# Optimal Control Techniques on a Mathematical Model for the Dynamics of Tungiasis in a Community

Authors: Jairos Kahuru; Livingstone S. Luboobi; Yaw Nkansah-Gyekye
Journal: International Journal of Mathematics and Mathematical Sciences
 (2017)

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## Abstract

Tungiasis is a permanent penetration of female sand flea“Tunga penetrans” into the epidermis of its host. It affects human beings and domestic and sylvatic animals. In this paper, we apply optimal control techniques to a Tungiasis controlled mathematical model to determine the optimal control strategy in order to minimize the number of infested humans, infested animals, and sand flea populations. In an attempt to reduce Tungiasis infestation in human population, the control strategies based on personal protection, personal treatment, educational campaign, environmental sanitation, and insecticidal treatments on the affected parts as well as on animal fur are considered. We prove the existence of optimal control problem, determine the necessary conditions for optimality, and then perform numerical simulations. The numerical results showed that the control strategy comprises all five control measures and that which involves the three control measures of insecticide control, insecticidal dusting on animal furs, and environmental hygiene has the significant impact on Tungiasis transmission. Therefore, fighting against Tungiasis infestation in endemic settings, multidimensional control process should be employed in order to achieve the maximum benefits.


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## Body

## 1. Introduction

Tungiasis is a skin disease caused by the sand flea“Tunga penetrans”; the disease is endemic in some poor resource communities where various domestic and sylvatic animals act as reservoirs for this zoonosis [1]. The flea infestation is associated with poverty and occurs in many resource-poor communities in the Caribbean, South America, and sub-Saharan Africa [2]. Transmission of Tungiasis is strictly by infestation of humans and animal reservoirs by“Tunga penetrans” when they are in contact with sandy soil in which female fleas are present or when in contact with infested animal reservoirs as it is known that the animal reservoirs harbor the fleas [3]. Tungiasis results in significant morbidity, manifesting itself in a number of symptoms such as severe local inflammation, autoamputation of digits, deformation and loss of nails, formation of fissures and ulcers, gangrene, and walking difficulties [4]. Moreover it may result into secondary infection caused by transmission of blood-borne pathogens such as hepatitis B and C virus and possibly also HIV/AIDS when a single nonsterile instrument is used to remove the jiggers from different affected individuals [5].Mathematical models have played a major role in increasing understanding of the underlying mechanisms which influence the spread of the diseases and provide guidelines as to how the spread can be controlled [6, 7]. Optimal control theory is a powerful mathematical tool which makes the decision involving complex dynamical systems. It is a standard method for solving dynamic optimization problems, when those problems are expressed in continuous time [8]. Optimal control theory was developed by the Russian mathematician Lev S. Pontryagin (1908–1988) and his coworkers with the formulation and proof of the Pontryagin Maximum Principle (Pontryagin et al., 1962). Optimal control is the process of determining control and state trajectories for a dynamic system over a period of time to minimize a performance index [9]. Optimal control problem is represented by a set of differential equations describing the paths of the control variables that minimize the cost functional and has been used successfully to make decisions involving biological or medical models [10]. The formulation of an optimal control problem requires a mathematical model of the system to be controlled, a specification of the performance index (cost function), and a specification of all boundary conditions on states and constraints to be satisfied by states and controls [11]. Pontryagin’s maximum (or minimum) principle of optimal control gives the fundamental necessary conditions for a controlled trajectory to be optimal [12]. The principle technique is to transform the constrained dynamic optimization problem into an unconstrained problem, by allowing each of the adjoint variables to correspond to each of the state variables accordingly and combining the results with the objective functional [13]. The resulting function is known as the Hamiltonian, which is used to solve a set of necessary conditions that an optimal control and corresponding state variables must satisfy. The necessary conditions are the optimality solutions and adjoint equations which form the optimality system. The optimality system consists of the state system and adjoint system with initial and transversal conditions together with characterization of optimal control.To the best of our knowledge Tungiasis dynamical model with application of optimal control technique has not been done. Therefore we are going to refer to other infectious diseases with similar characteristics where the optimal control theory has been applied. Bonyah et al. [20] applied optimal control theory to a Buruli ulcer model that takes into account human, water bug, and fish populations as well asMicrobacterium ulcerans in the environment. The control measures were applied on mass treatment, insecticide, and mass education to minimize the number of infected hosts, vectors, and infected fishes. The optimality system was determined and computed numerically for several scenarios. The results showed that the combination of all the control measures, mass treatment, insecticide, and mass education, is capable of helping reduce the number of infected humans, water bugs, small fishes, andMycobacterium ulcerans in the environment. Isere et al. [21] developed the optimal control model that includes two time dependent control functions with one minimizing the contact between the susceptible and the bacteria and the others, the population of bacteria in water. The results from the numerical solutions showed that increasing the susceptible pool and the infected populations above some threshold values were responsible for reducing cholera epidemic and the difference between the growth rate and the loss rate of the bacteria played a huge role in the outbreak of the disease. Devipriya and Kalaivani [22] conducted the study on “Optimal Control of Multiple Transmission of Water-Borne Disease.” A controlled SIWR model was considered. The control measures represented an immune boosting and pathogen suppressing drugs. Their objective function was based on a combination of minimizing the number of infected individuals and the cost of the drugs dose. The numerical results have shown that both the vaccines resulted in minimizing the number of infected individuals and at the same time in a reduction of the budget related to the disease.In this paper, the Tungiasis dynamical model with control measures is presented and a detailed qualitative optimal control model that minimizes the number of infested individuals (humans and animals) and sand fleas with minimal cost of implementing the control measures is developed. We establish the proof for existence of the optimal control and analyze the optimal control problem in order to determine the necessary conditions for optimality using the Pontryagin’s maximum principle (Pontryagins et al., 1962). We then determine numerically the optimality system for several scenarios. Our paper is arranged as follows. In Section2, we formulate an optimal control model. In Section 3, we analyze the optimal control model by determining the conditions for existence of optimal control and the necessary conditions for optimality. In Section 4, we carry out numerical simulations and discussion of the results and Section 5 is the conclusion.


## 2. Formulation of Optimal Control Problem

We formulate an optimal control model for Tungiasis disease in order to derive five optimal control measures with minimal implementation cost to eradicate the disease after a defined period of time. We employ the control effortswi(t) in human, animal reservoirs, and adult flea populations and 1-wi(t) is the failure rate for the control efforts wi(t) for i=1,…,5. We let w1(t) be the effort of controlling the flea infested soil environment with insecticides spraying, w2(t) be the efforts of controlling the flea infested animal reservoir through dusting them with ant-flea compounds, w3(t) be the efforts of controlling the transmission from flea infested environment to susceptible animals (this can be achieved by environmental hygiene and cementing the floors), w4(t) be the efforts of controlling the transmission from flea infested animals to susceptible humans (this can be achieved by educating people not to live with animals in the same quarters or sharing common resting places), and w5(t) be the efforts of controlling transmission from flea infested environment to susceptible humans (this can be achieved by environmental hygiene, cementing the floors, covering of feet with solid shoes, and application of plant based repellent (Zanzarin) within the time interval of [0,T]). Therefore we assume that the mortality rate of jigger fleas in the soil environment is increased by the factor μF+w1(t), on-host spraying of infested animals will reduce the shedding rate εA of adult jigger fleas into the environment by a fraction 1-w2(t), and the animal to animal effective contact rate ρA is reduced at the same fraction 1-w2(t) because spraying insecticides on animal fur will reduce the transmission of infestation within animal population. The transmission rate from the soil environment to animal hosts is reduced by the factor 1-w3(t), the factor 1-w4(t) reduces the transmission from severely infested animal reservoirs to susceptible humans, and the factor 1-w5(t) reduces the transmission from flea infested soil environment to susceptible humans. Here, we consider the model developed by Kahuru et al. [19] whereby we add a distinct epidemiological compartment TH which represents human beings under treatment and incorporate the five control measures w1, w2, w3, w4, and w5 as defined above. In the submodel of human population, the total human population NH is subdivided into susceptible population SH mildly infested population IHl, the severely infested population IHh, and the human treatment class denoted by TH; therefore we have NH=SH+IHl+IHh+TH. We assume that the humans are recruited into SH through birth by the adults at a rate bH. Individuals in class SH acquire infestation from the severely infested animal reservoirs IAh and move to class IHl at a rate 1-w4(t)ρAHIAh/NH and may also acquire infestation from the flea infested soil environment and move to class IHh at a rate 1-w5(t)αEHβEHrFFE/k+FE. IHl may as well acquire infestation from the flea infested soil environment and progresses to class IHh at a rate 1-w5(t)αEHβEHrFFE/k+FE. Classes IHh and IHl seek treatment at the respective rates p1 and p2 and join TH class, and eventually the treated individuals revert back to join SH at a progression rate ω. Individuals in compartments SH, IHl, and TH suffer a natural mortality rate μH and for the compartment IHh they suffer a natural mortality at a rate μH and the disease induced mortality at a rate σH. In the submodel of animal reservoir population, the total animal reservoir population NA is subdivided into susceptible population SA mildly infested population IAl and the severely infested population IAh; therefore we have NA=SA+IAl+IAh. We assume that the animals are recruited into SA through birth by the adults at a rate bA. Individuals in class SA acquire infestation from the severely infested animal reservoirs IAh and move to IAl at a rate 1-w2(t)ρAIAh/NA and also may acquire infestation from the flea infested environment and move to class IAh at a rate 1-w3(t)αEAβEArFFE/k+FE. IAl may as well acquire infestation from the flea infested soil environment and progresses to class IAh at a rate 1-w3(t)αEAβEArFFE/k+FE. Individuals in compartments SA and IAl suffer a natural mortality rate μA and for the compartment IAh they suffer a natural mortality at a rate μA and a disease induced mortality at a rate σA. The submodel of environmental component consists of two compartments, a compartment of larvae denoted by LE and a compartment of adult sand fleas denoted by FE. The larvae population are recruited into LE through shedding of jigger eggs by IHh and IAh at a constant rate δe; therefore we have the total contribution of δeIHh and δeIAh from infested humans and animal reservoir populations, respectively. The larvae in compartment LE mature into adult jigger fleas at a maturation rate γL and undergo a natural death at a rate μL. The adult jigger flea population are recruited into FE through maturation by larvae at a rate γL and from infested animal reservoirs who contributes the fleas into the soil environment at a rate 1-w2(t)εA. The adult fleas leave the compartment FE when they attack the hosts at a rate rFFE/k+FE and when they undergo a natural death at a rate μF and the additional death due to insecticides control at a rate w1(t), therefore we have the flea total death rate of μF+w1(t)FE.The variables and parameters that describe the flow rates between compartments are given, respectively, in Notations. The possible interactions between humans, animal reservoirs, and flea infested environment with control measures are presented by the model flow diagram in Figure1 and the differential equations describing the model are given in system (1).Figure 1
The flow chart showing the dynamics of Tungiasis with control measures incorporated.


