# Effect of Infection Hubs in District-Based Network Epidemic Spread Model +
## 3 Results
The graphs of the total infected over time for the degree of opening of the borders for the districts were similar to the dynamics with the case of symmetric links, despite the difference in the probability of infection inside the districts and were primarily determined by the degree of relationship with a hub, which can be seen for the first wave, but the flow and final values of the infected dependent from the internal probabilities of infection with small communication values a hub. The final amount of infected in the district increases with a degree of connection with a hub. The peak of the wave (maximal number) of infected individuals for secondary and subsequent waves 3 and 4 are shown in Figure 3. The graphs demonstrate growth for the communication values with the hub within the limits from \(p>0\) to \(p<2\). However, for the (\(p>2\)) maximal value of infected individuals, the second wave even decreases while peaks of subsequent waves barely change. For the relatively big values of \(p>8\), the maximal values converge to a constant. Figure 2: The time series of infected (**a**), susceptible (**b**) and recovered (**c**). The phase portrait for three districts (**d**), obtained for the values of the system parameters with the communication matrix (7): \(n=50,000\), \(\beta=1.9\), \(\gamma=1.5\), \(\delta=0.05\), \(b=0.01\), \(d=10^{-7}\); the violet line represents time series unaltered SIR model with the same parameters. The final number of infected demonstrates a slow upward slope on the dependence of the final number of infected in districts (red--1, green--2, blue--3) on the degree of openness of the boundaries between the hub and other districts, as shown in Figure 4. The numerical simulation showed that despite differences in infection probability, the final number of infected is quite similar. For the very low values of \(p\) (\(p<0.02\)), differences between the hub and rural districts are sufficient (more than 10 times) as the graphs demonstrate upward growth. However, for the values \(p>0.5\), the final number of infected in the hub area is only twice as large for for the and districts and keeps diminishing with the degree of openness of the boundaries between the hub and other districts. With a sufficiently large degree of openness, the whole city could be seen as one homogenous entity. Figure 4: The dependence of the final number of infected in districts (red—1, green—2, blue—3) on the degree of openness of the boundaries between the hub and other districts (p). Figure 3: The dependence of the maximum number of infected in the second and subsequent waves in districts (red—1, green—2, blue—3) on the degree of openness of the boundaries between the hub and other districts (p). As a demonstration of the practical application of the work, the situation on Shikoku Island was examined. Shikoku is the smallest of the four main islands of Japan and has a population of about 2.8 million people. This territory was chosen to reduce the possible effect of migration, as well as due to its territorial and administrative division into regions. Tokushima, Kagawa, Ehime and Kochi prefectures are located on the island. The infection rates for these regions were taken from public datasets [26; 27] for the last three months (22 October-30 December 2022). Through empirical fitting of model parameters, it was possible to qualitatively reproduce the dynamics of the number of infected using the communication matrix in Equation (8). \[P= \begin{bmatrix}0.1&0.2&0.1&0.05\\ 0.2&0.4&0.05&0.1\\ 0.1&0.05&0.5&0.15\\ 0.05&0.1&0.15&0.15\\ \end{bmatrix}  \tag{8}\]  
For comparison by region, the \(I(t)\) data are shown in Figure 5. Although the applicability of the model for the practical situation is demonstrated in a similar way, the model system, in this case, was applied without taking into account the current information on internal and external migration, as well as specific limiting measures expressed in quantitative terms due to the lack of open information.