Ring vaccination strategy in networks: A mixed percolation approach

Ring vaccination is a mitigation strategy that consists in seeking and vaccinating the contacts of a sick patient, in order to provide immunization and halt the spread of disease. We study an extension of the susceptible-infected-recovered (SIR) epidemic model with ring vaccination in complex and spatial networks. Previously, a correspondence between this model and a link percolation process has been established, however, this is only valid in complex networks. Here, we propose that the SIR model with ring vaccination is equivalent to a mixed percolation process of links and nodes, which offers a more complete description of the process. We verify that this approach is valid in both complex and spatial networks, the latter being built according to the Waxman model. This model establishes a distance-dependent cost of connection between individuals arranged in a square lattice. We determine the epidemic-free regions in a phase diagram based on the wiring cost and the parameters of the epidemic model (vaccination and infection probabilities and recovery time). In addition, we find that for long recovery times this model maps into a pure node percolation process, in contrast to the SIR model without ring vaccination, which maps into link percolation.

Figure 1

Schematic illustration showing the equivalence between (a) the SIR-V and (b) the mixed percolation process. In (a), R-R links (red) were used to spread the disease, while R-V links (light blue) were used to immunize. In (b), the gray area indicates the GC. Node I belongs to the GC because it is occupied (filled circle) and has an occupied link (solid line) with the GC. This type of node is equivalent to a recovered node (R). In contrast, node II is occupied but its link that leads to the GC is not (dashed line). Thus, it is equivalent to a susceptible node (S) because there was no interaction through the link. Node III is not occupied (open circle), but its link with the GC is, so its equivalence is a vaccinated node (V). Finally, node IV (unoccupied site and link) is also equivalent to a susceptible node.

Figure 2

Comparison between the simulation of the SIR-V process and the theoretical solutions for the number of recovered and vaccinated nodes. The epidemiological parameters used are $t_r=1$ and $\omega =0.1$. Note that the Erdös-Rényi (ER) and random regular (RR) types of networks are homogeneous, while the scale-free (SF) and log-normal (LN) networks are heterogeneous. For the ER network we use $\langle k\rangle =6$, while for the RR network we use $k=5$. For the SF network we choose an exponent of $\lambda =2$ and opt for an exponential cutoff with parameter $c=25$, while for the LN network the parameters are $\mu =1$ and $\sigma =0.7$. In all cases, $k_{\min }=2$ and $k_{\max }=250$.

Figure 3

Types of nodes in the perimeter of the GC of mixed percolation (equivalent to the cluster of recovered nodes). Filled and open circles represent occupied and unoccupied nodes, while solid and dashed lines represent occupied and unoccupied links, respectively.

Figure 4

Schematic diagram of a network built according to the Waxman model, where the length of the links follows an exponential distribution [Eq. (13)]. Circles represent nodes, which are arranged in a square lattice, while solid lines represent links of different lengths.

Figure 5

Structural properties of the Waxman model. (a) Degree distribution of the nodes. (b) Network average clustering coefficient.

Figure 6

Effect of varying $t_r$ on the fraction of (a) recovered and (b) vaccinated nodes as a function of $T_{\beta }$, for $\xi =1$, $\omega =0.1$. The network has $N={10}^6$ nodes and the cutoff is $s_c=300$. Different curves correspond to different values of $t_r$. Symbols represent the results of the SIR-V process, while dashed lines correspond to the results of mixed percolation.

Figure 7

Effect of varying $\xi$ on the fraction of (a) recovered and (b) vaccinated nodes as a function of $T_{\beta }$, for $t_r=1$, $\omega =0.1$. The network has $N={10}^6$ nodes and the cutoff is $s_c=300$. Different curves correspond to different values of $\xi$. Symbols represent the results of the SIR-V process, while dashed lines correspond to the results of mixed percolation. The first curve on the right (magenta $\times$'s) corresponds to a square lattice, for comparison, while the dash-dotted line on the left corresponds to the analytical solution for an ER network.

Figure 8

Relation between the fraction of recovered and the fraction of vaccinated nodes as a function of $\beta$. Dashed lines represent Eq. (5). From left to right, the values used are $\omega =0.1$, 0.2, 0.3, 0.4, and 0.5. The different symbols are the numerical results for $t_r=1$, 4, and 8 and $\xi =0.1$, 1, and 10, in all possible combinations. The dash-dotted line indicates the case $V=R$.

Figure 9

Phase diagram. The different curves represent the critical values that separate the epidemic phase (above the curves) from the epidemic-free phase (below the curves) for different values of $\xi$. If $C_{\beta }=1$ (top axis) the critical values correspond to link percolation, while if $T_t=1$ (right axis) they correspond to node percolation. In the ER limit ($\xi \to \infty$), the product of $C_{\beta }$ and $T_t$ equals ${\langle k\rangle }^{-1}$.

Figure 10

(a) Recovered and (b) vaccinated nodes as a function of $T_{\beta }$ in the limit of long recovery times. Symbols represent the results of the SIR-V process for $t_r=16$, while dashed lines correspond to the magnitudes of pure node percolation: (a) fraction of nodes in the giant component (GC); (b) fraction of nodes in the perimeter of the GC. Different curves correspond to different values of $\omega$.

Figure 11

Possible connections of Type II or III nodes. The central circles are half-filled because they represent either occupied or unoccupied nodes, while solid and dashed lines represent occupied and unoccupied links, respectively. If the link leads to the GC (represented by the $\infty$ symbol) it must be unoccupied. On the other hand, the $⨂$ symbol represents all cases that are not the giant component.