| Over the past quarter-century, considerable work has been invested in the study of the Human Immunodeficiency Virus (HIV). Within the mathematical arena, numerous models have been developed to reflect various phenomena associated with the virus. We construct a new ordinary differential equation model for the evolution of the CD4+ T cell population---the white blood cells principally targetted by the virus---in the presence of HIV, incorporating mutation of the wild-type virus and virus response to imperfect drug therapy. In so doing, we make the investigation of the model more tractable by eliminating an explicit reference to the virus population itself. We analyse this model both from a dynamical systems perspective and via numerical simulation, and show that the only possible long-term behaviours are the elimination of both forms of the virus, the elimination of the wild-type virus only, or the co-existence of both virus strains with the uninfected T cell population. We generalise this model to investigate the presence of multiple mutations, and demonstrate that the behaviour of this augmented model reduces naturally to the single-mutant case. Finally, we consider the possibility of imperfect adherence to drug therapy by the patient, by introducing impulsive differential equations into the original model. We determine the impulsive periodic orbits of this model and inspect it numerically. Finally, we use this impulsive model to consider different frequencies and patterns of non-adherence on the part of the HIV sufferer. We determine that as interruptions to drug therapy occur more closely together, they become less harmful to the patient with regard to the progression of the virus. |
