RT+dH+ε1+τ1.Differentiating equation (16) partially with respect to RT gives
(17)∂RH∂RT=θ3βHε1RTμMRTθ2RT+θ32RTdH+ε1+τ1−θ1RT+RTθ2RT+θ32.Now, whenever the right-hand side of equation (17) is strictly positive, an increase in the Mtb incidence in the society results in an increase in the incidence of helminth infection in the society.If RHS of equation (17) is equal to zero, it means that Mtb incidences have no influence on the spread of helminth infection.Also, expressingb in terms of RH from the expression of RH in equation (6), we get
(18)b=βHε1−θ6RHμMRH,where θ6=μMdH+τ1. Then, expressing RT in terms of RH, we have
(19)RT=βTk1+ηρ−1μM2RH2βHε1+μMk−θ6RHβHε1+μMdT+τ2−θ6RH,DifferentiatingRT with respect to RH, we get
(20)∂RT∂RH=kε1RHμM2βHβT1+ηρ−1RHμMdT+k+τ2−2θ6+2βHε1RHkμM−θ6+βHε12RHμMdT+τ2−θ6+βHε12.WheneverμMdT+k+τ2≥2θ6, then ∂RT/∂RH is strictly positive. This indicates that helminth infection incidence results in an increase in Mtb infection incidence in the society. Thus, helminth infection enhances Mtb infection in the society.


