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

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.


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 ) 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 2 International Journal of Mathematics and Mathematical Sciences 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 as Microbacterium 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, and Mycobacterium 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 Section 2, 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.

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 efforts   () in human, animal reservoirs, and adult flea populations and (1 −   ()) is the failure rate for the control efforts   () for  = 1, . . ., 5. We let  1 () be the effort of controlling the flea infested soil environment with insecticides spraying,  2 () be the efforts of controlling the flea infested animal reservoir through dusting them with ant-flea compounds,  3 () 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),  4 () 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  5 () 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, ]).Therefore we assume that the mortality rate of jigger fleas in the soil environment is increased by the factor (  +  1 ()), on-host spraying of infested animals will reduce the shedding rate   of adult jigger fleas into the environment by a fraction (1 −  2 ()), and the animal to animal effective contact rate   is reduced at the same fraction (1 −  2 ()) 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− 3 ()), the factor (1− 4 ()) reduces the transmission from severely infested animal reservoirs to susceptible humans, and the factor (1 −  5 ()) 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   which represents human beings under treatment and incorporate the five control measures  1 ,  2 ,  3 ,  4 , and  5 as defined above.In the submodel of human population, the total human population   is subdivided into susceptible population   mildly infested population   , the severely infested population  ℎ , and the human treatment class denoted by   ; therefore we have   =   +   +  ℎ +   .We assume that the humans are recruited into   through birth by the adults at a rate   .Individuals in class   acquire infestation from the severely infested animal reservoirs  ℎ and move to class   at a rate (1 −  4 ())   ℎ /  and may also acquire infestation from the flea infested soil environment and move to class  ℎ at a rate (1− 5 ())        /(+  ).  may as well acquire infestation from the flea infested soil environment and progresses to class  ℎ at a rate (1 −  5 ())        /( +   ).Classes  ℎ and   seek treatment at the respective rates  1 and  2 and join   class, and eventually the treated individuals revert back to join   at a progression rate .Individuals in compartments   ,   , and   suffer a natural mortality rate   and for the compartment  ℎ they suffer a natural mortality at a rate   and the disease induced mortality at a rate   .In the submodel of animal reservoir population, the total animal reservoir population   is subdivided into susceptible population   mildly infested population   and the severely infested population  ℎ ; therefore we have   =   +   +  ℎ .We assume that the animals are recruited into   through birth by the adults at a rate   .Individuals in class   acquire infestation from the severely infested animal reservoirs  ℎ and move to   at a rate (1 −  2 ())   ℎ /  and also may acquire infestation from the flea infested environment and move to class  ℎ at a rate (1 −  3 ())        /( +   ).  may as well acquire infestation from the flea infested soil environment and progresses to class  ℎ at a rate (1 −  3 ())        /( +   ).Individuals in compartments   and   suffer a natural mortality rate   and for the compartment  ℎ they suffer a natural mortality at a rate   and a disease induced mortality at a rate   .The submodel of environmental component consists of two compartments, a compartment of larvae denoted by   and a compartment of adult sand fleas denoted by   .The larvae population are recruited into   through shedding of jigger eggs by  ℎ and  ℎ at a constant rate   ; therefore we have the total contribution of    ℎ and    ℎ from infested humans and animal reservoir populations, respectively.The larvae in compartment   mature into adult jigger fleas at a maturation rate   and undergo a natural death at a rate   .The adult jigger flea population are recruited into   through maturation by larvae at a rate   and from infested animal reservoirs who contributes the fleas into the soil environment at a rate (1 −  2 ())  .The adult fleas leave the compartment   when they attack the hosts at a rate     /(+  ) and when they undergo a natural death at a rate   and the additional death due to insecticides control at a rate  1 (), therefore we have the flea total death rate of (  +  1 ())  .
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 Figure 1 and the differential equations describing the model are given in system (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.with initial condition

