Immunization of Epidemics in Multiplex Networks
Citation: Zhao D, Wang L, Li S, Wang Z, Wang L, et al. (
Immunization of Epidemics in Multiplex Networks
Ye Wu, Beijing University of Posts and Telecommunications, China
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 Erdo s-Re nyi (ER) random networks; while targeted immunization strategies provide better protection on multiplex scale-free (SF) networks.
Funding: This paper was supported by the National Natural Science Foundation of China (No. 61202362, 61262057, 61472433; URLs http://www.nsfc.gov.cn/; L.
H. W. and S. D. L. received the funding), the Project funded by China Postdoctoral Science Foundation (No. 2013M542560; URLs http://jj.chinapostdoctor.org.cn/
V1/Program1/Default.aspx; L. S. D. received the funding), the Natural Science Foundation of Shandong Province (No. ZR2013FM025, ZR2013FQ001, ZR2011FQ030;
URLs http://www.sdnsf.gov.cn/portal/; L. H. W. received the funding), and the Shandong Academy of Sciences Youth Fund Project (No. 2013QN007; URLs http://
www.sdnsf.gov.cn/portal/; L. H. W. received the funding). The funders had no role in study design, data collection and analysis, decision to publish, or preparation
of the manuscript.
Competing Interests: Z. W. is now a member of the Editorial Board of PLOS ONE. This does not alter the authors adherence to PLOS ONE Editorial policies and
The structure and dynamics of multiplex networks have
attracted much attention by the scientific communities .
Composed of a set of networks integrated by interconnected
layers, the multiplex networks well describe many real-world
complex systems, such as social networks, communication
networks and transportation networks (see  for a recent review).
Recently, an increasing number of works has tried to
understand the dynamics of epidemic spreading in multiplex
networks. Along this line, various mechanisms aiming at exploring
the disease propagation process in multiplex topology are
proposed and investigated. Examples include competing epidemics
, the effect of the interconnected network structure , joint
spreading of both information and disease [20,21], mutual
interaction of both social and epidemic spreading , the impact
of network correlation patterns , to name but a few. Looking
at some examples more specifically, in a recent research ,
where epidemic can only spread on partially overlapped networks,
the authors reveal that the epidemic threshold would decrease
monotonically with the increment of overlapped fraction.
Considering the SIR compartmental epidemic model in a multiplex
network composed of a virtual layer and a physical layer ,
Yagan et al. unfolds the prevalence of the disease in both layers,
even if the epidemiological parameters are assigned values lower
than the epidemic threshold of each layer. By superposition
processing of the network layers, Zhao et al. show that a strong
positive degree-degree correlation of nodes in different layers
could lead to a clearly low epidemic threshold and a relatively
smaller infection size . Interestingly, these measures are not
significantly affected by the average similarity of neighbors.
As above described, though there have been some achievements
focusing on the effect of multiplex architecture on the epidemic
dynamics and the resulting threshold, the impact of such increased
complexity on the immunization strategies is still virgin .
In the traditional study of network immunization, the vaccinated
candidate nodes are usually selected randomly, or chosen
intuitively according to their topological properties such as degree,
betweenness or k-shell, etc . Thus, an interesting question
naturally poses itself, which we aim to address in what follows. If
we consider the basic immunization cases in multiplex networks,
how do them affect the disease propagation?
Here, with the SIR epidemic model on multiplex networks ,
we explore the performance of several typical immunization
strategies, including multiplex node-based random (targeted)
immunization and layer node-based random (targeted)
immunization. Based on the theory of generating function, mathematical
analysis is utilized to distinguish the critical immunization
threshold. Extensive computational simulations are used to verify
our analysis. We reveal that the efficiency of proposed
immunization strategies rely on topology details of multiplex networks.
For simplicity (yet without loss of generality), we consider the
SIR dynamics as the epidemiology model, and then inspect the
effect of immunization strategies on disease propagation in
multiplex networks. With regard to networks, we select multiplex
Erdo s-Renyi (ER) random networks  and Barabasi-Albert
scale-free (SF) networks . For such a multiplex framework, it is
composed of m network layers, each of which contains N nodes
(namely, each node has the replica in different layers). At each
time step, every node can fall into one of three states: susceptible
(S), infected (I), or recovered (R). On each network layer i, the
infected node can infect its susceptible neighbors with
transmissibility probability li; and the infected node is also able to the
recovery with probability d. To be simple, we use the case of d~1.
