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1
Epidemic threshold in temporal multiplex networks
with individual layer preference
Cong Li, Member, IEEE, Yuan Zhang and Xiang Li, Senior Member, IEEE
Abstract—Many efforts have been devoted to understanding
how human behaviors induced by the awareness of disease influ-
ence the epidemic dynamics in multiplex networks. The aware-
ness and virus spreading is mostly studied with the unaware-
aware-unaware-susceptible-infected-susceptible (UAU-SIS) model
over a given and fixed (static) network. However, the role of the
spatial-temporal properties of multiplex networks to the epidemic
spreading is not fully understood. In this work, a temporal
multiplex network consists of a static information spreading
network and a temporal physical contact network with a layer-
preference walk. Defining the tendency of aware nodes moving
to a specified layer as the layer preference, we focus on two
scenarios: (i) the layer preference is constant, (ii) the layer
preference is a function of the degree of nodes in the information
layer. We deduce the epidemic threshold of such a temporal
multiplex network, and validate it through numerical simulations.
Moreover, we find that the epidemic threshold decreases with
the decrease of the effective information spreading rate and the
increase of the layer preference. Furthermore, we find that when
the node with a higher degree displays a higher layer preference,
the spreading process on the network is remarkably promoted.
The findings shed a new light on the control of the epidemic
spreading process.
Index Terms—Temporal Multiplex Network, Awareness
Spreading, Epidemic Spreading, Random Walk, Layer Prefer-
ence.
I. INTRODUCTION
Scientists have traditionally focused on the modelling and
analysis of epidemic spreading in single-layer networks, where
nodes represent individuals and links represent the contacts
between individuals [1], [2], [3]. Many authors have found
the existence of an epidemic threshold τcof the susceptible-
infected-suceptible (SIS) epidemic process [4], [5], [6], [7] in
such networks, where each individual has two possible states:
susceptible (S) or infected (I). Infected individuals could infect
their healthy neighbors with a probability β, and cure with
a probability µ. The effective spreading rate is defined as
τ=β/µ. If the effective spreading rate τ > τc, an infectious
disease will spread out in the network, otherwise, it will die
out exponentially fast [8]. However, societies are multiplex
This work was partly supported by the National Natural Science Foundation
of China (No. 71731004, No. 61603097, No. 61751303) and the Natural
Science Fund of Distinguished Young Scholarship of China (No. 61425019),
in part by Natural Science Foundation of Shanghai (No.16ZR1446400).
Cong Li, Yuan Zhang and Xiang Li (corresponding author) are with
the Adaptive Networks and Control Lab, Electronic Engineering Depart-
ment, and the Research Center of Smart Networks and Systems, School of
Information Science and Engineering Fudan University, Shanghai 200433,
China, as well as with the MOE Frontiers Center for Brain Science, Insti-
tutes of Brain Science, Fudan University Shanghai 200433, China (e-mail:
cong_li@fudan.edu.cn; 17210720049@fudan.edu.cn; lix@fudan.edu.cn).
networks [9], which are composed of many interrelated sub-
networks. For instance, individuals have physical contacts with
family members, friends, and colleagues in off-line social
networks, which are the so-called physical contact networks.
Meanwhile, individuals could be connected in the online social
networks, where information could spread. When a virus
spreads in a physical contact network, individuals who are
aware of the epidemics may share the information online.
The information propagates through the online social network,
and aware individuals may take actions, such as wearing an
anti-virus mask, washing hands carefully, and reducing the
frequency of going out to prevent the infection. The inter-
play between the two dynamical processes, i.e., the epidemic
spreading and information propagation, in multiplex networks
has not been fully understood yet.
Granell et al. [10] studied the dynamic processes on a
static two-layer network, where the SIS epidemic spreading
process in one layer is coupled with the awareness spreading
in the other layer, and found the awareness spreading threshold
can separate two distinct regimes: if the awareness spreading
rate is below the threshold, the epidemic threshold does not
depend on the awareness; otherwise, the epidemic is prevented.
Granell et al. [11] further investigated the influence of mass
media, which can broadcast information over the entire net-
work, on the epidemic threshold. They found that once a
mass media exists, the awareness spreading can inhibit the
epidemic consistently. Guo et al. [12] depicted the information
spreading process with a threshold model and revealed a two-
stage effect on epidemic threshold. They found that when the
local awareness is smaller than 0.5, the epidemic threshold is
a fixed large value, and when the local awareness is larger
than 0.5, the epidemic threshold is a fixed small value.
Most studies focus on the awareness-epidemic spreading
process on static multiplex networks. However, the interactions
in social networks change with time. Neglecting the time-
varying nature of connectivity pattern in real systems brings
strong biases in understanding of dynamical phenomena [13],
[14], [15], [16]. Guo et al. [17] utilized the activity-driven
model [18] to represent a temporal network on which the
information spreads, and a static network to represent the
physical contact network. They found that the time variability
of the awareness layer reduces the restraint effects of the
awareness on the epidemic spreading, compared with the static
multiplex network.
Moreover, the physical contact network is described as a
temporal network. For instance, individuals may visit different
locations and contact with different persons to form a temporal
social contact network. Mishkovski et al. [19] proposed a
2
multiplex physical contact network, which composes of many
physical contact layers representing the contacts in different
locations, such as the colleague relationship network in the
office, and the friendship network at school. An individual
could only present in one physical contact layer at a given
moment. Hence, the system can be regarded as a time-
dependent network. Mishkovski et al. studied the susceptible-
infected-susceptible spreading dynamics with a non-biased
random walk process, and analytically derived the epidemic
threshold in the multiplex network, which is larger than that
in each of the isolated layers. Furthermore, An et al. [20]
integrated the epidemics with a biased random walk process
that each individual prefers to move to the layer with more
infected neighbors. They found that a one-step discontinuous
phase transition exists due to the biased random walk process.
