Abstract:
The thesis addresses two main problems. The first is that of designing
reliable numerical method for approximating an SIS (susceptible-infectedsusceptible)
disease transmission model with discrete time delay. This is
achieved by using the theory and methodology of nonstandard finite difference
discretization which leads to a novel and robust numerical methods
which, unlike many other standard numerical integrators, were shown to be
dynamically consistent with the continuous delay SIS model.
The second problem is the mathematical modeling of the transmission
dynamics of bovine and mycobacterium tuberculosis in a human-buffalo population.
The buffalo-only component of the resulting deterministic model
undergo the phenomenon of backward bifurcation, due to the re-infection of
exposed and recovered buffalos. Furthermore, this sub-model has a unique
endemic equilibrium point which is shown to be globally asymptotically stable
for a special case, whenever the associated reproduction number exceeds
unity. Uncertainty and sensitivity analyses, using data relevant to the dynamics
of the two diseases in the Kruger National Park, South Africa, show
that the distribution of the associated reproduction number is less than unity
(hence, the diseases would not persist in the community). Crucial parameters
that influence the dynamics of the two diseases are also identified. The
human-buffalo model exhibit the same qualitative dynamics as the sub-model
with respect to the local and global asymptotic stability of their respective
disease free equilibrium, as well as the backward bifurcation phenomenon. Numerical simulations for the human-buffalo model show that the cumulative
number of mycobacterium tuberculosis cases in humans (buffalos) decreases
with increasing number of bovine tuberculosis infections in humans
(buffalos).
