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There are two main components of the immune system that are involved: Antibodies and T-cells.
T-cells are responsible for activating white blood cells to help fight off infection. These are the cells the HIV infects. Antibodies bond to the HIV virus and try to neutralize them before they can infect more cells. Unfortunately as the HIV virus replicates, there is too much virus and not enough antibodies to neutralize.
When a T-cell becomes infected with HIV, the virus takes over and causes it to start replicating more HIV. This eventually causes the T-cell to die. So T-cells keep dying while HIV loads keep increasing. When enough T-cells die, the infected person is considered to have AIDS.
It is particularly difficult for the immune system to fight off HIV infection for a number of reasons, including the following:
So conclusion, our bodies can fight HIV for awhile. It's just a matter of time before the viral load outweighs the capabilities of our immune system.
For an understandable explanation: The human body can defend against HIV - hence why an HIV infection does not immediately develop AIDS, and in fact for a long time the virus cannot be detected in blood. This is because the immune response is keeping them down, but slowly the virus works its way through the cells, proliferates and kills more and more T cells, until the immune response is too weak. That's when AIDS develops.