In this work, we study the effects of the awareness spreading
process on the epidemic spreading on temporal multiplex
networks, which consist of a static information spreading layer
and a temporal physical contact layer. Since the time-varying
nature of physical contact is mainly caused by the relocation
of individuals, we utilize a multiplex network to represent
the temporal physical contact network. Different from the
multiplex networks with the random walk process in [19],
[20], the temporal physical contact layer is formed in this
work by a multiplex network with a layer-preference walk,
which is controlled by the awareness spreading process and
the epidemic dynamics together. Meanwhile, we assume that
the on-line social network is a static network, for the time
variance of the on-line social is much slower than that of
the physical contact network. We deduce the expression of
epidemic threshold and analyze the influence of the individual
layer preference, which represents the probability of nodes
migrating to a particular layer, on the epidemic threshold.
The outline of the paper is listed as following. In Section
II, we give an introduction of the multiplex networks which
is formed by a static online social network and a temporal
physical contact network. We then describe the awareness and
epidemic spreading processes on the multiplex networks. In
Section III, we deduce the theoretical epidemic threshold in
the temporal multiplex networks with the microscopic Markov
chain approach (MMCA). In Section IV, we derive the expres-
sion of epidemic threshold with different layer preferences.
Moreover, in Section V, we validate the theoretical results
with simulations on network models and real-world networks.
Furthermore, the influence of the individual layer preference
on the epidemic threshold is studied in Section VI. Finally,
we conclude the paper in Section VII.
II. DYNAMIC PROCESSES OVER MULTIPLEX
NETWORKS
We here study the awareness-epidemic spreading on a
temporal multiplex network G, consisting of a static online
social network where the information spreads, and a temporal
physical contact network where the epidemic spreads. The
physical contact network has a layered network structure
with Dlayers. The schematic of the multiplex network is
shown in Figure 1. We consider that the network is made
Fig. 1. A scheme of the multiplex network model with the UAU dynamics in
the information spreading layer, and the SIS dynamics in the physical network
with D layers. Corresponding individuals on each layer are the same, and
each node can migrate among the physical contact layers. In the model, each
individual ihas three possible states: unaware and susceptible (US), aware
and susceptible (AS), as well as aware and infected (AI).
up of Nnodes, and there exists a one-to-one corresponding
interdependency between the Nnodes in the online social
network and the Dphysical contact layers. The online social
network and the Dphysical contact layers are all simple,
undirected, unweighted graphs. The online social network,
also called the information spreading network, is represented
by an adjacency matrix B, where the element bij = 1, if
individuals iand jare connected (i.e. in a friendship on
the online social network), otherwise bij = 0. The graph at
physical contact layer dis represented by an adjacency matrix
A(d), where the element a(d)
ij = 1, if individuals iand j
contact with each other at location d, otherwise, a(d)
ij = 0.
The transition of an individual iacross different physical
contact layers is represented by a D×Dlocation transition
matrix Li, where the element ljk is the probability that node
imigrates from layer jto layer k. We use the probability
vector wi(t)=[wi(t)]1×D= [w(1)
i(t), w(2)
i(t), ..., w(D)
i(t)] to
represent the probability that node iin each layer at time t,
and obtain
wi(t+ 1) = wi(t)Li.(1)
When the random walk process is in the steady-state, the
probability that node iin layer ddoes not change with time
tany more, and we have wi(t+ 1) = wi(t) = πi. We denote
the probability that node iin each layer in the steady-state by
vector πi= [π(1)
i, π(2)
i, ..., π(D)
i].
The information of the existence of the epidemics spreads
on the online social network. Each individual has only two
possible states: aware (A) and unaware (U) of the epidemics.
Aware individuals have the information of virus and con-
duct self-protection behaviors to reduce the risk of being
infected, while unaware individuals will not prevent infection.
An unaware individual becomes aware via two ways: (1)
being informed by an aware neighbor with probability λ,(2)
being infected already. An aware individual might forget the
information to become unaware again, with probability δ. The
information spreading is an unaware-aware-unaware (UAU)
process.
In the physical contact layers, we assume that the epidemic
spreading is a susceptible-infected-susceptible (SIS) model.
3
Individuals are in two possible states: susceptible (S) or
infected (I). An infected node infects its unaware susceptible
neighbors in the same layer with probability β, and cures with
probability µ. At the moment of being infected, the individual
changes the state in the information spreading network to
be aware. An aware susceptible individual could reduce its
probability of being infected by a factor γ. We denote the
infection probabilities of an aware individual and that of
an unaware individual as βAand βU, respectively. Thus,
we have βA=γβU. Therefore, in the multiplex network
scheme, each individual can be in one of the three possible
states: unaware and susceptible (US), aware and susceptible
(AS), as well as aware and infected (AI). We use vector
pi(t) = [pUS
i(t), pAS
i(t), pAI
i(t)] to denote the probabilities
that node iis unaware and susceptible (US), aware and
susceptible (AS), as well as aware and infected (AI) at time
t, respectively.
III. MICROSCOPIC MARKOV CHAIN APPROACH
We here study the interplay between the awareness and
epidemic spreadings on temporal multiplex networks with
microscopic Markov chain approach (MMCA), which could
quantify the microscopic dynamics at the individual level.