### 2.1. Model Flow Chart with Control Measures

The dynamical model with submodels of humans, animal reservoirs, and flea infested environment that incorporates time dependent control measures is presented hereunder.


### 2.2. Equations of the Model with Control Measures

From compartmental flow chart in Figure1 the nonlinear differential equations representing the controlled system of Tungiasis dynamical model are given by(1)DynamicsinhumanpopulationdSHtdt=bHNH+ωTH-1-w5tψEHSH-1-w4tψAHSH-μHSHdIHltdt=1-w4tψAHSH-1-w5tψEHIHl-p1+μHIHldIHhtdt=1-w5tψEHSH+1-w5tψEHIHl-p2+μH+σHIHhdTHtdt=p1IHl+p2IHh-ω+μHTHDynamicsinanimalreservoirpopulationdSAtdt=bANA-1-w2tψAASA-1-w3tψEASA-μASAdIAltdt=1-w2tψAASA-1-w3tψEAIAl-μAIAldIAhtdt=1-w3tψEASA+1-w3tψEAIAl-μA+σAIAhDynamicsinjiggerfleapopulationdLEtdt=δe1-LEKIHh+IAh-γL+μLLEdFEtdt=γLLE+1-w2tεAIAh-μF+w1tFE-rFFEk+FEwith initial condition(2)SH0>0,IHl0≥0,IHh0≥0,TH0≥0,SA0>0,IAl0≥0,IAh0≥0,LE0≥0,FE0>0,w1t≥0,w2t≥0,w3t≥0,w4t≥0,w5t≥0,where(3)ψAA=ρAIAhNA,ψEA=αEAβEArFFEk+FE,ψEH=αEHβEHrFFEk+FE,ψAH=ρAHIAhNH0≤w1t,w2t,w3t,w4t,w5t≤1.


### 2.3. Optimal Control Problem for Tungiasis Epidemic

In this section, we present the optimal control problem considering the performance index and the controlled model equations with initial state conditions whereby the goal is to find the optimal levels of the control measures needed to minimize the number of infested humans, animal reservoirs, and fleas as well as the cost of implementing the control strategies (wi for i=1,…,5). The objective functional Jw to be minimized is given by(4)Jw=min∫0TR1IHh+R2IHl+R3IAh+R4IAl+R5FE+12∑i=15Ziwi2dt.The terms IHh, IHl, IAh, IAl, FE in the objective functional Jw are the number of infested populations that need to be minimized. The terms Z1w12 represents the cost of off-host insecticides spraying, Z2w22 represents the cost of on-host spraying of infested domestic animals, Z3w32 represent the cost of implementing environmental hygiene, Z4w42 represents the cost of education campaign, and Z5w52 represents the cost of personal protection. R1, R2, R3, R4, R5, Z1, Z2, Z3, Z4, and Z5 are positive balancing coefficients (weights) which regularize the optimal control. Quadratic expressions of the controls are included to indicate nonlinear costs potentially arising at high intervention levels [23].The optimal control problem is formulated to obtain the minimum number of infested populationsIHh, IHl, IAh, IAl, FE under minimum cost. Therefore the objective function in (4) is minimized subject to the model equations in (1). We seek the optimal controls w1∗, w2∗, w3∗, w4∗, w5∗ such that(5)Jw1∗,w2∗,w3∗,w4∗,w5∗=min⁡Jw1,w2,w3,w4,w5∣wi∈Πfori=1,…,5subject  to  the  dynamical  system  in1and  the  control  set  given  byΠ=w1,w2,w3,w4,w5such  that;0≤wi≤1,fori=1,…,5;∀t∈0,T.


## 2.1. Model Flow Chart with Control Measures

The dynamical model with submodels of humans, animal reservoirs, and flea infested environment that incorporates time dependent control measures is presented hereunder.


## 2.2. Equations of the Model with Control Measures

From compartmental flow chart in Figure1 the nonlinear differential equations representing the controlled system of Tungiasis dynamical model are given by(1)DynamicsinhumanpopulationdSHtdt=bHNH+ωTH-1-w5tψEHSH-1-w4tψAHSH-μHSHdIHltdt=1-w4tψAHSH-1-w5tψEHIHl-p1+μHIHldIHhtdt=1-w5tψEHSH+1-w5tψEHIHl-p2+μH+σHIHhdTHtdt=p1IHl+p2IHh-ω+μHTHDynamicsinanimalreservoirpopulationdSAtdt=bANA-1-w2tψAASA-1-w3tψEASA-μASAdIAltdt=1-w2tψAASA-1-w3tψEAIAl-μAIAldIAhtdt=1-w3tψEASA+1-w3tψEAIAl-μA+σAIAhDynamicsinjiggerfleapopulationdLEtdt=δe1-LEKIHh+IAh-γL+μLLEdFEtdt=γLLE+1-w2tεAIAh-μF+w1tFE-rFFEk+FEwith initial condition(2)SH0>0,IHl0≥0,IHh0≥0,TH0≥0,SA0>0,IAl0≥0,IAh0≥0,LE0≥0,FE0>0,w1t≥0,w2t≥0,w3t≥0,w4t≥0,w5t≥0,where(3)ψAA=ρAIAhNA,ψEA=αEAβEArFFEk+FE,ψEH=αEHβEHrFFEk+FE,ψAH=ρAHIAhNH0≤w1t,w2t,w3t,w4t,w5t≤1.


## 2.3. Optimal Control Problem for Tungiasis Epidemic

In this section, we present the optimal control problem considering the performance index and the controlled model equations with initial state conditions whereby the goal is to find the optimal levels of the control measures needed to minimize the number of infested humans, animal reservoirs, and fleas as well as the cost of implementing the control strategies (wi for i=1,…,5). The objective functional Jw to be minimized is given by(4)Jw=min∫0TR1IHh+R2IHl+R3IAh+R4IAl+R5FE+12∑i=15Ziwi2dt.The terms IHh, IHl, IAh, IAl, FE in the objective functional Jw are the number of infested populations that need to be minimized. The terms Z1w12 represents the cost of off-host insecticides spraying, Z2w22 represents the cost of on-host spraying of infested domestic animals, Z3w32 represent the cost of implementing environmental hygiene, Z4w42 represents the cost of education campaign, and Z5w52 represents the cost of personal protection. R1, R2, R3, R4, R5, Z1, Z2, Z3, Z4, and Z5 are positive balancing coefficients (weights) which regularize the optimal control. Quadratic expressions of the controls are included to indicate nonlinear costs potentially arising at high intervention levels [23].The optimal control problem is formulated to obtain the minimum number of infested populationsIHh, IHl, IAh, IAl, FE under minimum cost. Therefore the objective function in (4) is minimized subject to the model equations in (1). We seek the optimal controls w1∗, w2∗, w3∗, w4∗, w5∗ such that(5)Jw1∗,w2∗,w3∗,w4∗,w5∗=min⁡Jw1,w2,w3,w4,w5∣wi∈Πfori=1,…,5subject  to  the  dynamical  system  in1and  the  control  set  given  byΠ=w1,w2,w3,w4,w5such  that;0≤wi≤1,fori=1,…,5;∀t∈0,T.


## 3. Model Analysis

The basic framework of an optimal control is to prove the existence of the optimal control and then characterize the optimal control through optimality system [24]. Given the optimal control problem in (5) we prove the existence of optimal control problem using the approach by Fleming and Rishel [25] and by Lukes [26] and then characterizing it for optimality.


### 3.1. The Existence of Optimal Control Problem

To prove the existence of optimal control the following conditions should be satisfied:(i)
The set of controls and corresponding state variables are nonempty.(ii)
The control set is convex and closed.(iii)
The right-hand side of the state system is bounded by a linear function in the state and control variables.(iv)
The integrand of the objective functional is convex.(v)
The integrand of the objective functional is bounded below byτ1∑i=15wi2ϖ/2-τ2, where τ1,τ2>0 and ϖ>1.Given that, SH, IHl, IHh, TH, SA, IAl, IAh, LE,  and  FE are the state variables of the controlled model system (1) and w1, w2, w3, w4, w5 are the control variables. We state and prove Theorem 1 as follows.Theorem 1.
There exists an optimal control variables setw∗=w1∗,w2∗,w3∗,w4∗,w5∗∈Π such that (6)Jw1∗,w2∗,w3∗,w4∗,w5∗=minJw1,w2,w3,w4,w5∣wi∈Πfori=1,…,5.Proof.
Using the results in Fleming and Rishel [25] and by Lukes [26], the control and the state variables are nonnegative values. In this minimization problem, the necessary convexity of the objective functional in w1, w2, w3, w4, w5 is satisfied. The set of admissible Lebesgue measurable control variables w1,w2,w3,w4,w5∈Π is also convex and closed by the definition. The optimal system is bounded which determines the compactness needed for the existence of optimal control. In order to verify this argument we use the approach adopted by Sadiq et al. [27] and Namawejje et al. [28], whereby system (1) is put in the following form: 7Y′=BY+GY,where(7a)Y=SH,IHl,IHh,TH,SA,IAl,IAh,LE,FET.
B is the 9×9 matrix(7b)GY=bHNH-1-w5tψEHSH-1-w4tψAHSH1-w4tψAHSH-1-w5tψEHIHl1-w5tψEHSH+1-w5tψEHIHl0bANA-1-w2tψAASA-1-w3tψEASA1-w2tψAASA-1-w3tψEAIAl1-w3tψEASA+1-w3tψEAIAl00(7c)B=-μH00ω000000-Ψ1000000000-Ψ20000000p1p2-Ψ3000000000-μA000000000-μA000000000-μA+σA0000δe000δe-Ψ50000000Ψ4γL-Ψ6,where(8)Ψ1=p1+μH,Ψ2=p2+μH+σH,Ψ3=ω+μH,Ψ4=1-w2εA,Ψ5=δeKIHh+IAh+γL+μL,Ψ6=μF+w1+rFk+FEand Y′ denotes the derivative of Y with respect to time t. System (4) is a nonlinear system with a bounded coefficient. We set7dRY=BY+GY.The second term on the right-hand side of 7d, GY, satisfies(9)GY1-GY2≤H1S1Ht-S2Ht+H2I1Hlt-I2Hlt+H3I1Hht-I2Hht+H4T1Ht-T2Ht+H5S1At-S2At+H6I1Alt-I2Alt+H7I1Aht-I2Aht+H8L1Et-L2Et+H9F1Et-F2Et≤HS1Ht-S2Ht+I1Hlt-I2Hlt+I1Hht-I2Hht+T1Ht-T2Ht+S1At-S2At+I1Alt-I2Alt+I1Aht-I2Aht+L1Et-L2Et-F1Et+F2Et,where the positive constant H=max⁡Hifori=1,…,9 is independent of the state variables. Also we have RY1-RY2≤HY1-Y2, where H=∑i=19Hi+M<∞. So, it follows that the function R is uniformly Lipchitz continuous. From the definition of control variables and nonnegative initial conditions we can see that a solution of the system (1) exists [29].
The integrand in the objective functional (4) which is given by the following equation Lt,y,w=R1IHh+R2IHl+R3IAh+R4IAl+R5FE+(1/2)∑i=15Ziwi2 is convex in the control set Π.
We must verify the condition that there exist a constantϖ>1 and positive numbers τ1 and τ2 such that(10)Lt,y,w≥τ1∑i=15wi2ϖ/2-τ2,R1IHh+R2IHl+R3IAh+R4IAl+R5FE+12∑i=15Ziwi2≥12∑i=15Ziwi212∑i=15Ziwi2≥τ1∑i=15wi2ϖ/2-τ2.The last condition is satisfied when ϖ=2, τ2>0, and τ1=min⁡Z1,Z2,Z3,Z4,Z5. This ends the proof.


