# Mathematical Modelling of Human African Trypanosomiasis Using Control Measures

Authors: Hamenyimana Emanuel Gervas; Nicholas Kwasi-Do Ohene Opoku; Shamsuddeen Ibrahim
Journal: Computational and Mathematical Methods in Medicine
 (2018)

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## Abstract

Human African trypanosomiasis (HAT), commonly known as sleeping sickness, is a neglected tropical vector-borne disease caused by trypanosome protozoa. It is transmitted by bites of infected tsetse fly. In this paper, we first present the vector-host model which describes the general transmission dynamics of HAT. In the tsetse fly population, the HAT is modelled by three compartments, while in the human population, the HAT is modelled by four compartments. The next-generation matrix approach is used to derive the basic reproduction number,R0, and it is also proved that if R0≤1, the disease-free equilibrium is globally asymptotically stable, which means the disease dies out. The disease persists in the population if the value of R0>1. Furthermore, the optimal control model is determined by using the Pontryagin’s maximum principle, with control measures such as education, treatment, and insecticides used to optimize the objective function. The model simulations confirm that the use of the three control measures is very efficient and effective to eliminate HAT in Africa.


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## Body

## 1. Introduction

Human African trypanosomiasis (HAT), commonly known as sleeping sickness, is a vector-borne tropical disease which is caused byTrypanosoma brucei protozoa species. It is one of the neglected tropical diseases which affect people in sub-Saharan Africa, specifically those living in rural areas. HAT is caused by two species of protozoa which are Trypanosoma brucei gambiense (TBG), which causes the chronic form of HAT in central and western Africa, and Trypanosoma brucei rhodesiense (TBR), which causes the acute form of the disease in eastern and southern Africa [1]. The HAT disease has killed millions of people since the beginning of 20th century and it is transmitted from one individual to another by tsetse flies (genus Glossina); TBG is transmitted by riverine tsetse species, while TBR is transmitted by savanna tsetse species [1]. Rhodesiense HAT is an acute disease that can lead to death if not treated within 6 months, while gambiense HAT is a slow chronic progressive disease which causes death with an average duration of 3 years [2]. The signs and symptoms for both forms of HAT are not specific and their appearances vary from one person to another; at the first stage of HAT, the disease is not severe and the signs and symptoms such as intermittent fever, headache, pruritus, lymphadenopathies, asthenia, anemia, cardiac disorders, endocrine disturbances, musculoskeletal pains, and hepatosplenomegaly may be observed, while in the second stage of HAT, sleep disorders and neuropsychiatric disorders are likely to dominate. The HAT disease can be treated by using drugs such as suramin, eflornithine, melarsoprol, and pentamidine.The disease is reported to affect about 37 sub-Saharan African countries; it affects much rural areas where there are suitable environments for the tsetse flies to live and reproduce, and the periurban areas can also be affected. The transmission of HAT can occur during human activities such as hunting, farming, as well as fishing [3]. The transmission of HAT needs the reservoir; reservoir is a species that can permanently maintain the pathogen and from which the pathogen can be transmitted to the target population [4]. Rhodesiense HAT is zoonotic which requires a nonhuman reservoir (animals) for maintaining its population, while in gambiense HAT, humans act as key reservoir [4].Mathematical models have been used to study the transmission and effective control of diseases simply and cheaply with no need of expensive and complicated experiments [5]. So far, different models have been developed and formulated by different researchers. One of the important modelling work on HAT was done by Rogers [6]; the model explained the mathematical framework on transmission of HAT in multiple host populations [6]. Rogers’ model was generalized by Hargrove et al [7], and a new parameter which allows the tsetse flies to feed off multiple hosts was introduced. The model compared the effectiveness of two methods used to control HAT: insecticide-treated cattle and the use of trypanocide drugs to treat cattle. They found out that treating cattle with insecticides is more effective and a cheaper approach to control HAT than using trypanocide drugs. Kajunguri [8] developed a model which was based on a constant population with a fixed number of domestic animals, human, and tsetse flies in one of the villages in West Africa. The major findings of their model estimated that the cattle population contributes to about 92% of the total TBR transmission, while the rest 8% is the contribution of human population in transmission of the disease. The study by Kajunguri [8], which also formulated a multihost model, was used to study the control of tsetse flies and TBR in southern Uganda. They found out that the effective application of insecticides brings about a cost-effective method of control and eliminating the disease. They realized that using insecticides for controlling HAT is more effective and efficient in the area where there are few wild hosts.Due to low mortality rate of the disease and poverty of its sufferers, the efforts toward the control of HAT has reduced. Most attention is given to popular diseases such as HIV/AIDS, tuberculosis, malaria, and ebola, although the disease is still a threat to the lives of sub-Saharan African people. Moreover, very few studies have been carried out on applying optimal control theory to HAT transmission models. In this paper, we use optimal control theory to study the transmission dynamics of HAT diseases by using education, treatment, and insecticides as the control measures.The rest of this paper is outlined as follows: Section2 represents the vector-host model and the underlying assumptions. In section 3, the model equilibria and stabilities are determined, whereas in Section 4, the optimal control model is analyzed by modifying the previous one to control the HAT by using control measures (education, insecticides, and treatment). In addition, the numerical simulations for the optimal control model are done in this section and we use the results obtained to compare the efforts of each control measure to control the HAT in Africa. Finally, we provide the conclusion in Section 5.


## 2. Model Formulation

In this section, the vector-host model as well as the necessary differential equations to describe the transmission of HAT from tsetse fly to human and vice versa are developed. The transmission of HAT in the human population is modelled using four subclasses: SusceptibleSH, Exposed EH, Infectious IH, and Recovered RH. The total human population, NH, is thus defined by(1)NH=SH+EH+IH+RH.The transmission of HAT in the vector (tsetse flies) population is also divided into Susceptible (SV), Exposed (EV), and Infectious (IV). The total population of the tsetse flies, NV, is also defined by(2)NV=SV+EV+IV.We assume a constant population for both host and vector. It is also assumed that the tsetse fly cannot recover from the disease and the infected tsetse fly remains infectious throughout the rest of its life; there is no disease-induced death rate for tsetse flies and the recruitment rates are assumed to be constant due to birth and immigration.In our model, the recruitment rate of hosts and vectors are represented byπH and πV, respectively. The susceptible host gets the disease when bitten by infectious tsetse fly, and susceptible tsetse fly gets the disease when it bites an infectious human at the rate a. The natural mortality rate for humans and vectors are represented by μH and μV, respectively. The parameter ω represents the disease-induced death rate for humans, while ξH and ξV are the force of infection for humans and vectors, respectively. The parameter σ represents per capita rate of a vector becoming infectious, and the rest of the parameters are explained in Table 1. Assuming that the transmission per bite from infectious tsetse fly to human is a, then the rate of infection per susceptible human is given by(3)ξH=apHIVNV,and also if we further assume that a is the tsetse-fly biting rate, that is, the average number of bites per tsetse fly per unit, then the rate of infection per susceptible tsetse fly can be represented by(4)ξV=apVIHNH.Table 1
The description of model variables and parameters.
Variable
Description
s
h
Susceptible human population
s
v
Susceptible tsetse fly population
e
h and ev
Exposed human and tsetse fly population, respectively
i
h and iv
Infectious human and tsetse population, respectively
r
h
Recovered human population
Parameter
Description
π
h
Recruitment rate for human population
π
v
Recruitment rate for tsetse fly population
p
h
Proportion of bites by the infectious vector on susceptible human population
p
v
Proportion of bites by susceptible vector on an infectious human population
a
The biting rate of the tsetse flies
σ
Per capita rate of a vector becoming infectious
ε
Per capita rate of human becoming infectious
ω
Disease induced death rate
ρ
The rate at which the recovered human can become susceptible again
τ
Recovery rate
μ
Natural death rate
ξ
h
Force of infection for human population
ξ
v
Force of infection for tsetse fliesFrom the model diagram in Figure1, the following differential equations are derived:(5)dSHdt=πHNH+ρRH−apHIVNVSH−μHSH,dEHdt=apHIVNVSH−εEH−μHEH,dIHdt=εEH−μHIH−ωIH−τIH,dRHdt=τIH−ρRH−μERH,dSVdt=πVNV−μVSV−apVIHNHSV,dEVdt=apVIHNHSV−μVEV−σEV,dIVdt=σEV−μVIV.Figure 1
Compartmental model for the transmission of human African trypanosomiasis.From system (5), the dimensionless technique is used to derive another equivalent differential equation; we denote sh=SH/NH, eh=EH/NH, ih=IH/NH, rh=RH/NH, sv=SV/NV, ev=EV/NV, and iv=IV/NV and substitute in system (5), to obtain the following new equivalent equations:(6)dshdt=πh+ρrh−aphivsh−μhsh,dehdt=aphivsh−εeh−μheh,dihdt=εeh−μhih−ωih−τih,drhdt=τih−ρrh−μhrh,dsvdt=πv−μvsv−apvihsv,devdt=apvihsv−μvev−σev,divdt=σev−μviv.Table1 shows the description of the model parameters and variables.


