# Contagion dynamics on adaptive multiplex networks with awareness-dependent
rewiring††thanks: Project supported by the National Natural Science Foundation
of China (Grant Nos. 11601294 and 61873154), Shanxi Scholarship Council of
China (Grant No. 2016-011), Shanxi Province Science Foundation for Youths
(Grant Nos. 201601D021012, 201801D221011, 201901D211159, 201801D221007,
201801D221003), and the 1331 Engineering Project of Shanxi Province.

Xiao-Long Peng1,2 Corresponding author. E-mail<EMAIL_ADDRESS>Yi-Dan
Zhang1,2,3
1Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shanxi,
China
2Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on
Disease
Control and Prevention, Shanxi University, Taiyuan 030006, Shanxi, China
3School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi,
China

###### Abstract

Over the last few years, the interplay between contagion dynamics of social
influences (e.g., human awareness, risk perception, and information
dissemination) and biological infections has been extensively investigated
within the framework of multiplex networks. The vast majority of existing
multiplex network spreading models typically resort to heterogeneous mean-
field approximation and microscopic Markov chain approaches. Such approaches
usually manifest richer dynamical properties on multiplex networks than those
on simplex networks; however, they fall short of a subtle analysis of the
variations in connections between nodes of the network and fail to account for
the adaptive behavioral changes among individuals in response to epidemic
outbreaks. To transcend these limitations, in this paper we develop a highly
integrated effective degree approach to modeling epidemic and awareness
spreading processes on multiplex networks coupled with awareness-dependent
adaptive rewiring. This approach keeps track of the number of nearest
neighbors in each state of an individual; consequently, it allows for the
integration of changes in local contacts into the multiplex network model. We
derive a formula for the threshold condition of contagion outbreak; also, we
provide a lower bound for the threshold parameter to indicate the effect of
adaptive rewiring. The threshold analysis is confirmed by extensive
simulations. Our results show that awareness-dependent link rewiring plays an
important role in enhancing the transmission threshold as well as lowering the
epidemic prevalence. Moreover, it is revealed that intensified awareness
diffusion in conjunction with enhanced link rewiring makes a greater
contribution to disease prevention and control. In addition, the critical
phenomenon is observed in the dependence of the epidemic threshold on the
awareness diffusion rate, supporting the metacritical point previously
reported in the literature. This work might shed light on the understanding of
the interplay between epidemic dynamics and social contagion on adaptive
networks.

Keywords: epidemic spreading, awareness diffusion, adaptive rewiring,
multiplex networks

PACS: 89.75.Hc, 87.23.Ge, 87.19.X-

##  1 Introduction

Nowadays, the communication networks—through media, such as telephones,
emails, Facebook, Twitter and Wechat—provide people easy access to information
about the emergence or outbreaks of infectious diseases. In many cases, public
health communications concerning the risk of infections have a beneficial
effect on human behavior [1]. For example, during the COVID-19 pandemic, the
World Health Organization has disseminated information on health issues such
as quarantining, social distancing, regulating international travelers,
closing schools, and operating public transportation and workplaces. People
who receive the disease information become aware of the transmission risk and
tend to change their behavior to reduce their susceptibility to infections
[2]. In this way, social contagion [3, 4, 5] of human awareness about
infections influences the biological contagion [1].

In the literature, many works have been devoted to understanding the impact of
human awareness on infectious disease spread. For example, Funk et al. [6]
formulated a network model to study how the spread of awareness, prompted by
first-hand information about a disease case, affects the disease spread. Along
the same lines, Wu et al. [7] proposed a continuous heterogeneous mean-field
model to study the impact of contact, local and global awareness on the
epidemic spread on scale-free networks. These models are based on a simplex
network structure by simply assuming that the physical contact network layer
(for disease transmission) and the information network layer (for awareness
propagation) are identical in that their connections among the individuals
coincide with each layer. In reality, however, the information network, where
the individuals communicate information about disease cases, is often
topologically different from the face-to-face contact network, where the
disease can be transmitted through infectious contacts. To address this
problem, Granell et al. [8] pioneered the study of competing dynamics between
awareness and disease spreading processes on a multiplex network consisting of
two network layers, in which the nodes represent the same entities in both
layers while edges in different layers stand for different types of social
relationships [9]. Their multiplex network framework was applied to a
susceptible-infected-susceptible (SIS) [10] disease transmission that confers
no immunity and a unaware-aware-unaware (UAU) information diffusion process.
Granell et al. [8] utilized a microscopic Markov chain approach to obtain the
threshold condition for the disease outbreak; interestingly, they discovered a
metacritical point for the epidemic threshold. They further considered the
general case where the awareness does not guarantee complete immunization and
infection does not confer instant awareness of it; also, the impact of mass
media on the epidemic prevalence was incorporated [11].

The pioneering work by Granell et al. has prompted a number of supplementary
studies on the competing processes of disease and awareness on multiplex
networks. For instance, Guo et al. [12] proposed a local awareness-controlled
SIS epidemic model on multiplex networks; interestingly, they observed two-
stage effects on epidemic threshold and epidemic prevalence. Kan and Zhang
[13] investigated the combined effects of awareness spread and self-initiated
awareness behavior on the SIS disease dynamics. More recently, Xia et al. [14,
15, 16] extended the multiplex network model of awareness-disease dynamics to
a susceptible-infected-recovered (SIR) [10] epidemic process that allows
permanent immunity after infection. All of these models have employed the
microscopic Markov chain approach by Granell et al. [8, 11] to identify the
epidemic threshold as the largest eigenvalue of a matrix operator that
includes information about the network structure and the density of aware
individuals (who know about the infection risk) in the steady state [12, 13,
14, 15, 16].

The above models considered the impact of behavioral response to disease
information by simply integrating the impact into a reduced susceptibility for
aware susceptibles (i.e., susceptible individuals who are aware of the
disease). Nevertheless, they did not actually account for the human behavioral
changes that can reshape the underlying contact network, which in turn results
in the coevolution of epidemics and network topology [17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28, 29]. The failure of the heterogeneous mean-field
approximation and microscopic Markov chain approaches to reflect the network
evolution (due to behavioral changes in response to infection threat) is
rooted in the lack of time evolution of edges in the total-degree-based
approximation frameworks. To eliminate this shortcoming, Gross et al. [17]
proposed an adaptive network epidemic model with adaptive rewiring of
dangerous links, in which susceptible nodes are allowed to break away from
their infected neighbors and reconnect towards other noninfected nodes.
Despite the low complexity in the analytical formalism, this adaptive epidemic
model revealed rich dynamical properties, such as bistability, oscillations
and first-order phase transitions [17, 28]. However, the low-dimensional
formalism, in general, does not provide with high accuracy the time evolution
of the disease progression and the underlying network topology. To this end,
Marceau et al. [20] introduced an improved compartmental formalism in which
nodes are classified into groups according to their total degree (namely, the
total number of their neighbors) and their infectious degree (namely, the
number of their infected neighbors). Since this formalism features the number
of active neighbors associated with the state of infectiousness, in the rest
of this paper we shall call it the effective degree formalism. Cai et al. [33]
successfully addressed the dynamical correlation problem of SIS epidemic model
on complex networks by combing the heterogeneous mean-field theory [30, 31,
32] and the effective degree model. Ma et al. [34] applied the effective
degree formalism to both the SIS and the SIR epidemic models and found that
these two effective degree epidemic models give rise to different epidemic
thresholds. Zhou et al. [35] has recently utilized the effective degree
formalism in the study of disease and awareness spreading dynamics on
multiplex networks, where excellent agreement was observed between theoretical
predictions and stochastic simulations for the spreading dynamics. The authors
further developed a partial effective degree theory with a mixture of the
effective degree formalism for the contact network layer and the heterogeneous
mean-field approximation for the information network layer [35]. In their
work, however, the behavioral adaptations of individuals that could lead to
network evolution have not been incorporated in the effective degree spreading
model on multiplex networks.

In view of the practical consideration that an individual’s behavioral
adaptations (e.g., activity reduction [37, 36]) could be affected by their own
infection state and their awareness of disease information [37, 38], it is of
significance to integrate the awareness-related behavioral adaptations into
the spreading dynamics of epidemic and awareness on multiplex networks. In
this paper, we apply the effective degree approach to modeling the
simultaneous contagion dynamics of an SIS epidemic transmission and a UAU
awareness propagation on adaptive multiplex networks with awareness-dependent
rewiring in the physical contact network layer. Our model is described in
section 2 and formalised in section 3. Threshold conditions of disease
outbreak and awareness information invasion are analyzed for the full
effective degree model in section 4 and the lower bounds of the threshold
conditions are derived for a reduced model with some special case in section
5. Simulation results are presented in section 6, where the effects of
adaptive rewiring and awareness diffusion on the epidemic threshold and
epidemic prevalence are analyzed and the critical phenomenon in the reliance
of epidemic onset on awareness diffusion is highlighted. Our conclusions are
drawn in section 7.

##  2 The SIS-UAU contagion model on adaptive multiplex networks

Figure 1: Flowchart of the multiplex network spreading model of an SIS
epidemic process and a UAU awareness propagation process, where the events of
adaptive rewiring of dangerous (infectious) contacts (that connect an infected
node and a susceptible one) randomly occurs in the epidemic layer. In the
awareness layer, unaware nodes become aware after they are informed of disease
information by an aware neighbor at a rate $\alpha$, whereas an aware node
returns to being unaware again after it forgets the disease information at a
rate $\mu$. In the epidemic layer, susceptible nodes are infected by their
infected neighbors at a transmission rate that depends on the awareness state
of the susceptible; here we denote by $\beta^{A}$ (resp., $\beta^{U}$) the
transmission rate for aware (resp., unaware) susceptible nodes. Meanwhile,
each infected node recovers to become susceptible again at a rate $\gamma$. In
addition, for each infectious connection linking an infected node and a
susceptible one, the susceptible will break the connection away from the
infected and rewire towards a randomly selected susceptible (other than the
original susceptible and its existing neighbors) at a rewiring rate, which is
dependent on the awareness state of the original susceptible node. Here we
denote by $\omega^{A}$ (resp., $\omega^{U}$) the rewiring rate for the aware
(resp., unaware) susceptible nodes.

We start with a two-layer multiplex network, as shown in Fig. 1, consisting of
an awareness (information) layer and an epidemic (physical contact) layer. The
awareness layer represents the online social (or communication) network where
individuals propagate disease information to their neighbors on the same
layer, while the epidemic layer stands for the physical contact network on
which individuals transmit disease when infected individuals have a physical
contact with susceptible ones. The nodes between the two layers form a one-to-
one correspondence as illustrated by the dashed lines in Fig. 1; each dashed
line indicates a correspondence for the same node. Namely, both layers contain
exactly the same nodes. However, the connectivity patterns in the two layers
are different from each other since the edges in different layers represent
different types of relationships. That means the degree distributions are
distinct between the two layers.

In the awareness layer we adopt the UAU model proposed by Granell et al. [8]
for the diffusion of awareness information on the disease, where each
individual is in one of the two discrete states: aware (A) or unaware (U),
representing whether or not they have acquired the disease information.
Unaware individuals become aware at a diffusion rate $\alpha$ by communication
with their aware neighbors (where the notations in the model definition are
summarized in Table 1). In addition, unaware individuals will directly become
aware with probability $1$ once they get infected via disease-causing physical
contacts. Meanwhile, each aware individual becomes unaware again at a rate
$\mu$ after it forgets the awareness or no longer cares about it.

Table 1: Notation used in the model definition.  Term |  Meaning
---|---
$\alpha$ |  awareness information diffusion rate
$\mu$ |  awareness information forgetting rate
$\beta^{U}$ |  disease transmission rate for unaware susceptible individuals
$\beta^{A}$ |  disease transmission rate for aware susceptible individuals
$\beta$ |  disease transmission rate (note that in this paper we simply assume a common transmission rate, i.e., $\beta^{U}=\beta^{A}=\beta$)
$\gamma$ |  recovery rate from infection
$\omega^{U}$ |  link rewiring rate on each risky contact between a unaware susceptible individual and an infected one
$\omega^{A}$ |  link rewiring rate on each risky contact between a aware susceptible individual and an infected one
$y$ |  ratio between $\omega^{U}$ and $\omega^{A}$, i.e., $y=\omega^{U}/\omega^{A}$

In the epidemic layer we consider an SIS epidemic process with awareness-
dependent link rewiring, where each individual is in one the the two distinct
sates: susceptible (S) or infected (I). The transmission rate for aware
(resp., unaware) susceptible nodes is $\beta^{A}$ (resp., $\beta^{U}$), the
recovery rate for each infected node to return to the susceptible sate is
$\gamma$, and the adaptive rewiring rate for the aware (resp., unaware)
susceptible node belonging to each susceptible-infected ($S$-$I$) edge to
rewire away from the infected node and connect towards another randomly
selected susceptible node is $\omega^{A}(\omega^{U})$. Since awareness of
disease infection often induces individuals to take preventive measures to
evade or reduce the risk of infection in the epidemic layer [39], it is
natural to assume $\beta^{A}\leq\beta^{U}$ and $\omega^{A}\geq\omega^{U}$. For
the sake of simplicity, we set $\omega^{U}=y\omega^{A}$ with $0\leq y\leq 1$;
here $y=0$ means unaware individuals take no self-protective measures at all,
whereas $y=1$ means the rewiring rate for susceptible nodes is independent of
their awareness state. It is worth mentioning that in the present paper we
simply assume $\beta^{A}=\beta^{U}=\beta$ and $\omega^{A}>\omega^{U}$. That
is, we don’t differentiate the transmission rate between aware and unaware
susceptibles, since the impact of awareness information on reducing the risk
of infection has already been reflected in the larger rewiring rate for aware
susceptibles than unaware ones. Moreover, in our model the link rewiring
process and the disease transmission process take place simultaneously but
independently.

