## 5 Conclusions  
In this paper, we use the method of cusp catastrophe theory to model the regional traffic flow in high-density areas under sudden fire disasters. The mathematical modeling method is proposed to study the changing process of traffic flow from the stable state to instable state based on cusp catastrophe theory to improve the understanding of traffic flow aggregation, propagation and distribution characteristics in disaster affected area under sudden fire disasters. This paper explores the catastrophe border of regional traffic flow as to provide a theoretical basis for traffic agents to manage and control regional traffic flow in disaster-affected area, to prevent the regional traffic flow being caught into disorder and congestion. Finally, we use the microcosmic traffic flow simulator VISSIM to simulate and collect the traffic data in high-density area under sudden fire disasters. Throughout the data normalization and analysis, the catastrophe border is identified as 439 pcu/lane/h, 529 pcu/lane/h, 377 pcu/lane/h at 5 s, 10 s, 15 s data collection interval respectively. The relative precision, which compares to the method of capacity assessment approach, is 89.1%, 92.7% and 76.5% respectively. It means that 10 s data collection interval may be the most suitable data collection interval in catastrophe border identification and regional traffic flow control in high-density area under sudden fire disasters. The low accuracy at 15 s data collection interval may infer that the stable state of traffic flow catastrophe changing to instable state may be happen in short time, coarse granularity data collection interval cannot capture the transformation process accurately.  
However, there are some topics remain to be studied, further research work includes: (1) The aggregation, propagation mechanism of reginal traffic flow under sudden disasters and their influence factors should be identified and analyzed. The changing of spatial-temporal distribution during the sudden disaster should be considered in the mathematical model. (2) In high-density areas of urban central, the types of land uses, traffic networks, and transportation patterns are different. In this paper, we just consider the vehicle stream in a simple Sudoku traffic network. Pedestrian flow and its disturbances to traffic flow should be considered and analyzed in a real traffic network. (3) At present, only the simulator-based tests are performed, the field traffic data in high-density area before, during and after sudden fire disasters should be carried out in the future. (4) The identified catastrophe border should be applied into traffic signal timing optimization, explores different types of signal timing (e.g., fixed-time, actuated control, adaptive control) or control model (e.g., isolated signal control, arterial coordination control, regional coordination control) impacting on the catastrophe border identification.  
In addition, how to match the signal timing with different traffic flow catastrophe borders should be simulated and calibrated to prevent the unstable state and disorder of traffic flow timely and effectively.  
Conceived and designed the experiments: [PERSON], [PERSON] Performed the experiments: B.G., Y.Y. Analyzed the data: [PERSON], Y.Y. Contributed reagents/materials/analysis tools: [PERSON], Y.Y., D.W., B.G. Wrote the paper: [PERSON], [PERSON], [PERSON] All authors have read and agreed to the published version of the manuscript.  
The authors acknowledge the National Natural Science Foundation of China (Grant No. 51408257 and 51308248), Youth Scientific Research Fund of Jilin (Grant No. 20180520075 JH), and the Science and technology project of Jilin Provincial Education Department (Grant No. JJKH20170810 KJ and JJKH20180150 KJ) are partly support this work.  
We are very thankful to the reviewers for their time and efforts, their comments and suggestions have greatly improved the quality of this paper.  
The authors declare no conflict of interest.