Equations of the
where (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 (  for  = 1, . . ., 5).The objective functional () to be minimized is given by International Journal of Mathematics and Mathematical Sciences 5 The terms  ℎ ,   ,  ℎ ,   ,   in the objective functional () are the number of infested populations that need to be minimized.The terms  1  2 1 represents the cost of offhost insecticides spraying,  2  2 2 represents the cost of onhost spraying of infested domestic animals,  3  2 3 represent the cost of implementing environmental hygiene,  4  2 4 represents the cost of education campaign, and  5  2  5 represents the cost of personal protection. 1 ,  2 ,  3 ,  4 ,  5 ,  1 ,  2 ,  3 ,  4 , and  5 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 populations  ℎ ,   ,  ℎ ,   ,   under minimum cost.Therefore the objective function in ( 4) is minimized subject to the model equations in (1).We seek the optimal controls subject to the dynamical system in (1) and the control set given by (5)

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.

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 by Given that,   ,   ,  ℎ ,   ,   ,   ,  ℎ ,   , and   are the state variables of the controlled model system (1) and  1 ,  2 ,  3 ,  4 ,  5 are the control variables.We state and prove Theorem 1 as follows.

Theorem 1. There exists an optimal control variables set 𝑤
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  1 ,  2 ,  3 ,  4 ,  5 is satisfied.The set of admissible Lebesgue measurable control variables ( 1 ,  2 ,  3 ,  4 ,  5 ) ∈ 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: where is the 9 × 9 matrix International Journal of Mathematics and Mathematical Sciences where and   denotes the derivative of  with respect to time .System ( 4) is a nonlinear system with a bounded coefficient.We set

𝑅 (𝑌) = 𝐵𝑌 + 𝐺 (𝑌) . (7d)
The second term on the right-hand side of (7d where the positive constant  = max(  for  = 1, . . ., 9) is independent of the state variables.Also we have |( 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 (, , ) = We must verify the condition that there exist a constant  > 1 and positive numbers  1 and  2 such that The last condition is satisfied when  = 2,  2 > 0, and  1 = min{ 1 ,  2 ,  3 ,  4 ,  5 }.This ends the proof.

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 , 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 where   for  = 1, . . ., 9 are the adjoint functions associated with the state equations in (1),   for  = 1, . . ., 9 is the right-hand side of the differential equations of th state variable in system (1).
The expanded form of Hamiltonian function in (11) is given by The optimality equations are obtained when taking the partial derivative of the Hamiltonian function  with respect to the control variables ( where 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 Therefore, the adjoint system evaluated at optimal controls  1 ,  2 ,  3 ,  4 , and  5 and the corresponding model state variables   ,   ,  ℎ ,   ,   ,   ,  ℎ ,   ,   is as given by where   =       /( +   ) 2 and   =       / ( +   ) 2 with transversality conditions, {  () for  = 1, 2, . . ., 9} = 0, and the characterization of optimal controls  * 1 ,  * 2 ,  * 3 ,  * 4 ,  * 5 ; that is, the optimality equations are based on the conditions: Subject to (17) If we set /  = 0 at  *  the results are the same as in characterization in (14).
The bounds and the impact notation for the control  1 are, respectively, given as Similar four-step arguments hold for optimal control schedules  * 2 ,  * 3 ,  * 4 , and  * 5 in the same way as in (19) based on the characterization in (14).

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) where   =       /( +   ) 2 and   =       / ( +   ) 2 with transversality conditions {  () for  = 1, 2, . . ., 9} = 0.

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 fourthorder procedure which is more accurate and elaborative technique.A Runge-Kutta method is a multiple-step method, where the solution at time  +1 is obtained from a defined set of previous values  − , . . .,   and  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.We use a set of parameter values whose sources are from literature and others are estimated as shown in Table 1.
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 Figures 2(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.

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 Figures 3-9.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.

When All the Control Strategies Are Not Implemented
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.,  1 ,  2 ,  3 ,  4 ,  5 = 0).

