Dynamic vaccination in partially overlapped multiplex network

In this work we propose and investigate a strategy of vaccination which we call “dynamic vaccination.” In our model, susceptible people become aware that one or more of their contacts are infected and thereby get vaccinated with probability $\omega$, before having physical contact with any infected patient. Then the nonvaccinated individuals will be infected with probability $\beta$. We apply the strategy to the susceptible-infected-recovered epidemic model in a multiplex network composed by two networks, where a fraction $q$ of the nodes acts in both networks. We map this model of dynamic vaccination into bond percolation model and use the generating functions framework to predict theoretically the behavior of the relevant magnitudes of the system at the steady state. We find a perfect agreement between the solutions of the theoretical equations and the results of stochastic simulations. In addition, we find an interesting phase diagram in the plane $\beta \text{−}\omega$, which is composed of an epidemic and a nonepidemic phase, separated by a critical threshold line ${\beta }_c$, which depends on $q$. As $q$ decreases, ${\beta }_c$ increases, i.e., as the overlap decreases, the system is more disconnected, and therefore more virulent diseases are needed to spread epidemics. Surprisingly, we find that, for all values of $q$, a region in the diagram where the vaccination is so efficient that, regardless of the virulence of the disease, it never becomes an epidemic. We compare our strategy with random immunization and find that, using the same amount of vaccines for both scenarios, we obtain that the spread of disease is much lower in the case of dynamic vaccination when compared to random immunization. Furthermore, we also compare our strategy with targeted immunization and we find that, depending on $\omega$, dynamic vaccination will perform significantly better and in some cases will stop the disease before it becomes an epidemic.

Figure 1

Schematic of the SI-R/V epidemic process in a multiplex network consisting of two networks, each of size $N=10$. A fraction $q$ of nodes in both networks represent the same individual acting in different environments. Here $q=0.7$. The colors of the nodes represent the following states: green ($●$) for susceptible (S), red ($\blacksquare$) for infected (I), blue ($⬟$) for recovered (R), and black ($▴$) for vaccinated individuals. In this case, we assume $t_r=1$. At $t=1$, the patient zero infects one of its neighbors in network $A$ and the other becomes vaccinated. Since this infected patient is present in both networks, it can also spread the disease in network $B$. The red lines indicate the direction of the branches of infection. In the next time step, $t=2$, this patient zero recover in both networks and the newly infected individuals continue to spread the disease. The process continues until the fourth temporary step, which is the steady state, where there are no more infected individuals that can continue spreading the disease.

Figure 2

Theoretical and simulation results of the total fraction of recovered ($R$) and vaccinated ($V$) nodes, for two different overlap scenarios ($q=0.1,0.9$), as a function of the virulence of the diseases $\beta$ and for different values of $\omega$, in the steady state of the process. We consider $t_r=1$, [(a) and (b)] $q=0.1$ and [(c) and (d)] $q=0.9$, and $\omega =0,0.1,0.3,0.5,0.7,0.8$ from left to right. The symbols correspond to the simulation results and the lines correspond to the theoretical evaluation of Eqs. (6). The multiplex network consists of two layers, each of size $N={10}^5$. Layer $A$ is an ER network with $\langle k_A\rangle =4$ with $k_{\min }=0$ and $k_{\max }=40$, and layer $B$ is an SF network with ${\lambda }_B=2.5$, $k_{\min }=2$, $k_{\max }=\sqrt{{10}^5}$ and exponential cutoff $c=50$, thus $\langle k_B\rangle \simeq 3.66$. Simulation results are averaged over ${10}^4$ realizations.

Figure 3

Difference between the fraction of recovered and vaccinated individuals in both layers as a function of the virulence of the disease for different values of the overlap $q$. We consider $\omega =0.1$ (left),0.7 (right), and $t_r=1$. The overlap $q$ varies from 0 ($•$) to 1 ($\_$) with $\mathrm{\Delta }q=0.1$. The curves correspond to the theoretical evaluation of Eqs. (6). In (a) and (b) we consider in layer $A$ an ER network with average degree $\langle k_A\rangle =4$, $k_{\min }=0$ and $k_{\max }=40$, and in layer $B$ a SF network with ${\lambda }_A={\lambda }_B=2.5$, $k_{\min }=2$ and $k_{\max }=\sqrt{{10}^5}$, and exponential cutoff $c=135$, thus $\langle k_B\rangle \simeq 4$. Both networks are of the same size $N={10}^5$. In (c) and (d) the system is composed by two ER, layer $A$ with $\langle k_A\rangle =4$ and layer $B$ with $\langle k_B\rangle =10$, both with $k_{\min }=0$ and $k_{\max }=40$. The insets show the fraction of recovered individuals in each layer for the case of $q=0$ (solid lines) and $q=0.5$ (dash-dotted lines).

Figure 4

Surface phase diagram obtained theoretically for the case of $t_r=1$. In the top panels we consider in layer $A$ an ER network with $\langle k_A\rangle =2$ with $k_{\min }=0$ and $k_{\max }=40$, and in layer $B$ a SF network with (a) ${\lambda }_B=1.5$ and (b) ${\lambda }_B=3.5$, $k_{\min }=2$, $k_{\max }=\sqrt{{10}^5}$ and exponential cutoff $c=135$. On the bottom we consider two ER networks with $\langle k_A\rangle =2$ in layer $A$ and (c) $\langle k_B\rangle =2$ (d) $\langle k_B\rangle =5$ in layer $B$. In all $k_{\min }=0$ and $k_{\max }=40$. In all networks, $N={10}^5$. We can see that there are two well marked regions: an epidemic and a nonepidemic phase, separated by a critical value of $\beta$. Increasing the overlapping $q$ facilitates the spread of the epidemic, thus, ${\beta }_c$ decreases and the epidemic phase regime is smaller. In all figures there is a critical value of $\omega$ above which, regardless of the virulence of the disease, there is no epidemic. In the case of no vaccination, $\omega =0$, we recover the regular SIR in multiplex networks and as $\omega$ increases, there will be more vaccinated individuals so ${\beta }_c$ increases.

Figure 5

Comparison among dynamic, random, and targeted vaccination. Theoretical results of the total fraction of recovered (R) nodes as a function of V. All dotted lines correspond to random vaccination (), the dashed lines to targeted immunization (), and the colored solid lines correspond to our strategy of dynamic vaccination. For the case of dynamic vaccination the colors in the curves indicate different values of $\omega$. Notice that $\omega$ decreases from top to bottom. We set $t_r=1$, $\beta =0.3$ and consider different values of overlap [(a)–(c)] $q=0.3$ and [(b)–(d)] $q=0.7$. In (a) and (b) we used two ER networks with $\langle k_A\rangle =\langle k_B\rangle =5$ with $k_{\min }=0$ and $k_{\max }=40$, and in (c) and (d) two scale-free networks, with ${\lambda }_A={\lambda }_B=2.5$, $k_{\min }=2$, $k_{\max }=\sqrt{N}$, $N={10}^5$, and exponential cutoff $c=50$.

Figure 6

Theoretical results of the total fraction of (a) recovered $R$, (b) $R/B_{\beta }$, (c) vaccinated $V$, and (d) $V/B_{\omega }$ nodes as a function of the overall transmissibility $T_{\beta }=(1-\omega )\beta$ when $t_r=1$ and for different values of $\omega$ in the steady state. The multiplex network consists of two ER networks and we set $q=0.3$. Each network size is $N={10}^5$ and $\langle k_A\rangle =\langle k_B\rangle =5$ with $k_{\min }=0$ and $k_{\max }=40$.