With assuming the absence of dynamical correlation [21], we
denote ri(t)the probability that an unaware node iin the in-
formation spreading layer is informed by its aware neighbors,
qU
i(t)the probability that an unaware and susceptible node i
in the epidemic spreading layer is infected by its infectious
neighbors, and qA
i(t)the probability an aware and susceptible
node iis infected by its infectious neighbors. The probability
of node iin layer dis represented by w(d)
i(t). The probabilities
for node icould be approximated as
ri(t)=1−
N
Y
j6=i=1
[1 −λbij pA
j(t)],(2)
qU
i(t) = 1 −
N
Y
j6=i=1
[1 −βUpI
j(t)
D
X
d=1
a(d)
ij w(d)
i(t)w(d)
j(t)],(3)
qA
i(t)=1−
N
Y
j6=i=1
[1 −βApI
j(t)
D
X
d=1
a(d)
ij w(d)
i(t)w(d)
j(t)],(4)
where pA
j(t) = pAS
j(t) + pAI
j(t),pI
j(t) = pAI
j(t), and wi(t+
1) = wi(t)Li. Hence, we could obtain the probabilities of
node iin each of the three states with the Microscopic Markov
Chain as
pUS
i(t+ 1) =δµpAI
i(t) + (1 −ri(t))(1 −qU
i(t))pUS
i(t)
+δ(1 −qU
i(t))pAS
i(t),(5)
pAS
i(t+ 1) =(1 −δ)µpAI
i(t) + ri(t)(1 −qA
i(t))pUS
i(t)
+ (1 −δ)(1 −qA
i(t))pAS
i(t),(6)
pAI
i(t+ 1) =(1 −µ)pAI
i(t)
+ [(1 −ri(t))qU
i(t) + ri(t)qA
i(t)]pUS
i(t)
+ [δqU
i(t) + (1 −δ)qA
i(t)]pAS
i(t).(7)
The coefficients in each term of Eqs. (5)-(7) are the Markov
transition probability. For instance, in Eq. (5), δµ represents
the probability for node iin the AI state curing from the virus
and forgetting the information, (1−ri(t))(1−qU
i(t)) represents
the probability for node iremaining in the same state US, and
δ(1 −qU
i(t)) represents the probability that node iin the AS
state will forget the information but not be infected in next
time step.
The steady-state of the epidemic spreading in multiplex
networks is obtained under the conditions that pAI
i(t+ 1) =
pAI
i(t)and wi(t+ 1) = wi(t) = πi= [π(1)
i, π(2)
i, ..., π(D)
i],
which is the probability vector that node iin each layer in
the steady-state. Consequently, with pUS
i(t+ 1) = pUS
i(t)and
pAS
i(t+ 1) = pAS
i(t), the steady-state fraction pI
iof infected
nodes can be calculated using
pAI
i=1
µ{[(1 −ri)qU
i+riqA
i]pUS
i
+ [δqU
i+ (1 −δ)qA
i]pAS
i}.(8)
Near the critical point, the probability that node iis infected
is pI
i=pAI
i=εi1for ∀i. Ignoring the high order
terms, such as the pI
1pI
2,pI
1pI
2pI
3, ..., pI
1pI
2...pI
N, the transition
probabilities for node iare written as
qU
i=βU(
N
X
j6=i=1
D
X
d=1
pI
ja(d)
ij π(d)
iπ(d)
j),(9)
qA
i=βA(
N
X
j6=i=1
D
X
d=1
pI
ja(d)
ij π(d)
iπ(d)
j).(10)
Since βA=γβU, we have
qA
i=γqU
i.(11)
Moreover, we obtain pU
i=pUS
iand pA
i=pAS
i+pAI
i≈pAS
i.
Removing O(εi)terms in the steady-state of Eqs. (5) and (6),
we have
pUS
i=pUS
i(1 −ri) + δpAS
i,(12)
pAS
i=pUS
iri+ (1 −δ)pAS
i.(13)
Substituting Eqs. (11), (12) and (13) into Eq.(7) leads to
µεi=[(1 −ri)qU
i+riqA
i]pUS
i+ [δqU
i+ (1 −δ)qA
i]pAS
i
=[pUS
i(1 −ri) + δpAS
i]qU
i+ [pUS
iri+ (1 −δ)pAS
i]qA
i
=pUS
iqU
i+pAS
iqA
i
=βU(pAS
iγ+pUS
i)
N
X
j6=i=1
D
X
d=1
εja(d)
ij π(d)
iπ(d)
j.(14)
Consequently,
N
X
j6=i=1
[(pUS
i+γpAS
i)
D
X
d=1
a(d)
ij π(d)
iπ(d)
j−µ
βUδij ]εj= 0,(15)
4
where δij is the element of the identity matrix I. Rewriting
in the matrix form, we have
[H−µ
βUI]ε= 0,(16)
where ε= [ε1, ε2,· · · , εN], and the element in matrix His
hij = (pUS
i+γpAS
i)
D
X
d=1
a(d)
ij π(d)
iπ(d)
j.(17)
By the Perron-Frobenius Theorem [22], matrix Hhas a pos-
itive largest eigenvalue Λmax(H)with corresponding eigen-
vector εwhose elements are all positive, and there is only one
eigenvector of Hwith non-negative components. Hence, the
nontrivial solutions of Eq.(15) are the elements of the largest
eigenvector εof matrix H, and
βU
c=µ
Λmax(H).(18)
We obtain the epidemic threshold
τc=βU
c
µ=1
Λmax(H).(19)
It is noting that the matrix Hdepends on the solutions of
Eqs.(12) and (13) or equivalently
pUS
i=pUS
i(1 −ri) + δ(1 −pUS
i),(20)
where
ri= 1 −
N
Y
j6=i=1
[1 −λbij pA
j].(21)
For example, for the multiplex graph with the identical topol-
ogy in each physical contact layer, i.e. A(d)=A(1), for ∀d,
in this case, the element hij of matrix Hin Eq. (17) reduces
to
hij = (pUS
i+γpAS
i)Da(1)
ij
D
X
d=1
π(d)
iπ(d)
j.(22)
IV. EPIDEMIC THRESHOLD WITH DIFFERENT
LAYER PREFERENCES
In this work, a temporal physical contact network is formed
with Dphysical contact layers, which represent Dlocations,
and individuals might transit among the layers. The probability
vector wi(t) = [wi(t)]1×D= [w(1)
i(t), w(2)
i(t), ..., w(D)
i(t)]
represents the probability that node iin each layer at time t.