### 3.2. Determination of the Necessary Conditions for Optimality

The necessary conditions include the optimality solutions and the adjoint equations that an optimal control must satisfy which come from Pontryagin’s maximum principle (Pontryagin’s et al., 1962). This principle converts systems (1) and (4) into a problem of minimizing pointwise Hamiltonian function H, which is formed by allowing each of the adjoint variables to correspond to each of the state variables accordingly and combining the results with the objective functional [21]. The resulting equation is as given by(11)Ht,y,w,λ=R1IHh+R2IHl+R3IAh+R4IAl+R5FE+12∑i=15Ziwi2+∑i=19λiQi,where λi for i=1,…,9 are the adjoint functions associated with the state equations in (1),Qi for i=1,…,9 is the right-hand side of the differential equations of ith state variable in system (1).The expanded form of Hamiltonian function in (11) is given by(12)Ht,y,w,λ=R1IHh+R2IHl+R3IAh+R4IAl+R5FE+12∑i=15Ziwi2+λ1bHNH+ωTH-1-w5ψEHSH-1-w4ψAHSH-μHSH+λ21-w4ψAHSH-1-w5ψEHIHl-p1IHl-μHIHl+λ31-w5ψEHSH+1-w5ψEHIHl-p2IHh-μH+σHIHh+λ4p1IHl+p2IHh-ωTH-μHTH+λ5bANA-1-w2ψAASA-1-w3ψEASA-μASA+λ61-w2ψAASA-1-w3ψEAIAl-μAIAl+λ71-w3ψEASA+1-w3ψEAIAl-μA+σAIAh+λ8δe1-LEKIHh+IAh-γL+μLLE+λ9γLLE+1-w2εAIAh-μF+w1FE-rFFEk+FE.The optimality equations are obtained when taking the partial derivative of the Hamiltonian function H with respect to the control variables w1,w2,w3,w4,w5, respectively, and the time derivative of adjoint equation λ′ which is obtained by taking the negative partial derivative of H with respect to the model state variables yt such that λ′=-Hy.Theorem 2.
There exist an optimal control setw1∗,w2∗,w3∗,w4∗,w5∗ and their corresponding state solutions SH∗, IHl∗, IHh∗, TH∗, SA∗, IAl∗, IAh∗, LE∗, and FE∗ that minimize Jw1,w2,w3,w4,w5, and therefore there exist adjoint functions λ1,λ2,…,λ9 such that (13)dλ1dt=λ1μH-1-w4tλ2-λ1ψAH-1-w5tλ3-λ1ψEHdλ2dt=1-w5tψEHλ2-λ3+λ2p1+μH-λ4p1-R2dλ3dt=λ3p2+μH+σH-λ8δe1-LEK-λ4p2-R1dλ4dt=λ4ω+μH-λ1ωdλ5dt=λ5μA-1-w2tλ6-λ5ψAA-1-w3tλ7-λ5ψEAdλ6dt=λ6-λ71-w3tψEA+λ6μA-R4dλ7dt=λ7μA+σA-1-w4tλ2-λ1ρAHSHNH-λ8δe1-LEK-λ9εA+λ6-λ5ρASANA1-w2t-R3dλ8dt=λ8IHh+IAhδeK+λ8γL+μL-λ9γLdλ9dt=λ9μF+w1t+rFkk+FE2-R5-χH1-w5tSHλ3-λ1+IHlλ3-λ2-χA1-w3tSAλ7-λ5+IAlλ7-λ6,where χH=αEHβEHrFk/k+FE2 and χA=αEAβEArFk/k+FE2 with transversality conditions, λiTfori=1,2,…,9=0, and the control variables w1∗,w2∗,w3∗,w4∗,w5∗ satisfy the following optimality conditions:(14)w1∗=minmax⁡0,λ9FEZ1,1,w2∗=minmax⁡0,ψAASAλ6-λ5+λ9εAIAhZ2,1,w3∗=minmax⁡0,ψEASAλ7-λ5+IAlλ7-λ6Z3,1,w4∗=minmax⁡0,ψAHSHλ2-λ1Z4,1,w5∗=minmax⁡0,ψEHSHλ3-λ1+IHlλ3-λ2Z5,1.Proof.
The differential equations for the adjoints are standard results from Pontryagin’s maximum principle (1962). Given the Hamiltonian function in (12) the adjoint equations can be easily computed by(15)dλ1dt=-dHdSH,dλ2dt=-dHdIHl,dλ3dt=-dHdIHh,dλ4dt=-dHdTH,dλ5dt=-dHdSA,dλ6dt=-dHdIAl,dλ7dt=-dHdIAh,dλ8dt=-dHdLE,dλ9dt=-dHdFE.Therefore, the adjoint system evaluated at optimal controls w1, w2, w3, w4, and w5 and the corresponding model state variables SH, IHl, IHh, TH, SA, IAl, IAh, LE, FE is as given by(16)dλ1dt=λ1μH-1-w4tλ2-λ1ψAH-1-w5tλ3-λ1ψEHdλ2dt=1-w5tψEHλ2-λ3+λ2p1+μH-λ4p1-R2dλ3dt=λ3p2+μH+σH-λ8δe1-LEK-λ4p2-R1dλ4dt=λ4ω+μH-λ1ωdλ5dt=λ5μA-1-w2tλ6-λ5ψAA-1-w3tλ7-λ5ψEAdλ6dt=λ6-λ71-w3tψEA+λ6μA-R4dλ7dt=λ7μA+σA-1-w4tλ2-λ1ρAHSHNH-λ8δe1-LEK-λ9εA+λ6-λ5ρASANA1-w2t-R3dλ8dt=λ8IHh+IAhδeK+λ8γL+μL-λ9γLdλ9dt=λ9μF+w1t+rFkk+FE2-R5-χH1-w5tSHλ3-λ1+IHlλ3-λ2-χA1-w3tSAλ7-λ5+IAlλ7-λ6,where χH=αEHβEHrFk/k+FE2 and χA=αEAβEArFk/k+FE2 with transversality conditions, λiTfori=1,2,…,9=0, and the characterization of optimal controls w1∗, w2∗, w3∗, w4∗, w5∗; that is, the optimality equations are based on the conditions:(17)∂H∂w1=∂H∂w2=∂H∂w3=∂H∂w4=∂H∂w5=0.Subject to (17) the optimality condition given the Lebesgue measurable control set Π=0≤wit≤1,fori=1,2,…,5andt∈[0,T], where the control variables w1, w2, w3, w4, w5 are measurable functions, is given by(18)∂H∂w1=Z1w1-λ9FE∂H∂w2=Z2w2+λ5ψAASA-λ6ψAASA-λ9εAIAh∂H∂w3=Z3w3+λ5ψEASA+λ6ψEAIAl-λ7ψEASA-λ7ψEAIAl∂H∂w4=Z4w4+λ1ψAHSH-λ2ψAHSH∂H∂w5=Z5w5+λ1ψEHSH+λ2ψEHIHl-λ3ψEHSH-λ3ψEHIHl.If we set ∂H/∂wi=0 at wi∗ the results are the same as in characterization in (14).
In order to satisfy the given bounds for the control functions (i.e.,0≤wi≤1 and t∈[0,T]) the optimal control is restricted to wi∗=min⁡max⁡0,wi,1. Therefore using the bounds for the controls w1, w2, w3, w4, w5 we have the following.
The bounds and the impact notation for the controlw1 are, respectively, given as(19)w1∗=λ9FE∗Z1if0≤λ9FE∗Z1≤1,0ifλ9FE∗Z1≤0,1ifλ9FE∗Z1≥1,w1∗=min⁡max⁡0,λ9FE∗Z1,1.Similar four-step arguments hold for optimal control schedules w2∗, w3∗, w4∗, and w5∗ in the same way as in (19) based on the characterization in (14).


