Microscopic hysteretic traffic model and stop-and-go waves

A microscopic particle model for hysteretic vehicular traffic flow is considered. The model is a system of possibly infinitely many ODEs for the positions of the cars, whose velocity depends not only on the discrete density in relation to the car ahead, but also on the sign of the acceleration, allowing hysteresis behaviors. This model's traveling wave profiles lead to a system of infinitely many delayed differential equations. Necessary and sufficient conditions are provided for the existence of traveling wave profiles in this microscopic hysteretic traffic model. The traveling wave profile is unique up to a shift. Both deceleration and acceleration shock profiles are observed, aligning with real-world traffic flow patterns. The existence of these shock types allows the construction of stop-and-go wave solutions.