Note that most existing concurrent spreading models on multiplex networks have
assumed a fixed network structure in each layer, whereas in our model the
network structure in the epidemic layer coevolves with the epidemic dynamics
due to adaptive rewiring of disease-causing contacts. In this sense we refer
to our model as the SIS-UAU contagion model on adaptive multiplex networks.
For the sake of simplicity, in the present work we only consider the network
evolution of the epidemic layer and hypothesize that the network topology in
the awareness layer does not change during the process of the concurrent
spreading of epidemic and awareness, though in reality the connectivity
pattern in the awareness layer may also evolve with time.

Table 2: Notation used in the model formulation and analytical approximation.  Term |  Meaning
---|---
$N$ |  number of nodes in each layer (note that each node in one layer forms a replica in the other layer)
$k$ |  total degree (i.e., the number of neighbors) of a node in the epidemic layer. $k=1,2,\cdots,K$, where $K$ is the maximal degree
$i$ |  infected degree (i.e., the number of infected neighbors) of a node in the epidemic layer, $0\leq i\leq k$
$n$ |  total degree (i.e., the number of neighbors) of a node in the awareness layer. $n=1,2,\cdots,M$, where $M$ is the maximal degree
$a$ |  aware degree (i.e., the number of aware neighbors) of a node in the epidemic layer, $0\leq a\leq n$
$S_{ki}^{U_{na}}$ |  fraction of unaware susceptible nodes both with $k$ total neighbors and $i$ infected neighbors in the epidemic layer but also with $n$ total neighbors and $a$ aware neighbors in the awareness layer
$S_{ki}^{A_{na}}$ |  fraction of aware susceptible nodes both with $k$ total neighbors and $i$ infected neighbors in the epidemic layer but also with $n$ total neighbors and $a$ aware neighbors in the awareness layer
$I_{ki}^{A_{na}}$ |  fraction of infected nodes both with $k$ total neighbors and $i$ infected neighbors in the epidemic layer but also with $n$ total neighbors and $a$ aware neighbors in the awareness layer
$S^{U}$ |  fraction of unaware susceptible nodes
$S^{A}$ |  fraction of aware susceptible nodes
$I^{A}$ |  fraction of infected nodes (that are automatically aware about the disease due to infection)
$\rho^{X}$ |  density of nodes in state $X$, where $X\in\\{S,I,U,A\\}$, note that $\rho^{S}=S^{A}+S^{U}=S$
$X_{Y}$ |  fraction of $X$-$Y$ links emanating from nodes in $X$ state towards nodes in $Y$ state, where $X,Y\in\\{S,I,U,A\\}$
$X_{YZ}$ |  fraction of triplets of type $Y$-$X$-$Z$ where a node in $Y$ is connected to a node in $X$ state, which in turn is connected to another node in $Z$ state, where $X,Y,Z\in\\{S,I,U,A\\}$

##  3 Effective degree model formulation

According to the evolution mechanism of the dynamical processes described in
the previous section, all the nodes in the multiplex network can be grouped
into three different states: susceptible and unaware (SU), susceptible and
aware (SA), or infected and aware (IA). Following the effective degree
approach by Marceau et al. [20] and Lindquist et al. [34], every node in the
multiplex network is classified by its disease state, awareness state, and its
total degree in each layer, as well as by its infected degree (in the epidemic
layer) and its aware degree (in the awareness layer). Here, by total degree in
each layer we mean the total number of links connected to a node in the layer
and by infected degree and aware degree we mean the number of links shared
with infected individuals and those shared with aware individuals,
respectively. Therefore, each node can be categorized into classes (also
called compartments in the literature) of ${\text{X}}_{ki}{{\text{Y}}_{na}}$,
where $\text{X}\in\\{\text{S},\text{I}\\}$ and
$\text{Y}\in\\{\text{U},\text{A}\\}$, subscripts $k$ and $i$ denote the node’s
total degree and infected degree, respectively, in the epidemic layer, and
subscripts $n$ and $a$ denote the node’s total degree and aware degree,
respectively, in the awareness layer. Obviously, one has $i\leq k$ and $a\leq
n$ according to the specific definition of the degrees. Note that the fake
class of $\text{I}_{ki}{\text{U}_{na}}$ actually does not exist in our model
because according to the definition of our model, once an individual gets
infected it becomes aware of the disease immediately. We denote by
$X_{ki}^{Y_{na}}(t)$ the fraction of nodes in class
$\text{X}_{ki}{\text{Y}_{na}}$ at time $t$ (where the notations used in the
model formulation are summarized in Table 2). For the sake of readability,
hereinafter we drop the explicit time dependence of $S_{ki}^{U_{na}}$,
$S_{ki}^{A_{na}}$, $I_{ki}^{A_{na}}$, and associated variables. Obviously, the
conservation law gives rise to

$\sum_{k=1}^{K}\sum_{i\leq k}\sum_{n=1}^{M}\sum_{a\leq
n}(S_{ki}^{U_{na}}+S_{ki}^{A_{na}}+I_{ki}^{A_{na}})=1,$ (1)

where $K$ (resp., $M$) represents the maximal degree in the epidemic (resp.,
awareness) layer. Following the notations by Marceau et al. [20], we define
the zeroth-order moments of the $S_{ki}^{U_{na}}$, $S_{ki}^{A_{na}}$,
$I_{ki}^{A_{na}}$ distributions by

$S^{U}\equiv\sum_{k,i}\sum_{n,a}S_{ki}^{U_{na}},\quad
S^{A}\equiv\sum_{k,i}\sum_{n,a}S_{ki}^{A_{na}},\quad
I^{A}\equiv\sum_{k,i}\sum_{n,a}I_{ki}^{A_{na}},$ (2)

the first-order moments by

$\displaystyle S_{S}$ $\displaystyle\equiv$
$\displaystyle\sum_{k,i}\sum_{n,a}(k-i)\big{(}S_{ki}^{U_{na}}+S_{ki}^{A_{na}}\big{)},\quad
S_{I}\equiv\sum_{k,i}\sum_{n,a}i\big{(}S_{ki}^{U_{na}}+S_{ki}^{A_{na}}\big{)},$
$\displaystyle I_{S}$ $\displaystyle\equiv$
$\displaystyle\sum_{k,i}\sum_{n,a}(k-i)I_{ki}^{A_{na}},\quad
I_{I}\equiv\sum_{k,i}\sum_{n,a}iI_{ki}^{A_{na}},$ $\displaystyle{U}_{U}$
$\displaystyle\equiv$
$\displaystyle\sum_{k,i}\sum_{n,a}(n-a)S_{ki}^{U_{na}},\quad{U}_{A}\equiv\sum_{k,i}\sum_{n,a}aS_{ki}^{U_{na}},$
$\displaystyle{A}_{U}$ $\displaystyle\equiv$
$\displaystyle\sum_{k,i}\sum_{n,a}(n-a)(S_{ki}^{A_{na}}+I_{ki}^{A_{na}}),\quad{A}_{A}\equiv\sum_{k,i}\sum_{n,a}a(S_{ki}^{A_{na}}+I_{ki}^{A_{na}}),$
(3)

and the second-order moments by

$\displaystyle S_{SI}$ $\displaystyle\equiv$
$\displaystyle\sum_{k,i}\sum_{n,a}i(k-i)\big{(}S_{ki}^{U_{na}}+S_{ki}^{A_{na}}\big{)},\quad
S_{II}\equiv\sum_{k,i}\sum_{n,a}i(i-1)\big{(}S_{ki}^{U_{na}}+S_{ki}^{A_{na}}\big{)},$
$\displaystyle{U}_{UA}$ $\displaystyle\equiv$
$\displaystyle\sum_{k,i}\sum_{n,a}a(n-a)S_{ki}^{U_{na}},\quad{U}_{AA}\equiv\sum_{k,i}\sum_{n,a}a(a-1)S_{ki}^{U_{na}}.$
(4)

Here, the physical meaning of the zeroth-order moments in Eq. (2) is in
essence the density of SU, SA and IA nodes. For convenience, let us denote by
$\rho^{S}$, $\rho^{I}$, $\rho^{U}$ and $\rho^{A}$ the density of susceptible,
infected, unaware and aware nodes, respectively, in the network. Then it is
straightforward to get the following relations:

$\rho^{S}=S=S^{U}+S^{A},\quad\rho^{I}=I^{A},\quad\rho^{U}=S^{U},\quad\rho^{A}=S^{A}+I^{A}.$
(5)

The first-order moments in Eq. (3) represent the density per node of the
various types of arcs. (Here, the arcs are defined as the directional links
between a pair of nodes. Therefore each undirected edge in the network
actually contains two directional arcs [20].) For example, the quantity
$S_{S}$ denotes the density of $S$-$S$ links emanating from susceptible nodes,
and $U_{A}$ denotes the density of $U$-$A$ links emanating from unaware nodes.
The second-order moments in Eq. (3) correspond to the density per node of the
various types of triplets in the entire network. A triplet is referred to as
the union of two directional links surrounding the same central node [20]. For
example, the quantities $S_{SI}$ and $U_{AA}$ denote the density of triplets
of types $S$-$S$-$I$ and $A$-$U$-$A$ with the central node in class S and U,
respectively. Clearly, it follows from Eqs. (1), (2) and (5) that the
conservation of nodes

$\rho^{S}+\rho^{I}=1,\quad\rho^{U}+\rho^{A}=1,$ (6)

and the conservation of links

$S_{S}+S_{I}+I_{S}+I_{I}=\langle k\rangle,\quad
U_{U}+U_{A}+A_{U}+A_{A}=\langle n\rangle,$ (7)

must hold at any time [17, 20], where $\langle k\rangle$ and $\langle
n\rangle$ are the average degree in the epidemic and awareness layer,
respectively. Furthermore, since the network under consideration is
undirected, the density of $S$-$I$ links must equal the density of $IS$ links;
similarly, the density of $UA$ links is equal to the density of $AU$ links.
This leads to the following constraints [20, 34]

$S_{I}=I_{S},\quad U_{A}=A_{U}.$ (8)

Based on our model definition stated in section 2, at each time step, nodes in
each class undergo the following three types of variations. First, nodes can
change their state and move to the corresponding class with the same
subscripts. Second, nodes can move to other classes due to changes of the
state of their neighbors in either network layer. Third, nodes can change to
other classes due the effects of adaptive rewiring of infectious contacts in
the epidemic layer.

Figure 2: (Color online) Flowchart of the effective degree SIS-UAU contagion
model on adaptive multiplex networks. Each arrow denotes the direction of an
event that causes a node to move from one compartment to another. The
mathematical expression labeled along each arrow is a per node rate for the
transition denoted by the arrow. Note that the notations $\bar{\beta}$ and
$\bar{\omega}$ are referred to as the average infection rate and average
rewiring rate, respectively, for a random susceptible node. Note that, the
infection rate of a randomly selected susceptible neighbor of a node in
compartment $S_{ki}^{U_{na}}$, depends on its awareness state, which is
unknown to us. Thus we approximate it by an average
$\bar{\beta}=\beta^{A}({S^{A}}/{S})+\beta^{U}({S^{U}}/{S})$, where the term
$S^{A}/S$ stands for the probability that the susceptible node is aware and
$S^{U}/S$ for the probability of being unaware. (Note that in this paper we
assume $\beta^{A}=\beta^{U}=\beta$ since the effects of awareness information
on reducing the transmission risk has been included in the larger rewiring
rate for aware nodes in comparison to unaware ones, $\omega^{A}>\omega^{U}$.
Thus we have $\bar{\beta}=\beta$.) Similarly, we have
$\bar{\omega}=\omega^{A}({S^{A}}/{S})+\omega^{U}({S^{U}}/{S})$. Moreover, some
notations are simplified by $G_{S}=S_{SI}/S_{S}$, $G_{I}=1+S_{II}/S_{I}$,
$H_{U}=U_{UA}/U_{U}$ and $H_{A}=1+U_{AA}/U_{A}$, which denote the average
number of infected (or aware) neighbors one node has given that the node is
connected with a node in a certain class.

