# Effect of Infection Hubs in District-Based Network Epidemic Spread Model +
## 2 Materials and Methods  
Schematically, we divide a territory into a set of interacting districts (in Figure 1).  
We consider the following network system describing \(k\) interacting districts:  
\[\begin{split} 1.&\dot{S}_{i}=-\beta\sum_{j}p_{j}S_{i}+ \delta R_{i}-dH_{i}S_{i}+\mu_{i}\,(i,j=1\dots k);\\ 2.&\dot{I}_{i}=\beta\sum_{j}p_{j}S_{i}+dH_{i}S_{i}-( \gamma+\epsilon)I_{i};\\ 3.&\dot{R}_{i}=\gamma I_{i}-\delta R_{i};\\ 4.&\dot{H}_{i}=\alpha I_{i}-bH_{i}.\end{split} \tag{1}\]here \(S_{i}(t)\) is the number of susceptible individuals of the i-th region, \(I(t)\) is the infected, \(R(t)\) is recovered and \(H(t)\) is the non-human carriers (viruses). The independent variable is time t, measured in days. The coefficient \(\alpha\) denotes the number of carriers (viruses) emitted by an infected individual, \(\beta\) is the probability of infection in the case of contact between a susceptible and an infected subject, \(\gamma\) is the probability of recovery (usually, it is interpreted as the rate of recovery, since the average duration of the subject is in the state of infection is \(1/\gamma\) days), \(\delta\) is the probability of losing immunity and the appearance of the re-infection possibility, \(\varepsilon\) is the probability of death, \(\mu_{i}\) is the immigration of the population for the \(i\)-th region and \(p_{j}\) is the interaction coefficient between the \(j\)-th and \(i\)-th districts given from the communication matrix (P). The relative degree of openness of the borders, which is relevant for modeling quarantine events [24], \(b\) denotes the coefficient of natural loss of carriers and \(d\) is the degree of contagiousness of the carrier (virulence). It should be noted that in the case of \(p=0\) (complete closure of boundaries), the system degenerates in the case of infecting one subject in the absence of the remaining infection [25]. In general, model Equation (1) includes four equations. The first describes the dynamics of the exposed, the second describes the dynamics of the infected, the third describes the dynamics of the recovered and the fourth describes the dynamics of the virus population. Compared to the classical SIR model, the first term of Equations (1)1 and (1)2 includes the dependence on the degree of interaction within the network \(\beta\sum_{j}p_{j}S\). The second term of Equation (1)2 \(\delta R\) describes the transition of the cured to the exposed population. \(dH_{i}S_{i}\) term in Equation (1)2 represents the dynamics of infecting people without contact with another infected person. \(\gamma I\) of Equation (1)3 represents the dynamics of recovery. The \(\alpha I\) term shows the production of viruses by the infected, and the term \(bH\) represents the natural death of the virus.  
If we consider the behavior of the model for a separate area (\(j=1\) and \(\sum_{j}p_{j}=1\)), then the matrices of infection and the transition could be calculated as follows:  
\[F=\begin{bmatrix}\beta&b\\ 0&0\end{bmatrix} \tag{2}\]  
\[V^{-1}=\begin{bmatrix}\dfrac{1}{\gamma+\varepsilon+\mu}&0\\ \dfrac{1}{\gamma+\varepsilon}&\dfrac{1}{b}\end{bmatrix} \tag{3}\]  
Figure 1: Schematic representation of a territory with interacting districts.  
The spectral radius of the matrix can be considered as an assessment of the main reproductive number:  
\[\mathfrak{R}_{0}=FV^{-1}=\frac{\beta+b}{\gamma+\epsilon+\mu}. \tag{4}\]  
Strictly speaking, there will be a separate reproductive number for each district. However, given the specifics of the model and the possibility of dividing a single entity into separate districts, the average number can be represented as follows:  
\[\mathfrak{R}_{0}=\frac{\beta_{F}^{\frac{1}{K}\sum_{i,j=1}^{k}p_{i,j}+b}}{ \gamma+\epsilon+\mu}. \tag{5}\]