## 2.7. Existence of Backward Bifurcation

To determine the local asymptotic stability of the endemic equilibrium, we apply the Center Manifold Theorem developed by [25]. To use the theory, we make the subsequent change of variables: let x1=s, x2=vT, x3=eT, x4=iT, x5=iH, x6=iHT, x7=rT, x8=rH, and x9=m. Then, model system (4) is often written within the following form:
(21)dXdt=Fx=f1x,f2x,f3x,f4x,f5x,f6x,f7x,f8x,f9xwhere
(22)dx1dt=f1x=1−ηb+γ1x8+γ2x7−b+βTx4+σx6−dTx4−dH+ε1x5−dHT+ε2x6x1,dx2dt=f2x=ηb−b+ρβTx4+σx6+βHx91+x9−dTx4−dH+ε1x5−dHT+ε2x6x2,dx3dt=f3x=βTx4+σx6x1+ρx2+x8−b+k+βHx91+x9−dTx4−dH+ε1x5−dHT+ε2x6x3,dx4dt=f4x=kx3+τ1x6−b+dT+τ2+βHx91+x9−dTx4−dH+ε1x5−dHT+ε2x6x4,dx5dt=f5x=βHx91+x9x1+x2+x7+τ2x6−b+dH+τ1+ε1+gβTx4+σx6−dTx4−dH+ε1x5−dHT+ε2x6x5,dx6dt=f6x=gβTx4+σx6x5+βHx91+x9x3+x4−b+τ1+τ2+ε2+dHT−dTx4−dH+ε1x5−dHT+ε2x6x6,dx7dt=f7x=τ2x4−b+γ2+βHx91+x9−dTx4−dH+ε1x5−dHT+ε2x6x7,dx8dt=f8x=τ1x5−b+γ1+βTx4+σx6−dTx4−dH+ε1x5−dHT+ε2x6x8,dx10dt=f9x=ε1x5+ε2x6−μMx10.So the Jacobian matrix of system (22) at the disease-free equilibrium (E0) is given by
(23)JE0=−b00J0−J1−J2γ2γ1−βHs00−b0ρβT−dTvT0−J3−J400−βHvT000−b+kβT∗s0+ρvT00βT∗σs0+ρvT000000k−J50τ10000000−J6τ200βH00000−J7000000τ200−J8000000τ100−J900000ε1ε200−μM,where J0=dT−βTs0, J1=dH+ε1s0, J2=σβT−dHT+ε2s0, J3=dH+ε1vT0, J4=ρσβT∗−dHT+ε2vT0, J5=b+dT+τ2, J6=b+dH+τ1, J7=b+τ1+τ2+ε2+dHT, J8=b+γ2, and J9=b+γ1.Recall thatRHT=maxRH,RT. If we consider RHT=1 (that is, RH<RT=1), let βT=βT∗ be a bifurcation parameter. Then, RT=1 gives
(24)βT=βT∗=b+kb+dT+τ2k1+ηρ−1.The JacobianJE0 has a simple zero eigenvalue whose related right eigenvectors are denoted by w=w1,w2,w3,w4,w5,w6,w7,w8,w9T.where w1=J8dT−βT∗s0+γ2τ2/bJ8w4+μMγ1τ1−J1J9−βT∗ε1s0J9/bμMJ9w5, w2=ρβT∗−dTvT0/bw4−μMdH+ε1+βHε1/bμMw5, w3=βT∗s0+ρvT0w4/b+k, w4=w4>0, w5=w5>0, w6=0, w7=τ2w4/J8, w8=τ1w5/J9, and w9=ε1w5/μM.The left eigenvectors ofJE0 associated with the simple eigenvalues are denoted by v=v1,v2,v3,v4,v5,v6,v7,v8,v9T, where v1=v2=v7=v8=0, v3=kv4/b+k, v4=v4>0, v5=v5>0, v6=kβT∗σs0+ρvT0+τ1b+k/J7b+kv4+τ2μM+ε2βH/J7μMv5, and v9=βHv5/μM.The coefficientsa and b that decide the local dynamics of the endemic equilibrium point are defined in equations (25) and (26):
(25)a=∑k,j,i=1nvkwiwj∂2fk∂xi∂xjE0,βT∗,(26)b=∑k,i=1nvkwi∂2fk∂xi∂βT∗E0,βT∗.From system (22), the nonzero partial derivatives of F related to a at DFE are given by
(27)∂2f1∂x4∂x1=dT−βT∗,∂2f1∂x5∂x1=dH+ε1,∂2f1∂x6∂x1=dH+ε2−σβT∗,∂2f2∂x4∂x2=dT−ρβT∗,∂2f2∂x5∂x2=dH+ε1,∂2f2∂x6∂x2=dHT+ε2−σρβT∗,∂2f2∂x9∂x2=−βH,∂2f3∂x4∂x3=dT,∂2f3∂x5∂x3=dH+ε1,∂2f3∂x6∂x3=dHT+ε2,∂2f3∂x9∂x3=−βH,∂2f3∂x4∂x1=βT,∂2f3∂x6∂x1=σβT,∂2f3∂x4∂x2=ρβT,∂2f3∂x6∂x2=ρσβT,∂2f3∂x4∂x8=βT,∂2f3∂x6∂x8=σβT,∂2f4∂x42=dT,∂2f4∂x5∂x4=dH+ε1,∂2f4∂x6∂x4=dHT+ε2,∂2f4∂x9∂x4=−βH,∂2f5∂x4∂x5=dT−gβT∗,∂2f5∂x52=dH+ε1,∂2f5∂x6∂x5=dHT+ε2−gσβT∗,∂2f5∂x9∂x1=βH,∂2f5∂x9∂x2=βH,∂2f5∂x9∂x7=βH,∂2f6∂x4∂x6=dT,∂2f6∂x5∂x6=dH+ε1+gσβT∗,∂2f6∂x62=dHT+ε2,∂2f6∂x4∂x5=gβT∗,∂2f6∂x9∂x3=βH,∂2f6∂x9∂x4=βH,∂2f7∂x9∂x7=−βH,∂2f7∂x4∂x7=dT,∂2f7∂x5∂x7=dH+ε1,∂2f7∂x6∂x7=dHT+ε2,∂2f8∂x4∂x8=dT−βT∗,∂2f8∂x5∂x8=dH+ε1,∂2f8∂x6∂x8=dHT+ε2−σβT∗.Therefore, equation (25) becomes
(28)a=a0w42bb+kJ8+a1w4w5bJμMJ8J9b+kv3+dTw42+μMdH+ε1−βHε1μMw4w5v4+a2w4w5bJ8μM2+a3w52bJ9μM2v5+a4w4w5μMb+kv6,where
(29)a0=bdTβT∗J8s0+ρvT0+J8ρβT∗vT0b+kρβT∗−dT+J8dT−βT∗s0+γ2τ2σβT∗,a1=βT∗s0+ρvT0bJ8J9μMdH+ε1−βHε1+βT∗μMb+kσJ9μMγ1τ1−J1J8−βT∗ε1s0J9−bτ1J8−ρβT∗J8J9bb+kμMdH+ε1+βHε1,a2=dT−gβT∗bJ8μM2+J8dT−βT∗s0+γ2τ2ε1βHμM+bμMε1βHτ2,a3=bJ9μM2dH+ε1+βHε1μMγ1τ1−J1J9−βT∗ε1s0J9,a4=μMβT∗b+k+βT∗s0+ρvT0ε1βH+βHε1b+k.To determine the sign ofb, we find the subsequent nonvanishing partial derivatives of F.
(30)∂2f1∂x4∂βT∗=−s0,∂2f1∂x6∂βT∗=−σs0,∂2f3∂x4∂βT∗=s0+ρvT0,∂2f3∂x6∂βT∗=σs0+ρvT0.Therefore,b=βT∗ks0+ρvT02/b+k2w4v4.Now, according to [25], the signs of a and b dictate the local dynamics; thus, we have the following lemma.Lemma 3.
Suppose thatb>0. Then, we have the following:
(i)
System (4) undergoes backward bifurcation if the coefficient a is positive(ii)
System (4) will exhibit transcritical bifurcation if the coefficient a is negative