According to the flow chart for the SIS-UAU transmission process presented in
Fig. 2, we obtain the set of
$\sum_{k=1}^{K}\sum_{n=1}^{M}3(k+1)(n+1)=3KM(K+3)(M+3)/4$ differential
equations that govern the temporal evolution of the nodes in each class for
$\\{(k,i;n,a):1\leq k\leq K,0\leq i\leq k;1\leq n\leq M,0\leq a\leq n\\}$ as
follows.

$\displaystyle\frac{{\rm d}S_{ki}^{U_{na}}}{{\rm d}t}$ $\displaystyle=$
$\displaystyle\mu S_{ki}^{A_{na}}-\alpha
aS_{ki}^{U_{na}}+\mu\bigg{[}(a+1)S_{ki}^{U_{n(a+1)}}-aS_{ki}^{U_{na}}\bigg{]}+\alpha\frac{U_{UA}}{U_{U}}\bigg{[}(n-a+1)S_{ki}^{U_{n(a-1)}}-(n-a)S_{ki}^{U_{na}}\bigg{]}$
(9) $\displaystyle-i\beta
S_{ki}^{U_{na}}+\gamma\bigg{[}(i+1)S_{k(i+1)}^{U_{na}}-iS_{ki}^{U_{na}}\bigg{]}+\beta\frac{S_{SI}}{S_{S}}\bigg{[}(k-i+1)S_{k(i-1)}^{U_{na}}-(k-i)S_{ki}^{U_{na}}\bigg{]}$
$\displaystyle+\omega^{U}\bigg{[}(i+1)S_{k(i+1)}^{U_{na}}-iS_{ki}^{U_{na}}\bigg{]}+\bigg{(}\omega^{A}\frac{S^{A}}{S}+\omega^{U}\frac{S^{U}}{S}\bigg{)}\frac{S_{I}}{S}\bigg{[}S_{(k-1)i}^{U_{na}}-S_{ki}^{U_{na}}\bigg{]},$
$\displaystyle\frac{{\rm d}S_{ki}^{A_{na}}}{{\rm d}t}$ $\displaystyle=$
$\displaystyle-\mu S_{ki}^{A_{na}}+\alpha
aS_{ki}^{U_{na}}+\mu\bigg{[}(a+1)S_{ki}^{A_{n(a+1)}}-aS_{ki}^{A_{na}}\bigg{]}+\alpha\bigg{(}1+\frac{U_{AA}}{U_{A}}\bigg{)}\bigg{[}(n-a+1)S_{ki}^{A_{n(a-1)}}-(n-a)S_{ki}^{A_{na}}\bigg{]}$
(10) $\displaystyle-i\beta
S_{ki}^{A_{na}}+\gamma\\*I_{ki}^{A_{na}}+\gamma\bigg{[}(i+1)S_{k(i+1)}^{A_{na}}-iS_{ki}^{A_{na}}\bigg{]}+\beta\frac{S_{SI}}{S_{S}}\bigg{[}(k-i+1)S_{k(i-1)}^{A_{na}}-(k-i)S_{ki}^{A_{na}}\bigg{]}$
$\displaystyle+\omega^{A}\bigg{[}(i+1)S_{k(i+1)}^{A_{na}}-iS_{ki}^{A_{na}}\bigg{]}+\bigg{(}\omega^{A}\frac{S^{A}}{S}+\omega^{U}\frac{S^{U}}{S}\bigg{)}\frac{S_{I}}{S}\bigg{[}S_{(k-1)i}^{A_{na}}-S_{ki}^{A_{na}}\bigg{]},$
$\displaystyle\frac{{\rm d}I_{ki}^{A_{na}}}{{\rm d}t}$ $\displaystyle=$
$\displaystyle\mu\bigg{[}(a+1)I_{ki}^{A_{n(a+1)}}-aI_{ki}^{A_{na}}\bigg{]}+\alpha\bigg{(}1+\frac{U_{AA}}{U_{A}}\bigg{)}\bigg{[}(n-a+1)I_{ki}^{A_{n(a-1)}}-(n-a)I_{ki}^{A_{na}}\bigg{]}$
(11)
$\displaystyle+i\beta\Big{(}S_{ki}^{U_{na}}+S_{ki}^{A_{na}}\Big{)}+\beta\bigg{(}1+\frac{S_{II}}{S_{I}}\bigg{)}\bigg{[}(k-i+1)I_{k(i-1)}^{A_{na}}-(k-i)I_{ki}^{A_{na}}\bigg{]}$
$\displaystyle-\gamma\\*I_{ki}^{A_{na}}+\gamma\bigg{[}(i+1)I_{k(i+1)}^{A_{na}}-iI_{ki}^{A_{na}}\bigg{]}+\bigg{(}\omega^{A}\frac{S^{A}}{S}+\omega^{U}\frac{S^{U}}{S}\bigg{)}\bigg{[}(k+1-i)I_{(k+1)i}^{A_{na}}-(k-i)I_{ki}^{A_{na}}\bigg{]}.$

We call the system of Eqs. (9)–(11) the SIS-UAU effective degree model. On the
right hand side (rhs) of Eq. (9), the first term accounts for the addition of
$S_{ki}^{U_{na}}$ at rate $\mu S_{ki}^{A_{na}}$ as nodes in compartment
$S_{ki}^{A_{na}}$ lose the awareness on disease and become unaware again. The
second term indicates that nodes are removed from $S_{ki}^{U_{na}}$ at rate
$\alpha aS_{ki}^{U_{na}}$ as they are informed of disease risk by their aware
neighbors. The third term accounts for a change in the state of the node’s
aware neighbors. Nodes from $S_{ki}^{U_{n(a+1)}}$ (resp., $S_{ki}^{U_{na}}$)
are transferred to $S_{ki}^{U_{na}}$ (resp., $S_{ki}^{U_{n(a-1)}}$) as one of
their aware neighbors becomes unaware again. The fourth term considers the
possibility that nodes from $S_{ki}^{U_{n(a-1)}}$ (resp., $S_{ki}^{U_{na}}$)
enters the compartment $S_{ki}^{U_{na}}$ (resp., $S_{ki}^{U_{n(a+1)}}$) as one
of the node’s unaware neighbors becomes aware due to information propagation.
Note that we have used here the quantity $U_{UA}/U_{U}$ to measure the average
number of aware neighbors that a unaware node reached by following a $U$-$U$
link has, for which we have assumed no degree correlation. The explanation for
this is as follows. Since we do not know the exact number of aware nodes that
are connected to each of the $(n-a)$ unaware neighbors of nodes in the
$S_{ki}^{U_{na}}$ compartment, one alternative is to take the average over the
whole network. Under the assumption of degree uncorrelation, a node from
$S_{k^{\prime}i^{\prime}}^{U_{n^{\prime}a^{\prime}}}$ with $a^{\prime}$ aware
neighbors will be reached by following a $U$-$U$ link with probability
$(n^{\prime}-a^{\prime})S_{k^{\prime}i^{\prime}}^{U_{n^{\prime}a^{\prime}}}/U_{U}$.
Consequently, it follows from Eq. (3) that the average number of aware
neighbors connected to a unaware node reached by following a $U$-$U$ link is
$U_{UA}/U_{U}$. Thus, the rate associated with the transition from
$S_{ki}^{U_{na}}$ to $S_{ki}^{U_{n(a-1)}}$ is
$\alpha(U_{UA}/U_{U})(n-a)S_{ki}^{U_{na}}$. The fifth term on the rhs of Eq.
(9) represents the reduction of $S_{ki}^{U_{na}}$ because each node in this
compartment is infected at rate $i\beta^{U}$. The sixth term refers to the
change as a result of recovery of one of the node’s infected neighbors. The
seventh term describes the infection of one of the node’s susceptible
neighbors. Nodes from $S_{k(i-1)}^{U_{na}}$ (resp., $S_{ki}^{U_{na}}$) are
moved to $S_{ki}^{U_{na}}$ (resp., $S_{k(i+1)}^{U_{na}}$) when one of their
susceptible neighbors contracts the disease. By analogy with the elucidation
for the fourth term, the quantity $S_{SI}/S_{S}$ gives the average number of
infected neighbors of a susceptible node that is reached by following a
$S$-$S$ link. Since we do not know the awareness state of each susceptible
neighbor of nodes in compartment $S_{ki}^{U_{na}}$, on which the infection
rate strongly depends, we considers an average infection rate $\bar{\beta}$
(see Fig. 2) approximated by an expectation over the entire network. That is,
$\bar{\beta}=\beta^{A}({S^{A}}/{S})+\beta^{U}({S^{U}}/{S})$, where the term
$S^{A}/S$ stands for the probability that the susceptible node is aware and
$S^{U}/S~{}(=1-S^{A}/S)$ for the probability of being unaware. The eighth term
is attributable to the fact that when one infected neighbor of a susceptible
node is rewired away from the susceptible node, then the susceptible node
decreases its infected degree by one. Each node from the compartment
$S_{k(i+1)}^{U_{na}}$ (resp., $S_{ki}^{U_{na}}$) is shifted to the compartment
$S_{ki}^{U_{na}}$ (resp., $S_{k(i-1)}^{U_{na}}$) at rate $(i+1)\omega^{U}$
(resp., $i\omega^{U}$). The ninth term depicts the changes of
$S_{ki}^{U_{na}}$ owing to their susceptible degree (i.e. the number of their
susceptible neighbors) increasing by one when the susceptible nodes are
randomly rewired towards by the susceptible endpoint on the $S$-$I$ link after
disconnecting from the infected endpoint. Namely, a node from compartment
$S_{ki}^{U_{na}}$ jumps to the compartment $S_{(k+1)i}^{U_{na}}$ as it is
randomly chosen as the “new neighbor” in a rewiring event. The average
strength of rewiring events is $\bar{\omega}S_{I}$, where the average rewiring
rate $\bar{\omega}$ (see Fig. 2) is approximated by taking an average over the
entire network since we do not know the awareness state of the susceptible
endpoint on the $S$-$I$ link that is to be rewired. In the same fashion as in
the approximation of $\bar{\beta}$, one has
$\bar{\omega}=\omega^{A}({S^{A}}/{S})+\omega^{U}({S^{U}}/{S})$. Note that a
node from the compartment $S_{ki}^{U_{na}}$ (resp., $S_{(k-1)i}^{U_{na}}$) is
randomly selected as a new connection target with probability
$S_{ki}^{U_{na}}/S$ (resp., $S_{(k-1)i}^{U_{na}}/S$), the event of transition
from $S_{ki}^{U_{na}}$ (resp., $S_{(k-1)i}^{U_{na}}$) to $S_{(k+1)i}^{U_{na}}$
(resp., $S_{ki}^{U_{na}}$) occurs at rate
$\bar{\omega}(S_{I}/S)S_{ki}^{U_{na}}$ (resp.,
$\bar{\omega}(S_{I}/S)S_{(k-1)i}^{U_{na}}$). Explanations for the derivation
of Eqs. (10) and (11) are similar except that in the fourth (resp., the
second) term on the rhs of Eq. (10) [resp., Eq. (11)], the quantity
$H_{A}=1+U_{AA}/U_{A}$ (see Fig. 2) denoting the average number of aware
degree of a unaware node reached by following an $A$-$U$ link has been taken
into account, and that in the fourth term on the rhs of Eq. (11), the variable
$G_{I}=1+S_{II}/S_{I}$ (see Fig. 2) measuring the average infected degree of a
susceptible reached by following an $I$-$S$ link has been included.

##  4 Threshold analysis for the full effective degree model

In what follows much attention will be paid on the threshold conditions for
disease invasion and awareness information outbreak. In the context of
epidemiology, one of the best known quantities that account for the threshold
condition for disease invasion is the basic reproduction number
$\mathcal{R}_{0}$, which is defined as the average number of new infections
produced by a single infected individual (during its entire infectious period)
in a completely susceptible population [10, 40, 41]. Typically, if
$\mathcal{R}_{0}>1$, the disease will break out and finally remains endemic in
the population; if $\mathcal{R}_{0}<1$, the disease will eventually die out.

