System-level stability and threshold dynamics of HIV-CTL interactions with intracellular and immune-activation delays

We formulate and analyze a within-host HIV-CTL model with two biologically motivated discrete delays: an intracellular eclipse delay associated with viral production and a delay in cytotoxic T-lymphocyte activation. While each delay has been studied separately, their combined influence on threshold structure and delay-dependent stability has not been systematically examined in a unified framework. We establish positivity and boundedness of solutions, characterize the infection-free and endemic equilibria, and derive the basic reproduction number together with delay-dependent stability conditions and a Hopf-type transition criterion for the endemic state. Numerically, we perform coarse and high-resolution delay sweeps and construct a two-parameter stability map using viral-load oscillation amplitude as the main diagnostic output. Under the biologically calibrated baseline parameter set, solutions converge to a non-oscillatory chronic state across the physiologically relevant delay ranges explored, and the computed oscillation amplitudes remain numerically negligible. Sensitivity screening and robustness checks further indicate that the infection rate and the CTL activation rate are the dominant drivers of long-term viral burden. These results clarify how intracellular and immune-activation delays shape HIV-CTL dynamics, while showing that delay-induced instability, although possible in principle, is not numerically detected in the baseline regime considered here.