We propose and analyse an age-structured model for within-host HIV virus dynamics which is incorporated with both virus-to-cell and cell-to-cell infection routes, and proliferations of both uninfected and infected cells in the form of logistic growth. The model turns out to be a hybrid system with two differential-integral equations and one first-order partial differential equation. We perform some rigorous analyses for the considered model. Among the interesting dynamical behaviours of the model is the occurrence of backward bifurcation in terms of the basic reproduction number R0 at R0 = 1, which raises new challenges for effective infection control. We also discuss the cause of such a backward bifurcation, based on our analytical results.