# Network Analysis to Identify the Risk of Epidemic Spreading
## 2 Materials and Methods
### Investigation of Epidemic Status Based on the Monte Carlo Method  
The epidemic simulation was performed for a time series event by constructing epidemic status matrix (\(z\)) to represent the status of the \(n\)th node at time step \(t\). For each node, the value of epidemic status matrix at time step t can be 0, 1, or 2, indicating that a node is susceptible, infective, or recovered, respectively. We initially (\(t=0\)) set every value of epidemic status matrix to 0 because all nodes are susceptible before the epidemic spreads. At the initial infection stage, randomly selected 0.05% of nodes were infected. At every time period, we performed immunization and observed the infection stages (Figure 3).  
At the immunization stage, we identified infective nodes and determined whether these nodes would be recovered in the next time step. To calculate the transition probability of infected and recovered phenomena, the Monte Carlo method was applied [35; 36]. When infection and recovery parameters are provided, it is possible to investigate whether a node transitions from an epidemic state to another state. To accomplish this, we compared the method revealing the change in each population in every compartment over time (Figure 4).  
Figure 3: Conceptual layout for status transition of nodes based on the Monte Carlo method. Status transition is determined by comparing the epidemic parameters (\(\beta\), \(\gamma\)) to random numbers (\(X\), \(Y\)) extracted from a uniform distribution.  
The final steady state of the epidemic spreading simulation model indicates the total number of casualties of the epidemic who either are dead or have recovered from the disease. Infective nodes at time \(t\) (\(z_{n}\left[t\right]=1\)) are transformed to recovered nodes at time \(t+1\) (\(z_{n}\left[t+1\right]=2\)) when \(1/\gamma_{d}\) is larger than a random real number between 0 and 1. We determined whether the neighbor nodes of the infection node would be infected by identifying susceptible nodes adjacent to the infective nodes at time \(t\) (\(z_{n}\left[t\right]=0\), with the adjacent infective node) (Figure 5). When \(\beta\) is larger than a random real number between 0 and 1, a susceptible node becomes an infective node at time \(t+1\) (\(z_{n}\left[t+1\right]=1\)); this scenario represents epidemic spread. For each time step, we recorded the number of susceptible, infective, and recovered nodes during epidemic spread.  
Figure 4: Time series analysis of an epidemic population of susceptible, infective, and recovered nodes. This example depicts the change of the susceptible, infective, and recovered proportion in a population with time as extracted from simulations with a scale-free network with \(k=5\) and infectious disease parameters \(\beta=0.35\) and \(\gamma=1\). Initially, a majority of the population is susceptible, whereas the remaining few are infective nodes. During the epidemic, infective nodes transmit their disease to susceptible nodes, and some of the infective nodes recover. Every infective node will recover after about \(\gamma_{d}\) steps. Eventually, two major compartments remain, that is, susceptible nodes and recovered nodes, without any infective nodes. No additional infective nodes are included because there are no infective nodes to transmit the disease. The recovered population includes both the immune living population and individuals who have died of the disease. Accordingly, the final steady state of the green curve indicates the total number of casualties. The fatality rate determines the ratio of the final population of deceased or recovered individuals. We assume that the fatality rate is 100% for the worst-case scenario, that is, extinction.  
Figure 5: Conceptual layout for the transmission of contagion between nodes based on the Monte Carlo method. When the transmission rate (\(\beta\)) is larger than a random real number (\(X\)), a susceptible node (blue) becomes an infective node (red) at time \(t+1\).