  We make a mathematical analysis of an age structured HIV infection model with
both virus-to-cell and cell-to-cell transmissions to understand the dynamical
behavior of HIV infection in vivo. In the model, we consider the proliferation
of uninfected CD4+ T cells by a logistic function and the infected CD4+ T cells
are assumed to have an infection-age structure. Our main results concern the
Hopf bifurcation of the model by using the theory of integrated semigroup and
the Hopf bifurcation theory for semilinear equations with nondense domain.
Bifurcation analysis indicates that there exist some parameter values such that
this HIV infection model has a non-trivial periodic solution which bifurcates
from the positive equilibrium. The numerical simulations are also carried out.
