Immunization of epidemics in multiplex networks

Abstract

Up to now, immunization of disease propagation has attracted great attention in both theoretical and experimental researches. However, vast majority of existing achievements are limited to the simple assumption of single layer networked population, which seems obviously inconsistent with recent development of complex network theory: each node could possess multiple roles in different topology connections. Inspired by this fact, we here propose the immunization strategies on multiplex networks, including multiplex node-based random (targeted) immunization and layer node-based random (targeted) immunization. With the theory of generating function, theoretical analysis is developed to calculate the immunization threshold, which is regarded as the most critical index for the effectiveness of addressed immunization strategies. Interestingly, both types of random immunization strategies show more efficiency in controlling disease spreading on multiplex Erdös-Rényi (ER) random networks; while targeted immunization strategies provide better protection on multiplex scale-free (SF) networks.

Figure 1. Relative size of infected clusters versus the fraction () of immunized nodes for multiplex node-based random or targeted immunization.

The dash lines denote theoretical immunization thresholds and . The networks used are multiplex ER networks with average degree (a) , (b) ; number of layers and the size . In all the figs, we use the value of transmission rate .

Figure 2. Theoretical immunization thresholds (a) and (b) versus average degree in multiplex ER and SF networks.

Networks have the same average degree (i.e. ) and the size of networks is .

Figure 3. The phase diagram for relative size of infected clusters for (a) layer node-based random immunization; (b) layer node-based targeted immunization on multiplex ER networks.

The black line indicates the theoretical immunization threshold of both immunization strategies. The networks have the average degree (i.e. ), size .

Figure 4. Theoretical immunization threshold for (a) layer node-based random immunization and (b) layer node-based targeted immunization.

The networks used are multiplex ER networks (solid line) and multiplex SF networks (dashed line) with average degree (red), (black), and (blue). The size of networks is .