  In this paper we study a nonlinear infection viral propagation model with
diffusion, in which, the left boundary is fixed and with homogeneous Dirichlet
boundary conditions, while the right boundary is free. We find that the habitat
always expands to the half line $[0, \infty)$, and that the virus and infected
cells always die out when the {\it Basic Reproduction Number} $\mathcal{R}_0\le
1$, while the virus and infected cells have persistence properties when
$\mathcal{R}_0>1$. To obtain the persistence properties of virus and infected
cells when $\mathcal{R}_0>1$, the most work of this paper focuses on the
existence and uniqueness of positive equilibrium solutions for subsystems and
the existence of positive equilibrium solutions for the entire system.