### 2.1. Positivity and Boundedness of the Solutions

In this subsection, we show that system (6) is epidemiologically and mathematically well defined in the positive invariant region:(7)D=sh,eh,ih,rh,sv,ev,iv∈R+7:nh≤πhμh;nv≤πvμv.Theorem 1.
There exists a domainD in which the solution sh,eh,ih,rh,sv,ev,iv is contained and bounded.Proof. We provide the proof following the idea by Olaniyi and Obabiyi [9]. Given the solution set sh,eh,ih,rh,sv,ev,iv with the positive initial conditions sh0,eh0,ih0,rh0,sv0,ev0,iv0, we define(8)nhsh,eh,ih,rh=sht+eht+iht+rhtandnvsv,ev,iv=svt+evt+ivt.The derivatives ofnh and nv with respect to time along the solution of system (6) for human and tsetse flies, respectively, are obtained by(9)nh′=dshdt+dehdt+dihdt+drhdt=πh−sh+eh+ih+rhμh−ωih=πh−nhμh−ωih,nv′=dsvdt+devdt+divdt=πv−sv+ev+ivμv=πv−nvμv.From these differential equations, it follows thatnh′≤πh−μhnh and nv′≤πv−μvnv. We obtain the solutions as follows:(10)nh≤πhμh1−exp−μht+nhsh0,eh0,ih0,rh0exp−μht,nv≤πvμv1−exp−μvt+nvsv0,ev0,iv0exp−μvt.By taking the limits of bothnh and nv above as t⟶∞, we obtain nh≤πh/μh and nv≤πv/μv; hence, the solutions are contained in the region D. This implies that all solutions of the human and tsetse fly population are contained in the region D and are nonnegative; this guarantees that the positive invariant region for system (6) exists and is given by(11)D=sh,eh,ih,rh,sv,ev,iv∈R+7:nh≤πhμh;nv≤πvμv.


## 2.1. Positivity and Boundedness of the Solutions

In this subsection, we show that system (6) is epidemiologically and mathematically well defined in the positive invariant region:(7)D=sh,eh,ih,rh,sv,ev,iv∈R+7:nh≤πhμh;nv≤πvμv.Theorem 1.
There exists a domainD in which the solution sh,eh,ih,rh,sv,ev,iv is contained and bounded.Proof. We provide the proof following the idea by Olaniyi and Obabiyi [9]. Given the solution set sh,eh,ih,rh,sv,ev,iv with the positive initial conditions sh0,eh0,ih0,rh0,sv0,ev0,iv0, we define(8)nhsh,eh,ih,rh=sht+eht+iht+rhtandnvsv,ev,iv=svt+evt+ivt.The derivatives ofnh and nv with respect to time along the solution of system (6) for human and tsetse flies, respectively, are obtained by(9)nh′=dshdt+dehdt+dihdt+drhdt=πh−sh+eh+ih+rhμh−ωih=πh−nhμh−ωih,nv′=dsvdt+devdt+divdt=πv−sv+ev+ivμv=πv−nvμv.From these differential equations, it follows thatnh′≤πh−μhnh and nv′≤πv−μvnv. We obtain the solutions as follows:(10)nh≤πhμh1−exp−μht+nhsh0,eh0,ih0,rh0exp−μht,nv≤πvμv1−exp−μvt+nvsv0,ev0,iv0exp−μvt.By taking the limits of bothnh and nv above as t⟶∞, we obtain nh≤πh/μh and nv≤πv/μv; hence, the solutions are contained in the region D. This implies that all solutions of the human and tsetse fly population are contained in the region D and are nonnegative; this guarantees that the positive invariant region for system (6) exists and is given by(11)D=sh,eh,ih,rh,sv,ev,iv∈R+7:nh≤πhμh;nv≤πvμv.


## 3. Model Equilibria and Stability Analysis

In this section, we give the model equilibria, the basic reproduction number,R0, and the stabilities at both disease-free and endemic equilibrium.


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

The DFE in system (6) is when there are no HAT infections within the human and tsetse fly population. Thus, the existence of the DFE is given by E0=πh/μh,0,0,0,πv/μv,0,0.


### 3.2. Endemic Equilibrium (EE)

The EE is the nontrivial equilibrium point at which the HAT disease persists in both human and tsetse fly population. Thus, the EE is obtained as follows:E∗=sh∗,eh∗,ih∗,rh∗,sv∗,ev∗,iv∗, where(12)sh∗=πhρ+μh+ρτih∗μvσ+μvapvih∗+μva2σphpvπvih∗+μvμhσ+μvapvih∗+μvμh+ρ,eh∗=ω+τ+μhih∗ε,rh∗=τih∗μh+ρ,sv∗=πvapvih∗+μv,ev∗=apvπvih∗σ+μvapvih∗+μv,iv∗=aσpvπvih∗σ+μvapvih∗+μvμv,ih∗=ρ+μha2εphpvπhπvσ−μv2μhε+μhσ+μvμh+τ+ωB,and the term(13)B=apvaσphπvερω+μhρτ+ω+ερ+τ+ω+μhε+ρ+τ+ω+μh+μhε+μhρ+μhτ+ω+μhμvσ+μv.