## 3. Sensitivity Analysis

In this section, we perform a sensitivity analysis of the effective reproduction number subject to each parameter in order to determine the impact of every parameter on the effective reproduction number. Therefore, we compute the forward sensitivity index of the effective reproduction numberRHT with respect to the parameters using the approach by Chitnis et al. [26]. Since RHT=maxRH,RT, then the sensitivity indices for RH and RT are computed. The normalized forward sensitivity index of a variable P with respect to parameter r is defined as ζrP=r/P∂P/∂r. For instance, the sensitivity index of RT with respect to βT is given by ζβTRT=βT/RT∂RT/βT=1, and other indices are evaluated using parameter values in Table 1. The remaining indices are given in Tables 2 and 3.Table 2
Sensitivity indices for the reproduction number of Mtb infectionRT.
ParameterDescriptionSensitivity indicesβTTransmission rate for Mtb+1.0000kProgression to active TB+0.9935ηRate of vaccination-0.5385ρVaccine wane rate+0.5385bNatural birth rate-0.9942dTMtb disease-induced death rate-0.9691τ2Natural recovery rate for Mtb-infected individuals-0.0303Table 3
Sensitivity indices for the reproduction number of helminth infectionRH.
ParameterDescriptionSensitivity indicesβHIngestion rate of parasitic worms+1ε1Shading rate for helminth-infected individuals+0.4402μMClearance rate of parasitic worms-1.0000bNatural birth rate-0.0003dHHelminth disease-induced death rate-0.2177τ1Natural recovery rate for helminth-infected individuals-0.2221From Tables2 and 3, the most positive parameter indices are βT, k, and βH while the most negative parameter indices are b, dT, and μM. This implies that when parametersβT,k, andβHare increased, and while the remaining are kept constant, the endemicity of the disease is increased, while when the parameterμMis increased, the endemicity of disease is decreased. Increasing the natural birth rate and Mtb mortality rate does not make biological sense that the disease would be contained. Thus, we suggest the application of optimal control to sensitive parameters such as βH, βT, and μM to study how effectively we can contain the helminth-Mtb infection.