(i) Disease threshold condition. In the immediate vicinity of the disease-free
equilibrium [40, 41] for our effective degree model, i.e., as the number of
infected individuals is rather small, the possibility for a susceptible
individual to have more than one infected neighbors is close to zero. In this
case we have $S_{ki}=\sum_{n,a}(S_{ki}^{U_{na}}+S_{ki}^{A_{na}})\approx 0$ for
all $i>1$ and

$1+\frac{S_{II}}{S_{I}}=\frac{\sum_{k=1}^{K}\sum_{i=0}^{k}i^{2}S_{ki}}{\sum_{k=1}^{K}\sum_{i=0}^{k}iS_{ki}}=\frac{\sum_{k=1}^{K}S_{k1}}{\sum_{k=1}^{K}S_{k1}}=1.$
(12)

Therefore, near the disease-free equilibrium the Eq. (11) can be written as

$\displaystyle\frac{{\rm d}I_{ki}^{A_{na}}}{{\rm d}t}$ $\displaystyle=$
$\displaystyle\mu\bigg{[}(a+1)I_{ki}^{A_{n(a+1)}}-aI_{ki}^{A_{na}}\bigg{]}+\alpha\bigg{(}1+\frac{U_{AA}}{U_{A}}\bigg{)}\bigg{[}(n-a+1)I_{ki}^{A_{n(a-1)}}-(n-a)I_{ki}^{A_{na}}\bigg{]}$
(13)
$\displaystyle+i\beta\Big{(}S_{ki}^{U_{na}}+S_{ki}^{A_{na}}\Big{)}+\beta\bigg{[}(k-i+1)I_{k(i-1)}^{A_{na}}-(k-i)I_{ki}^{A_{na}}\bigg{]}+\gamma\bigg{[}(i+1)I_{k(i+1)}^{A_{na}}-iI_{ki}^{A_{na}}\bigg{]}$
$\displaystyle-\gamma\\*I_{ki}^{A_{na}}+\bigg{(}\omega^{A}\frac{S^{A}}{S}+\omega^{U}\frac{S^{U}}{S}\bigg{)}\bigg{[}(k+1-i)I_{(k+1)i}^{A_{na}}-(k-i)I_{ki}^{A_{na}}\bigg{]}.$

It is to be noted that the above approximation is inappropriate when the
system is not close to the disease-free equilibrium. In order to obtain the
basic reproduction number $\mathcal{R}^{\rm dis}_{0}$ for disease outbreak
from the effective degree model, we collectively linearize the Eqs. (9), (10)
and (13) at the disease-free equilibrium and determine the its stability. In
particular, we apply the next generation matrix approach by van den Driessche
and Watmough [42] to derive the disease basic reproduction number as a
spectral radius of matrix ${\bm{F}}_{d}{\bm{V}}_{d}^{-1}$, i.e.,
$\mathcal{R}^{\rm dis}_{0}=\rho({\bm{F}}_{d}{\bm{V}}_{d}^{-1})$, where the
matrix ${\bm{J}}_{d}={\bm{F}}_{d}-{\bm{V}}_{d}$ is the Jacobian matrix at the
disease-free equilibrium, with the matrix ${\bm{F}}_{d}$ accounting for new
infections flowing from $S_{k0}^{U_{n0}}$ to $S_{k1}^{U_{n0}}$ and the matrix
$-{\bm{V}}_{d}$ involving transfers between classes $S_{ki}^{U_{na}}$ for
$i\geq 1$, and $S_{ki}^{A_{na}}$, $I_{ki}^{A_{na}}$ for all $k,i,n$ and $a$.
Observe from Eq. (9) that as new infections individuals in class
$S_{k0}^{U_{n0}}$ move into class $S_{k1}^{U_{n0}}$ at a rate

$\beta\frac{S_{SI}}{S_{S}}kS_{k0}^{U_{n0}}=\beta\frac{\sum_{k,i}\sum_{n,a}(k-i)i\Big{(}S_{ki}^{U_{na}}+S_{ki}^{A_{na}}\Big{)}}{\sum_{k,i}\sum_{n,a}(k-i)\Big{(}S_{ki}^{U_{na}}+S_{ki}^{A_{na}}\Big{)}}kS_{k0}^{U_{n0}},$
(14)

and then they can be further transmitted by their infectious neighbors.
According to the methodology in [42], the compartments $S_{k0}^{U_{na}}$ and
$S_{k0}^{A_{na}}$ do not belong to the disease class, thus they can be
disregarded for the threshold and stability calculation. For a fixed $k$
value, in the similar way to [34], we order the variables with
$S_{ki}^{U_{na}}$ equations (in increasing values of $n$) first, and
$S_{ki}^{A_{na}}$ equations (in increasing values of $n$) next, then
$I_{ki}^{A_{na}}$ equations (in increasing values of $n$) arranged
lexicographically, and then in increasing values of $k$. Consequently, the
variables are arranged in the order as follows: $S_{11}^{U_{10}}$,
$S_{11}^{U_{11}}$, $S_{11}^{U_{20}}$, …, $S_{11}^{U_{MM}}$, $S_{11}^{A_{10}}$,
…, $S_{11}^{A_{MM}}$, $I_{10}^{A_{10}}$, …, $I_{10}^{A_{MM}}$,
$I_{11}^{A_{10}}$, …, $I_{11}^{A_{MM}}$; $S_{21}^{U_{10}}$, …,
$S_{21}^{U_{MM}}$, $S_{22}^{U_{10}}$, …, $S_{22}^{U_{MM}}$, $S_{21}^{A_{10}}$,
…, $S_{21}^{A_{MM}}$, $S_{22}^{A_{10}}$, …, $S_{22}^{A_{MM}}$,
$I_{20}^{A_{10}}$, …, $I_{20}^{A_{MM}}$, $I_{21}^{A_{10}}$, …,
$I_{21}^{A_{MM}}$, $S_{22}^{A_{10}}$, …, $S_{22}^{A_{MM}}$;$\dots$,
$S_{K1}^{U_{10}}$, …, $S_{K1}^{U_{MM}}$, …, $S_{KK}^{U_{10}}$, …,
$S_{KK}^{U_{MM}}$, $S_{K1}^{A_{10}}$, …, $S_{K1}^{A_{MM}}$, …,
$S_{KK}^{A_{10}}$, …, $S_{KK}^{A_{MM}}$, $I_{K0}^{A_{10}}$, …,
$I_{K0}^{A_{MM}}$, …, $I_{KK}^{A_{10}}$, …, $I_{KK}^{A_{MM}}$. With some
elementary calculations, ${\bm{F}}_{d}$ is shown to be a
$\frac{K(3K+5)}{2}\times\frac{K(3K+5)}{2}$ matrix with rank $1$ and can be
expressed as

${\bm{F}}_{d}=\frac{1}{\sum_{k=1}^{K}\sum_{n=1}^{M}kS_{k0}^{U_{n0}}}\begin{pmatrix}{\bm{X}}_{1}\\\
{\bm{X}}_{2}\\\ \vdots\\\
{\bm{X}}_{K}\end{pmatrix}\Big{(}{\bm{Y}}_{1}^{T},{\bm{Y}}_{2}^{T},\cdots,{\bm{Y}}_{K}^{T}\Big{)},$
(15)

where ${\bm{X}}_{k}$ is a $(3k+1)\times 1$ vector with $kS_{k0}^{U_{n0}}$ in
its first entry and zeros elsewhere, and
${\bm{Y}}_{k}=\begin{pmatrix}{\bm{Y}}_{k}^{U}\\\ ~{}\\\
{\bm{Y}}_{k}^{A}\end{pmatrix}$ is a $(3k+1)\times 1$ vector with
${\bm{Y}}_{k}^{U}$ being a $k\times 1$ vector and ${\bm{Y}}_{k}^{A}$ a
$(2k+1)\times 1$ vector. For both vectors ${\bm{Y}}_{k}^{U}$ and
${\bm{Y}}_{k}^{U}$, their first $(k-1)$ entries are $(k-1)$, $2(k-2)$,
$3(k-3)$, $\dots$, $(k-1)$, while all of their remaining entries are zeros.
Moreover, since the degree of each node remains fixed, ${\bm{V}}_{d}$ is a
block diagonal matrix structured as
${\bm{V}}_{d}={\bm{V}}_{d1}\bigoplus{\bm{V}}_{d2}\bigoplus\dots\bigoplus{\bm{V}}_{dK}$,
where each block ${\bm{V}}_{dk}$ ($k=1,2,\cdots,K$) is a $(3k+1)\times(3k+1)$
square matrix depending on $\alpha$, $\mu$, $\beta^{U}$, $\beta^{A}$,
$\omega^{U}$, $\omega^{A}$, $n$ and $k$. Furthermore, it is easy to confirm
that for each $k$ all principal minors of ${\bm{V}}_{dk}$ are positive,
implying that the matrix block ${\bm{V}}_{dk}$ is a nonsingular $M$-matrix
[43]. Therefore, each inverse matrix ${\bm{V}}_{dk}^{-1}$ exists and
${\bm{V}}_{dk}^{-1}\geq 0$, that is, each entry of ${\bm{V}}_{dk}^{-1}$ is
nonnegative [43]. As a result, the threshold for disease outbreak is given by

$\mathcal{R}^{\rm
dis}_{0}=\rho\Big{(}{\bm{F}}_{d}{\bm{V}}_{d}^{-1}\Big{)}=\frac{1}{\sum_{k=1}^{K}\sum_{n=1}^{M}kS_{k0}^{U_{n0}}}\sum_{k=1}^{K}\sum_{n=1}^{M}{\bm{Y}}_{k}^{T}{\bm{V}}_{dk}^{-1}{\bm{X}}_{k}.$
(16)

If $\mathcal{R}^{\rm dis}_{0}<1$, then the disease-free equilibrium is stable
so the disease cannot invade; whereas if $\mathcal{R}^{\rm dis}_{0}>1$, then
the disease breaks out and spreads across the population. As shown in Fig. 3,
for a given set of parameters, the average of simulation results shows that
the density of infected nodes first increases and finally reaches an steady
(endemic) state. It is worth remarking that the threshold transmission rate
$\beta_{\rm c}$ for epidemic onset can be derived directly from Eq. (16) since
$\beta=\beta_{\rm c}$ is equivalent to the critical condition
$\mathcal{R}^{\rm dis}_{0}=1$. In fact, $\beta_{\rm
c}=\beta/\mathcal{R}_{0}^{\rm dis}$. Consequently, $\beta<\beta_{\rm c}$ is
equivalent to $\mathcal{R}_{0}^{\rm dis}<1$, while $\beta>\beta_{\rm c}$ is
equivalent to $\mathcal{R}_{0}^{\rm dis}>1$. Although complicated in form, the
formula in Eq. (16) will allow for numerical calculations of the transmission
threshold for disease onset, which will be demonstrated in Sec. 6.

(ii) Awareness threshold condition. Similarly to the derivation of the disease
threshold condition, in what follows we obtain the awareness basic
reproduction number $\mathcal{R}^{\rm awe}_{0}$ as the awareness threshold
condition for our effective degree model. Using the similar approximation near
the awareness information-free equilibrium (i.e., there is no awareness
information at all in the steady state), one has

$\displaystyle\frac{{\rm d}S_{ki}^{A_{na}}}{{\rm d}t}$ $\displaystyle=$
$\displaystyle-\mu S_{ki}^{A_{na}}+\alpha
aS_{ki}^{U_{na}}+\mu\bigg{[}(a+1)S_{ki}^{A_{n(a+1)}}-aS_{ki}^{A_{na}}\bigg{]}+\alpha\bigg{[}(n-a+1)S_{ki}^{A_{n(a-1)}}-(n-a)S_{ki}^{A_{na}}\bigg{]}$
(17) $\displaystyle-i\beta
S_{ki}^{A_{na}}+\gamma\\*I_{ki}^{A_{na}}+\gamma\bigg{[}(i+1)S_{k(i+1)}^{A_{na}}-iS_{ki}^{A_{na}}\bigg{]}+\beta\frac{S_{SI}}{S_{S}}\bigg{[}(k-i+1)S_{k(i-1)}^{A_{na}}-(k-i)S_{ki}^{A_{na}}\bigg{]}$
$\displaystyle+\omega^{A}\bigg{[}(i+1)S_{k(i+1)}^{A_{na}}-iS_{ki}^{A_{na}}\bigg{]}+\bigg{(}\omega^{A}\frac{S^{A}}{S}+\omega^{U}\frac{S^{U}}{S}\bigg{)}\frac{S_{I}}{S}\bigg{[}S_{(k-1)i}^{A_{na}}-S_{ki}^{A_{na}}\bigg{]},$
$\displaystyle\frac{{\rm d}I_{ki}^{A_{na}}}{{\rm d}t}$ $\displaystyle=$
$\displaystyle\mu\bigg{[}(a+1)I_{ki}^{A_{n(a+1)}}-aI_{ki}^{A_{na}}\bigg{]}+\alpha\bigg{[}(n-a+1)I_{ki}^{A_{n(a-1)}}-(n-a)I_{ki}^{A_{na}}\bigg{]}+i\beta\Big{(}S_{ki}^{U_{na}}+S_{ki}^{A_{na}}\Big{)}$
(18)
$\displaystyle+\beta\bigg{(}1+\frac{S_{II}}{S_{I}}\bigg{)}\bigg{[}(k-i+1)I_{k(i-1)}^{A_{na}}-(k-i)I_{ki}^{A_{na}}\bigg{]}+\gamma\bigg{[}(i+1)I_{k(i+1)}^{A_{na}}-iI_{ki}^{A_{na}}\bigg{]}$
$\displaystyle-\gamma\\*I_{ki}^{A_{na}}+\bigg{(}\omega^{A}\frac{S^{A}}{S}+\omega^{U}\frac{S^{U}}{S}\bigg{)}\bigg{[}(k+1-i)I_{(k+1)i}^{A_{na}}-(k-i)I_{ki}^{A_{na}}\bigg{]}.$