### 3.3. Basic Reproduction Number,R0

The basic reproduction number,R0, is defined as the number of secondary infections caused by one infected host or vector in a completely susceptible population [10]. The next-generation matrix approach as done by Van den Driessche and Watmough in [5, 11] is applied to derive(14)F=aphivsh0apvihsv,V=μheh+εeh−εeh+μhih+ωih+τihμvev+δev−δev+μviv.By denoting matrixF=∂F/∂xi and V=∂V/∂xi, where xi=eh,ih,ev,iv, the spectral radius of the next-generation matrix FV−1 gives the value of R0:(15)F=000aphsh00000apvsv000000,V=ε+μh000−εμh+ω+τ0000μv+δ000−δμv,FV−1=00aδphμv−δ+μv−aphμv0apvμh+τ+ω0000000000.The spectral radiusσFV−1 gives(16)R0=σFV−1=a2εphpvπhπvσμv2μhε+μhσ+μvμh+τ+ω.One infected human in a population of susceptible vectors will causeRv infected vectors; likewise, one infected vector in a population will cause Rh infected humans [5]. Therefore, the basic reproduction number can be rewritten as R0=RhRv, where Rh=aεphπh/μhμh+εμh+τ+ω and Rv=σapvπv/μv2σ+μv. Thus, R0 can also be defined as the square root of the product of the number of infected humans in the susceptible population caused by one infected tsetse fly in its infectious lifetime and the number of infected tsetse flies caused by one infected human during the infectious period [12].


### 3.4. Local Stability of Disease-Free Equilibrium (DFE)

Theorem 2.
IfR0≤1, the DFE given by E0 is locally asymptotically stable in the region defined by (7), and it is unstable when R0>1.Proof. The DFE is locally stable if all eigenvalues of Jacobian matrix JE0 are negative. The matrix has all eigenvalues negative only if the trace of JE0<0 and determinant of JE0>0. By linearizing system (6) around E0, we obtain the following Jacobian matrix:(17)JE0=−μh00ρ00−aphsh0−ε+μh0000aphsh0ε−ω+τ+μh000000τ−ρ+μh00000−apvsv0−μv0000apvsv00−σ+μv000000σ−μv.The trace of matrixJE0 is such that(18)trJE0=−μh+ε+μh+μh+ω+τ+ρ+μh+μv+σ+μv+μv=−4μh+3μv+ε+ω+τ+ρ+σ<0.Using the basic properties of matrix algebra as in [13], it is clear that the eigenvalues λ1=−μh and λ2=−μv of the matrix JE0 have negative real parts. The reduced matrix is(19)JE1=−ε+μh000aphshε−ω+τ+μh0000τ−ρ+μh000apvsv0−σ+μv0000σ−μv.From matrixJE1, the eigenvalue λ3=−ρ+μh has negative real part. The remaining matrix is further reduced by using the reduction techniques, and we obtain(20)JE2=−ε+μh00aphsh0−ω+τ+μh0aphshεε+μh0apvsv−σ+μv000σ−μv.Using the properties of matrix algebra, the matrixJE2 has eigenvalue −ε+μh which has negative real part. We further reduce to a 2×2 matrix by using the same reduction techniques. The matrix is(21)JE3=−ω+τ+μhaphshεε+μhapvsvσσ+μv−μv.From the reduced2×2 matrix, the trace is negative and the determinant is(22)DetJE3=ω+τ+μhμv−aphshεε+μh×apvsvσσ+μv=ω+τ+μhμv1−a2εphpvπhπvσμv2μhε+μhσ+μvμh+τ+ω.Since(23)R0=a2εphpvπhπvσμv2μhε+μhσ+μvμh+τ+ω,then, by letting RT=a2εphpvπhπvσ/μv2μhε+μhσ+μvμh+τ+ω, we find our determinant as(24)DetJE3=ω+τ+μhμv1−RT.The value ofRT can be seen to be positive because all the parameters are positive. As a result, the determinant in (24) is positive if and only if RT<1. Therefore, the DFE is locally stable if RT≤1.


### 3.5. Global Stability of Disease-Free Equilibrium (DFE)

To show that the DFE is globally stable, we apply Lyapunov’s theorem in [5].Theorem 3.
The DFE defined byE0 is globally asymptotically stable in the region defined by (7) if R0≤1. Otherwise unstable if R0>1.Proof. We define Lyapunov’s function as(25)V=k1sh−sh0−sh0lnshsh0+k2eh+k3ih+k4sv−sv0−sv0lnsvsv0+k5ev+k6iv,satisfying system (6), where k1,k2,k3,k4,k5,k6>0 are to be determined and sh0=πh/μh and sv0=πv/μv. We first show that V>0 for all E≠πh/μh,0,0,0,πv/μv,0,0. It is enough to check that(26)k1sh0shsh0−1−lnshsh0>0,k4sv0svsv0−1−lnsvsv0>0.The functiongm=m−1−lnm such that m=sh/sh0=sv/sv0 has minimum value equal to zero when m=1; hence, gm>0 for all m>0. Thus, Lyapunov’s function V>0. The function V is radially unbounded because as m⟶∞, the function gm⟶∞. We now take the derivative of V with respect to time and use system (6) to replace the derivatives in the right hand side such that(27)V′=k11−sh0shdshdt+k2dehdt+k3dihdt+k41−sv0svdsvdt+k5devdt+k6divdt,=k11−sh0shπh+ρrh−aphivsh−μhsh+k2aphivsh−εeh−μheh+k3εeh−μhih−ωih−τih+k41−sv0svπv−μvsv−apvihsv+k5apvihsv−μvev−σev+k6σev−μviv,=2k1πh−aphshk1iv+ρrhk1−k1μhsh−k1πh2μhsh+aphπhivμhk1−πhρrhμhshk1+k2aphshiv−k2ε+μheh+k3εeh−k3τ+μh+ωih+2k4πv−k4apvsvih−μvsvk4−πv2μvsvk4+apvihπvμvk4−k5μv+σev+k5apvsvih+k6σev−k6μviv.The terms withrh are ignored because if sh,eh,ih are globally stable, then rh⟶0 at any time t and the DFE for system (6) is globally stable. Taking k1=k2=1/μh+ε, k4=k5=1/μv+σ, k3=1/ε, and k6=1/σ, the derivative of V with respect to time becomes(28)V′=−πhμh+επhμhsh+μhshπh−2−τ+μh+ωεih+aphπhμh+εμhiv−πvμv+σπvμvsv+μvsvπv−2−μvσiv+apvπvμvμv+σih=−πhμh+επhμhsh+μhshπh−2+πvμv+σπvμvsv+μvsvπv−2+τ+μh+ωεεapvπvμvσ+μvμh+τ+ω−1ih+μvσaσphπhμvμhμh+ε−1iv=−πhμh+επhμhsh+μhshπh−2+πvμv+σπvμvsv+μvsvπv−2+τ+μh+ωεRv−1ih+μvσRh−1iv.The termsπh/μhsh+μhsh/πh−2 and πv/μvsv+μvsv/πv−2 are positive because if we suppose m=πh/μhsh=πv/μvsv, we have m+1/m−2=m2−2m+1/m=m−12/m>0 for all m>1 and since Rv≤1 and Rh≤1, then V′ is negative. Thus, we have V′<0 for all E0≠πh/μh,0,0,0,πv/μv,0,0.Thus, the largest compact invariant set inD is the singleton set E0. Hence, system (6) is globally asymptotically stable.


