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Abstract: For HIV patients, lifelong therapy is generally required to control their viral replication and it is very difficult to effectively eradicate HIV infection from the host. In this paper, using a mathematical model, we show that the stimulation of CD4+ T cells to proliferate in the presence of HIV is one possible factor causing such a difficulty. This is based on the fact that if the proliferation rate is non-negligible, then the model exhibits a backward bifurcation and hysteresis. Fitting to clinical data shows that the CD4+ T-cell proliferation does play an important role for HIV infection under antiretroviral therapy. The model and analysis are then extended to incorporate a bilinear term to account for the loss of free infectious virions during attacking the target cells. Our analysis suggests that, to effectively control HIV infection, antiretroviral therapies should aim at increasing the effectiveness of reverse transcriptase inhibitors and protease inhibitors, discouraging the stimulation of CD4+ T cells, and increasing the loss rate of virus particles when they infect the healthy CD4+ T cells. [Copyright &y& Elsevier]