# Network Analysis to Identify the Risk of Epidemic Spreading
## 3 Results  
Through our method, we obtained epidemic spreading data with various network and epidemic parameter sets. In the present study, we focused on the case where the epidemic infects all nodes and defined this phenomenon as \"extinctive spread\". Diseases causing extinctive spread are potential candidates of high pandemic risk. In the real world, extinctive spreading indicates that the disease will infect every person in the society. From the simulation data, we calculated the extinctive spread score by dividing the total number of simulation trials by the number of extinctive spread cases. Thereafter, we identified that the number of extinctive spread cases is mainly influenced by spreading speed, which is determined by \(\beta\), \(\gamma_{d}\), and \(k\) (Figure 6). The extinctive spread region (brown area in Figure 6) is expanded as the value of mean degree of network (\(k\)) is increased, thereby indicating that the area of extinctive spread becomes noticeably wider in a dense network than in a sparse network. Thus, the more contact between people, the higher the risk of epidemics. Moreover, high \(\gamma_{d}\) and high \(\beta\) cause extinctive spread across a large region, indicating that the high spreading rate and short time interval between infection and recovery are risk factors of epidemic diseases. In contrast, the infective nodes recover before they transmit the disease to their neighbors in low \(\beta\) and low \(\gamma_{d}\) scenarios, thus disconnecting the network and preventing extinctive spread. This occurs because the infective nodes need more time to transmit the disease in low \(\beta\) and high \(\gamma_{d}\) scenarios. Therefore, the disease begins to subside due to a lack of new infective nodes. Furthermore, we investigated the range of \(\beta\) and \(\gamma_{d}\) for existing epidemics of the common cold [40; 41] and fatal diseases, namely, cholera [42; 43], Marburg [44; 45], Ebola (Congo and Uganda) [46; 47; 48; 49], SARS [50], and MERS [51] (Table 1). We selected diseases with relatively well-known epidemic parameters, such as average duration of infection and basic number of reproductions from previous studies. Transmission rates were calculated using the mean duration of infectious periods and basic reproduction numbers of the epidemics. Different studies reveal multiple values of infectious period and transmission rate for some of these diseases; we considered these values separately [40; 41; 42; 43; 46; 47; 48; 49]. For example, the infectious period of a common cold is from \(3\) to \(7\) days and that of Ebola is \(6.5\) days. Next, we placed the possible regions of these epidemics as a disease band for various \(k\) values (colored lines in Figure 6). When \(k>5\), fatal diseases have an opportunity to cause a pandemic. Even when \(k=5\), diseases such as cholera and Ebola (Congo) can be threatening in regions of low \(\gamma_{d}\) and high, thus demonstrating that the knowledge of network parameters of the society and the characteristics of epidemic diseases can aid in quantifying the risk of epidemics. Figure 6: Extinctive spreading parameter maps (\(k\) = 2, 5, 7, and 10) reveal the relations between extinctive spread regions and the parameters \(\beta\), \(\gamma_{d}\). and \(k\), where the extinctive region is depicted in brown and the safe region is presented in blue. Colored bar represents extinctive spread score, which is calculated by dividing the total number of simulation trials by the number of extinctive spread cases. In low \(\beta\) and low \(\gamma_{d}\) cases, infective nodes become recovered nodes before transmitting the disease, causing disconnection within the network and preventing extinctive spread. In low \(\beta\) and high \(\gamma_{d}\) cases, infective nodes try to transmit the disease to adjacent susceptible nodes until these nodes become recovered; however, successful transmission does not occur as the transmission rate is too low. Moreover, the disease begins to subside due to the lack of new infective nodes.