### 3.6. Local Stability of Endemic Equilibrium (EE)

Theorem 4.
The unique endemic equilibrium defined byE∗ is locally asymptotically stable in the region defined by (7) if R0>1, but is unstable if R0≤1.Proof. We give the proof of this theorem based on the approach used by Olaniyi and Obabiyi [9, 14]. From the EE points defined in (12), since all values are positive, we express the value of ih∗ in terms of R0 to obtain(29)ih∗=μv2μhε+μhρ+μhσ+μvμh+τ+ωBa2εphpvπhπvσμv2μhε+μhσ+μvμh+τ+ω−1=μv2μhε+μhρ+μhσ+μvμh+τ+ωBR02−1,where(30)B=apvaσphπvερω+μhρτ+ω+ερ+τ+ω+μhε+ρ+τ+ω+μh+μhε+μhρ+μhτ+ω+μhμvσ+μv.Since the basic reproduction numberR0=a2εphpvπhπvσ/μv2μhε+μhσ+μvμh+τ+ω, if we let(31)RT=a2εphpvπhπvσμv2μhε+μhσ+μvμh+τ+ω,then we can find ih∗ as(32)ih∗=μv2μhε+μhρ+μhσ+μvμh+τ+ωBRT−1.The value ofB is clearly positive because all parameters are positive. Hence, ih∗>0 if and only if RT>1, implying that the EE is locally asymptotically stable if RT>1.


### 3.7. Global Stability of Endemic Equilibrium (EE)

To show the global stability of theEE, we use Lyapunov’s theorem together with the following lemma.Lemma 1.
Suppose thaty1,y2,⋯,yn are n positive numbers. Then, their arithmetic mean is greater than or equal to the geometric mean, that is y1+y2+⋯+yn/n≥y1y2⋯yn1/n.Theorem 5.
The EE defined byE∗ is globally asymptotically stable if R0>1, otherwise unstable.Proof. The proof is based on the idea as explained by Martcheva [5]. We define Lyapunov’s function as(33)V=k1sh−sh∗−sh∗lnshsh∗+k2eh−eh∗−eh∗lneheh∗+k3ih−ih∗−ih∗lnihih∗+k4sv−sv∗−sv∗lnsvsv∗+k5ev−ev∗−ev∗lnevev∗+k6iv−iv∗−iv∗lniviv∗,satisfying system (6) with k1,k2,k3,k4,k5,k6>0 to be determined. The function V is nonnegative for all sh,eh,ih,rh,sv,ev,iv≠sh∗,eh∗,ih∗,rh∗,sv∗,ev∗,iv∗ and radially unbounded.We need to prove thatV′<0 for all sh,eh,ih,rh,sv,ev,iv≠sh∗,eh∗,ih∗,rh∗,sv∗,ev∗,iv∗. We find the derivative of V with respect to time and replace the derivatives sh′,eh′,ih′,rh′,sv′,ev′,iv′ with system (6). We also ignore the rh terms because if sh,eh,ih are globally stable, then rh⟶0 at any time t and EE is globally stable:(34)V′=k11−sh∗shπh−aphivsh−μhsh+k21−eh∗ehaphivsh−μh+εeh+k31−ih∗ihεeh−ω+τ+uhih+k41−sv∗svπv−apvihsv−μvsv+k51−ev∗evapvihsv−μv+σev+k61−iv∗ivσev−uviv.We now substituteπh=aphiv∗sh∗+μhsh∗ and πv=apvih∗sv∗+μvsv∗ at the endemic equilibrium and then simplify and put similar terms together to obtain(35)V′=−k1sh−sh∗2μhsh+k1aphiv∗sh∗−k1aphivsh−k1aphsh∗2iv∗sh+k1aphivsh∗+k2aphshiv−k2ε+μheh−k2aphshiveh∗eh+k2ε+μheh∗+k3εeh−k3ω+τ+μhih∗−k3εehih∗ih+k3ω+τ+μhih∗−k4sv−sv∗2μvsv+k4apvih∗sv∗−k4apvihsv−k4apvsv∗2ih∗sv+k4apvihsv∗+k5apvsvih−k5σ+μvev−k5apvsvihev∗ev+k5σ+μvev∗+k6σev−k6μviv∗−k6σeviv∗iv+k6μviv∗.We supposek1=k2 and k4=k5 and multiply and divide the same equilibrium value to some of the fractions to obtain(36)V′=−k1sh−sh∗2μhsh+k1aphiv∗sh∗−k1aphsh∗2iv∗sh+k1aphivsh∗−k2ε+μheh−k2aphshiveh∗sh∗iv∗sh∗iv∗eh+k2ε+μheh∗+k3εeh−k3ω+τ+μhih−k3εehih∗eh∗iheh∗+k3ω+τ+μhih∗−k4sv−sv∗2μvsv+k4apvih∗sv∗−k4apvsv∗2ih∗sv+k4apvihsv∗−k5σ+μvev−k5apvsvihev∗sv∗ih∗sv∗ih∗ev+k5σ+μvev∗+k6σev−k6μviv∗−k6σevev∗iv∗ivev∗+k6μviv∗.We choosek3=k2μh+ε/ε such that k3ω+τ+μhih∗=k2ε+μheh∗ and choose k6=k5μh+σ/σ such that k6μviv∗=k5σ+μvev∗. Now, aphsh∗iv∗=μh+εeh∗ and apvsv∗iv∗=μv+σev∗ because k1=k2 and k4=k5, respectively. We now obtain(37)V′=−k1sh−sh∗2μhsh+k1aphiv∗sh∗3−sh∗sh−eh∗ivshsh∗ehiv∗−ehih∗iheh∗+k1aphivsh∗−k3ω+τ+μhih+k3ε−k2ε+μheh−k4sv−sv∗2μvsv+k4apvih∗sv∗3−sv∗sv−ev∗ihsvsv∗evih∗−eviv∗ivev∗+k4apvihsv∗−k6μviv+k6σ−k5σ+μvev.Supposek6=k1aphsh∗/μv and k3=k4apvsv∗/ω+τ+μh. We then substitute and simplify to get(38)V′=−k1sh−sh∗2μhsh+k1aphiv∗sh∗3−sh∗sh−eh∗ivshsh∗ehiv∗−ehih∗iheh∗−k4sv−sv∗2μvsv+k4apvih∗sv∗3−sv∗sv−ev∗ihsvsv∗evih∗−eviv∗ivev∗.From Lemma 1, the terms(39)k1aphiv∗sh∗3−sh∗sh−eh∗ivshsh∗ehiv∗−ehih∗iheh∗,k4apvih∗sv∗3−sv∗sv−ev∗ihsvsv∗evih∗−eviv∗ivev∗≤0.Therefore,V′<0 for all sh,eh,ih,rh,sv,ev,iv≠sh∗,eh∗,ih∗,rh∗,sv∗,ev∗,iv∗, implying that the endemic equilibrium is globally asymptotically stable if R0>1.


## 3.1. Disease-Free Equilibrium (DFE)

The DFE in system (6) is when there are no HAT infections within the human and tsetse fly population. Thus, the existence of the DFE is given by E0=πh/μh,0,0,0,πv/μv,0,0.