## 4. The Optimal Control Problem

In this section, the extension of model (1) is formed by including four time-dependent controls so as to decide the optimal strategy for controlling the two diseases. The controls are defined as follows:
(i)
Educational campaignu1t that sensitizes the parents to vaccinate more infants. Therefore, the recruitment for the Mtb-vaccinated individuals changes to ηN1+u1t meaning that when this control is 100% implemented, then the number of vaccinated babies doubles(ii)
Treatmentα1u2t of individuals with active TB, where α1 is the drug efficacy use for Mtb infection. Therefore, the Mtb recovery rate changes to τ2+α1u2t(iii)
Deworming at a rateα2u3t for the helminth-infected individuals and coinfected individuals, where α2 is the drug efficacy use for helminth infection. Therefore, the helminth recovery rate changes to τ1+α2u3t(iv)
Sanitation and proper hygieneu4 that reduces the ingestion rate of parasites. Therefore, the force of infection for the helminth infection is changed to λH=1−u4βHM/K+MAfter incorporating the controlsu1t, u2t, u3t, and u4t in the Mtb-helminth infection model (4), then the optimal control model becomes
(31)dSdt=1−ηbN+γ1RH+γ2RT−μ+λT+1−u4λHS,dVTdt=ηb1+u1N−μ+ρλT+1−u4λHVT,dETdt=λTS+ρλTVT+λTRH−μ+k+1−u4λHET,dITdt=kET+τ1+α2u3IHT−μ+dT+τ2+1−u4λHIT−τ2+α1u2IT,dIHdt=1−u4λHS+1−u4λHVT+1−u4λHRT+τ2IHT−gλT+μ+dH+ε1IH−τ1+α2u3IH,dIHTdt=gλTIH+1−u4λHET+IT−μ+dHT+τ2+ε2IHT−τ1+α2u3IHT,dRTdt=τ2+α1u2IT−1−u4λH+μ+γ2RT,RHdt=τ1+α2u3IH−λT+μ+γ1RH,dMdt=ε1IH+ε2IHT−μMM.We employ Pontryagin’s Maximum Principle to figure out the required conditions for the optimal control of helminth-Mtb coinfection. The control setU is Lebesgue measurable and is defined as U=u1t,u2t,u3t,u4t:0≤uit≤1,0<t≤tf. Then, the objective is to minimize individuals with active TB, individuals infested with helminth parasites, and coinfected individuals while keeping the cost low. For this problem, we consider the objective functional defined by
(32)J=∫t0tfC1IT+C2IH+C3IHT+12∑i=14wiui2dt,where C1IT, C2IH, are C3IHT are the costs related to active tuberculosis individuals, individuals infected with helminths, and coinfected individuals, respectively. The expression 1/2wiui2 is related to the background costs with relative cost weights wi for every control measure. The quadratic cost is tailored as utilized in other models with controls (see [16, 27]). The particular value of the weights requires intensive data processing; hence, we elect estimate values for theoretical purposes.The aim is to minimize the objective functionalJ as defined in (32) subject to model (31). Therefore, we seek to get the optimal controls u1∗, u2∗, u3∗, and u4∗ such that
(33)Ju1∗,u2∗,u3∗,u4∗=minu1,u2,u3,u4∈UJu1,u2,u3,u4,where U=u1,u2,u3,u4andthereforethecontrolssu1,u2,u3,areu4aremeasurablewith0≤u1≤1,0≤u2≤1,0≤u3≤1,and0≤u4≤1fort∈t0tf.Theorem 4.
Consider the control problem with the system of equations in (31). There exist u∗=u1∗,u2∗,u3∗,u4∗∈U such that
(34)minu1,u2,u3,u4∈UJu1,u2,u3,u4,=Ju1∗,u2∗,u3∗,u4∗.
The necessary conditions that an optimal solution must satisfy are derived from Pontryagin’s Maximum Principle [28]. The principle converts equations (31) and (32) into the problem of minimizing the pointwise Hamiltonian H, with reference to u1, u2, u3, and u4. The Hamiltonian is given by
