Epidemic spreading in a hierarchical social network

A model of epidemic spreading in a population with a hierarchical structure of interpersonal interactions is described and investigated numerically. The structure of interpersonal connections is based on a scale-free network. Spatial localization of individuals belonging to different social groups, and the mobility of a contemporary community, as well as the effectiveness of different interpersonal interactions, are taken into account. Typical relations characterizing the spreading process, like a range of epidemic and epidemic curves, are discussed. The influence of preventive vaccinations on the spreading process is investigated. The critical value of preventively vaccinated individuals that is sufficient for the suppression of an epidemic is calculated. Our results are compared with solutions of the master equation for the spreading process and good agreement of the character of this process is found.

Figure 1

An example of a network with $L=24$ and $L_G=8$ (nine local groups) from the point of view of the $S_{12,12}$ individual, who is connected with $k_{12,12}=6$ neighbors and four of whose connections are located in its local group (a). When a connection between two individuals $S_{ij}$ and $S_{nm}$ is created (solid line), the individual $S_{nm}$ is connected with the neighbors of the individual $S_{ij}$. Next, the connections between the neighbors $S_{nm}$ and the individual $S_{ij}$ are created (this is not shown in the figure). Each new connection (dashed line) is created with the probability $p_c$ (b).

Figure 2

Epidemic curves (the number of newly infected individuals per time step) as a function of time for different types of interpersonal interactions. Infections resulting from interpersonal connections, curve a; infections resulting from random contacts with the individuals from a local group, curve b; infections resulting from random contacts with individuals from the rest of the population, curve c. The values of the other parameters are $\gamma =3,L=500,L_G=20,W_{H\to I}=0.5,W_{I\to S}=0.5,W_{S\to R}=0.5,k_{\min }=8$, and $k_{\max }=24$.

Figure 3

The time $t_{\max }$ in which the number of ill individuals reaches a maximum value as a function of the clustering coefficient $C$ for different values of $W_{H\to I}$. Results were averaged over 100 independent simulations. The values of other parameters are $\gamma =3,L=200,L_G=20,W_{I\to S}=0.5,W_{S\to R}=0.5,k_{\min }=8$, and $k_{\max }=24$.

Figure 4

The influence of the time of incubation on the time $t_{\max }$ (a) and the maximum value of the number of ill individuals $w_{\max }$ (b) for different values of $W_{H\to I}$ and $p_c=0.5$. Results were averaged over 100 independent simulations. The values of the other parameters are as in Fig. 3.

Figure 5

The influence of the maximum value of the connectivity $k_{\max }$ on the range of the epidemic $V$ (a), the maximum value of the number of ill individuals $w_{\max }$ (b), and the time $t_{\max }$ (c), for different values of $W_{H\to I}$ and $p_c=0.5$. Results were averaged over 100 independent simulations. The values of the other parameters are as in Fig. 3.

Figure 6

The relation between time $t_{\max }$ and the number of preventively vaccinated individuals $N_{R0}$ for different values of $W_{H\to I},k_{\max }$, and $p_c=0.5$ (a),(b). The vaccinated individuals were randomly chosen (a) or those with the highest value of $k$ were vaccinated (b). The influence of the parameter $\gamma$ on the $t_{\max }\left(N_{R0}\right)$ relation for $W_{H\to I}=0.4,k_{\max }=64$, and $p_c=0.5$ is shown in (c). Results were averaged over 100 independent simulations. The values of the other parameters are as in Fig. 3.

Figure 7

Comparison of the relation $N_I\left(t\right)∕N$ obtained in the present model (solid line) with the solution of the master equation (dashed line) for $W_{H\to I}=0.5,W_{S\to R}=0.1$, and $p_c=0.5$. The values of the other parameters are as in Fig. 3.