## 4 Simulation Model of Metro Congestion Propagation Process
### ASEIR Model with Time Delays
The fraction of susceptible stations in ASEIR model:  
\[S_{ij}^{t+\Delta t}=S_{ij}^{t}-\lambda^{\varepsilon}S_{ij}^{t}I_{ij}^{t}-\sum_{(a,b)\in V}\frac{N_{ab}}{N_{ij}}\lambda^{\varepsilon}S_{ij}^{t}I_{ab}^{t} \tag{11}\]  
Exposed stations:  
\[ \begin{split} E_{ij}^{t+\Delta t}=E_{ij}^{t}+\lambda^{\varepsilon} S_{ij}^{t}I_{ij}^{t}+\sum_{(a,b)\in V}\frac{N_{ab}}{N_{ij}}\lambda^{ \varepsilon}S_{ij}^{t}I_{ab}^{t}\\ -\left(\lambda^{\varepsilon}S_{ij}^{t-\tau}I_{ij}^{t-\tau}+\sum_{( a,b)\in V}\frac{N_{ab}}{N_{ij}}\lambda^{\varepsilon}S_{ij}^{t-\tau}I_{ab}^{t-\tau} \right)-\varphi E_{ij}^{t}\end{split}  \tag{12}\]  
Infectious stations:  
\[I_{ij}^{t+\Delta t}=I_{ij}^{t}+\left(\lambda^{\varepsilon}S_{ij}^{t-\tau}I_{ij }^{t-\tau}+\sum_{(a,b)\in V}\frac{N_{ab}}{N_{ij}}\lambda^{\varepsilon}S_{ij}^ {t-\tau}I_{ab}^{t-\tau}\right)+\varphi E_{ij}^{t}-\varepsilon I_{ij}^{t} \tag{13}\]  
Due to the ridership in the metro network in time \(t\) is constant, the total amount of the stations in four states is constant too, so, \(S_{ij}^{t+\Delta t}+E_{ij}^{t+\Delta t}+I_{ij}^{t+\Delta t}+R_{ij}^{t+\Delta t }=S_{ij}^{t}+E_{ij}^{t}+I_{ij}^{t}+R_{ij}^{t}=1\), and the fractions of recovered stations can be calculated just using Equation (8). \(\lambda^{\varepsilon}\) is the congestion propagation rate in the ASEIR model, which changes susceptible stations into exposed stations. \(N_{ab}\) is the ridership transferred from station \((a,b)\) to station \((i,j)\), where station \((a,b)\) is the neighbor stations of station \((i,j)\). \(N_{ij}\) is the total ridership of station \((i,j)\). \(\varphi\) is the transformation rate for station states from exposed to infectious in per time step. \(\varepsilon\) is the recovery rate of stations changing from infectious to recovered. Unlike epidemic propagation, non-congested (susceptible) stations could transform into congested (infectious) stations directly due to particular conditions, like adjacent stations, particularly crowded space in the stations, and outburst passengers. Figure 6 shows the state transformation process in the ASEIR model in the congestion propagation. In order to compare the simulation results of the ASEIR model, we also implemented a more commonly used SIR model to simulate the congestion propagation in the metro network. SIR model consists of only three states, susceptible (S), infectious (I) and recovered (R). The propagation process in the SIR model is straight forward, where susceptible nodes will become infectious with a propagation rate once it contacts with infectious nodes in the network. Then, following a certain recover rate, the infectious nodes will become recovered nodes that will be immune to the epidemic forever. SIR model can be defined as \(Q_{ij}^{t}=\left(S_{ij}^{t},I_{ij}^{t},R_{ij}^{t}\right)\) in cellular space, and since the stations are consist of these 3 states, then the proportion of each state follows \(S_{ij}^{t}+I_{ij}^{t}+R_{ij}^{t}=1\). Moreover, without considering  
Figure 6: Congestion propagation process in the metro network using the ASEIR model. the postponed delays, the station state transformation function of the SIR model can be expressed as Equations (14) to (16). Fractions of susceptible stations in the SIR model:  
\[S_{ij}^{t+\Delta t}=S_{ij}^{t}-\lambda S_{ij}^{t}t_{ij}^{t}-\sum_{(a,b)\in V} \frac{N_{ab}}{N_{ij}}\lambda S_{ij}^{t}t_{ab}^{t} \tag{14}\]  
Infectious stations:  
\[I_{ij}^{t+\Delta t}=I_{ij}^{t}-\varepsilon I_{ij}^{t}+\lambda S_{ij}^{t}I_{ij} ^{t}+\sum_{(a,b)\in V}\frac{N_{ab}}{N_{ij}}\lambda S_{ij}^{t}I_{ab}^{t} \tag{15}\]  
Recovered stations:  
\[R_{ij}^{t+\Delta t}=R_{ij}^{t}+\varepsilon I_{ij}^{t} \tag{16}\]  
where, \(S_{ij}^{t+\Delta t}+I_{ij}^{t+\Delta t}+R_{ij}^{t+\Delta t}=S_{ij}^{t}+I_{ij}^ {t}+R_{ij}^{t}=1\), \(\lambda\) is the transformation rate of infectious stations.