Consequently, the probability that node iinteracts with node
jat time tis PD
d=1 a(d)
ij w(d)
i(t)w(d)
j(t), and the topology of
the instantaneous network changes with time.
We here assume that an aware individual ihas a layer
preference ξi, when the random walk process is in the steady-
state. The aware node itransits to one specific layer with a
given probability, which is a function of the layer preference
ξi, while, transits to other layers with the same probability.
An unaware node ihas no layer preference. The steady-state
probability vector πiis defined as
πi=
[1
D,1
D, ..., 1
D]if iis US,
[1
D,1
D, ..., 1
D] + ξi[x, x
1−D, ..., x
1−D]if iis AS,
[1
D,1
D, ..., 1
D] + ξi[x
1−D, x, ..., x
1−D]if iis AI,
(23)
where x∈[0,D−1
D], and the sum of elements of πiis one.
We could calculate the probability that node iinteracts with
node jin the same physical contact layer with the steady-state
probability vector πi. Since the state of node iis uncertain,
we use probabilities pUS
i,pAS
ito determine the value of πi.
We here assume the first layer is the preferred location for
an aware and susceptible node. The probability that node i
interacts with node jin the first physical contact layer at the
steady-state is
a(1)
ij π(1)
iπ(1)
j=a(1)
ij [pAS
ipAS
j(1
D+ξix)( 1
D+ξjx)
+pAS
ipUS
j(1
D+ξix)1
D
+pUS
ipAS
j(1
D+ξjx)1
D
+pUS
ipUS
j(1
D)( 1
D)].(24)
The probability that node iinteracts with node jin other
physical contact layers at the steady-state is
a(d)
ij π(d)
iπ(d)
j=a(d)
ij [pAS
ipAS
j(1
D+ξi
x
1−D)( 1
D+ξj
x
1−D)
+pAS
ipUS
j(1
D+ξi
x
1−D)1
D
+pUS
ipAS
j(1
D+ξj
x
1−D)1
D
+pUS
ipUS
j(1
D)( 1
D)].(25)
Hence, the element of matrix Hin Eq. (17) is expressed as
hij =(pUS
i+γpAS
i)
D
X
d=1
a(d)
ij π(d)
iπ(d)
j
=(pUS
i+γpAS
i){a(1)
ij [pAS
ipAS
j(1
D+ξix)( 1
D+ξjx)
+pAS
ipUS
j(1
D+ξix)1
D
+pUS
ipAS
j(1
D+ξjx)1
D
+pUS
ipUS
j(1
D)( 1
D)]
+
D
X
d=1
a(d)
ij [pAS
ipAS
j(1
D+ξi
x
1−D)( 1
D+ξj
x
1−D)
+pAS
ipUS
j(1
D+ξi
x
1−D)1
D
+pUS
ipAS
j(1
D+ξj
x
1−D)1
D
+pUS
ipUS
j(1
D)( 1
D)]}.(26)
Thus, the epidemic threshold is calculated with Eq. (19). In
order to simply Eq.(26), we set x=D−1
D. Since pUS
i+pAS
i+
5
pAI
i= 1 in the steady-state, and pAI
i=εi1, Eq.(26) is
simplified as
hij =(pUS
i+γpAS
i)
2
X
d=1
a(d)
ij π(d)
iπ(d)
j
=( 1
D)2(pUS
i+γpAS
i)
{a(1)
ij [(D−1)ξipAS
i+ 1][(D−1)ξjpAS
j+ 1]
+
D
X
d=1
a(d)
ij (ξipAS
i−1)(ξjpAS
j−1)}.(27)
We study the effect of the layer preference on the epidemic
threshold in two cases: (1) The layer preference is the same
for all aware nodes; (2) The layer preference ξiis influenced
by the degree of node iin the information spreading network.
A. Layer Preference Is a Constant
When the layer preference is a constant, i.e.,ξ1=ξ2=
· · · =ξN=ξ, where ξ∈[-1,1], the steady-state probability
vector πiis only determined by the state of node. In a
multiplex network with D= 2 and x=D−1
D=1
2, the element
of corresponding matrix H1is
hij = (pUS
i+γpAS
i)
2
X
d=1
a(d)
ij π(d)
iπ(d)
j
= (pUS
i+γpAS
i){1
4a(1)
ij (ξpAS
i+ 1)(ξpAS
j+ 1)
+1
4a(2)
ij (ξpAS
i−1)(ξpAS
j−1)}.(28)
The epidemic threshold is
τc1=1
Λmax(H1).(29)
B. Layer Preference is Determined by the Degree in the
Information Layer
In this case, the layer preference ξihas two types of
relationships with degree kiof node iin the information
spreading network.
(1) ξiis positively correlated with ki
The layer preference ξiof aware node iincreases with the
increase of degree ki. We assume the expression of ξias
ξi=
ki
kmax
if ki6= 0,
0if ki= 0.
(30)
where kmax is the maximum degree in the information spread-
ing network.
(2) ξiis negatively correlated with ki
The layer preference ξiof aware node iincreases with the
decrease of degree ki, thus ξiis expressed as
ξi=
kmin
ki
if ki6= 0,
1if ki= 0.
