## 1 Introduction
This system removes the traffic discontinuities with a second order spatial derivative of speed. This is a mathematical approach. Further, this system introduced a constant to effectively model driver anticipation [49]. [PERSON] et al. contemplated the road slope during traffic alignment with a constant driver sensitivity. This is an inadequate approach, as the driver is not constant for different traffic conditions and can provide unrealistic traffic prediction [18]. The forward looking effect is the perception of a driver to changes ahead. Whereas, with backward impact, a driver accelerates from the approaching vehicles behind to avoid accidents [50]. In 2008, [PERSON] et al. [51] considered the backward and forward impact with a constant driver sensitivity. Both the effects are devised with two different velocity systems. In 2018, [PERSON] et al. [52] developed a system that considers the backward looking effect in traffic. This system considers driver sensitivity with a constant greater than 0. According to this system, the speed during congestion reduces with a large driver sensitivity, whereas at a negligible driver sensitivity, the traffic follows average velocity. In 2023, [PERSON] et al. [53] developed a system for connected and autonomous vehicles for anticipation. This system considered headway and vehicle torque. Based on the given literature, the variable driver sensitivity for changes in the distance headway at transition in the second order traffic flow system has not been characterized. Therefore, in this paper, the impact of changes in distance headway between vehicles and driver sensitivity is considered in devising the second order (two-equation) traffic system. In this paper, the main contributions are  
1. The Payne Whitham system is improved. Unlike the constant \(A\) in the PW system, the changes in traffic transitions are based on the driver sensitivity, safe headway, and transition headway. This is significant, as the traffic evolution at transitions is influenced by these physical parameters. This evolution is rapid for larger driver sensitivity. Conversely, the evolution is slow for a larger distance headway and smaller sensitivity. Thus, the spatio-temporal traffic evolution with the proposed system comprehends driver sensitivity, transition distance, and safe distance headway. These parameters are based on the traffic physics. 2. The driver sensitivity is obtained from the data acquired over a 200 m road section. A regressive relation for velocity and traversed time is obtained, which helps in the driver sensitivity characterization. This is incorporated in the proposed system for accurate traffic evolution. 3. It has been proved that the proposed traffic system is hyperbolic. Unlike the PW system, the changes during congestion and smooth flow, respectively, down and up,are comprehended by the changes in transition distance about the safe headway and driver sensitivity. 4. The proposed system is assessed with the PW traffic system over a 2000 m circular road. The transition in density is considered at multiple locations with large changes in density over a circular road. At a location, density is 1, that is, 100%; 1 denotes that a road is at full capacity. This is the worst traffic scenario, and a system with a minor deviation from traffic physics can provide the density above the maximum and velocity below 0. This evolution is inappropriate traffic characterization, and the system cannot characterize the realistic practical traffic conditions during congestion. The performance results in this paper demonstrate that proposed traffic system can appropriately characterize traffic evolution as it considers the traffic physics and practical parameters. The traffic velocity with the proposed system does not go below 0 or above the maximum. The traffic density stays within the employed range. The velocity distribution is given in Section 2, and Section 3 gives proposed system. The hyperbolicity is demonstrated in Section 4.