ArticlePDF Available

Synchronized and mixed outbreaks of coupled recurrent epidemics

Springer Nature
Scientific Reports
Authors:

Abstract and Figures

Epidemic spreading has been studied for a long time and most of them are focused on the growing aspect of a single epidemic outbreak. Recently, we extended the study to the case of recurrent epidemics (Sci. Rep. {\bf 5}, 16010 (2015)) but limited only to a single network. We here report from the real data of coupled regions or cities that the recurrent epidemics in two coupled networks are closely related to each other and can show either synchronized outbreak phase where outbreaks occur simultaneously in both networks or mixed outbreak phase where outbreaks occur in one network but do not in another one. To reveal the underlying mechanism, we present a two-layered network model of coupled recurrent epidemics to reproduce the synchronized and mixed outbreak phases. We show that the synchronized outbreak phase is preferred to be triggered in two coupled networks with the same average degree while the mixed outbreak phase is preferred for the case with different average degrees. Further, we show that the coupling between the two layers is preferred to suppress the mixed outbreak phase but enhance the synchronized outbreak phase. A theoretical analysis based on microscopic Markov-chain approach is presented to explain the numerical results. This finding opens a new window for studying the recurrent epidemics in multi-layered networks.
Content may be subject to copyright.
arXiv:1610.02528v1 [physics.soc-ph] 8 Oct 2016
Synchronized and mixed outbreaks of coupled recurrent epidemics
Muhua Zheng,1, 2 Ming Zhao,3Byungjoon Min,2and Zonghua Liu1 ,
1Department of Physics, East China Normal University, Shanghai, 200062, P. R. China
2Levich Institute and Physics Department, City College of New York, New York, New York 10031, USA
3College of Physics and Technology, Guangxi Normal University, Guilin 541004, China
Epidemic spreading has been studied for a long time and most of them are focused on the growing aspect
of a single epidemic outbreak. Recently, we extended the study to the case of recurrent epidemics (Sci. Rep.
5, 16010 (2015)) but limited only to a single network. We here report from the real data of coupled regions
or cities that the recurrent epidemics in two coupled networks are closely related to each other and can show
either synchronized outbreak phase where outbreaks occur simultaneously in both networks or mixed outbreak
phase where outbreaks occur in one network but do not in another one. To reveal the underlying mechanism, we
present a two-layered network model of coupled recurrent epidemics to reproduce the synchronized and mixed
outbreak phases. We show that the synchronized outbreak phase is preferred to be triggered in two coupled
networks with the same average degree while the mixed outbreak phase is preferred for the case with different
average degrees. Further, we show that the coupling between the two layers is preferred to suppress the mixed
outbreak phase but enhance the synchronized outbreak phase. A theoretical analysis based on microscopic
Markov-chain approach is presented to explain the numerical results. This finding opens a new window for
studying the recurrent epidemics in multi-layered networks.
Epidemic spreading in complex networks has been well studied and a lot of great progress have been achieved such as the
infinitesimal threshold [1–6], reaction-diffusion model [7–10], flow driven epidemic [11–15], and objective spreading etc [16,
17], see the review Refs. [18–20] for details. Recently, the attention has been moved to the case of multilayer networks [18, 21–
31], which represent the interactions between different real-world networks such as critical infrastructure [32–34], transportation
networks [35, 36], living organisms [37–39], and social networks [21, 40] etc. These models enable us to determine how the
interplay between network structures influences the dynamic processes taking place on them [41–50]. For instance, a pathogen
spreads on a human contact network abetted by global and regional transportation networks [21]. Due to their ubiquitous
applications in complex systems [51–55], the understanding of the properties and dynamic processes in multilayered networks
carries great practical significance.
Two of the most successful models used to describe epidemic spreading are the susceptible-infected-susceptible (SIS) and
susceptible-infected-refractory (SIR) models. Mark et al used the SIR model to multilayered networks in 2012 [41]. Very
interesting, they found a mixed phase in weakly coupled networks where an epidemic occurs in one network but does not spread
to the coupled network. Saumell-Mendiola et al used the SIS model to multilayered networks also in 2012 [42]. However, they
found that such a mixing phase doesn’t exist in both analytic and simulation results. In their work, they mainly focused on the
epidemic threshold and studied how epidemics spread from one network to another.
All these studies are focused on the growing aspect of a single epidemic outbreak, no matter it is one layer or multilayered
networks. However, in realistic situations, the empirical data shows that epidemic is recurrent, i.e. with outbreaks from time to
time [56–60]. Thus, we recently extended the study to the case of recurrent epidemics [57], but limited only to a single network.
Considering that the interactions between different networks are ubiquitous, we here recheck several real data of coupled regions