## 3.2. Endemic Equilibrium (EE)

The EE is the nontrivial equilibrium point at which the HAT disease persists in both human and tsetse fly population. Thus, the EE is obtained as follows:E∗=sh∗,eh∗,ih∗,rh∗,sv∗,ev∗,iv∗, where(12)sh∗=πhρ+μh+ρτih∗μvσ+μvapvih∗+μva2σphpvπvih∗+μvμhσ+μvapvih∗+μvμh+ρ,eh∗=ω+τ+μhih∗ε,rh∗=τih∗μh+ρ,sv∗=πvapvih∗+μv,ev∗=apvπvih∗σ+μvapvih∗+μv,iv∗=aσpvπvih∗σ+μvapvih∗+μvμv,ih∗=ρ+μha2εphpvπhπvσ−μv2μhε+μhσ+μvμh+τ+ωB,and the term(13)B=apvaσphπvερω+μhρτ+ω+ερ+τ+ω+μhε+ρ+τ+ω+μh+μhε+μhρ+μhτ+ω+μhμvσ+μv.


## 3.3. Basic Reproduction Number,R0

The basic reproduction number,R0, is defined as the number of secondary infections caused by one infected host or vector in a completely susceptible population [10]. The next-generation matrix approach as done by Van den Driessche and Watmough in [5, 11] is applied to derive(14)F=aphivsh0apvihsv,V=μheh+εeh−εeh+μhih+ωih+τihμvev+δev−δev+μviv.By denoting matrixF=∂F/∂xi and V=∂V/∂xi, where xi=eh,ih,ev,iv, the spectral radius of the next-generation matrix FV−1 gives the value of R0:(15)F=000aphsh00000apvsv000000,V=ε+μh000−εμh+ω+τ0000μv+δ000−δμv,FV−1=00aδphμv−δ+μv−aphμv0apvμh+τ+ω0000000000.The spectral radiusσFV−1 gives(16)R0=σFV−1=a2εphpvπhπvσμv2μhε+μhσ+μvμh+τ+ω.One infected human in a population of susceptible vectors will causeRv infected vectors; likewise, one infected vector in a population will cause Rh infected humans [5]. Therefore, the basic reproduction number can be rewritten as R0=RhRv, where Rh=aεphπh/μhμh+εμh+τ+ω and Rv=σapvπv/μv2σ+μv. Thus, R0 can also be defined as the square root of the product of the number of infected humans in the susceptible population caused by one infected tsetse fly in its infectious lifetime and the number of infected tsetse flies caused by one infected human during the infectious period [12].


## 3.4. Local Stability of Disease-Free Equilibrium (DFE)

Theorem 2.
IfR0≤1, the DFE given by E0 is locally asymptotically stable in the region defined by (7), and it is unstable when R0>1.Proof. The DFE is locally stable if all eigenvalues of Jacobian matrix JE0 are negative. The matrix has all eigenvalues negative only if the trace of JE0<0 and determinant of JE0>0. By linearizing system (6) around E0, we obtain the following Jacobian matrix:(17)JE0=−μh00ρ00−aphsh0−ε+μh0000aphsh0ε−ω+τ+μh000000τ−ρ+μh00000−apvsv0−μv0000apvsv00−σ+μv000000σ−μv.The trace of matrixJE0 is such that(18)trJE0=−μh+ε+μh+μh+ω+τ+ρ+μh+μv+σ+μv+μv=−4μh+3μv+ε+ω+τ+ρ+σ<0.Using the basic properties of matrix algebra as in [13], it is clear that the eigenvalues λ1=−μh and λ2=−μv of the matrix JE0 have negative real parts. The reduced matrix is(19)JE1=−ε+μh000aphshε−ω+τ+μh0000τ−ρ+μh000apvsv0−σ+μv0000σ−μv.From matrixJE1, the eigenvalue λ3=−ρ+μh has negative real part. The remaining matrix is further reduced by using the reduction techniques, and we obtain(20)JE2=−ε+μh00aphsh0−ω+τ+μh0aphshεε+μh0apvsv−σ+μv000σ−μv.Using the properties of matrix algebra, the matrixJE2 has eigenvalue −ε+μh which has negative real part. We further reduce to a 2×2 matrix by using the same reduction techniques. The matrix is(21)JE3=−ω+τ+μhaphshεε+μhapvsvσσ+μv−μv.From the reduced2×2 matrix, the trace is negative and the determinant is(22)DetJE3=ω+τ+μhμv−aphshεε+μh×apvsvσσ+μv=ω+τ+μhμv1−a2εphpvπhπvσμv2μhε+μhσ+μvμh+τ+ω.Since(23)R0=a2εphpvπhπvσμv2μhε+μhσ+μvμh+τ+ω,then, by letting RT=a2εphpvπhπvσ/μv2μhε+μhσ+μvμh+τ+ω, we find our determinant as(24)DetJE3=ω+τ+μhμv1−RT.The value ofRT can be seen to be positive because all the parameters are positive. As a result, the determinant in (24) is positive if and only if RT<1. Therefore, the DFE is locally stable if RT≤1.


## 3.5. Global Stability of Disease-Free Equilibrium (DFE)

To show that the DFE is globally stable, we apply Lyapunov’s theorem in [5].Theorem 3.
The DFE defined byE0 is globally asymptotically stable in the region defined by (7) if R0≤1. Otherwise unstable if R0>1.Proof. We define Lyapunov’s function as(25)V=k1sh−sh0−sh0lnshsh0+k2eh+k3ih+k4sv−sv0−sv0lnsvsv0+k5ev+k6iv,satisfying system (6), where k1,k2,k3,k4,k5,k6>0 are to be determined and sh0=πh/μh and sv0=πv/μv. We first show that V>0 for all E≠πh/μh,0,0,0,πv/μv,0,0. It is enough to check that(26)k1sh0shsh0−1−lnshsh0>0,k4sv0svsv0−1−lnsvsv0>0.The functiongm=m−1−lnm such that m=sh/sh0=sv/sv0 has minimum value equal to zero when m=1; hence, gm>0 for all m>0. Thus, Lyapunov’s function V>0. The function V is radially unbounded because as m⟶∞, the function gm⟶∞. We now take the derivative of V with respect to time and use system (6) to replace the derivatives in the right hand side such that(27)V′=k11−sh0shdshdt+k2dehdt+k3dihdt+k41−sv0svdsvdt+k5devdt+k6divdt,=k11−sh0shπh+ρrh−aphivsh−μhsh+k2aphivsh−εeh−μheh+k3εeh−μhih−ωih−τih+k41−sv0svπv−μvsv−apvihsv+k5apvihsv−μvev−σev+k6σev−μviv,=2k1πh−aphshk1iv+ρrhk1−k1μhsh−k1πh2μhsh+aphπhivμhk1−πhρrhμhshk1+k2aphshiv−k2ε+μheh+k3εeh−k3τ+μh+ωih+2k4πv−k4apvsvih−μvsvk4−πv2μvsvk4+apvihπvμvk4−k5μv+σev+k5apvsvih+k6σev−k6μviv.The terms withrh are ignored because if sh,eh,ih are globally stable, then rh⟶0 at any time t and the DFE for system (6) is globally stable. Taking k1=k2=1/μh+ε, k4=k5=1/μv+σ, k3=1/ε, and k6=1/σ, the derivative of V with respect to time becomes(28)V′=−πhμh+επhμhsh+μhshπh−2−τ+μh+ωεih+aphπhμh+εμhiv−πvμv+σπvμvsv+μvsvπv−2−μvσiv+apvπvμvμv+σih=−πhμh+επhμhsh+μhshπh−2+πvμv+σπvμvsv+μvsvπv−2+τ+μh+ωεεapvπvμvσ+μvμh+τ+ω−1ih+μvσaσphπhμvμhμh+ε−1iv=−πhμh+επhμhsh+μhshπh−2+πvμv+σπvμvsv+μvsvπv−2+τ+μh+ωεRv−1ih+μvσRh−1iv.The termsπh/μhsh+μhsh/πh−2 and πv/μvsv+μvsv/πv−2 are positive because if we suppose m=πh/μhsh=πv/μvsv, we have m+1/m−2=m2−2m+1/m=m−12/m>0 for all m>1 and since Rv≤1 and Rh≤1, then V′ is negative. Thus, we have V′<0 for all E0≠πh/μh,0,0,0,πv/μv,0,0.Thus, the largest compact invariant set inD is the singleton set E0. Hence, system (6) is globally asymptotically stable.


