>Magister Scientiae - MSc / The influx of infecteds into a population plays a critical role in HIV transmission. These infecteds are known to migrate from one region to another, thereby having some interaction with a host population. This interactive mobility or migration causes serious public health problems. In a very insightful paper by Shedlin et al. [51], the authors discover risk factors but also beneficial factors with respect to fighting human immunodeficiency virus (HIV) transmission, in the lifestyles of immigrants from different cultural backgrounds. These associated behavioral factors with cross-cultural migrations have not received adequate theoretical a attention. In this dissertation we use the compartmental model of Bhunu et al. [6] to form a new model of the HIV epidemic, to include the effect of infective immigrants in a given population. In fact, we first produce a deterministic model and provide a detailed analysis. Thereafter we introduce stochastic perturbations on the new model and study stability of the disease-free equilibrium (DFE) state. We investigate theoretically and computationally how cross-cultural migrations and public health education impacts on the HIV transmission, and how best to intervene in order to minimize the spread of the disease. In order to understand the long-time progression of the disease, we calculate the threshold parameter, known as the basic reproduction number, R0. The basic reproduction number has the property that if R0 is sufficiently small, usually R0 < 1, then the disease eventually vanishes from the population, but if R0 > 1, the disease persists in the population. We study the sensitivity of the basic reproduction number with respect to model parameters. In this regard, if R0 < 1, we show that the DFE is locally asymptotically stable. We also show global stability of the DFE using the Lyapunov method. We derive the endemic equilibrium points of our new model. We intend to counteract the negative effect of the influx of infecteds into a population with educational campaigns as a control strategy. In doing so, we employ optimal control theory to find an optimal intervention on HIV infection using educational campaigns as a basic input targeting the host population. Our aim is to reduce the total number of infecteds while minimizing the cost associated with the use of educational campaign on [0, T ]. We use Pontryagin’s maximum principle to characterize the optimal level of the control. We investigate the optimal education campaign strategy required to achieve the set objective of the intervention. The resulting optimality system is solved numerically using the Runge-Kutta fourth order method. We present numerical results obtained by simulating the optimality system using ODE-solvers in MATLAB program. We introduce randomness known as white noise into our newly formed model, and discuss the almost sure exponential stability of the disease-free equilibrium. Finally, we verify the analytical results through numerical simulations.

Inspired by studies on the regularity of solutions to the fractional Navier-Stokes system and the impact of noise on singularity formation in hydrodynamic models, we investigated these issues within the framework of the fractional 1D Burgers equation. Initially, our research concentrated on the deterministic scenario, where we conducted precise numerical computations to understand the dynamics in both subcritical and supercritical regimes. We utilized a pseudo-spectral approach with automated resolution refinement for discretization in space combined with a hybrid Crank-Nicolson/ Runge-Kutta method for time discretization.We estimated the blow-up time by analyzing the evolution of enstrophy (H1 seminorm) and the width of the analyticity strip. Our findings in the deterministic case highlighted the interplay between dissipative and nonlinear components, leading to distinct dynamics and the formation of shocks and finite-time singularities. In the second part of our study, we explored the fractional Burgers equation under the influence of linear multiplicative noise. To tackle this problem, we employed the Milstein Monte Carlo approach to approximate stochastic effects. Our statistical analysis of stochastic solutions for various noise magnitudes showed that as noise amplitude increases, the distribution of blow-up times becomes more non-Gaussian. Specifically, higher noise levels result in extended mean blow-up time and increase its variability, indicating a regularizing effect of multiplicative noise on the solution. This highlights the crucial role of stochastic perturbations in influencing the behavior of singularities in such systems. Although the trends are rather weak, they nevertheless are consistent with the predictions of the theorem of [41]. However, there is no evidence for a complete elimination of blow-up, which is probably due to the fact that the noise amplitudes considered were not sufficiently large. This highlights the crucial role of stochastic perturbations in influencing the behavior of singularities in such systems. / Thesis / Master of Science (MSc)