(35)H=C1IT+C2IH+C3IHT+12w1u12+w2u22+w3u32+w4u42+λS1−ηbN+γ1RH+γ2RT−μ+λT+1−u4λHS+λVTηb1+u1N−μ+ρλT+1−u4λHVT+λETλTS+ρλTVT+λTRH−μ+k+1−u4λHET+λITkET+τ1+α2u3IHT−μ+dT+1−u4λHIT−τ2+α1u2IT+λIH1−u4λHS+VT+RT+τ2IHT−μ+dH+ε1+gλTIH−τ1+α2u3IH+λIHTgλTIH+1−u4λHET+IT−μ+dHT+τ2+ε2IHT−τ1+α2u3IHT+λRTτ2+α1u2IT−λH+μ+γ2RT+λRHτ1+α2u3IH−λT+μ+γ1RH+λMε1IH+ε2IHT−μMM.where λS, λVT, λET, λIT, λIH, λIHT, λRT, λRH, and λM are the costate variables or the adjoint variables. Now, applying Pontryagin’s Maximum Principle [28], the problem has a solution following the results of the optimal control problem in [29].Theorem 5.
Given the optimal controlui, fori=1,2,3,4, and the solutionsS,VT,ET,IT,IH,IHT,RT,RH, andM, and the state systems (31) and (32) that minimizeJu1,u2,u3,u4overU, there exist adjoint variablesλS,λVT,λET,λIT,λIH,λIHT,λRT,λRH, andλM satisfying
(36)−dλidt=∂H∂xiwhere xi=S,VT,IT,IH,IHT,RT,RH,M with transversality conditions λStf=λVTtf=λETtf=λITtf=λIHtf=λIHTtf=λRTtf=λRHtf=λMtf=0, and the subsequent characterization holds on the interior of the control set U:
(37)u1∗=max0,min1,−ηbNλVTw1,u2∗=max0,min1,α1ITλIT−λRTw2,u3∗=max0,min1,α2IHλIH−λRH+α2IHTλIHT−λITw3,u4∗=max0,min1,λHλIHS+VT+RT+λHλIHTET+ITw4−λHSλS+VTλVT+ETλET+ITλITw4.Proof.
Flemming and Rishel [23] provide the existence of an optimal control model (31) and the costate variables (36) due to the boundness of state equations and the Lipschitz structure of the ordinary differential equations. Therefore, applying the necessary conditions from Pontryagin’s Maximum Principle, we obtain the following system for costate variables:
(38)dλSdt=−1−ηbλS+βTIT+σIHTSN2λET−λS+βTIT+σIHTVTN2ρλET−λVT−ηb1+u1λVT+βTIT+σIHTRHN2λET−λRH+βTIT+σIHTNλS−λET+gβTIT+σIHTIHN2λIHT−λIH+1−u4βHMK+MλS−λIH+μλS,dλVTdt=−1−ηbλS+βTIT+σIHTVTN2ρλET−λVT+βTIT+σIHTRHN2λET−λRT−ηb1+u1λVT+βTIT+σIHTSN2λET−λS+βTIT+σIHTNλVT−ρλET+gβTIT+σIHTIHN2λIHT−λIH+1−u4βHMK+MλVT−λIH+μλVT,dλETdt=−1−ηbλS+βTIT+σIHTSN2λET−λS+ρβTIT+σIHTVTN2λET−λVT−ηb1+u1λVT+βTIT+σIHTRHN2λET−λRH+gβTIT+σIHTIHN2λIHT−λIH+1−u4βHMK+MλET−λIHT+kλET−λIT+μλET,dλITdt=−C1−1−ηbλS+βTIT+σIHTSN2λET−λS+ρβTIT+σIHTVTN2λET−λVT−ηb1+u1λVT+βTIT+σIHTRHN2λET−λRH+βTSλS+VTλVT+ITλRHN+gβTIT+σIHTIHN2λIHT−λIH+1−u4βHMK+MλIT−λIHT+μ+dTλIT+gβTIHNλIH−λIHT+τ2+α1u2λIT−λRH,dλIHdt=−C2−1−ηbλS+βTIT+σIHTSN2λET−λS+ρβTIT+σIHTVTN2λET−λVT−ηb1+u1λVT+βTIT+σIHTRHN2λET−λRH+gβTIT+σIHTNλIH−λIHT+gβTIT+σIHTIHN2λIHT−λIH+τ1+α2u3λIH−λRH+μ+dH+ε1λIH+ε1λM,dλIHTdt=−C3−1−ηbλS+βTIT+σIHTSN2λET−λS+ρβTIT+σIHTVTN2λET−λVT−ηb1+u1λVT+βTIT+σIHTRHN2λET−λRH+σgβTIT+σIHTNλIH−λIHT+gβTIT+σIHTIHN2λIHT−λIH+τ1+α2u3λIHT−λIT+μ+dHT+τ2+ε2λIHT+ε2λM−τ2λIH+βTσSλS+ρVTλVT+RHλRHN−σβTS+ρVT+RHN,dλRTdt=−1−ηbλS+βTIT+σIHTSN2λET−λS+ρβTIT+σIHTVTN2λET−λVT−ηb1+u1λVT+βTIT+σIHTRHN2λET−λRH+γ2λRT−λS+μλRT+gβTIT+σIHTIHN2λIHT−λIH+1−u4βHMK+MλλRT−λIH,dλRTdt=−1−ηbλS+βTIT+σIHTSN2λET−λS+ρβTIT+σIHTVTN2λET−λVT−ηb1+u1λVT+βTIT+σIHTRHN2λET−λRH+γ1λRH−λS+μλRH+gβTIT+σIHTIHN2λIHT−λIH+βTIT+σIHTNλλRH−λET,dλMdt=1−u4βHMSK+M2λIH−λS+1−u4βHMVTK+M2λIH−λVT+1−u4βHMRTK+M2λIH−λRT+1−u4βHMETK+M2λIHT−λET+1−u4βHMITK+M2λIHT−λIT+1−u4βHSλS+VTλVT+ETλET+ITλIT+RTλRTK+M+μMλM−1−u4βHS+VT+RTλIH+ET+ITλIHTK+M.
To get the controls, we solve the equations∂H/∂ui=0 at ui∗ for i=1,⋯,4 and obtained the following:
(39)u1∗=−ηbNλVTw1,u2∗=α1ITλIT−λRTw2,u3∗=α2IHλIH−λRH+α2IHTλIHT−λITw4,u4∗=λHλIHS+VT+RT+λHλIHTET+ITw4−λHSλS+VTλVT+ETλET+ITλIT.
Then, we write by standard control arguments involving the bounds on the controls as(40)u1∗=Θ1,if0<Θ1<1,0,ifΘ1≤0,1,ifΘ1≥1,u2∗=Θ2,if0<Θ2<1,0,ifΘ2≤0,1,ifΘ2≥1,u3∗=Θ3,if0<Θ3<1,0,ifΘ3≤0,1,ifΘ3≥1,u4∗=Θ4,if0<Θ4<1,0,ifΘ4≤0,1,ifΘ4≥1,where
(41)Θ1=−ηbNλVTw1,Θ2=α1ITλIT−λRTw2,Θ3=α2IHλIH−λRH+α2IHTλIHT−λITw4,Θ4=λHλIHS+VT+RT+λHλIHTET+ITw4−λHSλS+VTλVT+ETλET+ITλIT.