Linearizing Eqs. (9), (17) and (18) at the awareness information-free
equilibrium, we proceed as in the derivation of $\mathcal{R}^{\rm dis}_{0}$ by
writing the Jacobian matrix as ${\bm{J}}_{a}={\bm{F}}_{a}-{\bm{V}}_{a}$ and
using the same ordering of variables. For our model, matrix ${\bm{F}}_{a}$ is
shown to be a $\frac{M(3M+7)}{2}\times\frac{M(3M+7)}{2}$ matrix with rank $1$
and can be expressed as

${\bm{F}}_{a}=\frac{\alpha}{\sum_{n=1}^{M}\sum_{k=1}^{K}nS_{k0}^{U_{n0}}}\begin{pmatrix}{\bm{W}}_{1}\\\
{\bm{W}}_{2}\\\ \vdots\\\
{\bm{W}}_{M}\end{pmatrix}\Big{(}{\bm{Z}}_{1}^{T},{\bm{Z}}_{2}^{T},\cdots,{\bm{Z}}_{M}^{T}\Big{)},$
(19)

in which ${\bm{W}}_{n}$ is a $(3n+2)\times 1$ vector with $nS_{k0}^{U_{n0}}$
in its first entry and zeros elsewhere, and ${\bm{Z}}_{k}$ is also a
$(3n+2)\times 1$ vector with its first $(n-1)$ entries equal to $(n-1)$,
$2(n-2)$, $3(n-3)$, $\dots$, $(n-1)$, and all its remaining entries are zeros.
Moreover, matrix ${\bm{V}}_{a}$ is a block diagonal matrix structured as
${\bm{V}}_{a}={\bm{V}}_{a1}\bigoplus{\bm{V}}_{a2}\bigoplus\dots\bigoplus{\bm{V}}_{aM}$,
of which each block ${\bm{V}}_{an}$ ($n=1,2,\cdots,M$) is a
$(3n+2)\times(3n+2)$ nonsingular $M$-matrix [43]. Therefore, the awareness
threshold condition is given by

$\mathcal{R}^{\rm
awe}_{0}=\rho\Big{(}{\bm{F}}_{a}{\bm{V}}_{a}^{-1}\Big{)}=\frac{\alpha}{\sum_{n=1}^{M}\sum_{k=1}^{K}nS_{k0}^{U_{n0}}}\sum_{n=1}^{M}\sum_{k=1}^{K}{\bm{Z}}_{n}^{T}{\bm{V}}_{an}^{-1}{\bm{W}}_{n}.$
(20)

If $\mathcal{R}^{\rm awe}_{0}<1$, then the awareness information-free
equilibrium is stable so the awareness information cannot invade and in the
end all individuals are unaware of the disease information; whereas if
$\mathcal{R}^{\rm awe}_{0}>1$, then the awareness information breaks out and
eventually saturates in the population. In an analogous manner to the
deduction of $\beta_{\rm c}$, the critical diffusion rate $\alpha_{\rm c}$ for
awareness information outbreak can be derived as $\alpha_{\rm
c}=\alpha/\mathcal{R}_{0}^{\rm awe}$. Therefore, $\alpha>\alpha_{\rm c}$ is
equivalent to $\mathcal{R}_{0}^{\rm awe}>1$, whereas $\alpha<\alpha_{\rm c}$
is equivalent to $\mathcal{R}_{0}^{\rm awe}<1$.

##  5 Disease reproduction number in a reduced effective degree model

Similar to Ref. [34], in this section we consider a reduced SIS-UAU effective
degree model in which the term $\beta(k-i+1)I_{k(i-1)}^{A_{na}}$ in Eq. (13)
is set to zero. In this case the differential equations for $I_{k0}^{A_{na}}$
are decoupled from the other equations and hence they can be ignored from the
dynamical system for threshold and stability analysis. In this regard, we only
need to consider a reduced system of order
$\Big{(}\sum_{k=1}^{K}3k\Big{)}=3K(K+1)/2$. Rearrange the order of variables
as follows: $S_{11}^{U_{na}}$, $S_{11}^{A_{na}}$, $I_{11}^{A_{na}}$,
$S_{21}^{U_{na}}$, $S_{21}^{A_{na}}$, $I_{21}^{A_{na}}$, $S_{22}^{U_{na}}$,
$S_{22}^{A_{na}}$, $I_{22}^{A_{na}}$, $\dots$, $S_{KK}^{U_{na}}$,
$S_{KK}^{A_{na}}$, $I_{KK}^{A_{na}}$. It is to be noted that each of the above
variables is itself ordered in increasing value of $n$; for example,
$S_{11}^{U_{na}}$ is referred to as
$\big{(}S_{11}^{U_{10}},S_{11}^{U_{11}},S_{11}^{U_{20}},\dots,S_{11}^{U_{MM}}\big{)}$.
Note that
${\bm{V}}_{d}={\bm{V}}_{d1}\bigoplus{\bm{V}}_{d2}\bigoplus\dots\bigoplus{\bm{V}}_{dK}$,
where ${\bm{V}}_{dk}$ $(k=1,2,\cdots,K)$ are block matrices of ${\bm{V}}_{d}$.
In this reduced effective degree model, the blocks ${\bm{V}}_{dk}$ now turn to
be upper triangular with diagonal blocks of order $3$. Due to the sparsity of
matrix ${\bm{F}}_{d}$, only the blocks on variables $S_{k1}^{U_{na}}$,
$S_{k1}^{A_{na}}$, $I_{k1}^{A_{na}}$ contribute to matrix
${\bm{F}}_{d}{\bm{V}}_{d}^{-1}$. Consequently, the block matrices of
${\bm{V}}_{d}$ are

${\bm{V}}_{dk}=\begin{pmatrix}f_{1}&-\mu&0\\\ 0&f_{2}&-\gamma\\\
-\beta&-\beta&f_{3}\end{pmatrix},\qquad k=1,2,\cdots,K$ (21)

with inverse

${\bm{V}}_{dk}^{-1}=\frac{1}{\gamma\beta(\mu+f_{1})-f_{1}f_{2}f_{3}}\begin{pmatrix}\gamma\beta-
f_{2}f_{3}&-\mu f_{3}&-\mu\gamma\\\ -\gamma\beta&-f_{1}f_{3}&-\gamma f_{1}\\\
-\beta f_{2}&-\beta(\mu+f_{1})&-f_{1}f_{2}\end{pmatrix},$ (22)

where $f_{1}=\beta+\gamma+\omega^{U}$,
$f_{2}=n\alpha+\mu+\beta+\gamma+\omega^{A}$, and
$f_{3}=n\alpha+2\gamma+(k-1)(\beta+\omega^{U})$. In addition, the
corresponding blocks of ${\bm{F}}_{d}$ are

${\bm{F}}_{dk}=\frac{1}{\sum_{k=1}^{K}\sum_{n=1}^{M}kS_{k0}^{U_{n0}}}\begin{pmatrix}k(k-1)\beta
S_{k0}^{U_{n0}}&k(k-1)\beta S_{k0}^{U_{n0}}&0\\\ 0&0&0\\\ 0&0&0\end{pmatrix}.$
(23)

As a result, in the reduced effective degree model the basic reproduction
number $\mathcal{R}^{\rm dis}_{0}=\rho({\bm{F}}_{d}{\bm{V}}_{d}^{-1})$ reads

$\mathcal{R}^{\rm dis}_{0}=\sum_{k=1}^{K}\sum_{n=1}^{M}\frac{\beta
f_{2}f_{3}S_{k0}^{U_{n0}}}{f_{1}f_{2}f_{3}-\gamma\beta(\mu+f_{1})}\frac{k(k-1)}{\sum_{k=1}^{K}\sum_{n=1}^{M}kS_{k0}^{U_{n0}}}$
(24)

It is easy to check that in the denominator
$f_{1}f_{2}f_{3}-\gamma\beta(\mu+f_{1})>0$, implying
$f_{1}f_{2}f_{3}>f_{1}f_{2}f_{3}-\gamma\beta(\mu+f_{1})>0$. So, we have a
lower bound for the disease reproduction number, that is

$\mathcal{R}^{\rm dis}_{0}>\sum_{k=1}^{K}\sum_{n=1}^{M}\frac{\beta
S_{k0}^{U_{n0}}}{f_{1}}\frac{k(k-1)}{\sum_{k=1}^{K}\sum_{n=1}^{M}kS_{k0}^{U_{n0}}}=\frac{\beta}{\beta+\gamma+\omega^{U}}\frac{\langle
k(k-1)\rangle}{\langle k\rangle}\triangleq{\mathcal{R}}_{0}^{\rm LB},$ (25)

where $\langle\cdot\rangle$ denotes the expectation calculated from the given
degree distribution $P(k)=S_{k0}=\sum_{n=1}^{M}S_{k0}^{U_{n0}}$.

To compare our reduced model with the original SIS-UAU effective degree model
given by Eqs. (9)-(11), observe that every entry of ${\bm{V}}_{dk}$ in the
original effective degree model is smaller than or equal to the corresponding
entry of ${\bm{V}}_{dk}$ in the reduced model. Since ${\bm{V}}_{dk}$ is a
nonsingular $M$-matrix, by the theory of $M$-matrix [43], we can argue that
each entry of ${\bm{V}}_{dk}^{-1}$ in the original effective degree model is
greater than or equal to the corresponding ${\bm{V}}_{dk}^{-1}$ entry in the
reduced model. Note that matrix ${\bm{F}}_{d}{\bm{V}}_{d}^{-1}$ is
nonnegative, so the basic reproduction number $\mathcal{R}_{0}^{\rm dis}$ for
the original effective degree model is larger than or equal to
$\mathcal{R}_{0}^{\rm dis}$ for the reduced model. This implies that for the
same parameters $\beta$, $\gamma$ and $\omega^{U}$ and the degree distribution
$P(k)$, the disease basic reproduction number $\mathcal{R}_{0}^{\rm dis}$ of
the effective degree model always has a lower bound, i.e.,
$\mathcal{R}_{0}^{\rm dis}>{\mathcal{R}}_{0}^{\rm
LB}=\frac{\beta}{\beta+\gamma+\omega^{U}}\frac{\langle k(k-1)\rangle}{\langle
k\rangle}$.

(i) Effects of link rewiring on disease outbreak. Consider the case of no
rewiring, that is, when rewiring parameters $\omega^{U}$ and $\omega^{A}$ are
set to zero, the basic reproduction number for disease outbreak turns to

$\displaystyle\widetilde{\mathcal{R}}_{0}^{\rm
dis}=\rho(\widetilde{{\bm{F}}}_{d}\widetilde{{\bm{V}}}_{d}^{-1})$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{K}\sum_{n=1}^{M}\frac{\beta\widetilde{f}_{2}\widetilde{f}_{3}S_{k0}^{U_{n0}}}{\widetilde{f}_{1}\widetilde{f}_{2}\widetilde{f}_{3}-\gamma\beta(\mu+\widetilde{f}_{1})}\frac{k(k-1)}{\sum_{k=1}^{K}\sum_{n=1}^{M}kS_{k0}^{U_{n0}}}$
(26) $\displaystyle=$
$\displaystyle\sum_{k=1}^{K}\sum_{n=1}^{M}\frac{\beta\widetilde{f}_{2}\widetilde{f}_{3}S_{k0}^{U_{n0}}}{\widetilde{f}_{1}\widetilde{f}_{2}\widetilde{f}_{3}-\gamma\beta(\mu+\widetilde{f}_{1})}\frac{k(k-1)}{\langle
k\rangle},$

where $\widetilde{f}_{1}=\beta+\gamma$,
$\widetilde{f}_{2}=n\alpha+\mu+\beta+\gamma$, and
$\widetilde{f}_{3}=n\alpha+2\gamma+(k-1)\beta$.

As presented in Appendix A, the disease basic reproduction number
$\mathcal{R}_{0}^{\rm dis}$ in our effective degree model with rewiring is
proven to be smaller as compared with the model without rewiring, i.e.,
$\mathcal{R}_{0}^{\rm dis}<\widetilde{\mathcal{R}}_{0}^{\rm dis}$. This result
implies that the adaptive link rewiring scheme plays an important role in
preventing the disease outbreak by decreasing the basic reproduction number
$\mathcal{R}_{0}^{\rm dis}$.

