## 4 Simulation Model of Metro Congestion Propagation Process
### ASEIR Model with Time Delays  
Initially, the traditional SEIR model is used to depict epidemic propagation through complex networks, which contains 4 states: susceptible (S), exposed (E), infectious (I) and recovered (R). Instead of becoming into infectious nodes directly, some susceptible nodes contacted with infectious nodes will become exposed nodes first, before turning into infectious. The exposed nodes need an incubation period to obtain the ability to infect other susceptible nodes, and before the end of its incubation period, the exposed nodes cannot infect nodes connected with them. Therefore, in the SIR model, the congestion is directly transited through the network, whereas the influences of the emergency-caused delays are postponed by the exposed stations in SEIR models. When the traditional SEIR model is used to simulate travel congestion propagation, the infectious nodes are referred to as the congested to stations or links in the metro network. Once the emergency occurs while metro passengers are traveling inside the metro system, there is another key point to consider that the congestions propagate could be influenced by the network topology. Therefore, in the cell space of the graph-CA environment, the states of station \((i,j)\) in time \(t\) of the SEIR model can be defined as \(Q_{ij}^{t}=\left(S_{ij^{t}}^{t},E_{ij^{t}}^{t},l_{ij^{t}}^{t},R_{ij}^{t}\right)\). Moreover, \(S_{ij}^{t}\) represents the proportion of susceptible metro stations, where susceptible stations are easily transformed into exposed when connected to infectious stations in the metro network. \(E_{ij}^{t}\) is the proportion of exposed (congested) stations, while the exposed stations are unable to spread the congestions to other susceptible stations in the metro network. \(I_{ij}^{t}\) is the proportion of infected stations, and \(R_{ij}^{t}\) is the proportion of recovered stations that have got resistant to the congestions. Since the states of metro stations are all divided into 4 states in the SEIR model, then,  
\[S_{ij}^{t}+E_{ij}^{t}+I_{ij}^{t}+R_{ij}^{t}=1 \tag{8}\]  
We have to note that, on one hand, CA models are mathematic methods used to simulate discrete dynamic systems, therefore, in the metro system we investigated, the states of stations are also discrete between time \(t\) and time \(t+\Delta t\), where \(\Delta t\) is the time step we used to simulate the congestion propagation in the metro network. In our research, \(\Delta t\) equals the average section travel time (i.e., the average travel time on a link between two metro stations), which is 2 min averagely in GZM. On the other hand, the delay-caused incubation time is related to the conditions of each metro station's properties, where the stations' various facilities like platforms, staircases, and passageways are distinct from each other. To implement the differences of incubation time in different stations caused by the same metro emergency, we adjusted the traditional SEIR model (i.e., ASEIR model) by assuming the incubation time \(\tau\) follows gamma distribution, that is, \(\tau\ \sim\ \Gamma(\alpha,\ \beta)\), where \(\alpha\) is called shape parameter, and \(\beta\) is scale parameter. With a continuous random variable X, and \(\alpha>0\), \(\beta>0\), then, the probability distribution function can be given by Equation (9):  
\[f_{X}(x)=\left\{ \begin{array}{cc}\frac{\beta^{\alpha}x^{\alpha-1}e^{-\beta x }}{\Gamma(\alpha)},&x>0\\ 0&,\ \ otherwise\end{array} \right. \tag{9}\]  
where \(\Gamma(\alpha)\) is the Gamma function, it is a factorial function of positive integers, \(\Gamma(\alpha)=(\alpha-1)!\) However, to any positive number \(\alpha>0\), \(\Gamma(\alpha)\) can be calculated using Equation (10):  
\[\Gamma(\alpha)=\int_{0}^{\infty}x^{\alpha-1}e^{-x}dx \tag{10}\]  
Therefore, the ASEIR model with time-postponed \(\tau\) in the exposed state could be calculated by using Equations (11) to (13).