### 3.3. The Optimality System

The optimality system consists of the state system and adjoint system with initial and transversal conditions together with characterization of optimal control. Any optimal control pair must satisfy this optimality system as indicated in (20) and (22), respectively.(20)dSHtdt=bHNH-1-min⁡max⁡0,ψEH∗SHλ3-λ1+IHlλ3-λ2Z5,1ψEH∗SH+ωTH-1-min⁡max⁡0,ψAH∗SHλ2-λ1Z4,1ψAH∗SH-μHSHdIHltdt=1-min⁡max⁡0,ψAH∗SHλ2-λ1Z4,1ψAH∗SH-1-min⁡max⁡0,ψEH∗SHλ3-λ1+IHlλ3-λ2Z5,1ψEH∗IHl-p1IHl-μHIHldIHhtdt=1-min⁡max⁡0,ψEH∗SHλ3-λ1+IHlλ3-λ2Z5,1ψEH∗SH+IHl-p2IHh-μH+σHIHhdTHtdt=p1IHl+p2IHh-ωTH-μHTHdSAtdt=bANA-1-min⁡max⁡0,ψAA∗SAλ6-λ5+λ9εAIAhZ2,1ψAA∗SA-1-min⁡max⁡0,ψEA∗SAλ7-λ5+IAlλ7-λ6Z3,1ψEA∗SA-μASAdIAltdt=1-min⁡max⁡0,ψAA∗SAλ6-λ5+λ9εAIAhZ2,1ψAA∗SA-1-min⁡max⁡0,ψEA∗SAλ7-λ5+IAlλ7-λ6Z3,1ψEA∗IAl-μAIAldIAhtdt=1-min⁡max⁡0,ψEA∗SAλ7-λ5+IAlλ7-λ6Z3,1ψEA∗SA+IAl-μA+σAIAhdLEtdt=δe1-LEKIHh+IAh-γL+μLLEdFEtdt=γLLE+1-min⁡max⁡0,ψAA∗SAλ6-λ5+λ9εAIAhZ2,1εAIAh-μF+min⁡max⁡0,λ9FEZ1,1FE-rFFEk+FEwith initial conditions(21)SH0>0,IHl0≥0,IHh0≥0,TH0≥0,SA0>0,IAl0≥0,IAh0≥0,LE0≥0,FE0>0,(22)dλ1dt=λ1μH-1-min⁡max⁡0,ψEA∗SAλ7-λ5+IAlλ7-λ6Z3,1ψAH∗λ2-λ1-1-min⁡max⁡0,ψEH∗SHλ3-λ1+IHlλ3-λ2Z5,1ψEH∗λ3-λ1dλ2dt=1-min⁡max⁡0,ψEH∗SHλ3-λ1+IHlλ3-λ2Z5,1ψEH∗λ2-λ3+λ2p1+μH-λ4p1-R2dλ3dt=λ3p2+μH+σH-R1-λ4p2-λ8δe1-LEKdλ4dt=λ4ω+μH-λ1ωdλ5dt=λ5μA-1-min⁡max0,ψAA∗SAλ6-λ5+λ9εAIAhZ2,1ψAA∗λ6-λ5-1-minmax0,ψEA∗SAλ7-λ5+IAlλ7-λ6Z3,1ψEA∗λ7-λ5dλ6dt=λ6-λ71-min⁡max0,ψEA∗SAλ7-λ5+IAlλ7-λ6Z3,1ψEA+λ6μA-R4dλ7dt=λ7μA+σA-1-min⁡max⁡0,ψAH∗SHλ2-λ1Z4,1λ2-λ1ρAHSHNH-λ9εA+λ6-λ5ρASANA1-min⁡max⁡0,ψAA∗SAλ6-λ5+λ9εAIAhZ2,1-λ8δe1-LEK-R3dλ8dt=λ8IHh+IAhδeK+λ8γL+μL-λ9γLdλ9dt=λ9μF+μF+min⁡max⁡0,λ9FEZ1,1+rFkk+FE2-R5-χH1-min⁡max⁡0,ψEH∗SH∗λ3-λ1+IHl∗λ3-λ2Z5,1SHλ3-λ1+IHlλ3-λ2-χA1-min⁡max⁡0,ψEA∗SAλ7-λ5+IAlλ7-λ6Z3,1SAλ7-λ5+IAlλ7-λ6,where χH=αEHβEHrFk/k+FE2 and χA=αEAβEArFk/k+FE2 with transversality conditions λiTfori=1,2,…,9=0.


## 3.1. The Existence of Optimal Control Problem

To prove the existence of optimal control the following conditions should be satisfied:(i)
The set of controls and corresponding state variables are nonempty.(ii)
The control set is convex and closed.(iii)
The right-hand side of the state system is bounded by a linear function in the state and control variables.(iv)
The integrand of the objective functional is convex.(v)
The integrand of the objective functional is bounded below byτ1∑i=15wi2ϖ/2-τ2, where τ1,τ2>0 and ϖ>1.Given that, SH, IHl, IHh, TH, SA, IAl, IAh, LE,  and  FE are the state variables of the controlled model system (1) and w1, w2, w3, w4, w5 are the control variables. We state and prove Theorem 1 as follows.Theorem 1.
There exists an optimal control variables setw∗=w1∗,w2∗,w3∗,w4∗,w5∗∈Π such that (6)Jw1∗,w2∗,w3∗,w4∗,w5∗=minJw1,w2,w3,w4,w5∣wi∈Πfori=1,…,5.Proof.
Using the results in Fleming and Rishel [25] and by Lukes [26], the control and the state variables are nonnegative values. In this minimization problem, the necessary convexity of the objective functional in w1, w2, w3, w4, w5 is satisfied. The set of admissible Lebesgue measurable control variables w1,w2,w3,w4,w5∈Π is also convex and closed by the definition. The optimal system is bounded which determines the compactness needed for the existence of optimal control. In order to verify this argument we use the approach adopted by Sadiq et al. [27] and Namawejje et al. [28], whereby system (1) is put in the following form: 7Y′=BY+GY,where(7a)Y=SH,IHl,IHh,TH,SA,IAl,IAh,LE,FET.
B is the 9×9 matrix(7b)GY=bHNH-1-w5tψEHSH-1-w4tψAHSH1-w4tψAHSH-1-w5tψEHIHl1-w5tψEHSH+1-w5tψEHIHl0bANA-1-w2tψAASA-1-w3tψEASA1-w2tψAASA-1-w3tψEAIAl1-w3tψEASA+1-w3tψEAIAl00(7c)B=-μH00ω000000-Ψ1000000000-Ψ20000000p1p2-Ψ3000000000-μA000000000-μA000000000-μA+σA0000δe000δe-Ψ50000000Ψ4γL-Ψ6,where(8)Ψ1=p1+μH,Ψ2=p2+μH+σH,Ψ3=ω+μH,Ψ4=1-w2εA,Ψ5=δeKIHh+IAh+γL+μL,Ψ6=μF+w1+rFk+FEand Y′ denotes the derivative of Y with respect to time t. System (4) is a nonlinear system with a bounded coefficient. We set7dRY=BY+GY.The second term on the right-hand side of 7d, GY, satisfies(9)GY1-GY2≤H1S1Ht-S2Ht+H2I1Hlt-I2Hlt+H3I1Hht-I2Hht+H4T1Ht-T2Ht+H5S1At-S2At+H6I1Alt-I2Alt+H7I1Aht-I2Aht+H8L1Et-L2Et+H9F1Et-F2Et≤HS1Ht-S2Ht+I1Hlt-I2Hlt+I1Hht-I2Hht+T1Ht-T2Ht+S1At-S2At+I1Alt-I2Alt+I1Aht-I2Aht+L1Et-L2Et-F1Et+F2Et,where the positive constant H=max⁡Hifori=1,…,9 is independent of the state variables. Also we have RY1-RY2≤HY1-Y2, where H=∑i=19Hi+M<∞. So, it follows that the function R is uniformly Lipchitz continuous. From the definition of control variables and nonnegative initial conditions we can see that a solution of the system (1) exists [29].
The integrand in the objective functional (4) which is given by the following equation Lt,y,w=R1IHh+R2IHl+R3IAh+R4IAl+R5FE+(1/2)∑i=15Ziwi2 is convex in the control set Π.
We must verify the condition that there exist a constantϖ>1 and positive numbers τ1 and τ2 such that(10)Lt,y,w≥τ1∑i=15wi2ϖ/2-τ2,R1IHh+R2IHl+R3IAh+R4IAl+R5FE+12∑i=15Ziwi2≥12∑i=15Ziwi212∑i=15Ziwi2≥τ1∑i=15wi2ϖ/2-τ2.The last condition is satisfied when ϖ=2, τ2>0, and τ1=min⁡Z1,Z2,Z3,Z4,Z5. This ends the proof.