## 5. Numerical Simulation

In this section, different optimal control strategies are used to investigate numerically their effect on the spread of helminth and Mtb coinfection. The state system (31), the adjoint system (38), and therefore the characterization in equation (37) are solved numerically using an iterative scheme so as to attain the optimal control. Using the initial conditions for the state system and the transversality conditions for the adjoint system with the initial guess of the controls, we solve the state system forward in time and the adjoint system backward in time using the fourth-order Runge-Kutta scheme with the current solutions of the state system. Then, the controls are updated using a convex combination of the controls and the values from characterization (37). The solution of the state system and the adjoint system is repeated until the present iteration is close to the previous iteration [30].In numerical simulation, the weights are assumed to depend on the cost of investment and importance of the controls. For example, the cost associated with controlu2 requires huge investment that includes clinical examination of Mtb cases and procurement and transportation of the drugs. The cost associated with control u2 requires procurement and transportation of antihelminthic drugs and monitoring of the distribution of drugs. We choose the weights as C1=5822, C2=0.89, C3=5822.89, w1=20, w2=40, w3=30, and w4=10 and the initial state variables as S=1500, VT=2000, ET=200, IT=150, IH=100, IHT=80, RT=5, RH=5, and M=200. The efficacy parameters are α1=0.75 and α2=0.8, and the remaining parameter values are given in Table 1.


### 5.1. Control with Educational Campaignu1 and Treatment of Individuals with Active TB u2

The educational campaign controlu1 and treatment of individuals with active TB control u2 are used to optimize the objective functional J while the other controls u3 and u4 related to helminths were set to zero. We observed that in Figure 2(b) the number of individuals with active TB was controlled but starts to increase again until the end of the intervention period. This may be related to our previous analysis that helminth infection enhances Mtb infection. In this strategy, helminth infection is not controlled; Figures 2(c) and 2(d) indicate that there is slight significance in the case with controls and without controls for helminth-infected individuals and coinfected individuals, respectively. Figure 2(e) indicates that the parasite population is not controlled with this strategy while Figure 2(f) indicates that the educational campaign control should be seized after 158 days while treatment of individuals with active TB control should be kept maximum in the entire period of intervention.Figure 2
Simulations of the helminth-Mtb coinfection model with the effect of educational campaign and treatment of Mtb-infected individuals.
(a)(b)(c)(d)(e)(f)


### 5.2. Control with Mass Drug Administrationu3 and Sanitation u4

The mass drug administration (MDA) controlu3 and sanitation control u4 were used to optimize the objective functional J. Again, we observe that Figure 3(a) indicates the increase in vaccinated individuals. Also, Figure 3(b) indicates that the number of people with active TB increases rapidly and is eventually controlled after 55 days. An equivalent scenario is observed in Figures 3(c) and 3(d) where the helminth-infected individuals and coinfected individuals were effectively controlled. Figure 3(e) suggests that the parasite population is substantially controlled at the end of the intervention period. This strategy seems to be effective in controlling the two diseases from the community. Figure 3(f) indicates that MDA and sanitation controls should be kept maximum throughout the intervention period.Figure 3
Simulations of the helminth-Mtb coinfection model with the effect of MDA and sanitation.
(a)(b)(c)(d)(e)(f)


