HIV or Human Immunodeficiency virus describes a viral infection most commonly found in humans which infects and destroys the immune system of the host. In this report we examine several simple mathematical models for the spread of the HIV virus. In our report we define a parameter R0, such that R0 =βλ k/(adu), as the reproductive ratio of the viral infection. This condition is suggestive of whether or not a viral infection will establish. In later models we explore the effects of both immune response and anti-viral drug therapy on the reduction of viral load within an individual.
We also investigate anomalies that occur within an individual's particular immune response. Most notably the situation that describes the event in which a given immune response can never ultimately eradicate a viral infection, but remains limited by its own constraints. In this report we also show that with the onset of anti-viral drug therapy the population abundance of free virus particles is massively reduced within infected individuals. As a result of this investigation we wish to show that simple mathematical techniques can be applied to this model to obtain important results and allow us to understand the behaviour of the virus, and to propose methods of combating the viral infection.