## 3.2. Determination of the Necessary Conditions for Optimality

The necessary conditions include the optimality solutions and the adjoint equations that an optimal control must satisfy which come from Pontryagin’s maximum principle (Pontryagin’s et al., 1962). This principle converts systems (1) and (4) into a problem of minimizing pointwise Hamiltonian function H, which is formed by allowing each of the adjoint variables to correspond to each of the state variables accordingly and combining the results with the objective functional [21]. The resulting equation is as given by(11)Ht,y,w,λ=R1IHh+R2IHl+R3IAh+R4IAl+R5FE+12∑i=15Ziwi2+∑i=19λiQi,where λi for i=1,…,9 are the adjoint functions associated with the state equations in (1),Qi for i=1,…,9 is the right-hand side of the differential equations of ith state variable in system (1).The expanded form of Hamiltonian function in (11) is given by(12)Ht,y,w,λ=R1IHh+R2IHl+R3IAh+R4IAl+R5FE+12∑i=15Ziwi2+λ1bHNH+ωTH-1-w5ψEHSH-1-w4ψAHSH-μHSH+λ21-w4ψAHSH-1-w5ψEHIHl-p1IHl-μHIHl+λ31-w5ψEHSH+1-w5ψEHIHl-p2IHh-μH+σHIHh+λ4p1IHl+p2IHh-ωTH-μHTH+λ5bANA-1-w2ψAASA-1-w3ψEASA-μASA+λ61-w2ψAASA-1-w3ψEAIAl-μAIAl+λ71-w3ψEASA+1-w3ψEAIAl-μA+σAIAh+λ8δe1-LEKIHh+IAh-γL+μLLE+λ9γLLE+1-w2εAIAh-μF+w1FE-rFFEk+FE.The optimality equations are obtained when taking the partial derivative of the Hamiltonian function H with respect to the control variables w1,w2,w3,w4,w5, respectively, and the time derivative of adjoint equation λ′ which is obtained by taking the negative partial derivative of H with respect to the model state variables yt such that λ′=-Hy.Theorem 2.
There exist an optimal control setw1∗,w2∗,w3∗,w4∗,w5∗ and their corresponding state solutions SH∗, IHl∗, IHh∗, TH∗, SA∗, IAl∗, IAh∗, LE∗, and FE∗ that minimize Jw1,w2,w3,w4,w5, and therefore there exist adjoint functions λ1,λ2,…,λ9 such that (13)dλ1dt=λ1μH-1-w4tλ2-λ1ψAH-1-w5tλ3-λ1ψEHdλ2dt=1-w5tψEHλ2-λ3+λ2p1+μH-λ4p1-R2dλ3dt=λ3p2+μH+σH-λ8δe1-LEK-λ4p2-R1dλ4dt=λ4ω+μH-λ1ωdλ5dt=λ5μA-1-w2tλ6-λ5ψAA-1-w3tλ7-λ5ψEAdλ6dt=λ6-λ71-w3tψEA+λ6μA-R4dλ7dt=λ7μA+σA-1-w4tλ2-λ1ρAHSHNH-λ8δe1-LEK-λ9εA+λ6-λ5ρASANA1-w2t-R3dλ8dt=λ8IHh+IAhδeK+λ8γL+μL-λ9γLdλ9dt=λ9μF+w1t+rFkk+FE2-R5-χH1-w5tSHλ3-λ1+IHlλ3-λ2-χA1-w3tSAλ7-λ5+IAlλ7-λ6,where χH=αEHβEHrFk/k+FE2 and χA=αEAβEArFk/k+FE2 with transversality conditions, λiTfori=1,2,…,9=0, and the control variables w1∗,w2∗,w3∗,w4∗,w5∗ satisfy the following optimality conditions:(14)w1∗=minmax⁡0,λ9FEZ1,1,w2∗=minmax⁡0,ψAASAλ6-λ5+λ9εAIAhZ2,1,w3∗=minmax⁡0,ψEASAλ7-λ5+IAlλ7-λ6Z3,1,w4∗=minmax⁡0,ψAHSHλ2-λ1Z4,1,w5∗=minmax⁡0,ψEHSHλ3-λ1+IHlλ3-λ2Z5,1.Proof.
The differential equations for the adjoints are standard results from Pontryagin’s maximum principle (1962). Given the Hamiltonian function in (12) the adjoint equations can be easily computed by(15)dλ1dt=-dHdSH,dλ2dt=-dHdIHl,dλ3dt=-dHdIHh,dλ4dt=-dHdTH,dλ5dt=-dHdSA,dλ6dt=-dHdIAl,dλ7dt=-dHdIAh,dλ8dt=-dHdLE,dλ9dt=-dHdFE.Therefore, the adjoint system evaluated at optimal controls w1, w2, w3, w4, and w5 and the corresponding model state variables SH, IHl, IHh, TH, SA, IAl, IAh, LE, FE is as given by(16)dλ1dt=λ1μH-1-w4tλ2-λ1ψAH-1-w5tλ3-λ1ψEHdλ2dt=1-w5tψEHλ2-λ3+λ2p1+μH-λ4p1-R2dλ3dt=λ3p2+μH+σH-λ8δe1-LEK-λ4p2-R1dλ4dt=λ4ω+μH-λ1ωdλ5dt=λ5μA-1-w2tλ6-λ5ψAA-1-w3tλ7-λ5ψEAdλ6dt=λ6-λ71-w3tψEA+λ6μA-R4dλ7dt=λ7μA+σA-1-w4tλ2-λ1ρAHSHNH-λ8δe1-LEK-λ9εA+λ6-λ5ρASANA1-w2t-R3dλ8dt=λ8IHh+IAhδeK+λ8γL+μL-λ9γLdλ9dt=λ9μF+w1t+rFkk+FE2-R5-χH1-w5tSHλ3-λ1+IHlλ3-λ2-χA1-w3tSAλ7-λ5+IAlλ7-λ6,where χH=αEHβEHrFk/k+FE2 and χA=αEAβEArFk/k+FE2 with transversality conditions, λiTfori=1,2,…,9=0, and the characterization of optimal controls w1∗, w2∗, w3∗, w4∗, w5∗; that is, the optimality equations are based on the conditions:(17)∂H∂w1=∂H∂w2=∂H∂w3=∂H∂w4=∂H∂w5=0.Subject to (17) the optimality condition given the Lebesgue measurable control set Π=0≤wit≤1,fori=1,2,…,5andt∈[0,T], where the control variables w1, w2, w3, w4, w5 are measurable functions, is given by(18)∂H∂w1=Z1w1-λ9FE∂H∂w2=Z2w2+λ5ψAASA-λ6ψAASA-λ9εAIAh∂H∂w3=Z3w3+λ5ψEASA+λ6ψEAIAl-λ7ψEASA-λ7ψEAIAl∂H∂w4=Z4w4+λ1ψAHSH-λ2ψAHSH∂H∂w5=Z5w5+λ1ψEHSH+λ2ψEHIHl-λ3ψEHSH-λ3ψEHIHl.If we set ∂H/∂wi=0 at wi∗ the results are the same as in characterization in (14).
In order to satisfy the given bounds for the control functions (i.e.,0≤wi≤1 and t∈[0,T]) the optimal control is restricted to wi∗=min⁡max⁡0,wi,1. Therefore using the bounds for the controls w1, w2, w3, w4, w5 we have the following.
The bounds and the impact notation for the controlw1 are, respectively, given as(19)w1∗=λ9FE∗Z1if0≤λ9FE∗Z1≤1,0ifλ9FE∗Z1≤0,1ifλ9FE∗Z1≥1,w1∗=min⁡max⁡0,λ9FE∗Z1,1.Similar four-step arguments hold for optimal control schedules w2∗, w3∗, w4∗, and w5∗ in the same way as in (19) based on the characterization in (14).


## 3.3. The Optimality System

The optimality system consists of the state system and adjoint system with initial and transversal conditions together with characterization of optimal control. Any optimal control pair must satisfy this optimality system as indicated in (20) and (22), respectively.(20)dSHtdt=bHNH-1-min⁡max⁡0,ψEH∗SHλ3-λ1+IHlλ3-λ2Z5,1ψEH∗SH+ωTH-1-min⁡max⁡0,ψAH∗SHλ2-λ1Z4,1ψAH∗SH-μHSHdIHltdt=1-min⁡max⁡0,ψAH∗SHλ2-λ1Z4,1ψAH∗SH-1-min⁡max⁡0,ψEH∗SHλ3-λ1+IHlλ3-λ2Z5,1ψEH∗IHl-p1IHl-μHIHldIHhtdt=1-min⁡max⁡0,ψEH∗SHλ3-λ1+IHlλ3-λ2Z5,1ψEH∗SH+IHl-p2IHh-μH+σHIHhdTHtdt=p1IHl+p2IHh-ωTH-μHTHdSAtdt=bANA-1-min⁡max⁡0,ψAA∗SAλ6-λ5+λ9εAIAhZ2,1ψAA∗SA-1-min⁡max⁡0,ψEA∗SAλ7-λ5+IAlλ7-λ6Z3,1ψEA∗SA-μASAdIAltdt=1-min⁡max⁡0,ψAA∗SAλ6-λ5+λ9εAIAhZ2,1ψAA∗SA-1-min⁡max⁡0,ψEA∗SAλ7-λ5+IAlλ7-λ6Z3,1ψEA∗IAl-μAIAldIAhtdt=1-min⁡max⁡0,ψEA∗SAλ7-λ5+IAlλ7-λ6Z3,1ψEA∗SA+IAl-μA+σAIAhdLEtdt=δe1-LEKIHh+IAh-γL+μLLEdFEtdt=γLLE+1-min⁡max⁡0,ψAA∗SAλ6-λ5+λ9εAIAhZ2,1εAIAh-μF+min⁡max⁡0,λ9FEZ1,1FE-rFFEk+FEwith initial conditions(21)SH0>0,IHl0≥0,IHh0≥0,TH0≥0,SA0>0,IAl0≥0,IAh0≥0,LE0≥0,FE0>0,(22)dλ1dt=λ1μH-1-min⁡max⁡0,ψEA∗SAλ7-λ5+IAlλ7-λ6Z3,1ψAH∗λ2-λ1-1-min⁡max⁡0,ψEH∗SHλ3-λ1+IHlλ3-λ2Z5,1ψEH∗λ3-λ1dλ2dt=1-min⁡max⁡0,ψEH∗SHλ3-λ1+IHlλ3-λ2Z5,1ψEH∗λ2-λ3+λ2p1+μH-λ4p1-R2dλ3dt=λ3p2+μH+σH-R1-λ4p2-λ8δe1-LEKdλ4dt=λ4ω+μH-λ1ωdλ5dt=λ5μA-1-min⁡max0,ψAA∗SAλ6-λ5+λ9εAIAhZ2,1ψAA∗λ6-λ5-1-minmax0,ψEA∗SAλ7-λ5+IAlλ7-λ6Z3,1ψEA∗λ7-λ5dλ6dt=λ6-λ71-min⁡max0,ψEA∗SAλ7-λ5+IAlλ7-λ6Z3,1ψEA+λ6μA-R4dλ7dt=λ7μA+σA-1-min⁡max⁡0,ψAH∗SHλ2-λ1Z4,1λ2-λ1ρAHSHNH-λ9εA+λ6-λ5ρASANA1-min⁡max⁡0,ψAA∗SAλ6-λ5+λ9εAIAhZ2,1-λ8δe1-LEK-R3dλ8dt=λ8IHh+IAhδeK+λ8γL+μL-λ9γLdλ9dt=λ9μF+μF+min⁡max⁡0,λ9FEZ1,1+rFkk+FE2-R5-χH1-min⁡max⁡0,ψEH∗SH∗λ3-λ1+IHl∗λ3-λ2Z5,1SHλ3-λ1+IHlλ3-λ2-χA1-min⁡max⁡0,ψEA∗SAλ7-λ5+IAlλ7-λ6Z3,1SAλ7-λ5+IAlλ7-λ6,where χH=αEHβEHrFk/k+FE2 and χA=αEAβEArFk/k+FE2 with transversality conditions λiTfori=1,2,…,9=0.