Strategy 1: All Control Measures
Are Implemented (i.e.,  1 ,  2 ,  3 ,  4 ,  5 ̸ = 0).Under strategy 1, all the control measures ( 1 ,  2 ,  3 ,  4 ,  5 ) are used to optimize the objective functional ().In Figure 3(d), the control measure  5 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  3 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  1 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  4 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  2 starts at the upper bound at the beginning and remains there until it drops to the lower bound.

Strategy 2:
The Control Measures (i.e.,  2 ,  3 ,  4 ,  5 ̸ = 0 and  1 = 0).Under strategy 2, the control measures ( 2 ,  3 ,  4 ,  5 ) are used to optimize the objective functional ().In Figure 4(d), the control measure  4 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  5 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  3 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  2 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  1 is at the lower bound at the beginning and remains there till the final time.().In Figure 5(d), the control measure  4 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  5 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  3 starts at the upper bound at the beginning and remains there until it drops to the lower bound.The control measures  1 and  2 start at the lower bound at the beginning and remains there till the final time.

Strategy 4:
The Control Measures (i.e.,  1 ,  2 ̸ = 0 and  3 ,  4 ,  5 = 0).Under strategy 4, the control measures ( 1 ,  2 ) are used to optimize the objective functional ().In Figure 6(d), the control measure  1 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  2 starts at the upper bound at the beginning and remains there until it drops to the lower bound.The control measures,  3 ,  4 , and  5 , start at the lower bound at the beginning and remain there till the final time.

Strategy 5:
The Control Strategies (i.e.,  1 ,  2 ,  3 ̸ = 0 and  4 ,  5 = 0).Under strategy 5, the control measures ( 1 ,  2 ,  3 ) are used to optimize the objective functional ().In Figure 7(d), the control measure  3 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  1 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  2 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  4 and  5 start at the lower bound at the beginning and remains there till the final time.

Strategy 6:
The Control Measures (i.e.,  2 ,  3 ,  4 ̸ = 0 and  1 ,  5 = 0).Under strategy 6, the control measures ( 2 ,  3 ,  4 ) are used to optimize the objective functional ().In Figure 8(d), the control measure  4 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  2 and  3 start at the upper bound at the beginning and remain there until they drop to the lower bound.The control measures  1 and  5 start at the lower bound at the beginning and remain there till the final time.

Strategy 7:
The Control Measures (i.e.,  1 ,  2 ,  4 ,  5 ̸ = 0 and  3 = 0).Under strategy 7, the control measures ( 1 ,  2 ,  4 ,  5 ) are used to optimize the objective functional ().In Figure 9(d), the control measure  5 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  1 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  4 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  2 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  3 starts at the lower bound at the beginning and remains there till the final time.From Figures 10(a)-10(c), it is observed that the control strategy involving the combination of all the five control measures  1 ,  2 ,  3 ,  4 ,  5 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 ( 1 ), yield better results.These are control strategies 5, 6, and 7.But control strategy 5 denoted by the control measures ( 1 ,  2 ,  3 ) performs better in the same way as the control strategy 1 denoted by the control measures ( 1 ,  2 ,  3 ,  4 ,  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 International Journal of Mathematics and Mathematical Sciences performances.The numerical results showed that the control strategy that comprises all five control measures and that with control measures ( 1 ,  2 ,  3 ) have the significant impact on reduced Tungiasis transmission and those control strategies involving insecticides control to the premises ( 1 ) 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 ( 1 ,  2 ,  3 ̸ = 0) and the strategy that involves the combination of the five control measures ( 1 ,  2 ,  3 ,  4 ,  5 ̸ = 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.

Notations
The State Variables of the Model with Control Measures (Source: Kahuru et al. [19]) The density of fleas population in the environment at time    (): The density of larvae population in the environment at time    (),   (): Total human and animal reservoirs populations at time

Figure 1 :
Figure 1: The flow chart showing the dynamics of Tungiasis with control measures incorporated.

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).

Table 1 :
Parameter values for numerical simulation of optimal control model.