Mathematical models for the study of buruli ulcer dynamics in Ghana
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Date
2015-03-31
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Abstract
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
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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.
Description
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