## 3.6. Local Stability of Endemic Equilibrium (EE)

Theorem 4.
The unique endemic equilibrium defined byE∗ is locally asymptotically stable in the region defined by (7) if R0>1, but is unstable if R0≤1.Proof. We give the proof of this theorem based on the approach used by Olaniyi and Obabiyi [9, 14]. From the EE points defined in (12), since all values are positive, we express the value of ih∗ in terms of R0 to obtain(29)ih∗=μv2μhε+μhρ+μhσ+μvμh+τ+ωBa2εphpvπhπvσμv2μhε+μhσ+μvμh+τ+ω−1=μv2μhε+μhρ+μhσ+μvμh+τ+ωBR02−1,where(30)B=apvaσphπvερω+μhρτ+ω+ερ+τ+ω+μhε+ρ+τ+ω+μh+μhε+μhρ+μhτ+ω+μhμvσ+μv.Since the basic reproduction numberR0=a2εphpvπhπvσ/μv2μhε+μhσ+μvμh+τ+ω, if we let(31)RT=a2εphpvπhπvσμv2μhε+μhσ+μvμh+τ+ω,then we can find ih∗ as(32)ih∗=μv2μhε+μhρ+μhσ+μvμh+τ+ωBRT−1.The value ofB is clearly positive because all parameters are positive. Hence, ih∗>0 if and only if RT>1, implying that the EE is locally asymptotically stable if RT>1.


## 3.7. Global Stability of Endemic Equilibrium (EE)

To show the global stability of theEE, we use Lyapunov’s theorem together with the following lemma.Lemma 1.
Suppose thaty1,y2,⋯,yn are n positive numbers. Then, their arithmetic mean is greater than or equal to the geometric mean, that is y1+y2+⋯+yn/n≥y1y2⋯yn1/n.Theorem 5.
The EE defined byE∗ is globally asymptotically stable if R0>1, otherwise unstable.Proof. The proof is based on the idea as explained by Martcheva [5]. We define Lyapunov’s function as(33)V=k1sh−sh∗−sh∗lnshsh∗+k2eh−eh∗−eh∗lneheh∗+k3ih−ih∗−ih∗lnihih∗+k4sv−sv∗−sv∗lnsvsv∗+k5ev−ev∗−ev∗lnevev∗+k6iv−iv∗−iv∗lniviv∗,satisfying system (6) with k1,k2,k3,k4,k5,k6>0 to be determined. The function V is nonnegative for all sh,eh,ih,rh,sv,ev,iv≠sh∗,eh∗,ih∗,rh∗,sv∗,ev∗,iv∗ and radially unbounded.We need to prove thatV′<0 for all sh,eh,ih,rh,sv,ev,iv≠sh∗,eh∗,ih∗,rh∗,sv∗,ev∗,iv∗. We find the derivative of V with respect to time and replace the derivatives sh′,eh′,ih′,rh′,sv′,ev′,iv′ with system (6). We also ignore the rh terms because if sh,eh,ih are globally stable, then rh⟶0 at any time t and EE is globally stable:(34)V′=k11−sh∗shπh−aphivsh−μhsh+k21−eh∗ehaphivsh−μh+εeh+k31−ih∗ihεeh−ω+τ+uhih+k41−sv∗svπv−apvihsv−μvsv+k51−ev∗evapvihsv−μv+σev+k61−iv∗ivσev−uviv.We now substituteπh=aphiv∗sh∗+μhsh∗ and πv=apvih∗sv∗+μvsv∗ at the endemic equilibrium and then simplify and put similar terms together to obtain(35)V′=−k1sh−sh∗2μhsh+k1aphiv∗sh∗−k1aphivsh−k1aphsh∗2iv∗sh+k1aphivsh∗+k2aphshiv−k2ε+μheh−k2aphshiveh∗eh+k2ε+μheh∗+k3εeh−k3ω+τ+μhih∗−k3εehih∗ih+k3ω+τ+μhih∗−k4sv−sv∗2μvsv+k4apvih∗sv∗−k4apvihsv−k4apvsv∗2ih∗sv+k4apvihsv∗+k5apvsvih−k5σ+μvev−k5apvsvihev∗ev+k5σ+μvev∗+k6σev−k6μviv∗−k6σeviv∗iv+k6μviv∗.We supposek1=k2 and k4=k5 and multiply and divide the same equilibrium value to some of the fractions to obtain(36)V′=−k1sh−sh∗2μhsh+k1aphiv∗sh∗−k1aphsh∗2iv∗sh+k1aphivsh∗−k2ε+μheh−k2aphshiveh∗sh∗iv∗sh∗iv∗eh+k2ε+μheh∗+k3εeh−k3ω+τ+μhih−k3εehih∗eh∗iheh∗+k3ω+τ+μhih∗−k4sv−sv∗2μvsv+k4apvih∗sv∗−k4apvsv∗2ih∗sv+k4apvihsv∗−k5σ+μvev−k5apvsvihev∗sv∗ih∗sv∗ih∗ev+k5σ+μvev∗+k6σev−k6μviv∗−k6σevev∗iv∗ivev∗+k6μviv∗.We choosek3=k2μh+ε/ε such that k3ω+τ+μhih∗=k2ε+μheh∗ and choose k6=k5μh+σ/σ such that k6μviv∗=k5σ+μvev∗. Now, aphsh∗iv∗=μh+εeh∗ and apvsv∗iv∗=μv+σev∗ because k1=k2 and k4=k5, respectively. We now obtain(37)V′=−k1sh−sh∗2μhsh+k1aphiv∗sh∗3−sh∗sh−eh∗ivshsh∗ehiv∗−ehih∗iheh∗+k1aphivsh∗−k3ω+τ+μhih+k3ε−k2ε+μheh−k4sv−sv∗2μvsv+k4apvih∗sv∗3−sv∗sv−ev∗ihsvsv∗evih∗−eviv∗ivev∗+k4apvihsv∗−k6μviv+k6σ−k5σ+μvev.Supposek6=k1aphsh∗/μv and k3=k4apvsv∗/ω+τ+μh. We then substitute and simplify to get(38)V′=−k1sh−sh∗2μhsh+k1aphiv∗sh∗3−sh∗sh−eh∗ivshsh∗ehiv∗−ehih∗iheh∗−k4sv−sv∗2μvsv+k4apvih∗sv∗3−sv∗sv−ev∗ihsvsv∗evih∗−eviv∗ivev∗.From Lemma 1, the terms(39)k1aphiv∗sh∗3−sh∗sh−eh∗ivshsh∗ehiv∗−ehih∗iheh∗,k4apvih∗sv∗3−sv∗sv−ev∗ihsvsv∗evih∗−eviv∗ivev∗≤0.Therefore,V′<0 for all sh,eh,ih,rh,sv,ev,iv≠sh∗,eh∗,ih∗,rh∗,sv∗,ev∗,iv∗, implying that the endemic equilibrium is globally asymptotically stable if R0>1.