### 5.3. Control with Educational Campaignu1 and Sanitation u4

The educational campaign controlu1 and sanitation control u4 are employed to optimize the objective functional J. This strategy has the same profiles as the strategy with mass drug administration and sanitation controls but takes a long time to control the two diseases. Figure 4(f) indicates that the educational campaign must be stopped after 30 days whereas sanitation control must be kept maximum in the entire period of intervention.Figure 4
Simulations of the helminth-Mtb coinfection model with the effect of educational campaign and sanitation.
(a)(b)(c)(d)(e)(f)


### 5.4. Control with Treatment of Individuals with Active TBu2 and MDA u3

The treatment of individuals with active TB controlu2 and MDA control u3 are used to optimize the objective functional J while the other controls u1 and u4 are set to zero. Figure 5(a) indicates a slight increase in vaccinated individuals compared to the other strategies (Sections 5.1, 5.2, and 5.3). Figures 5(b) and 5(d) indicate that the treatment of individuals with active TB and coinfected individuals is effectively controlled while Figure 5(c) indicates a decrease in the helminth-infected individuals but a rise at the final intervention period. This is due to the absence of preventive measures in this strategy. Figure 5(f) indicates that both controls should be kept maximum until the final intervention period.Figure 5
Simulations of the helminth-Mtb coinfection model with the effect of treatment of Mtb-infected individuals and MDA.
(a)(b)(c)(d)(e)(f)


### 5.5. Control with Educational Campaignu1, Treatment of Mtb-Infected Individuals u2, MDA u3, and Sanitation u4

With this strategy, all four controls are employed to optimize the objective functionalJ. We observed that Figure 6(a) indicates that the number of vaccinated individuals increases more at the end of the intervention period. Figures 6(b)–6(d) indicate that individuals with active TB, helminth-infected individuals, and coinfected individuals are all effectively and efficiently controlled at the end of the intervention period. Figure 6(e) depicts that the parasite population is decreased to a number that cannot harm the human population. This strategy indicates that optimal educational campaign, treatment of individuals with active TB, MDA, and sanitation would effectively control the helminth-Mtb infection at the end of the intervention period. Figure 6(f) suggests that the educational campaign should be stopped after 115 days whereas the remaining controls should be maintained maximum for the whole intervention period.Figure 6
Simulations of the helminth-Mtb coinfection model with the effect of educational campaign, treatment of Mtb-infected individuals, MDA, and sanitation.
(a)(b)(c)(d)(e)(f)


## 5.1. Control with Educational Campaignu1 and Treatment of Individuals with Active TB u2

The educational campaign controlu1 and treatment of individuals with active TB control u2 are used to optimize the objective functional J while the other controls u3 and u4 related to helminths were set to zero. We observed that in Figure 2(b) the number of individuals with active TB was controlled but starts to increase again until the end of the intervention period. This may be related to our previous analysis that helminth infection enhances Mtb infection. In this strategy, helminth infection is not controlled; Figures 2(c) and 2(d) indicate that there is slight significance in the case with controls and without controls for helminth-infected individuals and coinfected individuals, respectively. Figure 2(e) indicates that the parasite population is not controlled with this strategy while Figure 2(f) indicates that the educational campaign control should be seized after 158 days while treatment of individuals with active TB control should be kept maximum in the entire period of intervention.Figure 2
Simulations of the helminth-Mtb coinfection model with the effect of educational campaign and treatment of Mtb-infected individuals.
(a)(b)(c)(d)(e)(f)


## 5.2. Control with Mass Drug Administrationu3 and Sanitation u4

The mass drug administration (MDA) controlu3 and sanitation control u4 were used to optimize the objective functional J. Again, we observe that Figure 3(a) indicates the increase in vaccinated individuals. Also, Figure 3(b) indicates that the number of people with active TB increases rapidly and is eventually controlled after 55 days. An equivalent scenario is observed in Figures 3(c) and 3(d) where the helminth-infected individuals and coinfected individuals were effectively controlled. Figure 3(e) suggests that the parasite population is substantially controlled at the end of the intervention period. This strategy seems to be effective in controlling the two diseases from the community. Figure 3(f) indicates that MDA and sanitation controls should be kept maximum throughout the intervention period.Figure 3
Simulations of the helminth-Mtb coinfection model with the effect of MDA and sanitation.
(a)(b)(c)(d)(e)(f)