For networked immunization, there is usually one critical index,
immunization threshold wc (namely, the required minimum
fraction of immunized nodes) , which elevates the
efficiency of immunization strategies. Above this threshold, the
number of infected nodes is null. Up to now, immunization
strategies of single network have been numerously proposed to
lower the value of wc . However, different from the single
network, each node of multiplex networks has a replica in each
network layer. To distinguish the node of multiplex networks and
its replica in each network layer, we define the terminology:
multiplex node and layer node, respectively. Naturally,
immunization of multiplex networks can be classified into multiplex
nodebased immunization and layer node-based immunization. It is
worth mentioning the difference of two immunization scenarios:
the former means that all the replicas of the same node take
immunization, while the latter could just provide protection for
one replica in the certain network layer. In what follows, we will
investigate the multiplex node-based and layer node-based
immunization strategies in multiplex networks, and provide
theoretical frame to calculate the critical immunization threshold
of different immunization strategies.
Multiplex node-based immunization
Multiplex node-based immunization refers to the case that a
fraction of multiplex nodes is random or targeted immunized. If
we use w(!kj ), where !kj ~(kj 1,kj2,:::,kj m) is the degree of a
multiplex node j in each layer, to denote the probability that a
multiplex node with degree !kj is immunized, then the generating
function  of the joint degree distribution, for multiplex
nodebased immunization, could be defined as
where !x~(x1,x2,:::xm) is used to denote the auxiliary variables
coupled to !kj and p(!kj ) indicates the probability that a randomly
chosen multiplex node has degree !kj . The generating function of
remaining joint degree distribution by following a randomly
chosen link of network layer i is given by
where zi is the average degree of network layer i.
Then, the probability ui (i~1,2,:::,m) that a multiplex node
connects to a link of the chosen network layer i and belongs to the
infected cluster is given by the coupled self-consistency equations
Thus, the existence of epidemic regime under multiplex
nodebased immunization requires the largest eigenvalue L of the
Jacobian matrix of Eq. (3) at (0,0,,0) to be larger than unity
[11,12]. For multiplex networks formed by two network layers
(duplex networks), L can be expressed as
Multiplex node-based random immunization. For
multiplex node-based random immunization, each node has the same
probability to be immunized, so we can write w(!kj )~wMR for
j~1,2,:::,N. Furthermore, the critical immunization threshold
wcMR will be the value of wMR which satisfies L~1 in Eq. (4).
Multiplex node-based targeted immunization. For
multiplex node-based targeted immunization, the nodes are generally
immunized according to their degree, betweeness or k-shell, etc.
However, since the transmissibility of epidemics in each layer may
be different, the role of multiplex nodes may be not identical (even
if they have the same degree kj ~kj1zkj 2z:::zkj m). Thus, we
define a new index, spreading degree Kj , which takes the
transmissibility of epidemics into account, to evaluate the
importance of multiplex nodes as follows
Under this immunization framework, the immunized
probability of a multiplex node with spreading degree Kj could be
where Kc is the cutoff spreading degree for immunization, and f
is the immunized probability of nodes with spreading degree Kc.
Consequently, the total fraction of immunized nodes is given by
where p(Kj) indicates the fraction of nodes with spreading
degree Kj . Thus, in the case of multiplex node-based targeted
immunization, the critical immunization threshold wcMT will be the
value of wMT satisfying L~1 when w(!kj )~w(Kj ) in Eq. (4).
Fig. 1 shows the theoretical immunization thresholds wcMR and
wcMT (vertical dash lines) of multiplex node-based random and
targeted immunization on the multiplex ER networks with
different average degree. Besides, we also show how the relative
size R=R(0) of infected clusters varies in dependence on the
fraction wMR (wMT ) of immunized multiplex nodes, where R
denotes the size of infected clusters and R(0) is the value of R
when no node takes immunization. It is clear that the theoretical
immunization thresholds are accurate in evaluating the existence
of epidemic regime, irrespective of immunization strategies and
average degree of networks. More interestingly, random
immunization strategy shows larger threshold than that of targeted
immunization, which means that the complete eradication of
infection risk needs more chosen nodes to take immunization
under the framework of random immunization. To compare the
efficiency of multiplex node-based immunization strategies, we
also introduce them into multiplex SF networks. As shown in
Fig. 2(a), the critical immunization threshold wcMR of multiplex ER
networks is always lower than that of multiplex SF networks in the
case of random immunization. This means multiplex node-based
random immunization is more efficiency in multiplex ER
networks. At variance, multiplex node-based targeted
immunization can provide better protection in multiplex SF networks (see
Layer node-based immunization
Besides above proposal, the objects of immunization now turn
to layer nodes. If we define wi(kji) as the probability that a layer
node with degree kj i is immunized, then the generating function of
the joint degree distribution could be expressed as
The generating function of remaining joint degree distribution
by following a randomly chosen link of network layer i will be
For layer node-based immunization, the probability vi
(i~1,2,:::,m) that a multiplex node connects to the chosen
network layer i and belongs to the infected cluster is given by
Similarly, the existence of epidemic regime requires the largest
eigenvalue L of the Jacobian matrix of Eq. (10) at (0,0,,0) to be
larger than unity. For multiplex networks, L can be expressed as
Figure 1. Relative size R=R(0) of infected clusters versus the fraction wMR (wMT ) of immunized nodes for multiplex node-based
random or targeted immunization. The dash lines denote theoretical immunization thresholds wcMR and wcMT . The networks used are multiplex
ER networks with average degree (a) z1~z2~1, (b) z1~z2~2; number of layers m~2 and the size N~2,000. In all the figs, we use the value of
transmission rate l1~l2~1:0.