## 4. Numerical Simulation of the Optimal Control Model, Results, and Discussion

Sometimes it may not be possible to solve the optimality system analytically; instead numerical methods are used to approximate the solutions and display the results. The optimality system is a two-point boundary problem, because of the initial condition of the state system and the terminal condition of the adjoint system [30]. To solve the optimality system with initial conditions for the states and final time conditions for the adjoints, we use the Runge-Kutta fourth-order procedure which is more accurate and elaborative technique. A Runge-Kutta method is a multiple-step method, where the solution at time tk+1 is obtained from a defined set of previous values tn-k,…,tk and n is the number of steps. This method is described in a book by Lenhart and Workman [8] and it is known as forward-backward sweep method. The process begins with an initial guess on the control variable and given initial conditions for states, we approximate solutions for state equations using Runge-Kutta forward sweep method. Given the state solutions from previous step and the final time conditions for adjoints, we approximate solutions for adjoint equations using Runge-Kutta backward sweep method. The value of control variables is updated by averaging the previous value and the new value arising from the control characterization. The process is repeated for forward numerical scheme and updating of the controls until successive values of all states, adjoints, and controls are sufficiently close or converge.


### 4.1. Numerical Simulation of Optimal Control Model

We conduct numerical simulation in order to investigate the effects of the control strategies on the transmission dynamics of Tungiasis. The simulations are performed using MALAB, and we set time in days. The estimated initial values of model state variables areSH=1000, IHl=200, IHh=300, TH=0, SA=900, IAl=200, IAh=300, LE=10000, and FE=4000 and for the adjoint system we have terminal conditions λiTfori=1,…,9=0, where we set T=200 days. The cost coefficients corresponding to state variables are estimated to be R1=20, R2=10, R3=15, R4=20, and R5=25. The quadratic cost coefficient corresponding to control measures is as well estimated to be Z1=50, Z2=60, Z3=80, Z4=10, and Z5=15. We use a set of parameter values whose sources are from literature and others are estimated as shown in Table 1.Table 1
Parameter values for numerical simulation of optimal control model.
Parameter
Value/range
Source/references
k
1
×
1
0
4
c
e
l
l
/
m
3
Estimated
K
1
×
1
0
5
c
e
l
l
/
m
3
Estimated
γ
L
0.0105 per day
Estimated
σ
H
0.011 per day
Estimated
σ
A
0.037 per day
Estimated
μ
H
0.000045 per day
UNICEF. [14]
μ
A
0.0028
360
–
3600
-
1 per day
Gaff et al. [15]; Radostits. (2001)
μ
F
0.04 per day
Eisele et al. [16],
r
F
0.58 per day
Estimated
μ
L
0.08 per day
Estimated
β
E
H
0.19 per day
Estimated
β
E
A
0.48 per day
Estimated
ρ
A
H
0.052 per day
Gaff et al. [15]
ρ
A
0.26 (0.091–0.9) per day
Allerson et al. [17]
b
H
0.00011 per day
TP, [18].
b
A
0.022 per day
Gaff et al. [15]
ε
A
0.40 per day
Estimated
δ
e
0.12 per day
Estimated
α
E
H
0.4
Estimated
α
E
A
0.6
Estimated
p
1
0.15
Estimated
p
2
0.15
Estimated
ω
0.09
Estimated
Source: Kahuru et al. [19].We then plot the graphs to show the effects of the control measures when implemented under different combination options. We first illustrate the situation when no optimal control strategy is implemented as shown in Figures2(a)–2(d) and then we suggest seven control strategies with different combinations of control measures and compare their performance in order to determine the best option to control the disease for maximum benefit.Figure 2
Simulations of model variablesIHh, IAh, FE without intervention.
(a)
(b)
(c)
(d)


### 4.2. Discussion of the Results

In this section, we present the results of the numerical simulation of our optimal control problem by discussing the implications of implementing the seven optimal control strategies on the Tungiasis dynamical model. To observe the effects of the optimal control strategies, we plot the results from simulation of the uncontrolled system and that from the controlled system together in Figures3–9. To compare the effects of these options, we plot the results together in Figures 10(a)–10(c). Thus we consider the following seven combination options.Figure 3
Optimal solutions for model variablesIHh, IAh, FE and the control profiles for w1, w2, w3, w4, w5 with w1,w2,w3,w4,w5≠0.
(a)
(b)
(c)
(d)Figure 4
Optimal solutions for model variablesIHh, IAh, FE and the control profiles for w1, w2, w3, w4, w5 with w2,w3,w4,w5≠0.
(a)
(b)
(c)
(d)Figure 5
Optimal solutions for model variablesIHh, IAh, FE and the control profiles for w1, w2, w3, w4, w5 with w3,w4,w5≠0.
(a)
(b)
(c)
(d)Figure 6
Optimal solutions for model variablesIHh, IAh, FE and the control profiles for w1, w2, w3, w4, w5 with w1,w2≠0.
(a)
(b)
(c)
(d)Figure 7
Optimal solutions for model variablesIHh, IAh, FE and the control profiles for w1, w2, w3, w4, w5 with w1,w2,w3≠0.
(a)
(b)
(c)
(d)Figure 8
Optimal solutions for model variablesIHh, IAh, FE and the control profiles for w1, w2, w3, w4, w5 with w2,w3,w4≠0.
(a)
(b)
(c)
(d)Figure 9
Optimal solutions for model variablesIHh, IAh, FE and the control profiles for w1, w2, w3, w4, w5 with w1,w2,w4,w5≠0.
(a)
(b)
(c)
(d)Figure 10
(a)
The comparison of the control strategies on severely infested humans
(b)
The comparison of the control strategies on severely infested animals
(c)
The comparison of the control strategies on the adult flea population


#### 4.2.1. When All the Control Strategies Are Not Implemented (i.e.,w1,w2,w3,w4,w5=0)

Figures2(a)–2(c) show the situation where no control strategy is implemented to the dynamical system whereby the model trajectories represented by a dotted lines for severely infested humans, severely infested animal reservoir, and flea populations remain unchanged and the control profiles in Figure 2(d) show that all control measures are at the lower bound (i.e., w1,w2,w3,w4,w5=0).


#### 4.2.2. Strategy  1: All Control Measures Are Implemented (i.e.,w1,w2,w3,w4,w5≠0)

Under strategy  1, all the control measuresw1,w2,w3,w4,w5 are used to optimize the objective functional Jw. In Figure 3(d), the control measure w5 is at upper bound at the beginning and after 40 days it gradually drops to the lower bound at the final time. The control measure w3 is at the upper bound at the beginning and after 45 days it gradually drops to the lower bound at the final time. The control measure w1 is at the upper bound at the beginning and after 60 days it drops to the lower bound at the final time. The control measure w4 is at the upper bound at the beginning and after 140 days it rapidly drops to the lower bound at the final time and the control measure w2 starts at the upper bound at the beginning and remains there until it drops to the lower bound.


#### 4.2.3. Strategy  2: The Control Measures (i.e.,w2,w3,w4,w5≠0 and w1=0)

Under strategy  2, the control measuresw2,w3,w4,w5 are used to optimize the objective functional Jw. In Figure 4(d), the control measure w4 is at upper bound at the beginning and after 140 days it rapidly drops to the lower bound at the final time. The control measure w5 is at the upper bound at the beginning and after 175 days it rapidly drops to the lower bound at the final time. The control measure w3 is at the upper bound at the beginning and after 195 days it drops to the lower bound at the final time. The control measure w2 starts at the upper bound at the beginning and remains there until it drops to the lower bound at the final time. The control measure w1 is at the lower bound at the beginning and remains there till the final time.


#### 4.2.4. Strategy  3: The Control Measures (i.e.,w3,w4,w5≠0 and w1,w2=0)

Under strategy  3, the control measuresw3,w4,w5 are used to optimize the objective functional Jw. In Figure 5(d), the control measure w4 is at upper bound at the beginning and after 140 days it rapidly drops to the lower bound at the final time. The control measure w5 is at the upper bound at the beginning and after 190 days it rapidly drops to the lower bound at the final time. The control measure w3 starts at the upper bound at the beginning and remains there until it drops to the lower bound. The control measures w1 and w2 start at the lower bound at the beginning and remains there till the final time.


#### 4.2.5. Strategy  4: The Control Measures (i.e.,w1,w2≠0 and w3,w4,w5=0)

Under strategy  4, the control measuresw1,w2 are used to optimize the objective functional Jw. In Figure 6(d), the control measure w1 is at upper bound at the beginning and after 80 days it gradually drops to the lower bound at the final time. The control measure w2 starts at the upper bound at the beginning and remains there until it drops to the lower bound. The control measures, w3, w4, and w5, start at the lower bound at the beginning and remain there till the final time.


#### 4.2.6. Strategy  5: The Control Strategies (i.e.,w1,w2,w3≠0 and w4,w5=0)

Under strategy  5, the control measuresw1,w2,w3 are used to optimize the objective functional Jw. In Figure 7(d), the control measure w3 is at upper bound at the beginning and after 50 days it gradually drops to the lower bound at the final time. The control measure w1 is at the upper bound at the beginning and after 60 days it gradually drops to the lower bound at the final time. The control measure w2 starts at the upper bound at the beginning and remains there until it drops to the lower bound at the final time. The control measures w4 and w5 start at the lower bound at the beginning and remains there till the final time.


#### 4.2.7. Strategy  6: The Control Measures (i.e.,w2,w3,w4≠0 and w1,w5=0)

Under strategy  6, the control measuresw2,w3,w4 are used to optimize the objective functional Jw. In Figure 8(d), the control measure w4 is at upper bound at the beginning and after 140 days it rapidly drops to the lower bound at the final time. The control measures w2 and w3 start at the upper bound at the beginning and remain there until they drop to the lower bound. The control measures w1 and w5 start at the lower bound at the beginning and remain there till the final time.