## 4. Analysis of Optimal Control Model

In this section, we formulate the optimal control model by modifying system (6) to an optimal control problem. We thus define some linear functions cit=1, for i=1,2,3. It is important to note that controls are fully effective when cit=1 and not effective when cit=0. The forces of infection ξh and xiv, which correspond to the human and vector population, respectively, are reduced by the factor 1−c1, where c1 measures the level of success obtained due to the effort of educating people on the dangers of exposing their skin, and encouraging them to wear long sleeves and long pants during the day to minimize tsetse fly-human contacts. The factor c2 represents the effort of treatment to control the disease, and the factor c3 also represents the effort of using insecticides to ensure that the breeding sites of the tsetse fly are minimized. Hence, taking into account the assumptions and extensions made, we try to find the most effective strategy that reduces the HAT infection in the population at a very minimum cost. With the use of bounded Lebesgue measurable control, we define the objective function to be minimized as(40)Jc1,c2,c3=∫0tFM1eh+M2ih+M3ev+M4iv+12k1c12+12k2c22+12k3c32dt.Thus, the dynamics of the controls that minimizes the objective function is given by(41)dshdt=πh+ρrh−1−c1aphivsh−μhsh,dehdt=1−c1aphivsh−εeh−μheh,dihdt=εeh−μhih−ωih−c2τih,drhdt=c2τih−ρrh−μhrh,dsvdt=πv−c3μvsv−1−c1apvihsv,devdt=1−c1apvihsv−c3μvev−σev,divdt=σev−c3μviv,subject to the initial conditions sh≥0,eh≥0,ih≥0,rh≥0,sv≥0,ev≥0,andiv≥0. The associated effective reproduction number for (41) denoted by RE is obtained as(42)RE=a2ε1−c12phπhπvσc3μhε+μhμv2σ+c3μvμh+c2τ+ω=Rc.The goal is to minimize the exposed and infectious human populationseh,ih, the exposed and infectious vector populations ev,iv, and the cost of implementing the control by the use of possible ci,i=1,2,3. The functional objective includes the social cost which relates to the resources that are needed for educating people on personal protection 1/2k1c12, the application of treatment 1/2k2c22, and spraying of tsetse fly operations 1/2k3c32. The quantities M1 and M2, respectively, represent the associated cost with minimizing the exposed and infected human population, while M3 and M4 also represent the cost associated with minimizing the exposed and infected vectors, respectively. The quantity tF is the time period of intervention. As explained in [15], the costs corresponding to M1eh,M2ih,M3ev, and M4iv are linear, while the cost control functions 1/2k1c12,1/2k2c22, and 1/2k3c32 should be nonlinear and take a quadratic form. Therefore, we seek to minimize the objective function over the given time interval 0,tF. Pontryagin’s maximum principle is used to solve this optimal control problem and the derivation of the necessary conditions. The Lagrangian of the optimal control problem is given by(43)L=M1eh+M2ih+M3ev+M4iv+12k1c12+12k2c22+12k3c32.To determine the Lagrangian minimum value, we define the Hamiltonian,H, for the control problem as(44)H=M1eh+M2ih+M3ev+M4iv+12k1c12+12k2c22+12k3c32+λshdshdt+λehdehdt+λihdihdt+λrhdrhdt+λsvdsvdt+λevdevdt+λivdivdt,where λsh,λeh,λih,λrh,λsv,λev,andλiv are adjoint variables or costate variables. The differential equations of adjoint variables are obtained by taking the partial derivatives of the Hamiltonian equation with respect to the state variables, which gives(45)dλshdt=λshμh+λsh−λeh1−c1aphiv,dλehdt=−M1+λeh−λihε+λehμh,dλihdt=−M2+λih−λrhc2τ+λihω+μh+λsv−λev1−c1apvsv,dλrhdt=λrh−λshρ+λrhμh,dλsvdt=λsv−λev1−c1apvih+λsvμvc3,dλevdt=−M3+λev−λivσ+λevμvc3,dλivdt=−M4+λsh−λeh1−c1aphsh+λivμvc3.Theorem 6.
Given the optimal controlsc1∗,c2∗,c3∗ and the solutions sh,eh,ih,rh,sv,ev,iv of the corresponding state equations (41) and (40) which minimize Jc1,c2,c3 over the region Ω, then there exist adjoint variables λsh,λeh,λih,λrh,λsv,λev,λiv satisfying(46)−dλidt=∂H∂i,i∈sh,eh,ih,rh,sv,ev,iv,and the optimal solution c1∗,c2∗,c3∗ is given by(47)c1∗=min1,max0,c1^,c2∗=min1,max0,c2^,c3∗=min1,max0,c3^.Proof. The Pontryagin’s maximum principle described in [16, 17] is applied. Corollary 4.1 in [17] shows the existence of an optimal control due to the convexity of the integrand J with respect to c1,c2,c3, and Lipschitz property of the state system with respect to the state variables. By using the optimal conditions(48)∂H∂c1=0,∂H∂c2=0,∂H∂c3=0,we obtain,(49)∂H∂c1=k1c1+λsh−λehaphivsh+λsv−λevapvihsv=0,∂H∂c2=k2c2+λrh−λihτih=0,∂H∂c3=k3c3−λsvμvsv+λevμvev+λivμviv=0.Solving (49), we have(50)c1^=λeh−λshaphivsh+λev−λsvapvihsvk1,c2^=λih−λrhτihk2,c3^=λsvμvsv+λevμvev+λivμvivk3.As stated earlier, the lower and upper boundaries for the control parameters are 0 and 1, respectively. Ifc1^,c2^,c3^<1, then c1=c2=c3=0 and if c1^,c2^,c3^>1, then c1=c2=c3=1, otherwise c1=c1^,c2=c2^,c3=c3^. Therefore, for the control parameters c1∗,c2∗,c3∗, we obtain the optimum value of the function Jc1,c2,c3.


