  We consider the standard three-component differential equation model for the
growth of an HIV virion population in an infected host in the absence of drug
therapy. The dynamical properties of the model are determined by the set of
values of six parameters which vary across host populations. There may be one
or two critical points whose natures play a key role in determining the outcome
of infection and in particular whether the HIV population will persist or
become extinct. There are two cases which may arise. In the first case, there
is only one critical point P_1 at biological values and this is an
asymptotically stable node. The system ends up with zero virions and so the
host becomes HIV-free. In the second case, there are two critical points P_1
and P_2 at biological values. Here P_1 is an unstable saddle point and P_2 is
an asymptotically stable spiral point with a non-zero virion level. In this
case the HIV population persists unless parameters change. We let the parameter
values take random values from distributions based on empirical data, but
suitably truncated, and determine the probabilities of occurrence of the
various combinations of critical points. From these simulations the probability
that an HIV infection will persist, across a population, is estimated. It is
found that with conservatively estimated distributions of parameters, within
the framework of the standard 3-component model, the chances that a within host
HIV population will become extinct is between 0.6% and 6.9%. With less
conservative parameter estimates, the probability is estimated to be as high as
24%. The many factors related to the transmission and possible spontaneous
elimination of the virus are discussed.
