International Journal of Mathematics and Mathematical Sciences

Volume 2017 (2017), Article ID 4804897, 19 pages

https://doi.org/10.1155/2017/4804897

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

^{1}School of Computational and Communication Science and Engineering, Nelson Mandela African Institution of Science and Technology, P.O. Box 447, Arusha, Tanzania^{2}Department of Mathematics, Makerere University, P.O. Box 7062, Kampala, Uganda

Correspondence should be addressed to Jairos Kahuru

Received 20 March 2017; Accepted 2 July 2017; Published 14 August 2017

Academic Editor: Nawab Hussain

Copyright © 2017 Jairos Kahuru et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 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 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.

#### 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 efforts in human, animal reservoirs, and adult flea populations and is the failure rate for the control efforts for . We let be the effort of controlling the flea infested soil environment with insecticides spraying, be the efforts of controlling the flea infested animal reservoir through dusting them with ant-flea compounds, 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), 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 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 ). Therefore we assume that the mortality rate of jigger fleas in the soil environment is increased by the factor , on-host spraying of infested animals will reduce the shedding rate of adult jigger fleas into the environment by a fraction , and the animal to animal effective contact rate is reduced at the same fraction 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 , the factor reduces the transmission from severely infested animal reservoirs to susceptible humans, and the factor 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 , , , , and 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 , and the human treatment class denoted by ; therefore we have . 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 and move to class at a rate and may also acquire infestation from the flea infested soil environment and move to class at a rate . may as well acquire infestation from the flea infested soil environment and progresses to class at a rate . Classes and seek treatment at the respective rates and 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 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 ; therefore we have . 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 and move to at a rate and also may acquire infestation from the flea infested environment and move to class at a rate . may as well acquire infestation from the flea infested soil environment and progresses to class at a rate . Individuals in compartments and suffer a natural mortality rate and for the compartment 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 and at a constant rate ; therefore we have the total contribution of and 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 . 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 , therefore we have the flea total death rate of .

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).