#### 4.2.8. Strategy  7: The Control Measures (i.e.,w1,w2,w4,w5≠0 and w3=0)

Under strategy  7, the control measuresw1,w2,w4,w5 are used to optimize the objective functional Jw. In Figure 9(d), the control measure w5 is at the upper bound at the beginning and after 40 days it gradually drops to the lower bound at the final time. The control measure w1 is at the upper bound at the beginning and after 75 days it gradually drops to the lower bound at the final time. The control measure w4 is at the upper bound at the beginning and after 170 days it rapidly drops to the lower bound at the final time. The control measure w2 starts at the upper bound at the beginning and remains there until it drops to the lower bound at the final time and the control measure w3 starts at the lower bound at the beginning and remains there till the final time.Generally, it is observed in Figures3(a)–3(c), 4(a)–4(c), 5(a)–5(c), 6(a)–6(c), 7(a)–7(c), 8(a)–8(c), and 9(a)–9(c) that the controlled trajectories represented by the solid lines decrease compared to uncontrolled trajectories represented by the dotted lines. This implies that the control strategies have the positive impacts which lead to a reduction in the number of infested humans and animals and also reduces the sand flea population in the soil environment.


#### 4.2.9. The Comparison of the Control Strategies

To compare the performance of the control strategies under consideration, we plot the results on same graphs indicating the effects of the control strategies as indicated in Figures10(a)–10(c).From Figures10(a)–10(c), it is observed that the control strategy involving the combination of all the five control measures w1, w2, w3, w4, w5 has the significant impact on the reduction of disease transmission because it lowers the severely infested humans and animals and the sand flea populations to minimal levels compared to other control strategies. The red solid line as depicted in Figures 10(a), 10(b), and 10(c) represents the effect caused by the control strategy  1: with the combination of all five control measures. This is the best control strategy because it dominates both graphs as it decreases rapidly compared to other trajectories. Moreover we have observed that those control strategies, whose combinations involve the control measure based on insecticides applications to the premises w1, yield better results. These are control strategies  5, 6, and 7. But control strategy  5 denoted by the control measures w1,w2,w3 performs better in the same way as the control strategy  1 denoted by the control measures w1,w2,w3,w4,w5.


## 4.1. Numerical Simulation of Optimal Control Model

We conduct numerical simulation in order to investigate the effects of the control strategies on the transmission dynamics of Tungiasis. The simulations are performed using MALAB, and we set time in days. The estimated initial values of model state variables areSH=1000, IHl=200, IHh=300, TH=0, SA=900, IAl=200, IAh=300, LE=10000, and FE=4000 and for the adjoint system we have terminal conditions λiTfori=1,…,9=0, where we set T=200 days. The cost coefficients corresponding to state variables are estimated to be R1=20, R2=10, R3=15, R4=20, and R5=25. The quadratic cost coefficient corresponding to control measures is as well estimated to be Z1=50, Z2=60, Z3=80, Z4=10, and Z5=15. We use a set of parameter values whose sources are from literature and others are estimated as shown in Table 1.Table 1
Parameter values for numerical simulation of optimal control model.
Parameter
Value/range
Source/references
k
1
×
1
0
4
c
e
l
l
/
m
3
Estimated
K
1
×
1
0
5
c
e
l
l
/
m
3
Estimated
γ
L
0.0105 per day
Estimated
σ
H
0.011 per day
Estimated
σ
A
0.037 per day
Estimated
μ
H
0.000045 per day
UNICEF. [14]
μ
A
0.0028
360
–
3600
-
1 per day
Gaff et al. [15]; Radostits. (2001)
μ
F
0.04 per day
Eisele et al. [16],
r
F
0.58 per day
Estimated
μ
L
0.08 per day
Estimated
β
E
H
0.19 per day
Estimated
β
E
A
0.48 per day
Estimated
ρ
A
H
0.052 per day
Gaff et al. [15]
ρ
A
0.26 (0.091–0.9) per day
Allerson et al. [17]
b
H
0.00011 per day
TP, [18].
b
A
0.022 per day
Gaff et al. [15]
ε
A
0.40 per day
Estimated
δ
e
0.12 per day
Estimated
α
E
H
0.4
Estimated
α
E
A
0.6
Estimated
p
1
0.15
Estimated
p
2
0.15
Estimated
ω
0.09
Estimated
Source: Kahuru et al. [19].We then plot the graphs to show the effects of the control measures when implemented under different combination options. We first illustrate the situation when no optimal control strategy is implemented as shown in Figures2(a)–2(d) and then we suggest seven control strategies with different combinations of control measures and compare their performance in order to determine the best option to control the disease for maximum benefit.Figure 2
Simulations of model variablesIHh, IAh, FE without intervention.
(a)
(b)
(c)
(d)


## 4.2. Discussion of the Results

In this section, we present the results of the numerical simulation of our optimal control problem by discussing the implications of implementing the seven optimal control strategies on the Tungiasis dynamical model. To observe the effects of the optimal control strategies, we plot the results from simulation of the uncontrolled system and that from the controlled system together in Figures3–9. To compare the effects of these options, we plot the results together in Figures 10(a)–10(c). Thus we consider the following seven combination options.Figure 3
Optimal solutions for model variablesIHh, IAh, FE and the control profiles for w1, w2, w3, w4, w5 with w1,w2,w3,w4,w5≠0.
(a)
(b)
(c)
(d)Figure 4
Optimal solutions for model variablesIHh, IAh, FE and the control profiles for w1, w2, w3, w4, w5 with w2,w3,w4,w5≠0.
(a)
(b)
(c)
(d)Figure 5
Optimal solutions for model variablesIHh, IAh, FE and the control profiles for w1, w2, w3, w4, w5 with w3,w4,w5≠0.
(a)
(b)
(c)
(d)Figure 6
Optimal solutions for model variablesIHh, IAh, FE and the control profiles for w1, w2, w3, w4, w5 with w1,w2≠0.
(a)
(b)
(c)
(d)Figure 7
Optimal solutions for model variablesIHh, IAh, FE and the control profiles for w1, w2, w3, w4, w5 with w1,w2,w3≠0.
(a)
(b)
(c)
(d)Figure 8
Optimal solutions for model variablesIHh, IAh, FE and the control profiles for w1, w2, w3, w4, w5 with w2,w3,w4≠0.
(a)
(b)
(c)
(d)Figure 9
Optimal solutions for model variablesIHh, IAh, FE and the control profiles for w1, w2, w3, w4, w5 with w1,w2,w4,w5≠0.
(a)
(b)
(c)
(d)Figure 10
(a)
The comparison of the control strategies on severely infested humans
(b)
The comparison of the control strategies on severely infested animals
(c)
The comparison of the control strategies on the adult flea population


### 4.2.1. When All the Control Strategies Are Not Implemented (i.e.,w1,w2,w3,w4,w5=0)

Figures2(a)–2(c) show the situation where no control strategy is implemented to the dynamical system whereby the model trajectories represented by a dotted lines for severely infested humans, severely infested animal reservoir, and flea populations remain unchanged and the control profiles in Figure 2(d) show that all control measures are at the lower bound (i.e., w1,w2,w3,w4,w5=0).


### 4.2.2. Strategy  1: All Control Measures Are Implemented (i.e.,w1,w2,w3,w4,w5≠0)

Under strategy  1, all the control measuresw1,w2,w3,w4,w5 are used to optimize the objective functional Jw. In Figure 3(d), the control measure w5 is at upper bound at the beginning and after 40 days it gradually drops to the lower bound at the final time. The control measure w3 is at the upper bound at the beginning and after 45 days it gradually drops to the lower bound at the final time. The control measure w1 is at the upper bound at the beginning and after 60 days it drops to the lower bound at the final time. The control measure w4 is at the upper bound at the beginning and after 140 days it rapidly drops to the lower bound at the final time and the control measure w2 starts at the upper bound at the beginning and remains there until it drops to the lower bound.


### 4.2.3. Strategy  2: The Control Measures (i.e.,w2,w3,w4,w5≠0 and w1=0)

Under strategy  2, the control measuresw2,w3,w4,w5 are used to optimize the objective functional Jw. In Figure 4(d), the control measure w4 is at upper bound at the beginning and after 140 days it rapidly drops to the lower bound at the final time. The control measure w5 is at the upper bound at the beginning and after 175 days it rapidly drops to the lower bound at the final time. The control measure w3 is at the upper bound at the beginning and after 195 days it drops to the lower bound at the final time. The control measure w2 starts at the upper bound at the beginning and remains there until it drops to the lower bound at the final time. The control measure w1 is at the lower bound at the beginning and remains there till the final time.


### 4.2.4. Strategy  3: The Control Measures (i.e.,w3,w4,w5≠0 and w1,w2=0)

Under strategy  3, the control measuresw3,w4,w5 are used to optimize the objective functional Jw. In Figure 5(d), the control measure w4 is at upper bound at the beginning and after 140 days it rapidly drops to the lower bound at the final time. The control measure w5 is at the upper bound at the beginning and after 190 days it rapidly drops to the lower bound at the final time. The control measure w3 starts at the upper bound at the beginning and remains there until it drops to the lower bound. The control measures w1 and w2 start at the lower bound at the beginning and remains there till the final time.


### 4.2.5. Strategy  4: The Control Measures (i.e.,w1,w2≠0 and w3,w4,w5=0)

Under strategy  4, the control measuresw1,w2 are used to optimize the objective functional Jw. In Figure 6(d), the control measure w1 is at upper bound at the beginning and after 80 days it gradually drops to the lower bound at the final time. The control measure w2 starts at the upper bound at the beginning and remains there until it drops to the lower bound. The control measures, w3, w4, and w5, start at the lower bound at the beginning and remain there till the final time.


### 4.2.6. Strategy  5: The Control Strategies (i.e.,w1,w2,w3≠0 and w4,w5=0)

Under strategy  5, the control measuresw1,w2,w3 are used to optimize the objective functional Jw. In Figure 7(d), the control measure w3 is at upper bound at the beginning and after 50 days it gradually drops to the lower bound at the final time. The control measure w1 is at the upper bound at the beginning and after 60 days it gradually drops to the lower bound at the final time. The control measure w2 starts at the upper bound at the beginning and remains there until it drops to the lower bound at the final time. The control measures w4 and w5 start at the lower bound at the beginning and remains there till the final time.


### 4.2.7. Strategy  6: The Control Measures (i.e.,w2,w3,w4≠0 and w1,w5=0)

Under strategy  6, the control measuresw2,w3,w4 are used to optimize the objective functional Jw. In Figure 8(d), the control measure w4 is at upper bound at the beginning and after 140 days it rapidly drops to the lower bound at the final time. The control measures w2 and w3 start at the upper bound at the beginning and remain there until they drop to the lower bound. The control measures w1 and w5 start at the lower bound at the beginning and remain there till the final time.