Figure 2. Theoretical immunization thresholds (a) wcMR and (b) wcMT versus average degree z in multiplex ER and SF networks.
Networks have the same average degree z1~z2~z (i.e. m~2) and the size of networks is N~2,000.
Layer node-based random immunization. For layer
node-based random immunization, each node in the same
network layer has equal probability of taking immunization. So,
we can get wi(kji)~wiLR (i~1,2,:::,m, j~1,2,:::,N). For duplex
networks, the critical immunization threshold can be calculated via
(w1LR,w2LR)c~f(w1LR,w2LR)DL~1 when wi(kj i)~wiLRg,
where wLR and wLR are the fraction of immunized nodes on
both network layers, respectively.
Layer node-based targeted immunization. For the layer
node-based targeted immunization, the immunized probability of
a layer node with degree kji is determined by its degree as follows
where kci is the cutoff degree for immunization in layer i, and fi
is the immunized probability of nodes with degree kci.
Consequently, the total fraction of immunized nodes in network layer i is
where pi(kji) indicates the fraction of nodes with degree kj i in
network layer i.
Thus, for layer node-based targeted immunization of duplex
networks, the critical immunization threshold is given by
(w1LT ,w2LT )c~f(w1LT ,w2LT )jL0~1 when wi(kji)
is set based on Eq: (13)g:
In Fig. 3, we present the color code of relative size R=R(0) of
infected clusters, and use the black line to indicate the theoretical
immunization thresholds of layer node-based random [Fig. 3(a)]
and targeted immunization [Fig. 3(b)]. Together with the results of
Fig. 1, we could prove that the proposed theoretical framework
allows us to accurately calculate the immunization thresholds of
multiplex networks under different immunization strategies.
Moreover, we also notice that layer node-based targeted
immunization can eradicate the disease even with lower fraction
of immunized nodes in multiplex ER networks, which is similar to
the observation of Fig. 1.
Subsequently, we further extend the layer node-based
immunization proposals to multiplex SF networks and compare its
Figure 3. The phase diagram for relative size R=R(0) 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 z1~z2~2 (i.e. m~2), size N~2,000.
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 z1~z2~1
(red), z1~z2~2 (black), and z1~z2~3 (blue). The size of networks is N~2,000.
efficiency with case of multiplex ER networks in Fig. 4. It is
interesting that we find that the layer node-based random
immunization of multiplex ER networks is more effective than
that of multiplex SF networks. However in the case of layer
nodebased targeted immunization, the efficiency of multiplex SF
networks is better. Combining with foregoing results, we can get
that random immunization is better in multiplex ER networks and
targeted has higher efficiency in multiplex SF networks,
irrespective of multiplex node or layer node.
To sum, we propose four kinds of immunization strategies of
multiplex networks, including multiplex node-based random
immunization and targeted immunization, and layer node-based
random immunization and targeted immunization. By using
generating function methods, we provide one new theoretical
framework which allows us to accurately calculate the critical
immunization thresholds of different immunization strategies. We
also evaluate the efficiency of the proposed immunization
strategies for multiplex ER networks and multiplex SF networks.
We show that both multiplex node-based and layer node-based
random immunization has higher efficiency in multiplex ER
networks, while two types of targeted immunization strategies can
provide better protection in multiplex SF networks.
Conceived and designed the experiments: DWZ LHW ZW. Performed the
experiments: DWZ LHW. Analyzed the data: SDL LW BG. Wrote the
paper: ZW LW.
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