Stability and bifurcation analysis of some HIV/AIDS models

The aim of this thesis is to investigate the biological dynamics between HIV, tumor and immune system cells using mathematical modelling. For this purpose, the thesis consists of two different mathematical models that describe this biological dynamic. Mathematical models are obtained through a dynamical system approach. The change of the populations in the models is described using ordinary differential equations. As a first step, the positivity of the solutions of the differential equation systems is proved, as desired in any population dynamics. Then, positive equilibrium points of dynamical systems is determined. The global stability of equilibrium points of the first system are analyzed. The basic reproduction number is determined depending on the parameters. Then, the system has been improved by integrating two different time delays into the system. Here, the dynamical system is aimed to better describe the biological dynamics. In the analyses of the delayed system, the delay termis chosen as bifurcation parameter. Then, the conditions under which fold-Hopf bifurcation will occur in the system is determined. Thus, the conditions in which periodic solutions are observed in the system have been determined. For the second system, we analyse the local stability of equilibrium points of the model. In addition, the basic reproduction number is determined depending on the parameters in the system. the existence of transcriptional bifurcation is investigated. Using the normal form theory, the normal form for the transcriptional bifurcation of the system is obtained. Similar to the point of view in the first model, the system is improved by incorporating a discreate time delay into the system. Then, tby choosing the delay term as bifurcation parameter we determine the conditions under which Hopf bifurcation will occur in the system. Thus, the conditions in which periodic solutions are seen in the system are determined. In addition, in order to determine the properties of the periodic solutions whose existence guaranteed in this way the system is reduced to the center manifold. Finally, the theoretical and analytical results obtained for all the studied systems is numerically supported and expanded using the MATLAB program.