(31)
Based on the assumptions, set D= 2,x=D−1
D=1
2,
matrix H2is written with element
hij = (pUS
i+γpAS
i)
2
X
d=1
a(d)
ij w(d)
iw(d)
j
= (pUS
i+γpAS
i){1
4a(1)
ij (ξipAS
i+ 1)(ξjpAS
j+ 1)
+1
4a(2)
ij (ξipAS
i−1)(ξjpAS
j−1)},(32)
and the epidemic threshold is
τc2=1
Λmax(H2).(33)
V. COMPARISON OF THE FRACTION AWARENESS
NODES AND INFECTED NODES VERSUS τ
In this section, we use the Monte Carlo (MC) simulations as
a benchmark to verify the theoretical results of the microscopic
Markov chain approach (MMCA). We calculate the fraction
ρAof awareness nodes and the fraction ρIof infected nodes
in the steady-state of the spreading processes with the MC
and MMCA, and compare the epidemic threshold in Eq. (19)
with that of the MC and MMCA on both multiplex network
models and real-world networks. We investigate the condition
under which the theoretical epidemic threshold provides a
better approximation.
In the MMCA, the probability pI
i(∞)of node ibeing in-
fected and the probability pA
i(∞)of node ibeing aware in the
steady-state are obtained by the iterative computation of Eqs.
(2) - (7). We here start with pI
i(0) = 0.001 and pA
i(0) = 0.001,
and stop the iteration at time step 3000 to guarantee that
the epidemic process reaches steady-state. Accordingly, the
fraction of infected nodes is ρI=1
NPN
i=1 pI
i(∞), and the
fraction of aware nodes is ρA=1
NPN
i=1 pA
i(∞)in the steady-
state.
In the MC simulations, utilizing the UAU-SIS model, we
obtain the state of each node at each time step, and calculate
the number NIof infected nodes and the number NAof
aware nodes in the steady-state. In detail, we randomly select
an infected node at the initial time. At each time step, an
unaware node will be informed the virus and become aware
with probability 1−(1−λ)NANB , where NANB is the number
of aware neighbors, and each aware node will forget the
information to become unaware with probability δin the
information spreading network. In each physical contact layer,
an unaware susceptible node and aware susceptible nodes
will be infected with probabilities 1−(1 −βU)NIN B and
1−(1 −βA)NIN B , where NIN B is the number of infected
neighbors, and each infected node will cure with probability
µ. Once a node is infected, it will change the corresponding
state in the information spreading network to be aware. Then,
we update the location of each node according to the cor-
responding location transition matrix Liand start next time
step. We performed all the simulations for 3000 time steps
at each realization and multiple realizations to calculate the
steady-state fraction ρI=E[NI
N]of infected nodes and the
steady-state fraction ρA=E[NA
N]of aware nodes, where E[·]
is the expectation.
6
In a multiplex network with Dphysical contact layers, we
set wi(0) = [ 1
D,1
D,· · · ,1
D],∀i. With a given layer preference
ξi, we write the vector πi=wi(t+ 1) = wi(t), when the
random walk process is in the steady-state. The corresponding
location transition matrix Liis obtained with wi(t+ 1) =
wi(t)Li,Σkljk = 1 and the given (D−1)2independent ljk .
Here, we will explain the MC and MMCA simulations. In
the MC simulation, we randomly select an infected node at
the initial time, and its corresponding node in the information
spreading network is aware of the epidemic. Correspondingly,
pI
i(0) = 0.001 and pA
i(0) = 0.001 in the MMCA. At each
time step, the aware nodes will spread the awareness to their
neighbors, and the probability of its neighbors becomes aware
is λ. At the next time step, all aware nodes will recover to
be unaware with probability δ. After the UAU information
spreading process, the infected node infects its unaware sus-
ceptible neighbors and aware susceptible neighbors in the same
layer with probability βUand βA, respectively. After this, all
infected nodes will recover to be susceptible with probability
µ. Once the nodes get infected, they change their state in
the information spreading network to be aware. When the
spreading processes end, we update the location of each node
according to the corresponding location transition matrix Li.
We simulate for 3000 time steps to make sure the spreading
processes and the random walk process all become stable.
The analytical results derived in Eq. (19) are valid for
arbitrary multiplex network structures. For simulations, two
classic network models could be GER and GBA : (i) the Erdös-
Rényi (ER) random networks [23], which have a binomial
degree distribution, and (ii) the Barabási-Albert (BA) scale-
free networks [24] with a pow-law degree distribution, which
are observed in many real-world networks. In numerical simu-
lations, we consider two multiplex network models G1and G2:
(i) G1is formed by one information spreading network GIN F
and two GBA physical contact layers; (ii) G2is formed by one
information spreading network GIN F , one GBA physical con-
tact layer and one GER physical contact layers. Specifically,
the network size N= 1000, in the BA scale-free networks
m= 3 and the link density is pB A '0.006. We consider the
ER networks with the same link density pER =pBA = 0.006,
and the information spreading network owning the topology as
the BA networks with 20 extra random links. The real-world
network, denoted by G3, is a four-layer multiplex network
[19] which consists of the relationships among 61 employees
at the department of Computer Science at Aarhus University.
The information spreading network is the F acebook online
social network, and three physical contact layers are the face-
to-face social networks at the Leisure scenario, the W ork
scenario and the Lunch scenario, respectively.
A. Constant Case for Layer Preference
We consider the constant case for layer preference in the
awareness-epidemic spreading on multiplex networks G1,G2
and G3. We set the effective information spreading rate η=
λ
σ= 0.3, and infection reducing factor γ= 0.1. We simulate
the UAU-SIS awareness-epidemic spreading with the MC and
the MMCA. The steady-state fraction ρIof infected nodes and
0.0 0.2 0.4 0.6 0.8 1.0
τ
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
ρA
MMCA
ξ
=0.4
MMCA
ξ
=-1
MC
ξ
=0.4
MC
ξ
=-1
0.0 0.2 0.4 0.6 0.8 1.0
τ
0.0
0.1
0.2
0.3
0.4
0.5
ρI
τc
= 0.276
τc
= 0.392
MMCA
ξ
=0.4
MMCA
ξ
=-1
MC
ξ
=0.4
MC
ξ
=-1
Fig. 2. Top: comparison of the steady-state fraction ρAof aware nodes using
the MC simulation (marks) and the MMCA (solid line). Bottom: comparison
of the steady-state fraction ρIof infected individuals using MC simulation
(marks) and the MMCA (solid line). The layer preference ξ= 0.4(in blue)
and ξ=−1(in red). The initial number of infected nodes is 1, and the
effective information spreading rate η= 0.3. The MC simulation is performed
on the multiplex network G1with 400 realizations.