### 4.2.8. Strategy  7: The Control Measures (i.e.,w1,w2,w4,w5≠0 and w3=0)

Under strategy  7, the control measuresw1,w2,w4,w5 are used to optimize the objective functional Jw. In Figure 9(d), the control measure w5 is at the upper bound at the beginning and after 40 days it gradually drops to the lower bound at the final time. The control measure w1 is at the upper bound at the beginning and after 75 days it gradually drops to the lower bound at the final time. The control measure w4 is at the upper bound at the beginning and after 170 days it rapidly drops to the lower bound at the final time. The control measure w2 starts at the upper bound at the beginning and remains there until it drops to the lower bound at the final time and the control measure w3 starts at the lower bound at the beginning and remains there till the final time.Generally, it is observed in Figures3(a)–3(c), 4(a)–4(c), 5(a)–5(c), 6(a)–6(c), 7(a)–7(c), 8(a)–8(c), and 9(a)–9(c) that the controlled trajectories represented by the solid lines decrease compared to uncontrolled trajectories represented by the dotted lines. This implies that the control strategies have the positive impacts which lead to a reduction in the number of infested humans and animals and also reduces the sand flea population in the soil environment.


### 4.2.9. The Comparison of the Control Strategies

To compare the performance of the control strategies under consideration, we plot the results on same graphs indicating the effects of the control strategies as indicated in Figures10(a)–10(c).From Figures10(a)–10(c), it is observed that the control strategy involving the combination of all the five control measures w1, w2, w3, w4, w5 has the significant impact on the reduction of disease transmission because it lowers the severely infested humans and animals and the sand flea populations to minimal levels compared to other control strategies. The red solid line as depicted in Figures 10(a), 10(b), and 10(c) represents the effect caused by the control strategy  1: with the combination of all five control measures. This is the best control strategy because it dominates both graphs as it decreases rapidly compared to other trajectories. Moreover we have observed that those control strategies, whose combinations involve the control measure based on insecticides applications to the premises w1, yield better results. These are control strategies  5, 6, and 7. But control strategy  5 denoted by the control measures w1,w2,w3 performs better in the same way as the control strategy  1 denoted by the control measures w1,w2,w3,w4,w5.


## 4.2.1. When All the Control Strategies Are Not Implemented (i.e.,w1,w2,w3,w4,w5=0)

Figures2(a)–2(c) show the situation where no control strategy is implemented to the dynamical system whereby the model trajectories represented by a dotted lines for severely infested humans, severely infested animal reservoir, and flea populations remain unchanged and the control profiles in Figure 2(d) show that all control measures are at the lower bound (i.e., w1,w2,w3,w4,w5=0).


## 4.2.2. Strategy  1: All Control Measures Are Implemented (i.e.,w1,w2,w3,w4,w5≠0)

Under strategy  1, all the control measuresw1,w2,w3,w4,w5 are used to optimize the objective functional Jw. In Figure 3(d), the control measure w5 is at upper bound at the beginning and after 40 days it gradually drops to the lower bound at the final time. The control measure w3 is at the upper bound at the beginning and after 45 days it gradually drops to the lower bound at the final time. The control measure w1 is at the upper bound at the beginning and after 60 days it drops to the lower bound at the final time. The control measure w4 is at the upper bound at the beginning and after 140 days it rapidly drops to the lower bound at the final time and the control measure w2 starts at the upper bound at the beginning and remains there until it drops to the lower bound.


## 4.2.3. Strategy  2: The Control Measures (i.e.,w2,w3,w4,w5≠0 and w1=0)

Under strategy  2, the control measuresw2,w3,w4,w5 are used to optimize the objective functional Jw. In Figure 4(d), the control measure w4 is at upper bound at the beginning and after 140 days it rapidly drops to the lower bound at the final time. The control measure w5 is at the upper bound at the beginning and after 175 days it rapidly drops to the lower bound at the final time. The control measure w3 is at the upper bound at the beginning and after 195 days it drops to the lower bound at the final time. The control measure w2 starts at the upper bound at the beginning and remains there until it drops to the lower bound at the final time. The control measure w1 is at the lower bound at the beginning and remains there till the final time.


## 4.2.4. Strategy  3: The Control Measures (i.e.,w3,w4,w5≠0 and w1,w2=0)

Under strategy  3, the control measuresw3,w4,w5 are used to optimize the objective functional Jw. In Figure 5(d), the control measure w4 is at upper bound at the beginning and after 140 days it rapidly drops to the lower bound at the final time. The control measure w5 is at the upper bound at the beginning and after 190 days it rapidly drops to the lower bound at the final time. The control measure w3 starts at the upper bound at the beginning and remains there until it drops to the lower bound. The control measures w1 and w2 start at the lower bound at the beginning and remains there till the final time.


## 4.2.5. Strategy  4: The Control Measures (i.e.,w1,w2≠0 and w3,w4,w5=0)

Under strategy  4, the control measuresw1,w2 are used to optimize the objective functional Jw. In Figure 6(d), the control measure w1 is at upper bound at the beginning and after 80 days it gradually drops to the lower bound at the final time. The control measure w2 starts at the upper bound at the beginning and remains there until it drops to the lower bound. The control measures, w3, w4, and w5, start at the lower bound at the beginning and remain there till the final time.


## 4.2.6. Strategy  5: The Control Strategies (i.e.,w1,w2,w3≠0 and w4,w5=0)

Under strategy  5, the control measuresw1,w2,w3 are used to optimize the objective functional Jw. In Figure 7(d), the control measure w3 is at upper bound at the beginning and after 50 days it gradually drops to the lower bound at the final time. The control measure w1 is at the upper bound at the beginning and after 60 days it gradually drops to the lower bound at the final time. The control measure w2 starts at the upper bound at the beginning and remains there until it drops to the lower bound at the final time. The control measures w4 and w5 start at the lower bound at the beginning and remains there till the final time.


## 4.2.7. Strategy  6: The Control Measures (i.e.,w2,w3,w4≠0 and w1,w5=0)

Under strategy  6, the control measuresw2,w3,w4 are used to optimize the objective functional Jw. In Figure 8(d), the control measure w4 is at upper bound at the beginning and after 140 days it rapidly drops to the lower bound at the final time. The control measures w2 and w3 start at the upper bound at the beginning and remain there until they drop to the lower bound. The control measures w1 and w5 start at the lower bound at the beginning and remain there till the final time.


## 4.2.8. Strategy  7: The Control Measures (i.e.,w1,w2,w4,w5≠0 and w3=0)

Under strategy  7, the control measuresw1,w2,w4,w5 are used to optimize the objective functional Jw. In Figure 9(d), the control measure w5 is at the upper bound at the beginning and after 40 days it gradually drops to the lower bound at the final time. The control measure w1 is at the upper bound at the beginning and after 75 days it gradually drops to the lower bound at the final time. The control measure w4 is at the upper bound at the beginning and after 170 days it rapidly drops to the lower bound at the final time. The control measure w2 starts at the upper bound at the beginning and remains there until it drops to the lower bound at the final time and the control measure w3 starts at the lower bound at the beginning and remains there till the final time.Generally, it is observed in Figures3(a)–3(c), 4(a)–4(c), 5(a)–5(c), 6(a)–6(c), 7(a)–7(c), 8(a)–8(c), and 9(a)–9(c) that the controlled trajectories represented by the solid lines decrease compared to uncontrolled trajectories represented by the dotted lines. This implies that the control strategies have the positive impacts which lead to a reduction in the number of infested humans and animals and also reduces the sand flea population in the soil environment.


## 4.2.9. The Comparison of the Control Strategies

To compare the performance of the control strategies under consideration, we plot the results on same graphs indicating the effects of the control strategies as indicated in Figures10(a)–10(c).From Figures10(a)–10(c), it is observed that the control strategy involving the combination of all the five control measures w1, w2, w3, w4, w5 has the significant impact on the reduction of disease transmission because it lowers the severely infested humans and animals and the sand flea populations to minimal levels compared to other control strategies. The red solid line as depicted in Figures 10(a), 10(b), and 10(c) represents the effect caused by the control strategy  1: with the combination of all five control measures. This is the best control strategy because it dominates both graphs as it decreases rapidly compared to other trajectories. Moreover we have observed that those control strategies, whose combinations involve the control measure based on insecticides applications to the premises w1, yield better results. These are control strategies  5, 6, and 7. But control strategy  5 denoted by the control measures w1,w2,w3 performs better in the same way as the control strategy  1 denoted by the control measures w1,w2,w3,w4,w5.


## 5. Conclusion

In this paper, the optimal control techniques have been applied on Tungiasis dynamical model with control strategies. We defined the control set including controlling the transmission of infestation from flea infested soil environment to human population, from the flea infested soil environment to animal population, and from flea infested animal to human population, controlling flea infested soil environment, and controlling the flea infested animal population. We proved the existence of optimal control problem and determined the necessary conditions for optimality using Pontryagin’s maximum principle which converts constrained optimization problem into unconstrained Hamiltonian function whereby optimality and adjoint equations are obtained. We, lastly, performed numerical simulations of the resulting control problem to investigate the effects of the control strategies under consideration and compare their performances. The numerical results showed that the control strategy that comprises all five control measures and that with control measuresw1,w2,w3 have the significant impact on reduced Tungiasis transmission and those control strategies involving insecticides control to the premises w1 yielded better results, which implies that the insecticides application control is more effective than other individual control measures. In poor rural communities where resources are always scarce, we suggest that the combination option involving the controls of focal insecticides spraying, insecticidal dusting on animal furs, and environmental hygiene should be adopted, having observed from the comparison of all seven control strategies in Figures 10(a), 10(b), and 10(c) that there is no significant difference between this strategy w1,w2,w3≠0 and the strategy that involves the combination of the five control measures w1,w2,w3,w4,w5≠0. Among others, poor housing conditions and the presence of domestic and sylvatic animals on the home compound are risk factors. Therefore, controlling of infested soils and animal reservoirs with insecticides control, ant-flea compounds or animal furs, environmental hygiene, and cementing the floors of houses may serve as a possible approach to control the epidemic and, thus, to fight against Tungiasis infestation in endemic settings multidimensional control process should be employed in order to achieve the maximum benefits.



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*Source: 4804897-2017-08-14.xml*