## 5 Empirical Study
The infectious stations increase quickly and obvious double peaks appear before time step 40, which is consistent with the analysis we made in Section 4. The first maximum value of infectious stations appears at time step 10 (about 20 min), and, coincidentally, the assumption of delay time is 20 min. The variation tendency of exposed stations is in line with infectious stations, which indicates that congestion spread rapidly in the metro network during the time of emergency-caused delays. At about 35 time steps (about 50 min after the first extreme value), as the delay has already ended, the number of exposed stations and infectious stations have another increase and reaches a similar value of the first peak value. Moreover, at time step 49 (less than 2 h), all the stations in the network have been influenced by the congestion, because the amount of susceptible stations has become 0. During nonpeak periods, the intensity of congestion propagation in the network is obviously moderate comparing with peak periods, and all of the susceptible stations are transferred into exposed stations or infectious stations at time step 60, which is clearly slower than that in peak periods. Moreover, the number of infectious stations reaches its maximum only once, at time step 60, and it is later than peak periods as well. While during nonpeak periods, it only takes about 100 time steps (less than 3 and a half hours), if there are no intervention measures. Therefore, during nonpeak periods, the duration of congestion propagation is also shorter than the duration in peak periods. In order to investigate the proportions of each state in the entire propagation process, Figure 12 shows the simulation results of all the stations, and each horizontal stripe represents a station's state. We have to note that as transfer stations contain more than one virtual station, the states of virtual stations determine the state of the transfer station, where the state of infectious is the highest that if  
Figure 11: Congestion propagation during peak and nonpeak periods. one virtual station turns into infectious, then the transfer station is infectious. The priority of the rest states are as follows: exposed > recovered > susceptible. In Figure 12, there are 3 stations (QZ station, Zoo station, and YJ station) congested in the beginning, and these stations in the interrupted area are adjacent to each other. Due to the high congestion propagation rate during peak periods, the exposed (congested) state in most of the stations are very short, or even in some stations, they just omit the exposed state and immediately transfer into infectious from the susceptible state, such as HS station, Yide Road station, etc. Meanwhile, with more passengers abandon the metro, which diminished the spread of congestion, the duration for stations in the state of congest is shorter than nonpeak periods, as shown in Figure 12b. The detailed congestion propagation process can be seen in Figure 13, where the green nodes and lines are standing for susceptible stations and links, yellow ones represent exposed stations (and links), the red ones are infectious (congested) and the gray stations and links are in the recovered state. Figure 12: Congestion states in the GZM network. Figure 13: Congestion propagation process in GZM. From the analysis above, we can get a clear picture of the metro congestion propagation process under emergencies. Without intervention measures, congestion caused by emergencies could last for over 3 h in the metro network. During peak hours, congestion spread more intensively both in duration and in scale. Due to a higher transmission intensity, a double peak phenomenon appears in the peak periods. While the congestion propagation during nonpeak periods is much moderate and the congestion only shows one peak in the entire propagation process.