0.0 0.2 0.4 0.6 0.8 1.0
τ
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
ρA
MMCA
ξ
=0.4
MC
ξ
=0.4
0.0 0.2 0.4 0.6 0.8 1.0
τ
0.0
0.1
0.2
0.3
0.4
0.5
ρI
τc
= 0.44
MMCA
ξ
=0.4
MC
ξ
=0.4
Fig. 3. Top: comparison of the steady-state fraction ρAof aware individuals
using the MC simulation (marks) and the MMCA (solid line). Bottom:
comparison of the steady-state fraction ρIof infected individuals using the
MC simulation (marks) and the MMCA (solid line). The initial number of
infected nodes is 1, the layer preference ξ= 0.4and the effective information
spreading rate η= 0.3. The MC simulation is performed on the multiplex
network G2with 400 realizations.
the steady-state fraction ρAof aware nodes are calculated for
increasing effective spreading rate τ.
First, we set the layer preference ξi=ξ= 0.4and −1,
to study the influence of the ξion the difference between
the MMCA and MC simulation. Fig. 2 depicts the steady-
state fraction ρAof aware nodes and steady-state fraction ρI
7
0.0 0.5 1.0 1.5 2.0 2.5
τ
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
ρA
MMCA
ξ
=0.4
MC
ξ
=0.4
0.0 0.5 1.0 1.5 2.0 2.5
τ
0.0
0.1
0.2
0.3
0.4
0.5
ρI
τc
=0.72
MMCA
ξ
=0.4
MC
ξ
=0.4
Fig. 4. The steady-state fraction ρAof aware nodes and the steady-state
fraction ρIof infected nodes for the real-world multiplex network G3. The
initial number of infected nodes is 1, the layer preference ξi= 0.4and the
effective information spreading rate η= 0.3. The MC simulation is performed
on the multiplex network G3with 1000 realizations.
of infected nodes versus the effective spreading rate τon
multiplex network G1. As shown in Fig. 2, the difference
between the MMCA and the MC is rather small, and the
difference is smaller when the layer preference |ξi|is larger.
The epidemic threshold τccalculated by Equation (29) is
confirmed by the MC simulation. An interesting finding is
that the epidemic threshold with ξi= 0.4is much larger than
that with ξi=−1.
Then, we set the layer preference ξi=ξ= 0.4to calculate
the steady-state fraction ρAof aware nodes and steady-state
fraction ρIof infected nodes versus the effective spreading
rate τon multiplex network G2. Fig. 3 illustrates that the
MMCA approximates the MC simulation well in network G2.
Figures 2 and 3 show that the epidemic threshold in G1is
smaller than that in G2, when the layer preference ξiis the
same. It could be explained by the fact that virus more easily
propagates throughout the BA scale-free networks than the
case with the ER networks [25].
We further compare the MMCA and the MC, and verify
the epidemic threshold in a real-world network G3. Since the
number of nodes is relatively small, we simulate the UAU-
SIS awareness-epidemic spreading with 1000 realizations of
the MC. We set ξi=ξ= 0.4,i.e.,πi= [ 3
5,1
5,1
5]for AS
nodes, πi= [1
5,3
5,1
5]for AI nodes, and πi= [1
3,1
3,1
3]for
US nodes. As shown in Fig. 4, the accuracy of our MMCA
analysis is confirmed by the MC simulation. However, there
still exists a tiny inaccuracy, which might be caused by the
finite network size [19].
B. Degree Related Case for Layer Preference
We consider the case that the layer preference ξiis a
function of the degree of nodes in the awareness-epidemic
spreading on multiplex network G1. We assume that the layer
0.0 0.2 0.4 0.6 0.8 1.0
τ
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
ρA
MMCA
ξi
=
ki
kma
x
MC
ξi
=
ki
kma
x
0.0 0.2 0.4 0.6 0.8 1.0
τ
0.0
0.1
0.2
0.3
0.4
0.5
ρI
τc
= 0.38
MMCA
ξi
=
ki
kma
x
MC
ξi
=
ki
kma
x
Fig. 5. The steady-state fraction ρAof aware nodes and the steady-state
fraction ρIof infected nodes when ξiis positively correlated with the
degree of nodes in the information spreading network. The MC simulation is
performed on the multiplex network G1with 400 realizations. The effective
information spreading rate η= 0.3.
preference ξiof node ihas two types of correlations with its
degree kiin the information layer: ξiis positively correlated
with ki, and ξiis negatively correlated with ki, respectively.
We set η=λ
σ= 0.3,γ= 0.1. We perform the UAU-SIS
spreading on the network G1, and calculate the fraction ρAof
awareness nodes and the fraction ρIof infected nodes in the
steady-state with the MC and the MMCA.
Figs. 5 and 6 depict the steady-state fraction ρAof aware
nodes and the steady-state fraction ρIof infected nodes
versus the effective spreading rate τ, when ξiis positively
and negatively correlated with ki, respectively. The epidemic
threshold τccalculated by Eq. (33) is confirmed by the MC
simulation.
It is worth noting that, compared with the positive case,
when ξiis negatively correlated with ki, the epidemic thresh-
old is relatively large. The phenomenon demonstrates that the
AI nodes with large degree preferring to the same layer could
promote the spreading, even when the AS nodes with large
degrees prefer to another layer.
