American Journal of Computational and Applied Mathematics 2014, 4(3): 61-76 DOI: 10.5923/j.ajcam.20140403.01 Optimal Control Applied to the Spread of Buruli Uclcer Disease Ebenezer Bonyah 1,* , Isaac Dontwi 1 , Farai Nyabadza 2 1 Department of Mathematics, Kwame Nkumah University of Science and Technology, Kumasi, Ghana 2 Department of Mathematical Science, University of Stellenbosch, Private Bag X1, Matieland, 7602, South Africa Abstract Optimal control theory is applied to a system of ordinary differential equations modeling Buruli ulcer transmission in population. We apply controls on mass treatment, insecticide and mass education to minimize the number of infected hosts and infected vectors as well as infected fishes. The model takes into account human, water bug and fish populations as well as MU in the environment. The host, vector and small fish are all assumed constant. First, we investigated the existence and stability of equilibria of the model without control based on the basic reproduction ratio. We then, applied Pontryagins maximum principle to characterize the optimal control. The optimality system is determined and computed numerically for several scenarios. Keywords Mathematical model, Optimal control, Buruli ulcer, Basic reproduction ratio, Stability, Mass treatment, Insecticide 1. Introduction Buruli ulcer is a debilitating human skin disease caused by Mycobacterium ulcerous [16].This infection is a neglected emerging disease that has recently been reported in some countries as the second most frequent mycobacterial disease in human tuberculosis [1]. A large number of lesions often result in scarring, contractual deformities amputations and disabilities [3]. In Africa, all ages and sexes are affected, but most cases of the disease occur in children between the ages of 4-15years [6]. It has emerged dramatically over the past two decades especially in Central and West Africa and has been confirmed by laboratory tests in 26 countries with reports in other countries around the world [2, 4]. Burului ulcer has been reported in over 30 countries mainly with tropical and subtropical climates but it may also occur in the same countries where it has not yet been recognized such as Burkina Faso and Guinea [2]. Notable contributing factors for Buruli ulcer outbreak include deforestation, eutrophication, dam construction, farming and habitat fragmentation. When treatments are delayed often in West Africa where due to lack of adequate knowledge about the disease they normally reported late, infections could lead to ulcerated lesions[16].When data from Ghana were analysed it was revealed that socioeconomic impact had a huge economic burden on affected countries. For example, gross domestic product per capital (Ghana) in 1998 was estimated * Corresponding author: ebbonya@yahoo.com (Ebenezer Bonyah) Published online at http://journal.sapub.org/ajcam Copyright © 2014 Scientific & Academic Publishing. All Rights Reserved at USD 399.41. The cost of surgical excisions of a large ulcer including 3 months inpatient expenses, transportation, food and income lost was USD 783. The current strategy of controlling the disease includes the use of drugs, surgery, heat, hyperbaric oxygen and traditional medicine. These are strong social economic ways to the burden of the BU disease, which in different ways affects fertility, population growth, saving and investment, absenteeism, stigmatization of women in adulthood for marriage and medical cost [19]. Modeling of epidemiological phenomena has a very long story with the first model for smallpox formulated by Daniel Bernoulli in 1760. Mathematical modeling of the population models continues to provide vital insights into population behaviour and control. In the past years this has become an essential tool in understanding the dynamics of diseases, and the decision which has to do with process regarding intervention programs for controlling population and disease problems and in many countries. Similarly, different techniques have been employed to study vital optimal control problem related to dynamical systems. In particular, [10] developed a continuous model for malaria vector control with the objective of studying how genetically modified mosquitoes should be introduced in the environment using optimal control problem strategies. [13] applied optimal control theory to model malaria disease that includes treatment and vaccination with warning immaturity, to study the impact of a possible vaccination with treatment strategies in controlling the spread of malaria. Time- -depend-control strategies have been applied for the study of HIV model [11, 12] for strain tuberculosis models. [14] also examined SIR epidemiological models numerically and obtains controls that exert maximum efforts on some initial