Dynamics and reliable numerics for some epidemic models with and without delay

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).