This paper addresses the asymptotic stability of a continuous-time deterministic car-following model with respect to different driving strategies. More precisely, the model takes into account a constant time headway driving behavior as well as the presence of multiple decision-making and actuation delays in describing the drivers' actions. The stability analysis of the corresponding derived models is challenging and we will focus on characterizing the stability regions in the delay parameter space. Such a problem depicts some interesting properties in terms of delays that simplify the overall analysis leading to some simple frequency-sweeping based algorithms as well as to various practical interpretations of the results in terms of drivers' behaviors. Illustrative examples complete the presentation.