  In this work we introduce a differential equation model with time-delay that
describes the three-stage dynamics and the two time scales observed in HIV
infection. Assuming that the virus has high mutation and rapid reproduction
rates that stress the immune system throughout the successive activation of new
responses to new undetectable strains, the delay term describes the time
interval necessary to mount new specific immune responses. This single term
increases the number of possible solutions and changes the phase space dynamics
if compared to the model without time delay. We observe very slow transits near
the unstable fixed point, corresponding to a healthy state, and long time decay
to the stable fixed point that corresponds to the infected state. In contrast
to the results obtained for models using regular ODE, which only allow for
partial descriptions of the course of the infection, our model describes the
entire course of infection observed in infected patients: the primary
infection, the latency period and the onset of acquired immunodeficiency
syndrome (AIDS). The model also describes other scenarios, such as the very
fast progression to the disease and the less common outcome in which, although
the patient is exposed to HIV, he/she does not develop the disease.
