  In this paper, we investigate a novel 3-compartment model of HIV infection of
CD4$^+$ T-cells with a mass action term by including two versions: one baseline
ODE model and one delay-differential equation (DDE) model with a constant
discrete time delay. Similar to various endemic models, the dynamics within the
ODE model is fully determined by the basic reproduction term $R_0$. If $R_0<1$,
the disease-free (zero) equilibrium will be asymptotically stable and the
disease gradually dies out. On the other hand, if $R_0>1$, there exists a
positive equilibrium that is globally/orbitally asymptotically stable within
the interior of a predefined region. To present the incubation time of the
virus, a constant delay term $\tau$ is added, forming a DDE model. In this
model, this time delay (of the transmission between virus and healthy cells)
can destabilize the system, arising periodic solutions through Hopf
bifurcation. Finally, numerical simulations are conducted to illustrate and
verify the results.
