A close look at the viral reduction rate in target cell limited models

The target cell limited (TCL) model is a cornerstone of within-host viral dynamics and is widely used to interpret viral load (VL) data and assess antiviral efficacy. We examine how TCL models predict VL reduction under drug treatments, thereby focusing on parameters that govern infection and viral production. Our simulations and analyses reveal a non-monotonic relationship: moderate drug effects can slow VL decline and prolong persistence above the limit of quantification. We provide a rigorous analytical characterization of the conditions under which this non-monotonic behavior arises. We derive an explicit formula for the critical parameter value that delineates monotonicity and show that this phenomenon broadly arises in generalized TCL models. Additionally, we propose a modified formulation with a sigmoid infection term, which reduces the extent of non-monotonicity while preserving the analytical tractability. These findings contribute to a deeper understanding of the TCL model behavior and provide a practical alternative formulation when a monotonic dose-response relationship is desired.