# Effect of Infection Hubs in District-Based Network Epidemic Spread Model +
## 3 Results  
To study the basic effects of the spread of infection in the case of various interactions between districts, we used a simplified model with zero levels of immigration and mortality. To check the main effects of the net with several interacting districts, the communication matrix was set to be symmetrical as follows:  
\[P= \begin{bmatrix}1&p&p\\ p&1&p\\ p&p&1\end{bmatrix}  \tag{6}\]  
where the initial appearance of infected individuals in the first district was implied, the spread of infection in the districts interacting with the first one was identical. This matrix reflects a homogeneous but densely populated area, including three districts. Three closely spaced homogeneous settlements can be considered to be of this type. Since the interaction within a district is conditionally considered active, the values on the main diagonal are set to 1. The interaction between districts is considered equal and given \(p\) due to less active interaction between districts, both due to natural causes and possible restrictions. The dynamics of Equation (1) were identical for districts 2 and 3. The number of infected individuals demonstrated a big first wave in the time period from 20 to 30 days, with a recovery period from 30 to 40 days and a second lower wave from 30 to 50. Initially, infected district 1 demonstrated the maximal value of infected in the first wave, while the second and third districts were only slightly lower due to the high interaction within each district. We observed significant reductions in both the maximum and the end values in the case of even partial closure of boundaries. At the same time, the apex of the maximum also falls on later values, and at boundaries that are more open, the occurrence of secondary and subsequent infection waves begins. The difference between the infection waves in the first district and bound districts \(\Delta I(p,t)=I_{1}(p,t)-I_{2}(p,t)\) was especially noticeable in the first infection wave when the degree of openness of the boundaries between the districts is lower than 0.4. A lower degree causes a bigger delay between the maximums of the infected reached by the first wave. To check the effect of the presence of the central district (a hub), as well as the observed difference between the second and third districts, the communication matrix was taken in the following form:  
\[P= \begin{bmatrix}1&p&p\\ p&0.5&0.05\\ p&0.05&0.1\end{bmatrix}  \tag{7}\]  
The assumptions underlying the construction of this matrix include the homogeneity of districts, as well as the varying degrees of interaction within districts. The second and third districts were located on the outskirts and had weak interaction between themselves. Most of the population is assumed to be interested in spending time in the hub rather than an outskirt. The first district was treated as a hub, and its level of exposure to other districts can vary. Another reason for the significant difference in the main diagonal numbersof Equation (7) is the desire to demonstrate the difference in dynamics for objects with medium and weak interactions. Typical curves for a strong bond (\(p=1.1\)) are shown in Figure 2. The number of infected demonstrated a big first wave in the time period from 30 to 40 days, a recovery period from 40 to 60 days and a second lower wave from 180 to 220. For comparison, the figure also shows a graph for the time series of an unaltered SIR model that does not demonstrate any secondary or consequent waves. The numerical simulation showed that despite the fact that the probability of internal infection in the third area is five times lower than in the second, the final number of infected differs only by 10%.