### 4.1. Optimal Control Simulations

The Octave programming language is used to simulate the optimal control model using the set of parameters obtained from previously reported studies and datasets, which have been cited. Some of these parameters are assumed for the sake of illustrations. Table2 represents the values of the model parameters used for simulations. The following initial conditions were considered:(51)sh0=30,eh0=7,ih0=2,rh0=0,sv0=40,ev0=10,iv0=3,and the weight constants were assumed to be(52)M1=1,M2=2,M3=2,M4=2,k1=2,k2=10,k3=5.Table 2
Parameters values used for simulations.
Parameter
Value
Reference
π
h
0.000215/day
[14]
π
v
0.07/day
[14]
p
h
0.62
[6]
p
v
0.065
[6, 15]
a
Varying
Assumed
σ
0.001
Assumed
ε
0.083
[18]
ω
0.004
[3]
ρ
0.02
[6]
τ
0.125
[3]
μ
h
0.00044
Assumed
μ
v
0.034
[15]Figures2 and 3 represent the control profiles at different values of c1,c2, and c3, while the rest of the plots are the graphs of infectious human and vector population plotted against time in days and they represent the effect of optimal controls c1,c2, and c3 in reducing the number of individuals infected. From Figure 4, we observe that the use of treatment and insecticides only has a significant impact in reducing the number of infectious individuals and they show that this strategy is effective to control tsetse flies and infected human populations. In Figure 5, we observe that the use of education and insecticides reduce the number of infectious individuals but the results depicted in Figure 5(a) shows that this strategy is not effective and efficient to control the infectious human population. From Figure 6, we observe that the use of education and treatment reduces the number of infectious individuals but the results from Figure 6(b) shows that this strategy is not effective and efficient to control the population of infectious tsetse flies. Lastly, the results depicted from Figure 7 shows that the strategy of using education, treatment, and insecticides is very efficient and effective to reduce the number of infected individuals. Therefore, the use of education, treatment, and insecticides simultaneously is very efficient and effective to eliminate HAT in Africa.Figure 2
(a) Control profile whenc1=0, c2≠0, and c3≠0 at Rc = 0.0075. (b) Control profile when c1≠0,c2=0, and c3≠0 at Rc = 0.00013.
(a)
(b)Figure 3
(a) Control profile whenc1≠0,c2≠0, and c3=0 at Rc = 0.044. (b) Control profile when c1≠0,c2≠0, and c3≠0 at Rc = 0.000075.
(a)
(b)Figure 4
Simulations of the model showing the efforts of treatment and insecticides only on infectious individuals atRc = 0.0075.
(a)
(b)Figure 5
Simulations of the model showing the efforts of education and insecticides only on infectious individuals atRc = 0.00013.
(a)
(b)Figure 6
Simulations of the model showing the efforts of education and treatment only on infectious individuals atRc = 0.044.
(a)
(b)Figure 7
Simulations of the model showing the efforts of education, treatment, and insecticides on infectious individuals atRc = 0.000075.
(a)
(b)In epidemiology, a reproduction number less than unity implies that the disease can be eradicated in the long run. Hence, choosing suitable parameters for the controlsc1,c2, and c3, it was observed that the effective reproduction numbers obtained for Figures 2–7 were 0.0075, 0.00013, 0.044, 0.000075, 0.0075, 0.00013, 0.044, and 0.000075, respectively. This shows that incorporating all the control measures, that is, educating individuals, giving treatment, and applying insecticides, is an effective method to help reduce the number secondary infections in the population which corresponds with eradicating the disease in the long run.


## 4.1. Optimal Control Simulations

The Octave programming language is used to simulate the optimal control model using the set of parameters obtained from previously reported studies and datasets, which have been cited. Some of these parameters are assumed for the sake of illustrations. Table2 represents the values of the model parameters used for simulations. The following initial conditions were considered:(51)sh0=30,eh0=7,ih0=2,rh0=0,sv0=40,ev0=10,iv0=3,and the weight constants were assumed to be(52)M1=1,M2=2,M3=2,M4=2,k1=2,k2=10,k3=5.Table 2
Parameters values used for simulations.
Parameter
Value
Reference
π
h
0.000215/day
[14]
π
v
0.07/day
[14]
p
h
0.62
[6]
p
v
0.065
[6, 15]
a
Varying
Assumed
σ
0.001
Assumed
ε
0.083
[18]
ω
0.004
[3]
ρ
0.02
[6]
τ
0.125
[3]
μ
h
0.00044
Assumed
μ
v
0.034
[15]Figures2 and 3 represent the control profiles at different values of c1,c2, and c3, while the rest of the plots are the graphs of infectious human and vector population plotted against time in days and they represent the effect of optimal controls c1,c2, and c3 in reducing the number of individuals infected. From Figure 4, we observe that the use of treatment and insecticides only has a significant impact in reducing the number of infectious individuals and they show that this strategy is effective to control tsetse flies and infected human populations. In Figure 5, we observe that the use of education and insecticides reduce the number of infectious individuals but the results depicted in Figure 5(a) shows that this strategy is not effective and efficient to control the infectious human population. From Figure 6, we observe that the use of education and treatment reduces the number of infectious individuals but the results from Figure 6(b) shows that this strategy is not effective and efficient to control the population of infectious tsetse flies. Lastly, the results depicted from Figure 7 shows that the strategy of using education, treatment, and insecticides is very efficient and effective to reduce the number of infected individuals. Therefore, the use of education, treatment, and insecticides simultaneously is very efficient and effective to eliminate HAT in Africa.Figure 2
(a) Control profile whenc1=0, c2≠0, and c3≠0 at Rc = 0.0075. (b) Control profile when c1≠0,c2=0, and c3≠0 at Rc = 0.00013.
(a)
(b)Figure 3
(a) Control profile whenc1≠0,c2≠0, and c3=0 at Rc = 0.044. (b) Control profile when c1≠0,c2≠0, and c3≠0 at Rc = 0.000075.
(a)
(b)Figure 4
Simulations of the model showing the efforts of treatment and insecticides only on infectious individuals atRc = 0.0075.
(a)
(b)Figure 5
Simulations of the model showing the efforts of education and insecticides only on infectious individuals atRc = 0.00013.
(a)
(b)Figure 6
Simulations of the model showing the efforts of education and treatment only on infectious individuals atRc = 0.044.
(a)
(b)Figure 7
Simulations of the model showing the efforts of education, treatment, and insecticides on infectious individuals atRc = 0.000075.
(a)
(b)In epidemiology, a reproduction number less than unity implies that the disease can be eradicated in the long run. Hence, choosing suitable parameters for the controlsc1,c2, and c3, it was observed that the effective reproduction numbers obtained for Figures 2–7 were 0.0075, 0.00013, 0.044, 0.000075, 0.0075, 0.00013, 0.044, and 0.000075, respectively. This shows that incorporating all the control measures, that is, educating individuals, giving treatment, and applying insecticides, is an effective method to help reduce the number secondary infections in the population which corresponds with eradicating the disease in the long run.


## 5. Conclusion

In this paper, we studied and analyzed the model for transmission of HAT, and determined the basic reproduction number. The local and global stabilities of disease-free equilibrium and endemic equilibrium were also proved. For the optimal control model, education, treatment, and insecticides as control measures were used to optimize the objective function defined by Equation (40). The numerical simulations of the optimal control model show that the best strategy to reduce the number of infected individuals is through the use of education, treatment, and insecticides. This is the effective and efficient method to eliminate the disease. Furthermore, the national authorities, nongovernmental organizations (NGOs), and stakeholders must not lose their interest in controlling the disease because neglecting this disease may cause rapid reoccurrence and much effect to the people who are at risk.



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*Source: 5293568-2018-11-22.xml*