VI. EFFECT OF LAYER PREFERENCE ON THE
EPIDEMIC THRESHOLD
We here study how the effective information spreading rate
ηand the layer preference ξaffect the epidemic threshold.
We perform the UAU-SIS awareness-epidemic spreading on
multiplex networks G1,G2and G3. When the layer preference
ξi=ξis a constant, we calculate the epidemic threshold τc
with given η= 0 : 0.05 : 0.4and ξ=−1:0.1:1.
Fig. 7 shows that the epidemic threshold τcis small, which
is almost not affected by the layer preference ξ, when η
is small in G1. In other words, the virus may spread out
exponentially even when the effective spreading rate τis small
in the early stage of the epidemic, when the information of
8
0.0 0.2 0.4 0.6 0.8 1.0
τ
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
ρA
MMCA
ξi
=
kmin
ki
MC
ξi
=
kmin
ki
0.0 0.2 0.4 0.6 0.8 1.0
τ
0.0
0.1
0.2
0.3
0.4
0.5
ρI
τc
= 0.41
MMCA
ξi
=
kmin
ki
MC
ξi
=
kmin
ki
Fig. 6. The steady-state fraction ρAof aware nodes and the steady-state
fraction ρIof infected nodes when ξiis negatively correlated with the
degree of nodes in the information spreading network. The MC simulation is
performed on the multiplex network G1with 400 realizations. The effective
information spreading rate η= 0.3.
virus has not spread out among the individuals. Moreover,
the epidemic threshold τcincreases with the increase of
the effective information spreading rate η, when the layer
preference |ξ|is fixed. However, the epidemic threshold τc
decreases with the increase of the layer preference |ξ|, when
the effective information spreading rate ηis fixed.
The results in network G2are similar to that in network
G1(see Figs. 7 and 8). The main difference is that Fig. 8 is
asymmetric with ξ= 0. The epidemic threshold τcdecreases
with the increase of ξwhen ηis small, and the epidemic
threshold τc(ξi=−ξ)> τc(ξi=ξ)when ηis large and ξ
is a positive constant. The observation is mainly influenced
by the network topology, and the theoretically explanation is
given in Appendix. Moreover, we verify the analysis with real-
world network G3in Fig. 9.
The results bring a new perspective on the control of
epidemics. The emergence of the novel corona virus COVID-
19 has considerably influenced the public health and global
economy [26], [27], [28]. In the beginning stage of COVID-
19, few news or information about the corona virus makes
the virus spread relatively fast, which matches the results that
the epidemic threshold τcis small when ηis small. Then,
the growing information regarding the COVID-19 leads to the
increase number of AS nodes, thus the epidemic threshold
increases. However, our results show that the spreading of
information causes the infected nodes preferring to the same
location, i.e., hospital, which inversely makes the virus spreads
more easily. The results suggest that in the early stage of an
unknown virus like COVID-19, patients with mild symptoms
are not recommended to go to hospital before the building
of mobile cabin hospitals. Therefore, we suggest that the
government should not only distribute to the society the alert
information of a new virus like COVID-19, but also tell the
ξ
−1.00
−0.75
−0.50
−0.25
0.00
0.25
0.50
0.75
1.00
η
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
τC
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.0
0.2
0.4
0.6
Fig. 7. Epidemic threshold τcas a function of ξiand ηon the multiplex
network G1.
public that the irrational preference of rushing into hospital
may cause their unexpected infection. The suggestion could
help suppress the rapidly growing of infected individuals and
avoid the ineffective allocation of the scarce medical resources
in the early stage.
Furthermore, we compare the epidemic threshold τcversus
ηon multiplex network G1in three cases: (i) ξiis positively
correlated with ki, (ii) ξiis negatively correlated with ki, (iii)
ξi= 0 for ∀i. Fig. 10 shows that the epidemic threshold is
almost the same when ηis small. However, the difference
among the three epidemic thresholds increases with η, and the
epidemic threshold in case (i) is the smallest, even when ηis
the same in the three cases. The result implies that avoiding the
gathering of high-degree nodes in the same layer is important
for increasing the epidemic threshold, although the number of
high-degree nodes is small in scale-free networks. The finding
is consistent with the result in [29], that the epidemic threshold
of a multi-layer network is determined by the network with the
strongest degree distribution heterogeneity. Since low-degree
nodes do not have a layer preference, and high-degree nodes
with a strong preference will gather in a specific layer, which
increases the heterogeneity of the specific layer and thereby
decreases the epidemic threshold.
VII. CONCLUSIONS
Epidemic threshold plays an important role in the virus
spreading processes in networks. We have studied a cou-
pled spreading process of awareness and epidemic (UAU-SIS
model) in temporal multiplex networks with a spatial-temporal
property, i.e., the layer preference. We have theoretically
shown that the epidemic threshold τc=1
λ(H), where H
could be obtained with the iterative computation of dynamic
equations, and verified the epidemic threshold with numerical
simulations. The influences of the layer preference on the
epidemic threshold reveal that the awareness of virus increases
the epidemic threshold, while the human behavior induced by
the awareness, such as preferring to the same location like
9
ξ
−1.00
−0.75
−0.50
−0.25
0.00
0.25
0.50
0.75
1.00
η
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
τC
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Fig. 8. Epidemic threshold τcas a function of ξiand ηon the multiplex
network G2.
ξ
0.0 0.2 0.4 0.6 0.8 1.0
η
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
τC
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Fig. 9. Epidemic threshold τcas a function of ξiand ηon the real-world
multiplex network G3.
hospital, inversely decreases the epidemic threshold. Specially,
it is observed that avoiding high degree nodes in the same
location could help suppress the prevalence of virus. Our
results provide essential insights for controlling the epidemic
spreading. However, it is still challenging to theoretically study
the epidemic spreading in multiplex networks with various
curing rates. It may be worthwhile that further investigations
compute or estimate the epidemic threshold of temporal multi-
plex networks with different curing rates in different physical
contact layers.
