## 4 Simulation Model of Metro Congestion Propagation Process
### Quantify Method for Congestion Propagation Rate  
Although the ASEIR model is capable of describing the process of congestion propagation caused by the metro emergencies in the metro network, it is still worth noting that the congestion propagation rate in the aforementioned SIR model and ASEIR model are constant to different stations and ridership situations. It seems unreasonable that the congestion in the metro network spread through different stations at the same speed. Intuitively, stations adjacent to where the metro emergency occurs are more susceptible to the congestions. Meanwhile, they will be the first to become congested stations that further spread the congestion to the entire network. Transfer stations are also vulnerable under this particular condition where the transfers between different lines will be much more frequent than usual. [PERSON] et al. [52] and [PERSON] [53] quantified the metro congestion propagation rate with aggregated ridership data in the framework of the SIR model.  
Figure 9: Stations’ states in ASEIR simulation.  
However, another vital element should be taken into account that passengers routing behaviors also have significant influences on the propagation of congestion. Annoyed and may be frustrated by the metro emergencies, the stations on the route that the influenced passengers choose, are the ones that more easily generate congestions alongside the stations on the route. Not to mention that the normally functioning metro lines have already packed with passengers, once encounter the metro emergency influenced passengers, the congestions generate inconvenience to both metro operators and passengers in the crowded underground space.  
Therefore, in order to dynamically simulating the congestion propagation process, with the analyzed elements above, the infectious rate \(\lambda_{ij}^{k}\) of each station (\(i,j\)) on route \(k\) can be described by Equation (17)  
\[\lambda_{ij}^{k}=1-\exp\left[-\left(\frac{H_{ij}^{k,w}}{\sum\limits_{w}\left( H_{ij}^{k,w}+H_{ij}^{w,k}\right)}\cdot x_{1}+\frac{C_{ij}^{k}-V_{ij}^{k}+O_{ij}-I_{ ij}}{C_{ij}^{k}}+\frac{DT_{k}}{T_{k}}\cdot x_{2}+(1-x_{2})RSC_{k}\right)\right] \tag{17}\]  
where, \(H_{ij}^{k,w}\) is the transfer ridership switching route \(k\) to route \(w\) in transfer station (\(i,j\)), on the contrary, \(H_{ij}^{w,k}\) is the transfer ridership switch from route \(w\) to route \(k\) in transfer station (\(i,j\)). \(x_{1}\) is 1 when the station (\(i,j\)) is a transfer station, otherwise \(x_{1}\) equals to 0. \(C_{ij}^{k}\) is maximum transportation capability on the section before station (\(i,j\)) on route \(k\). \(V_{ij}^{k}\) is the current section flow on the section before station (\(i,j\)) on route \(k\). \(O_{ij}\) and \(I_{ij}\) are outbound ridership and inbound ridership in the station (\(i,j\)). \(x_{2}\) equals 1 when route \(k\) is the shortest route of travel time (route \(l\)) among all the routes between a certain origin \(o\) and destination \(d\), otherwise, \(x_{2}\) equals 0.  
The first term in the exponential terms, the transfer rate, is determined by passengers' route choices where the influenced passengers could choose to switch their routes to avoid further disturbance of the metro emergency-caused delays. With the estimated parameters in Table 2, the product of route \(k\)'s choice probability \(P(k|C)\) and the ridership \(Rd_{od}\) between \(o\) and \(d\) can be used to represent the ridership on route \(k\), \(G_{ad}^{k}\), between origin \(o\) and destination \(d\), as shown in Equation (18)  
\[G_{od}^{k}=P(k|C)\cdot Rd_{ad} \tag{18}\]  
Therefore, the transfer ridership can be calculated by Equation (19)  
\[H_{ij}^{k,w}=\frac{C_{od}^{k}}{N_{ad}} \tag{19}\]