>Magister Scientiae - MSc / Tuberculosis (TB) is currently one of the major public health challenges in South Africa, and in many countries. Mycobacterium tuberculosis is among the leading causes of morbidity and mortality. It is known that tuberculosis is a curable infectious disease. In the case of incomplete treatment, however, the remains of Mycobacterium tuberculosis in the human system often results in the bacterium developing resistance to antibiotics. This leads to relapse and treatment against the resistant bacterium is extremely expensive and difficult. The aim of this work is to present and analyse mathematical models of the population dynamics of tuberculosis for the purpose of studying the effects of efficient treatment versus incomplete treatment. We analyse the spread, asymptotic behavior and possible eradication of the disease, versus persistence of tuberculosis. In particular, we consider inflow of infectives into the population, and we study the effects of screening. A sub-model will be studied to analyse the transmission dynamics of TB in an isolated population. The full model will take care of the inflow of susceptibles as well as inflow of TB infectives into the population. This dissertation enriches the existing literature with contributions in the form of optimal control and stochastic perturbation. We also show how stochastic perturbation can improve the stability of an equilibrium point. Our methods include Lyapunov functions, optimal control and stochastic differential equations. In the stability analysis of the DFE we show how backward bifurcation appears. Various phenomena are illustrated by way of simulations.

We introduce random fluctuations on contact and recovery rates in three basic deterministic models in mathematical epidemiology and obtain stochastic counterparts. This paper addresses qualitative and quantitative analysis of stochastic SIS model with disease deaths and demographic effects, and stochastic SIR models with/without disease deaths and demographic effects. We prove the global existence of a unique strong solution and discuss stochastic asymptotic stability of disease free and endemic equilibria. We also investigate numerical properties of these models and prove the convergence of the Balanced Implicit Method approximation to the analytic solution. We simulate the models with fairly realistic parameters to visualize our conclusions.

The technological changes and educational expansion have created the heterogeneity in the human species. Clearly, this heterogeneity generates a structure in the population dynamics, namely: citizen, permanent resident, visitor, and etc. Furthermore, as the heterogeneity in the population increases, the human mobility between meta-populations patches also increases. Depending on spatial scales, a meta-population patch can be decomposed into sub-patches, for examples: homes, neighborhoods, towns, etc. The dynamics of human mobility in a heterogeneous and scaled structured population is still its infancy level. We develop and investigate (1) an algorithmic two scale human mobility dynamic model for a meta-population. Moreover,the two scale human mobility dynamic model can be extended to multi-scales by applying the algorithm. The subregions and regions are interlinked via intra-and inter regional transport network systems. Under various types of growth order assumptions on the intra and interregional residence times of the residents of a sub region, different patterns of static behavior of the mobility process are studied. Furthermore, the human mobility dynamic model is applied to a two-scale population dynamic exhibiting a special real life human transportation network pattern. The static evolution of all categories of residents of a given site ( homesite, visiting sites within the region, and visiting sites in other regions) over continuous changes in the intra and inter-regional visiting times is also analyzed. The development of the two scale human mobility dynamic model provides a suitable approach to undertake the study of the non-uniform global spread of emergent infectious diseases of humans in a systematic and unified way. In view of this, we derive (2) a SIRS stochastic epidemic dynamic process in a two scale structured population. By defining a positively self invariant set for the dynamic model the stochastic asymptotic stability results of the disease free equilibrium are developed(2). Furthermore, the significance of the stability results are illustrated in a simple real life scenario that is under controlled quarantine disease strategy. In addition, the epidemic dynamic model (2) is applied to a SIR influenza epidemic in a two scale population that is under the influence of a special real life human mobility pattern. The simulated trajectories for the different states (susceptible, Infective, Removal) with respect to current location in the two-scale population structure are presented. The simulated findings reveal comparative evolution patterns for the different states and current locations over time. The SIRS stochastic epidemic dynamic model (2) is extended to a SIR delayed stochastic epidemic dynamic model(3). The delay effects in the dynamic model (3) is temporary and account for natural or infection acquired immunity conferred by the disease after disease recovery. Again, we justify the model validation as a prerequisite for the dynamic modeling. Moreover, we also exhibit the real life scenario under controlled quarantine disease strategy.In addition, the developed delayed SIR dynamic model is also applied to SIR influenza epidemic with temporary immunity to an influenza disease strain. The simulated results reveal an oscillatory effect in the trajectory of the naturally immune population. Moreover, the oscillations are more significant at the homesite. We further extended the stochastic temporary delayed epidemic dynamic model (3) into a stochastic delayed epidemic dynamic model with varying immunity period(4). The varying immunity period accounts for the varying time lengths of natural immunity against the infectious agent exhibited within the naturally immune population. Obviously, the stochastic dynamic model with varying immunity period generalizes the SIR temporary delayed dynamic.