Dynamical immunization strategy for seasonal epidemics

The topic of finding an effective strategy to halt virus in a complex network is of current interest. We propose an immunization strategy for seasonal epidemics that occur periodically. Based on the local information of the infection status from the previous epidemic season, the selection of vaccinated nodes is optimized gradually. The evolution of vaccinated nodes during iterations demonstrates that the immunization tends to locate in both global hubs and local hubs. We analyze the epidemic prevalence using a heterogeneous mean-field method, and we present numerical simulations of our model. This immunization performs better than some other previously known strategies. Our work highlights an alternative direction in immunization for seasonal epidemics.

Figure 1

An illustration of our dynamical immunization strategy. First in (a), we randomly assign node 1 as the vaccinated node. After the epidemic spreading process, the system state is shown in (b). White nodes stand for susceptible nodes while black nodes denote recovered nodes. We calculate the $W$ value of node 1 and its neighbors, i.e., node 2 and node 3. Since $W\left(1\right)=2,W\left(2\right)=4$, and $W\left(3\right)=2$, node 2 replaces node 1 as the new vaccinated node in the next epidemic season, as shown in (c).

Figure 2

The comparison between simulation values and theoretical values for $r_{\infty }^{\left(S\right)}$. Results of the BA network with $N=100$, 500, and 1000 are shown in (a), (b), and (c), respectively. The black circles denote the simulation value while the blue dashed line denotes the theoretical value. Inset of (a): The relationship of $r_k^{\left(S\right)}$ and time step $t$. The left figure is for $S=1$ while the right figure is for $S=5$. Top curves (red squares) stand for $k=20$, and bottom curves (green triangles) stand for $k=2$. Epidemic parameters are $\beta =0.1$ and $v=0.1$. Each simulation point is the average value of ${10}^2$ experiments.

Figure 3

A demonstration of immunization for $S=1$–6 [(a)–(f)]. Here we adopt a BA network with $N=100$ and $\langle k\rangle =3.94$. Epidemic parameters are $\beta =0.1$ and $v=0.1$. The size of the node implies its degree. Dark red (solid) nodes stand for vaccinated nodes, while light blue (hollow) nodes stand for unvaccinated nodes. We merely show the immunization situation, neglecting the epidemic spreading results.

Figure 4

The curves of average degree (a), average $k$-shell (b), and average path length (c) in the BA network. The dashed lines represent the value of the entire network, and the circles represent the corresponding value among vaccinated nodes. The epidemic prevalence is shown in (d). Network structure is shown in Fig. 3. Epidemic parameters are $\beta =0.1$ and $v=0.1$.

Figure 5

The relationship between epidemic prevalence $r_{\infty }^{\left(S\right)}$ and epidemic season $S$ for different immunization strategies. Black circles stand for our dynamical immunization method. Three lines represent uniform immunization (blue, solid), acquaintance immunization (green, dashed), and targeted immunization (red, dash-dotted) from top to bottom. Results of Wiki-Vote network, Epinions network, Slashdot network, Enron network, BA network with $N=100$, and BA network with $N={10}^6$ are shown in (a)–(f), respectively. Epidemic parameters are $\beta =0.1$ and $v=0.1$. Each point is the average value of ${10}^2$ experiments.

Figure 6

The relationship between recurrence rate $Q_1\left(S\right)$ [$Q_2\left(S\right)$] and epidemic season $S$. The red circles stand for $Q_1\left(S\right)$ and the blue triangles for $Q_2\left(S\right)$. Results of Wiki-Vote network, Epinions network, Slashdot network, and Enron network are shown in (a), (b), (c), and (d), respectively. Epidemic parameters are $\beta =0.1$ and $v=0.1$. Each point is the average value of ${10}^2$ experiments.

Figure 7

The relationship between $A_{10}\left(S\right)$ and epidemic season $S$. Results of Wiki-Vote network, Epinions network, Slashdot network, and Enron network are shown in (a), (b), (c), and (d), respectively. Epidemic parameters are $\beta =0.1$ and $v=0.1$. Each point is the average value of ${10}^2$ experiments.

Figure 8

The relationship between $F_{10}\left(i\right)$ and the number of epidemic seasons $i$. Results of Wiki-Vote network, Epinions network, Slashdot network, and Enron network are shown in (a), (b), (c), and (d), respectively. Epidemic parameters are $\beta =0.1$ and $v=0.1$. Each point is the average value of ${10}^2$ experiments.

Figure 9

Main plot: The relationship between epidemic prevalence $r_{\infty }^{\left(S\right)}$ and immunization proportion $v$ for different infected rates $\beta$. Here $S=5$. Blue triangles, green circles, and red squares stand for the case of $\beta =0.10,0.05$, and $0.02$, respectively. Inset: The relationship between vaccinated threshold $v_c$ and $\beta$. Here $v_c$ is the value of immunization proportion that makes $r_{\infty }^{\left(5\right)}<0.005$. Results of Wiki-Vote network, Epinions network, Slashdot network, and Enron network are shown in (a), (b), (c), and (d), respectively. Each point is the average value of ${10}^2$ experiments.