(ii) Special case—the model without awareness contagion. Consider the special
case when the awareness layer is absent, namely, there is no awareness
information propagating among the population, then the original two-layer
network effective degree model is degenerated into a single-layer network
effective degree model (which was studied numerically by Marceau et al. [20],
where the theoretical disease outbreak threshold was not provided). In
addition, each node only has one of the two states: susceptible or infected,
and does not possess the awareness information states. Moreover, there is only
one parameter $\omega$ for link rewiring for all susceptible nodes (i.e.,
$\omega^{U}=\omega^{A}=\omega$) and there is no parameters for information
contagion (i.e., $\alpha=\mu=0$). Thus, in this case, the basic reproduction
number for disease outbreak becomes

$\widehat{\mathcal{R}}_{0}^{\rm
dis}(\omega)=\sum_{k=1}^{K}\sum_{n=1}^{M}\frac{\beta\widehat{f}_{2}\widehat{f}_{3}S_{k0}^{U_{n0}}}{\widehat{f}_{1}\widehat{f}_{2}\widehat{f}_{3}-\gamma\beta\widehat{f}_{1}}\frac{k(k-1)}{\langle
k\rangle},$ (27)

where $\widehat{f}_{1}=\widehat{f}_{2}=\beta+\gamma+\omega$, and
$\widehat{f}_{3}=2\gamma+(k-1)(\beta+\omega)$. By substituting
$\widehat{f}_{1}$, $\widehat{f}_{2}$ and $\widehat{f}_{3}$ into Eq. (27) it is
easy to obtain

$\displaystyle\widehat{\mathcal{R}}_{0}^{\rm dis}(\omega)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{K}\sum_{n=1}^{M}\frac{\beta\widehat{f}_{3}S_{k0}^{U_{n0}}}{\widehat{f}_{2}\widehat{f}_{3}-\gamma\beta}\frac{k(k-1)}{\langle
k\rangle}=\sum_{k=1}^{K}\frac{\beta\widehat{f}_{3}P(k)}{\widehat{f}_{2}\widehat{f}_{3}-\gamma\beta}\frac{k(k-1)}{\langle
k\rangle},$ (28) $\displaystyle=$
$\displaystyle\sum_{k=1}^{K}\frac{\beta\Big{[}2\gamma+(k-1)(\beta+\omega)\Big{]}}{(\beta+\gamma+\omega)\Big{[}2\gamma+(k-1)(\beta+\omega)\Big{]}-\gamma\beta}\frac{k(k-1)P(k)}{\langle
k\rangle}.$

As the rewiring process is removed from the model, i.e., if $\omega=0$, then
the basic reproduction number turns into

$\widehat{\mathcal{R}}_{0}^{\rm
dis}(0)=\sum_{k=1}^{K}\frac{2\beta\gamma+(k-1)\beta^{2}}{2\gamma^{2}+(k-1)\beta^{2}+k\beta\gamma}\frac{k(k-1)P(k)}{\langle
k\rangle},$ (29)

which reproduces the result obtained by Lindquist et al. (see Appendix C in
Ref. [34]). Moreover, since $k\geq 1$, it can be easily verified that for any
positive rewiring rate $\omega$, the following inequality holds:

$\frac{2\beta\gamma+(k-1)\beta^{2}}{2\gamma^{2}+(k-1)\beta^{2}+k\beta\gamma}>\frac{\beta\Big{[}2\gamma+(k-1)(\beta+\omega)\Big{]}}{(\beta+\gamma+\omega)\Big{[}2\gamma+(k-1)(\beta+\omega)\Big{]}-\gamma\beta}.$
(30)

Therefore, one has for $\omega>0$

$\widehat{\mathcal{R}}_{0}^{\rm dis}(0)>\widehat{\mathcal{R}}_{0}^{\rm
dis}(w),$ (31)

which highlights the important role of the adaptive rewiring in decreasing the
disease reproduction number and hence hampering the disease outbreak.

Figure 3: (color online) Time series of the density of infected nodes $\rho^{I}$. The simulation results are obtained by running the disease-awareness contagion model with $10\%$ of infected seeds on an initial scale-free multiplex network, in which the disease transmission layer is initially assigned with a power-law degree distribution $P(k)\propto k^{-2.5}$, whilst the awareness information layer is initially assigned a power-law degree distribution $Q(n)\propto n^{-3.5}$, both with degree values ranging from $1$ to $10$. Each grey (thin) curve denotes an individual simulation result and the red (thick) dash-dotted curve marks the average of the $100$ individual simulations. Other parameters are $\alpha=0.04$, $\beta=0.2$, $\mu=0.1$, $\gamma=0.02$, $\omega^{A}=0.02$, $\omega^{U}=0.004$. |
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Figure 4: (color online) The fraction $\rho^{I}$ of infected nodes in the stationary state as a function of the disease transmission rate $\beta$ for different values of (a) the awareness information forgetting rate $\mu$, (b) the awareness information spreading rate $\alpha$, (c) the recovery rate $\gamma$ from disease, and (d) the rewiring rate $\omega^{A}$ for aware susceptible individuals. In this plot, the results are obtained on an initial Poisson multiplex network, in which the disease transmission layer is initially assigned with a Poisson degree distribution $P(k)\propto 2^{k}e^{-2}/k!$, whilst the awareness information layer is initially assigned a power-law degree distribution $Q(n)\propto 3^{n}e^{-3}/n!$, both with degree values ranging from $1$ to $10$. Each point is obtained by averaging over $200$ independent realizations. Other parameters are set in panels (a) $\alpha=0.02$, $\gamma=0.08$, $\omega^{A}=0.05$, $\omega^{U}=0.01$; (b) $\gamma=0.04$, $\mu=0.06$, $\omega^{A}=0.08$, $\omega^{U}=0.0016$; (c) $\alpha=0.02$, $\mu=0.06$, $\omega^{A}=0.02$, $\omega^{U}=0.004$; (d) $\alpha=0.02$, $\gamma=0.04$, $\mu=0.04$, $y=\omega^{U}/\omega^{A}=0.2$. |
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|
Figure 5: (color online) The fraction $\rho^{I}$ of infected nodes in the
stationary state as a function of the disease transmission rate $\beta$ for
different values of (a) the awareness information forgetting rate $\mu$, (b)
the awareness information spreading rate $\alpha$, (c) the recovery rate
$\gamma$ from disease, and (d) the rewiring rate $\omega^{A}$ for aware
susceptible individuals. In this plot, the results are obtained on an initial
scale-free multiplex network, in which the disease transmission layer is
initially assigned with a power-law degree distribution $P(k)\propto
k^{-2.5}$, whilst the awareness information layer is initially assigned a
power-law degree distribution $Q(n)\propto n^{-3.5}$, both with degree values
ranging from $1$ to $10$. Each point is obtained by averaging over $200$
independent realizations. Other parameters are set in panels (a)
$\alpha=0.08$, $\gamma=0.04$, $\omega^{A}=0.02$, $\omega^{U}=0.0004$; (b)
$\gamma=0.02$, $\mu=0.02$, $\omega^{A}=0.05$, $\omega^{U}=0.01$; (c)
$\alpha=0.02$, $\mu=0.06$, $\omega^{A}=0.02$, $\omega^{U}=0.0004$; (d)
$\alpha=0.04$, $\gamma=0.02$, $\mu=0.04$, $y=\omega^{U}/\omega^{A}=0.2$.

##  6 Simulation results

In this section, numerical computation and stochastic simulation have been
carried out to verify the theoretical analysis for our effective degree model.
In the simulation, the network size (i.e., the number of nodes) is set to
$N=10^{4}$. Initially, all nodes are set to be susceptible and unaware of
disease information, and then $10\%$ of nodes are randomly chosen as the
infected seeds that will trigger the contagion process across the network in
both layers.

|
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|
Figure 6: (color online) The fraction $\rho^{I}$ (of which the simulation
results are represented by the color bar in the heat map) of infected
individuals in the stationary state as a function of $\beta$ and (a) $\alpha$,
(b) $\gamma$, (c) $\mu$, and (d) $\omega^{A}$. All full phase diagram are
obtained for the same initial multiplex network structure as in Fig. 5. For
comparison, the theoretical threshold conditions for disease outbreak obtained
from our effective degree model [Eq. (16)], the reduced model [Eq. (24)], and
the lower bound [Eq. (25)] are indicated by solid curves, empty circles, and
dashed curves, respectively. The red filled circle in panel (a) marks the
metacritical point $(\alpha_{\rm c},\beta_{\rm c})$. Note that the threshold
transmission rate for disease outbreak is $\beta_{\rm
c}=\beta/\mathcal{R}_{0}^{\rm dis}$, above which the disease breaks out.
Analogously, the threshold diffusion rate for awareness information outbreak
is $\alpha_{\rm c}=\alpha/\mathcal{R}_{0}^{\rm awe}$, above which the
awareness information prevails. The simulation results are obtained by
averaging over $100$ independent realizations. Other parameters are used for
panels: (a) $\mu=0.02$, $\gamma=0.01$, $\omega^{A}=0.01$, $\omega^{U}=0.002$;
(b) $\alpha=0.08$, $\mu=0.09$, $\omega^{A}=0.05$, $\omega^{U}=0.01$; (c)
$\alpha=0.02$, $\gamma=0.01$, $\omega^{A}=0.02$, $\omega^{U}=0.004$; (d)
$\alpha=0.04$, $\mu=0.06$, $\gamma=0.02$, $y=\omega^{U}/\omega^{A}=0.2$.

In Fig. 3, we present the results of $100$ stochastic simulations for the
temporal evolution of the fraction of infected nodes on an artificial
multiplex networks with both layers initially set as scale-free. Each grey
solid curve represents an independent simulation trajectory, while the red
dash-dotted curve is the average simulation result. It is illustrated that
although there are evident fluctuations in the time required to reach the
expansion phase and the final size (which is largely due to the impact of
random events in the early stage) , the proportion of infected nodes
eventually reaches the stationary state, meaning the disease becomes endemic.

To further investigate the effects of parameters on disease transmission, we
report the density $\rho^{I}$ of infected nodes [given by Eq. (5)] as a
function of the disease transmission rate $\beta$ with different values of
other parameters in an initial multiplex network with Poisson degree
distributions and power-law degree distributions in Fig. 4 and Fig. 5,
respectively. Direct comparison between the results in Fig. 4 and Fig. 5
reveals that the initial network structures have little impact on the
qualitative behavior of the $\rho^{I}$ versus $\beta$ bifurcation dynamics. It
is worth emphasizing that the effect of initial network setting can be ignored
since the process of link rewiring causes the network to converge to an
randomly connected network with a Poisson degree distribution [17, 26]. As
shown in Fig. 9 in Appendix B, in the case of adaptive rewiring, the resulting
network will converge to a narrowly distributed degree distribution even if
the initial network setting is a perfect power law. Therefore, from now on, in
our simulations we will simply apply an initial scale-free network setting
just as done in Refs. [8, 11, 13]. As shown in Fig. 4(a) and Fig. 5(a), with
the increase of the information forgetting rate $\mu$ from $0.04$ to $0.1$,
the disease outbreak threshold decreases slightly and the value of $\rho^{I}$
rises correspondingly when the disease breaks out. This suggests that the
disease outbreak threshold is not much sensitive (i.e., it is relatively
robust) to the information forgetting rate. As illustrated in Fig. 4(b) and
Fig. 5(b), with the increase of the information propagating rate $\alpha$, the
disease prevalence $\rho^{I}$ increases accordingly when the disease breaks
out. However, the disease outbreak threshold almost remains unchanged for the
small values of $\alpha=0$ and $\alpha=0.01$; however, as $\alpha$ becomes
large enough (in this case $\alpha=0.04$ and $\alpha=0.08$), the disease
threshold grows evidently with $\alpha$. This implies a possible critical
phenomenon in the dependence of the disease threshold $\beta_{\rm c}$ on the
information spreading rate $\alpha$. That is, there exists a critical value
$\alpha_{\rm c}$ such that only when $\alpha<\alpha_{\rm c}$ will $\beta_{\rm
c}$ grow with $\alpha$, otherwise $\beta_{\rm c}$ keeps unchanged. This meta-
critical behavior is further verified in Fig.6(a). The enhancement of disease
information propagation would help reduce the infectious disease prevalence to
some extent; nevertheless, from the perspective of disease prevention, only a
large enough information diffusion rate will help enlarge the disease
threshold to curb the onset of disease. In Fig. 4(c) and Fig. 5(c), it is
shown that, as expected, increasing the recovery rate $\gamma$ from disease
infection will help impede the disease transmission by enlarging the disease
threshold as well as reducing the epidemic prevalence $\rho^{I}$. The larger
$\gamma$, the more infected individuals turning back to the aware susceptible
state, and hence the lower infection risk in the population. What is more, it
follows from Eq. 4(d) and Fig. 5(d) that enhancing the adaptive rewiring
mechanism for informed susceptible individuals would help prevent the disease
outbreak by raising the disease threshold and mitigate the epidemic
transmission by diminishing the disease prevalence $\rho^{I}$. This can be
expounded as follows. According to the adaptive rewiring mechanism, parts of
susceptible individuals are separated from infected ones. The greater the
rewiring rate is, the more pathway for disease transmission will be
interrupted, therefore the higher disease transmission threshold and the lower
epidemic prevalence in case the disease prevails.