## 5.3. Control with Educational Campaignu1 and Sanitation u4

The educational campaign controlu1 and sanitation control u4 are employed to optimize the objective functional J. This strategy has the same profiles as the strategy with mass drug administration and sanitation controls but takes a long time to control the two diseases. Figure 4(f) indicates that the educational campaign must be stopped after 30 days whereas sanitation control must be kept maximum in the entire period of intervention.Figure 4
Simulations of the helminth-Mtb coinfection model with the effect of educational campaign and sanitation.
(a)(b)(c)(d)(e)(f)


## 5.4. Control with Treatment of Individuals with Active TBu2 and MDA u3

The treatment of individuals with active TB controlu2 and MDA control u3 are used to optimize the objective functional J while the other controls u1 and u4 are set to zero. Figure 5(a) indicates a slight increase in vaccinated individuals compared to the other strategies (Sections 5.1, 5.2, and 5.3). Figures 5(b) and 5(d) indicate that the treatment of individuals with active TB and coinfected individuals is effectively controlled while Figure 5(c) indicates a decrease in the helminth-infected individuals but a rise at the final intervention period. This is due to the absence of preventive measures in this strategy. Figure 5(f) indicates that both controls should be kept maximum until the final intervention period.Figure 5
Simulations of the helminth-Mtb coinfection model with the effect of treatment of Mtb-infected individuals and MDA.
(a)(b)(c)(d)(e)(f)


## 5.5. Control with Educational Campaignu1, Treatment of Mtb-Infected Individuals u2, MDA u3, and Sanitation u4

With this strategy, all four controls are employed to optimize the objective functionalJ. We observed that Figure 6(a) indicates that the number of vaccinated individuals increases more at the end of the intervention period. Figures 6(b)–6(d) indicate that individuals with active TB, helminth-infected individuals, and coinfected individuals are all effectively and efficiently controlled at the end of the intervention period. Figure 6(e) depicts that the parasite population is decreased to a number that cannot harm the human population. This strategy indicates that optimal educational campaign, treatment of individuals with active TB, MDA, and sanitation would effectively control the helminth-Mtb infection at the end of the intervention period. Figure 6(f) suggests that the educational campaign should be stopped after 115 days whereas the remaining controls should be maintained maximum for the whole intervention period.Figure 6
Simulations of the helminth-Mtb coinfection model with the effect of educational campaign, treatment of Mtb-infected individuals, MDA, and sanitation.
(a)(b)(c)(d)(e)(f)


## 6. Discussions and Conclusions

In this paper, we described and formulated a deterministic model for the transmission dynamics and control of helminth-Mtb coinfection. The effective reproduction number was computed, and we explored the steadiness of the disease-free equilibrium point. We again established the existence of backward or forward bifurcation using the Center Manifold Theorem. Then, we investigated the impact the two diseases have on each other and found that helminth infection cases increase Mtb infection cases. In Section4, we modified the helminth-Mtb model by incorporating four control measures: educational campaign to sensitize the parents to take babies to the BCG vaccine clinic, treatment of individuals with active TB, mass drug administration for helminth and coinfected individuals, and sanitation to reduce the intake rate of parasites. The optimal control was analyzed using Pontryagin’s Maximum Principle [28], by first finding the Hamiltonian, the costate variables, the characterization of the controls, and the optimality system.Then, we solved numerically the optimality system by the iterative scheme using the Forward-Backward Sweep Method (FBSM) with the combination of the following control measures:(i)
By applying a combination of educational campaign and treatment of individuals with active TB(ii)
By applying a combination of mass drug administration (MDA) and sanitation(iii)
By applying a combination of educational campaign and sanitation(iv)
By applying a combination of treatment of individuals with active TB and mass drug administration(v)
By applying a combination of educational campaign, treatment of individuals with active TB, mass drug administration, and sanitationThe control strategies which focus on TB only while helminths are not controlled would not lead to the efficient and effective control of TB or helminth infection. Moreover, control strategies that focus on helminths only while TB is not controlled would control both TB and helminth infections. This is linked to the increased susceptibility of Mtb for individuals infested with helminths. However, control strategies that include all control measures would effectively control the helminth-Mtb coinfection at the end of the intervention period. Thus, we suggest to the public health stakeholders that in order to control the helminth-Mtb coinfection, the intervention strategies that target controlling helminth infection and vaccination of babies with BCG after birth and treatment of individuals with active tuberculosis should be emphasized. One of the limitations of this study is the number of compartments involved that hinders some analytical analyses; however, numerical simulation shows the convergence.



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