Appendix
EFFECT OF THE LAYER PREFERENCE ON THE
EPIDEMIC THRESHOLD ON G2
When the layer preference is a constant, we have ξ1=
ξ2=· · · =ξN=ξ, where ξ∈[-1,1]. Since D= 2 and
0.0 0.2 0.4 0.6 0.8 1.0
η
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
τC
case (i)
case (ii)
case (iii)
Fig. 10. Comparison of the epidemic threshold τcversus ηon the multiplex
network G1in three cases: (i) ξiis positively correlated with kiin blue dash
line, (ii) ξiis negatively correlated with kiin purple dot line, (iii) ξi= 0 for
all nodes in black line.
x=D−1
D=1
2, the element of corresponding matrix H1is
hij = (pUS
i+γpAS
i)
2
X
d=1
a(d)
ij π(d)
iπ(d)
j
= (pUS
i+γpAS
i){1
4a(1)
ij (ξpAS
i+ 1)(ξpAS
j+ 1)
+1
4a(2)
ij (ξpAS
i−1)(ξpAS
j−1)}.(34)
When ηis relatively small, pAS
i≈0, thus, ξpAS
i≈0,
indicating that the influence of ξon the matrix H1is small,
and H1is determined by both [a(1)
ij ]N×Nand [a(2)
ij ]N×N. In
the multiplex network G2,[a(1)
ij ]N×Nis the adjacency matrix
of the BA scale-free network, and [a(2)
ij ]N×Nis the adjacency
matrix of the ER random network. Therefore, as ξincreases,
the influence of the ER random network decreases, while the
BA scale-free network obviously influences the matrix H1.
Accordingly, the epidemic threshold τcdecreases with the
increase of ξwhen ηis small.
When ηis relatively large, we ignore the terms whose
product is approximate to 0and write the matrix with element
hij = (pUS
i+γpAS
i)
2
X
d=1
a(d)
ij π(d)
iπ(d)
j
=1
4(pUS
i+γpAS
i)a∗
ij (|ξ|pAS
i+ 1)(|ξ|pAS
j+ 1),
where a∗
ij =a(1)
ij when ξ > 0, and a∗
ij =a(2)
ij when ξ < 0.
The element of matrix H1satisfies that
1
4(pUS
i+γpAS
i)a∗
ij (|ξ|pAS
min + 1)2
≤1
4(pUS
i+γpAS
i)a∗
ij (|ξ|pAS
i+ 1)(|ξ|pAS
j+ 1)
≤1
4(pUS
i+γpAS
i)a∗
ij (|ξ|pAS
max + 1)2.(35)
Defining max
1≤i≤N(pAS
i) = pAS
max,min
1≤i≤N(pAS
i) = pAS
min, and
Q=1
4[a∗
ij ]N×N[pUS
i+γpAS
i]N×1, we obtain the inequations
10
(|ξ|pAS
min + 1)2Λmax(Q)≤Λmax (H)≤(|ξ|pAS
max + 1)2Λmax(Q).
(36)
Accordingly,
1
(|ξ|pAS
max + 1)2Λmax(Q)≤τc≤1
(|ξ|pAS
min + 1)2Λmax(Q).
(37)
We arrive that the upper and lower bounds of the epidemic
threshold τcdecrease with the increase of the layer preference
|ξ|, when ηis relatively large.
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Cong Li (M’15) received the Ph.D. degree in intel-
ligent systems from Delft University of Technology
(TUDelft), Delft, The Netherlands, in 2014. She
is currently an associate professor in Electronic
Engineering Department at Fudan University, where
she involves in complex network theory and applica-
tions. Her research focuses on analysis and modeling
of complex networks, including network properties,
dynamic processes, network of networks etc. Her
work on these subjects include 30 international jour-
nal papers and conference papers.
Yuan Zhang received the B.S. degree at Zhengzhou
University, in 2016. She is currently a postgraduate
student in Electronic Engineering Department at Fu-
dan University. Her research focuses on the epidemic
spreading dynamics on complex networks.
11
Xiang Li (M’05–SM’08) received the BS and PhD
degrees in control theory and control engineering
from Nankai University, China, in 1997 and 2002,
respectively. Before joining Fudan University as a
professor of the Electronic Engineering Department
in 2008, he was with City University of Hong Kong,
Int. University Bremen and Shanghai Jiao Tong
University, as post-doc research fellow, Humboldt
research fellow and an associate professor in 2002-
2004, 2005-2006 and 2004-2007, respectively. He
served as head of the Electronic Engineering De-
partment at Fudan University in 2010-2015. Currently, he is a distinguished
professor of Fudan University, and chairs the Adaptive Networks and Control
(CAN) group and the Research Center of Smart Networks & Systems,
School of Information Science & Engineering, Fudan University. He served as
associate editor for the IEEE Transactions on Circuits and Systems-I: Regular
Papers (2010-2015), and serves as associate editor for the IEEE Transactions
on Network Science and Engineering, the Journal of Complex Networks and
the IEEE Circuits and Systems Society Newsletter. His main research interests
cover network science and systems control in both theory and applications.
He has (co-)authored 5 research monographs, 7 book chapters, and more than
200 peer-refereed publications in journals and conferences. He received the
IEEE Guillemin-Cauer Best Transactions Paper Award from the IEEE Circuits
and Systems Society in 2005, Shanghai Natural Science Award (1st class)
in 2008, Shanghai Science and Technology Young Talents Award in 2010,
National Science Foundation for Distinguished Young Scholar of China in
2014, National Natural Science Award of China (2nd class) in 2015, Ten
Thousand Talent Program of China in 2017, TCCT CHEN Han-Fu Award of
Chinese Automation Association in 2019, among other awards and honors.