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Figure 7: (color online) Critical transmission rate $\beta_{\rm c}$ for
disease outbreak as a function of the awareness information spreading rate
$\alpha$ with different values of (a) the information forgetting rate $\mu$,
(b) the recovery rate $\gamma$ from infectious disease, (c) the rewiring rate
$\omega^{A}$ for aware susceptible individuals, and (d) the ratio
$y=\omega^{A}/\omega^{U}$. All the results are obtained based on the basic
reproduction number for disease transmission [Eq. (16)]. The initial network
setting is the same as in Fig. 5. Other parameters are set for panels (a)
$\gamma=0.02$, $\omega^{A}=0.05$, $\omega^{U}=0.01$; (b) $\mu=0.04$,
$\omega^{A}=0.05$, $\omega^{U}=0.01$; (c) $\mu=0.06$, $\gamma=0.03$,
$y=\omega^{U}/\omega^{A}=0.2$; (d) $\mu=0.06$, $\gamma=0.03$,
$\omega^{A}=0.05$.

In Fig. 6, we provide a clear comparison for the disease outbreak condition
between theoretical analysis and stochastic simulations. In particular, as
denoted by the color bar in the heat map, the density $\rho^{I}$ of infected
nodes in the steady state is mapped onto a two-parameter phase plane. In each
panel of Fig. 6, the theoretical results of disease outbreak threshold are
tripartite. The disease threshold derived from Eq. (16) in the effective
degree model is marked by solid curves, the disease threshold obtained from
Eq.(24) in the reduced model is represented by empty circles, and the lower
bound of disease threshold calculated based on Eq. (25) is indicated by dashed
curves. As shown in Fig. 6(a), in the case of relatively low rewiring rate
($\omega^{A}=0.01$), the information diffusion rate $\alpha$ does not have
much impact on the density $\rho^{I}$ of infected nodes, which is greatly
affected by the disease transmission rate $\beta$. By contrast, in the case of
relatively large rewiring rates [$\omega^{A}=0.08$ in Fig. 4(b) and
$\omega^{A}=0.05$ in Fig. 5(b)], the information diffusion rate $\alpha$ has
substantial effects on $\rho^{I}$. This suggests that the combination of
enhanced information propagation with forceful adaptive rewiring would play an
important role in disease prevention and control. More importantly, though the
lower bound of disease threshold is independent of $\alpha$, both the disease
transmission thresholds $\beta_{\rm c}$ derived from our effective degree
model and the reduced model demonstrate a metacritical point ($\alpha_{\rm
c},\beta_{\rm c}$) [marked by the red filled circle in Fig. 6(a)] as
previously reported in [8]. As indicated by solid curves and empty circles in
Fig. 6(a), there is a critical phenomenon in the dependence of the disease
outbreak threshold $\beta_{\rm c}$ on the information spreading rate $\alpha$.
Namely, there exists a critical value $\alpha_{\rm c}$ for information
spreading rate, above which $\beta_{\rm c}$ nonlinearly grows with the
increase of $\alpha$, whereas below which $\beta_{\rm c}$ does not vary with
$\alpha$. In Fig. 6(b), the value of $\rho^{I}$ is plotted in the
$\beta-\gamma$ phase plane, where the disease-free region (marked in blue)
accounts for a major part, which highlights the great contribution of
enlarging the recovery rate $\gamma$ to preventing the disease outbreak. The
solid curve indicates that the disease transmission threshold $\beta_{\rm c}$
obtained from effective degree model [Eq. (16)] grows nonlinearly with
$\gamma$, whilst the disease transmission thresholds derived from the reduced
model [Eq. (24)] (empty circles) and the lower bound [Eq. (25)] (dashed
curves) increase almost linearly with $\gamma$. Only in the case of an
extremely small $\gamma$ with a relatively large $\beta$ can the value
$\rho^{I}$ reach the maximum. In Fig. 6(c), it is shown that for the given
parameters the disease transmission rate $\beta$ exhibits a dominant influence
on the density $\rho^{I}$ of infected nodes over the information forgetting
rate $\mu$, especially for large values of $\mu$. In addition, the disease
outbreak threshold shows a negligible reliance on $\mu$ , which has also been
demonstrated in Fig. 4(a) and Fig. 5(a). Furthermore, it is shown in Fig. 6(d)
that all the three theoretical disease thresholds grows almost linearly with
the adaptive rewiring rate $\omega^{A}$. The higher the disease transmission
rate, the more frequent adaptive rewiring needed for disease extinction. In
general, as shown in all panels in Fig. 6, the theoretical result of disease
outbreak threshold matches well with the stochastic simulations. What is more,
as expected, the threshold transmission rate $\beta_{\rm c}$ obtained by
effective degree model is smaller than the one obtained in the reduced model,
which is in turn less than that derived from the lower bound
$\mathcal{R}_{0}^{\rm LB}$. It is worth noting that $\mathcal{R}_{0}^{\rm
LB}>1$ ensures $\mathcal{R}_{0}^{\rm dis}>1$. Therefore, the lower bound can
be used as an early warning signal of epidemic risk as well as a preliminary
guidance for potential intervention measures for disease prevention.

Figure 8: (color online) Heat map for the density $\rho^{A}$ (denoted by the
color bar) of informed individuals (including infected individuals and aware
susceptible individuals) who are aware of disease information in the
stationary state as plotted in the $\alpha-\beta$ phase diagram. The orange
solid curve is for the disease outbreak threshold ($\beta=\beta_{\rm c}$)
derived from Eq. (16), while the black dashed curve corresponds to the
threshold condition ($\alpha=\alpha_{\rm c}$) for awareness information
outbreak obtained based on Eq. (20). The $\alpha-\beta$ phase plane is divided
by the threshold curves into four regions: the grey region A denotes the
disease- and information-free steady state (i.e., there are only unaware
susceptible individuals in the end); the blue region B denotes the disease-
free but information-present steady state (i.e., both aware and unaware
susceptible individuals exist without disease in the end); and the polychrome
regions ${\rm C}_{1}$ and ${\rm C}_{2}$ mark the disease endemic state where
infected individuals prevail in the population. The initial multiplex network
structure is the same as in Fig. 5. The simulation results are obtained by
averaging over $100$ independent realizations. Other parameters are
$\mu=0.02$, $\gamma=0.01$, $\omega^{A}=0.05$, $\omega^{U}=0.001$.

In Fig. 7, we present the disease transmission threshold $\beta_{\rm c}$ as a
function of the the awareness information diffusion rate $\alpha$ with
different values of other parameters. In all panes of Fig. 7, there exhibits a
metacritical point $(\alpha_{\rm c},\beta_{\rm c})$ such that the value of
$\beta_{\rm c}$ grows nonlinearly with $\alpha$ when $\alpha>\alpha_{\rm c}$;
otherwise, $\beta_{\rm c}$ is independent of $\alpha$. This critical
phenomenon in the reliance of disease transmission threshold on the awareness
information diffusion rate coincides with the findings in [8, 44, 45]. As
shown in Fig. 7(a), with parameters $\gamma=0.02$, $\omega^{A}=0.05$ and
$\omega^{U}=0.01$, the disease threshold $\beta_{\rm c}$ decreases slightly as
$\mu$ rises, which is consistent with the results in Figs. 4(a), 5(a) and
6(c). Moreover, for any fixed $\alpha$, as displayed in Fig. 7(a), the larger
the information forgetting rate $\mu$, the smaller the disease threshold
$\beta_{\rm c}$. Reversely, it is shown from Eq. 7(b-d) that the disease
transmission threshold $\beta_{\rm c}$ grows significantly with the rise of
parameters $\gamma$, $\omega^{A}$ and $y$. That is, for any fixed information
propagating rate $\alpha$, the larger the disease recovery rate and adaptive
rewiring rates, the bigger the critical transmission rate for the onset of
epidemic. In addition, in combination with a large $\alpha$, increasing the
rates of recovery and rewiring leads to a greater $\beta_{\rm c}$, which
highlights the combined effects of awareness information spreading and
adaptive rewiring in preventing and suppressing the disease transmission.
Furthermore, it is worth noting that the critical value $\alpha_{\rm c}$ of
awareness spreading rate (for the increase of the epidemic onset $\beta_{\rm
c}$) grows accordingly with the increase of $\mu$ and with the decrease of
$\gamma$, $\omega^{A}$ and $y$. In particular, as shown in each panel of Fig.
7, the smaller $\beta_{\rm c}$, the larger $\alpha_{\rm c}$.

According to our theoretical analysis on the threshold conditions for disease
outbreak and awareness information invasion in 4, it follows that if
$\beta>\beta_{\rm c}$ (which is equivalent to $\mathcal{R}_{0}^{\rm dis}>1$)
then the disease will break out; otherwise, the disease will die out. On the
other hand, if $\alpha>\alpha_{\rm c}$ (which is equivalent to
$\mathcal{R}_{0}^{\rm awe}>1$) then the awareness information will prevail in
the population; otherwise, the awareness information can not invade through
the information layer. Besides, based on the stability analysis by Kiss et al.
[44] and Huang et al. [45] on the similar awareness-disease model, it is
claimed that when $\mathcal{R}_{0}^{\rm dis}\leq 1$ and $\mathcal{R}_{0}^{\rm
awe}\leq 1$, the disease- and information-free equilibrium is stable; when
$\mathcal{R}_{0}^{\rm dis}\leq 1$ and $\mathcal{R}^{\rm awe}_{0}>1$, the
disease-free and information statured equilibrium is stable; in the case of
$\mathcal{R}_{0}^{\rm dis}>1$, whether the disease eventually dies out or not
depends on the competition between the two processes of awareness information
diffusion and disease transmission. In particular, if $\mathcal{R}^{\rm
dis}_{0}>1$ and $\mathcal{R}_{0}^{\rm awe}\leq 1$, then the disease spreading
outweighs the information diffusion process, the disease prevails; otherwise,
if the information diffusion outweighs the disease spreading, then awareness
information prevails. Note that if the disease prevails in the population
then, based on our model definition, the infected individuals will immediately
become aware of the disease information. Therefore, in the case of
$\mathcal{R}_{0}^{\rm dis}>0$ the awareness information prevails in the
population.

In order to verify the above statement, we present in Fig. 8 the heat map for
the density $\rho^{A}$ of aware individuals who know of the disease
information as a function of spreading rates $\alpha$ of awareness information
and $\beta$ of disease. Note that the grey region A marks the information-free
state (i.e., $\rho^{A}=0$) and the $\rho^{A}$ value goes from $0.0001$ to
$0.96$ as the color changes gradually from blue to red, as denoted in the
color bar. For comparison, the critical conditions for the onset of disease
and awareness information have been depicted in the $\alpha-\beta$ plane. The
orange solid curve represents the disease threshold $\beta_{\rm c}$ below
which the disease will die out, whereas the black dashed curve stands for the
awareness information threshold $\alpha_{\rm c}$ above which the awareness
information prevails in the population. Consequently, the $\alpha-\beta$ plane
is divided by the threshold curves into four areas A, B, ${\rm C}_{1}$ and
${\rm C}_{2}$. The simulation results agree well with our theoretical analysis
in such a way that in area A, there is no disease and no awareness information
at all; in area B, the disease dies out but the information persists in the
population; in both areas ${\rm C}_{1}$ and ${\rm C}_{2}$, the awareness
information prevails with a relative higher level, albeit whether the disease
dies out or not in area ${\rm C}_{2}$ depends on the outcome of the
competition between disease transmission and awareness information diffusion
[44, 45]. It can be observed from Fig. 8 that the critical value of
information diffusion rate in the metacritical point mentioned in Fig. 6(a)
and Fig. 7 is exactly the awareness information threshold $\alpha_{\rm c}$,
above which the onset of disease grows.

##  7 Conclusions

Summarizing, in this paper we have investigated the effects of awareness
propagation and individuals’ contact behavioral changes on infectious disease
transmission over multiplex networks. The risk-avoiding contact changes are
characterized by adaptive link rewiring that is supposed to be dependent on
individuals’ awareness state about the disease. To account for the structural
evolution of the contact network caused by the adaptive link rewiring process,
we have developed an effective-degree contagion dynamics model incorporating
an SIS epidemic spreading process and a UAU awareness diffusion process.
Employing the next generation matrix approach [42], we derived the disease
reproduction number $\mathcal{R}_{0}^{\rm dis}$ and awareness information
reproduction number $\mathcal{R}_{0}^{\rm awe}$ for our effective degree
model, based on which we obtained the threshold rates $\beta_{\rm c}$ for
disease transmission and $\alpha_{\rm c}$ for awareness information diffusion.
In addition, through a reduced model, two lower bounds of the basic
reproduction number have been provided, which could be used as a warning
signal to assess the risk of infections and to provide early preparedness for
disease prevention. Moreover, the effects of various parameters on the disease
threshold as well as the epidemic prevalence have been explored via extensive
simulations. The critical phenomenon is observed in the dependence of the
onset of epidemics on the awareness information diffusion rate, which supports
the emergence of a metacritical point reported by Granell et al. [8].
Furthermore, it is shown from our results that the awareness-dependent link
rewiring contributes to enlarging the disease transmission threshold
$\beta_{\rm c}$ and lowering the epidemic prevalence, which highlights the
effects of adaptive rewiring on preventing the disease spread. Besides, the
results also suggest that the combination of intensified awareness diffusion
and enhanced link rewiring will lead to greater contribution for disease
prevention and control. Finally, we have also presented a two-parameter
bifurcation diagram for the density of aware individuals in the stationary
state, providing a numerical support for the stability analysis in the SIS-UAU
contagion models [44, 45]. It is to be noted that although the proposed
effective-degree model allows for detailed description of changes in the
connectivity structure, its drawback mainly lies in the computational
complexity due to a complicated high-dimensional dynamical systems. Therefore,
it is not appropriate to apply this model to a complex network with a number
of high-degree nodes. It has long been challenging to strike the right balance
between fine characterizations of dynamical processes with feasible
computation complexities.

