Mathematical models for the study of buruli ulcer dynamics in Ghana

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Mycobacterium ulcerans (MU) is known to cause Buruli ulcer (BU). The association between the ulcer and environmental exposure has been fairly documented, however, the epidemiology of the ulcer is not well understood. The hypothesised transmission involves humans being bitten by the water bugs that prey on mollusks, snails and young fishes. In this thesis, deterministic, optimal control and age dependent models were developed for the study of the dynamics of the disease. The models equilibria are determined and conditions for the existence of the equilibria established. The transmission dynamics of BU model of the Susceptible, Infected and Recovered (SIR) type showed that the infected humans increased as long as there are enough infected water bugs to sustain the epidemic. Sensitivity analysis showed that the BU epidemic is highly influenced by the shedding of MU into the environment. The model SIR is found to fit reasonably well to data from Ashanti region of Ghana and projections on the future of the BU epidemic are also made. Again, a deterministic of the model Susceptible,Infected, under Treatment and Recovered (SITR) type with saturation treatment is formulated. The suggestion that giving the patients timely treatment, improving the cure efficiency and decreasing the infective coefficient are all valid methods for the control of disease. It was also found that increasing the density of Mycobacterium ulcerans in the environment led to an increase in the number of infected water bugs. Furthermore, model was modified to incorporate treatment and preventive measures. The SIR model was analysed without treatment and preventive measures and investigated its stability at both disease free and endemic steady states. Furthermore, treatment and preventive measures were incorporated (mass treatment, spaying of insecticides and provision of mass education) and investigated the effects of different control strategies on the spread of Buruli ulcer. Further, we used optimal control methods iii to determine the necessary conditions for the optimality of the disease eradication or control. The best strategies in fighting Buruli ulcer disease was determined and obtained that a combination of all the three strategies are the most effective way to manage BU disease. On the age model SIR dynamics, a representation from the method of characteristics and fixed point theory was applied to determine the existence and uniqueness of solutions to the nonlinear system of the age model. The simulations revealed an increase in recovered humans and this is attributed to antibiotic treatment and few people getting recovered naturally. It was found that there is a peak for MU spread, which subsequently reduces as more susceptible get awareness of the disease in both two and three dimensional plots.
A thesis submitted to the Department of Mathematics, Kwame Nkrumah University of Science and Technology in partial fufillment of the requirement for the degree of Doctor of Philosophy in Applied Mathematics