## Appendix A: Proof of inequality $\mathcal{R}_{0}^{\rm
dis}<\widetilde{\mathcal{R}}_{0}^{\rm dis}$

To proof $\mathcal{R}_{0}^{\rm dis}<\widetilde{\mathcal{R}}_{0}^{\rm dis}$, it
is equivalent to show

$\frac{\widetilde{f}_{2}\widetilde{f}_{3}}{\widetilde{f}_{1}\widetilde{f}_{2}\widetilde{f}_{3}-\gamma\beta(\mu+\widetilde{f}_{1})}>\frac{f_{2}f_{3}}{f_{1}f_{2}f_{3}-\gamma\beta(\mu+f_{1})}.$
(A1)

Since the denominators on both sides of the above inequality (A1) are
positive, what we need is to show

$\Delta_{k}\triangleq
f_{1}f_{2}\Big{[}{\widetilde{f}_{1}\widetilde{f}_{2}\widetilde{f}_{3}-\gamma\beta(\mu+\widetilde{f}_{1})}\Big{]}-{\widetilde{f}_{2}\widetilde{f}_{3}}\Big{[}{f_{1}f_{2}f_{3}-\gamma\beta(\mu+f_{1})}\Big{]}<0.$
(A2)

Substituting the expressions of $\widetilde{f}_{1}$, $\widetilde{f}_{2}$ and
$\widetilde{f}_{3}$ into $\Delta_{k}$, then expanding all the terms, and
letting $\Lambda=k-1\geq 0$ gives rise to

$\displaystyle\Delta_{k}$ $\displaystyle=$
$\displaystyle-\Lambda^{2}\alpha^{2}\beta^{2}n^{2}\omega^{U}-\Lambda^{2}\alpha^{2}\beta
n^{2}(\omega^{U})^{2}-2\Lambda\alpha^{3}\beta{n^{3}}\omega^{U}-\Lambda\alpha^{3}{n^{3}}(\omega^{U})^{2}-\alpha^{4}{n^{4}}\omega^{U}-2\Lambda^{2}\alpha\beta^{3}n\omega^{U}-2\Lambda^{2}\alpha\beta^{2}\gamma
n\omega^{U}$ (A3) $\displaystyle-2\Lambda^{2}\alpha\beta^{2}\mu
n\omega^{U}-\Lambda^{2}\alpha\beta^{2}n\omega^{A}\omega^{U}-2\Lambda^{2}\alpha\beta^{2}n(\omega^{U})^{2}-2\Lambda^{2}\alpha\beta\gamma
n(\omega^{U})^{2}-2\Lambda^{2}\alpha\beta\mu
n(\omega^{U})^{2}-\Lambda^{2}\alpha\beta n\omega^{A}(\omega^{U})^{2}$
$\displaystyle-4\Lambda\alpha^{2}\beta^{2}n^{2}\omega^{U}-8\Lambda\alpha^{2}\beta\gamma
n^{2}\omega^{U}-4\Lambda\alpha^{2}\beta\mu
n^{2}\omega^{U}-2\Lambda\alpha^{2}\beta
n^{2}\omega^{A}\omega^{U}-2\Lambda\alpha^{2}\beta
n^{2}(\omega^{U})^{2}-4\Lambda\alpha^{2}\gamma n^{2}(\omega^{U})^{2}$
$\displaystyle-2\Lambda\alpha^{2}\mu
n^{2}(\omega^{U})^{2}-\Lambda\alpha^{2}n^{2}\omega^{A}(\omega^{U})^{2}-2\alpha^{3}\beta
n^{3}\omega^{U}-6\alpha^{3}\gamma n^{3}\omega^{U}-2\alpha^{3}\mu
n^{3}\omega^{U}-\alpha^{3}n^{3}\omega^{A}\omega^{U}-\Lambda^{2}\beta^{4}\omega^{U}$
$\displaystyle-2\Lambda^{2}\beta^{3}\gamma\omega^{U}-2\Lambda^{2}\beta^{3}\mu\omega^{U}-\Lambda^{2}\beta^{3}\omega^{A}\omega^{U}-\Lambda^{2}\beta^{3}(\omega^{U})^{2}-\Lambda^{2}\beta^{2}\gamma^{2}\omega^{U}-2\Lambda^{2}\beta^{2}\gamma\mu\omega^{U}-\Lambda^{2}\beta^{2}\gamma\omega^{A}\omega^{U}$
$\displaystyle-2\Lambda^{2}\beta^{2}\gamma(\omega^{U})^{2}-\Lambda^{2}\beta^{2}\mu^{2}\omega^{U}-\Lambda^{2}\beta^{2}\mu\omega^{A}\omega^{U}-2\Lambda^{2}\beta^{2}\mu(\omega^{U})^{2}-\Lambda^{2}\beta^{2}\omega^{A}(\omega^{U})^{2}-\Lambda^{2}\beta\gamma^{2}(\omega^{U})^{2}$
$\displaystyle-2\Lambda^{2}\beta\gamma\mu(\omega^{U})^{2}-\Lambda^{2}\beta\gamma\omega^{A}(\omega^{U})^{2}-\Lambda^{2}\beta\mu^{2}(\omega^{U})^{2}-\Lambda^{2}\beta\mu\omega^{A}(\omega^{U})^{2}-2\Lambda\alpha\beta^{3}n\omega^{U}-12\Lambda\alpha\beta^{2}\gamma
n\omega^{U}$ $\displaystyle-4\Lambda\alpha\beta^{2}\mu
n\omega^{U}-2\Lambda\alpha\beta^{2}n\omega^{A}\omega^{U}-\Lambda\alpha\beta^{2}n(\omega^{U})^{2}-11\Lambda\alpha\beta\gamma^{2}n\omega^{U}-13\Lambda\alpha\beta\gamma\mu
n\omega^{U}-6\Lambda\alpha\beta\gamma n\omega^{A}\omega^{U}$
$\displaystyle-6\Lambda\alpha\beta\gamma
n(\omega^{U})^{2}-2\Lambda\alpha\beta\mu^{2}n\omega^{U}-2\Lambda\alpha\beta\mu
n\omega^{A}\omega^{U}-2\Lambda\alpha\beta\mu
n(\omega^{U})^{2}-\Lambda\alpha\beta
n\omega^{A}(\omega^{U})^{2}-5\Lambda\alpha\gamma^{2}n(\omega^{U})^{2}$
$\displaystyle-6\Lambda\alpha\gamma\mu n(\omega^{U})^{2}-3\Lambda\alpha\gamma
n\omega^{A}(\omega^{U})^{2}-\Lambda\alpha\mu^{2}n(\omega^{U})^{2}-\Lambda\alpha\mu
n\omega^{A}(\omega^{U})^{2}-\alpha^{2}\beta^{2}n^{2}\omega^{U}-9\alpha^{2}\beta\gamma
n^{2}\omega^{U}$ $\displaystyle-2\alpha^{2}\beta\mu
n^{2}\omega^{U}-\alpha^{2}\beta
n^{2}\omega^{A}\omega^{U}-13\alpha^{2}\gamma^{2}n^{2}\omega^{U}-10\alpha^{2}\gamma\mu
n^{2}\omega^{U}-5\alpha^{2}\gamma
n^{2}\omega^{A}\omega^{U}-\alpha^{2}\mu^{2}n^{2}\omega^{U}-\alpha^{2}\mu
n^{2}\omega^{A}\omega^{U}$
$\displaystyle-\Lambda\beta^{3}\gamma\omega^{A}-4\Lambda\beta^{3}\gamma\omega^{U}-\Lambda\beta^{2}\gamma^{2}\omega^{A}-9\Lambda\beta^{2}\gamma^{2}\omega^{U}-\Lambda\beta^{2}\gamma\mu\omega^{A}-9\Lambda\beta^{2}\gamma\mu\omega^{U}-5\Lambda\beta^{2}\gamma\omega^{A}\omega^{U}-2\Lambda\beta^{2}\gamma(\omega^{U})^{2}$
$\displaystyle-5\Lambda\beta\gamma^{3}\omega^{U}-10\Lambda\beta\gamma^{2}\mu\omega^{U}-5\Lambda\beta\gamma^{2}\omega^{A}\omega^{U}-4\Lambda\beta\gamma^{2}(\omega^{U})^{2}-5\Lambda\beta\gamma\mu^{2}\omega^{U}-5\Lambda\beta\gamma\mu\omega^{A}\omega^{U}-4\Lambda\beta\gamma\mu(\omega^{U})^{2}$
$\displaystyle-2\Lambda\beta\gamma\omega^{A}(\omega^{U})^{2}-2\Lambda\gamma^{3}(\omega^{U})^{2}-4\Lambda\gamma^{2}\mu(\omega^{U})^{2}-2\Lambda\gamma^{2}\omega^{A}(\omega^{U})^{2}-2\Lambda\gamma\mu^{2}(\omega^{U})^{2}-2\Lambda\gamma\mu\omega^{A}(\omega^{U})^{2}$
$\displaystyle-\alpha\beta^{2}\gamma n\omega^{A}-3\alpha\beta^{2}\gamma
n\omega^{U}-\alpha\beta\gamma^{2}n\omega^{A}-13\alpha\beta\gamma^{2}n\omega^{U}-\alpha\beta\gamma\mu
n\omega^{A}-7\alpha\beta\gamma\mu n\omega^{U}-4\alpha\beta\gamma
n\omega^{A}\omega^{U}$
$\displaystyle-12\alpha\gamma^{3}n\omega^{U}-16\alpha\gamma^{2}\mu
n\omega^{U}-8\alpha\gamma^{2}n\omega^{A}\omega^{U}-4\alpha\gamma\mu^{2}n\omega^{U}-4\alpha\gamma\mu
n\omega^{A}\omega^{U}-2\beta^{2}\gamma^{2}\omega^{A}-2\beta^{2}\gamma^{2}\omega^{U}$
$\displaystyle-2\beta\gamma^{3}\omega^{A}-6\beta\gamma^{3}\omega^{U}-2\beta\gamma^{2}\mu\omega^{A}-6\beta\gamma^{2}\mu\omega^{U}-4\beta\gamma^{2}\omega^{A}\omega^{U}-4\gamma^{4}\omega^{U}-8\gamma^{3}\mu\omega^{U}-4\gamma^{3}\omega^{A}\omega^{U}$
$\displaystyle-4\gamma^{2}\mu^{2}\omega^{U}-4\gamma^{2}\mu\omega^{A}\omega^{U}.$

Evidently, $\Delta_{k}<0$ holds for any $k\geq 1$. The proof of the inequality
$\mathcal{R}_{0}^{\rm dis}<\widetilde{\mathcal{R}}_{0}^{\rm dis}$ is complete.

## Appendix B: Impact of rewiring on degree distribution of the epidemic-layer
network

According to the adaptive rewiring rule in our model, the underlying network
structure in the epidemic layer—especially the degree distribution—will evolve
along with the rewiring process. Adaptive rewiring of high-risk links leads to
the breakdown of edges that connect susceptible nodes and infected ones, and
meanwhile, gives rise to the formation of low-risk links connecting towards a
randomly chosen susceptible node. The random attaching mechanism that has been
widely used in random graph models [46] will typically generate a relatively
narrow degree distribution. As demonstrated in Fig. 9 the degree distribution
of underlying contact network in the epidemic layer exhibits different scaling
behaviors: in the case without rewiring, the node degree follows a perfect
power law with exponent $\nu=2.5$; while in the case with rewiring, the degree
values are narrowly distributed along the average degree, approximating to a
Poisson distribution.

Figure 9: (color online) Degree distributions of the underlying network
capable of disease transmission in the case without (blue solid circles) and
with (red empty squares) adaptive rewiring, respectively, at time $t=10^{4}$
when the system is in the stationary state. Black solid line denotes the exact
power law with exponent $\nu=2.5$. The simulation results (denoted by symbols)
are obtained by starting with the same initial network architecture as in Fig.
3. To make a direct comparison between the cases without and with adaptive
rewiring, the results are plotted in the log-log scale, while the inset
displays the same results in a linear-linear scale. Parameters are
$\alpha=0.04$, $\mu=0.1$, $\gamma=0.02$, $\beta=0.2$, $\omega^{A}=0.02$,
$\omega^{U}=0.004$.

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