or cities such as the General Out-Patient Clinics (GOPC) network and its coupled General Practitioners (GP) network of Hong
Kong (see Fig. 1(a) and (b)), the coupled regional networks of California and Nevada (see Fig. 1(c) and (d)), the coupled
regional networks of Arizona and California (see Fig. 1(a) and (b) in SI), the coupled city networks of Boston and Fall River
(see Fig. 1(c) and (d) in SI), and the coupled city networks of Los Angeles and Sacramento (see Fig. 1(e) and (f) in SI). We
interestingly find that their recurrent epidemics are closely related to each other. Moreover, we find that the coupled time series
of recurrent epidemics can show either synchronized outbreak phase where outbreaks occur simultaneously in both networks or
mixed outbreak phase where outbreaks occur only in one network but do not in another one. This finding calls our great interest
and motivates us to study its underlying mechanism. In this sense, we believe that it is very necessary to further extend the study
of recurrent epidemics to the case of multilayered networks.
In this work, we present a two-layered network model of coupled recurrent epidemics to reproduce the synchronized and
mixed outbreak phases. To guarantee the appearance of recurrent outbreaks, we choose the susceptible-infected-refractory-
susceptible (SIRS) model for each node of network and let the infectious rate be time dependent, symbolizing the larger annual
variation of environment. By this model, we find that the average degrees of both the intra- and inter-networks play key roles on
the emergence of synchronized and mixed outbreak phases. The synchronized outbreak phase is preferred to be triggered in two
coupled networks with the same average degree while the mixed outbreak phase is preferred for the case with different average
Electronic address: zhliu@phy.ecnu.edu.cn
2
FIG. 1: (color online). Time series of recurrent epidemics in two coupled regions or cities. (a) and (b) represent the weekly consultation
rates of influenza-like illness (per 1000 consultations) from 1999 to 2013 in Hong Kong for the General Out-Patient Clinics (GOPC) and the
General Practitioners (GP), respectively, where the data from 2009/6/13 to 2010/5/23 in (a) are not available and the value of Cin (b) is from
0to 150. (c) and (d) represent the time series of reported weekly measles infective cases Iin California and Nevada, respectively.
degrees. Further, we show that the increasing of coupling strength, i.e. either the inter-layer infection rate or inter-layer average
degree, will prefer to suppress the mixed outbreak phase but enhance the synchronized outbreak phase. A theoretical analysis
based on microscopic Markov-chain approach is presented to explain the numerical results. This finding may be of significance
to the long-term prediction and control of recurrent epidemics in multi-areas or cities.
Results
The synchronized and mixed outbreak phases of recurrent epidemics in real data. Monitoring epidemic spreading is vital
for us to prevent and control infectious diseases. For this purpose, Hong Kong Department of Health launched a sentinel surveil-
lance system to collect data of infectious diseases, aiming to analyze and predict the trend of infectious spreading in different
regions of Hong Kong. In this system, there are about 64 General Out-Patient Clinics (GOPC) and 50 General Practitioners
(GP), which form two sentinel surveillance networks of Hong Kong [56], respectively. By these two networks, we can obtain
the weekly consultation rates of influenza-like illness (per 1000 consultations). Fig. 1(a) and (b) show the collected data from
1999 to 2013 for the GOPC and GP, respectively, where the data from 2009/6/13 to 2010/5/23 in (a) was not collected by Hong
Kong Department of Health and the value of Cin (b) is from 0to 150. This weekly consultation rates of influenza-like illness
can well reflect the overall influenza-like illness activity in Hong Kong. From the data in Fig. 1(a) and (b) we easily find that
there are intermittent peaks, marking the recurrent outbreaks of epidemics. By a second check on the data in Fig. 1(a) and (b)
we interestingly find that some peaks occur simultaneously in the two networks at the same time, indicating the appearance of
synchronized outbreak phase. While other peaks appear only in Fig. 1(a) but not in Fig. 1(b), indicating the existence of mixed
outbreak phase (see the light yellow shaded areas).
Is this finding of synchronized and mixed outbreak phases in recurrent epidemics a specific phenomenon only in Hong Kong?
3
FIG. 2: (color online). Schematic figure of the epidemic model to reproduce the synchronized and mixed outbreak phases. (a) Schematic
figure of the two-layered network, where the “black”, “blue” and “red” lines represent the links of the networks A,Band the inter-network
AB, respectively. βab denotes the infectious rate through one interconnection between Aand B. (b) Schematic figure of the extended SIRS
model for each node in Aand B, where the symbols S, I and Rrepresent the susceptible, infectious, and refractory states, respectively, and
the parameters β, γ and δrepresent the infectious, refractory and recovery rates, respectively. p0denotes the probability for a susceptible node
to be naturally infected by environment or other factors.
To figure out its generality, we have checked a large number of other recurrent infectious data in different pathogens and in
different states and cities in the United States and found the similar phenomenon. Fig. 1(c) and (d) show the data of weekly
measles infective cases Iin the states of California and Nevada, respectively, which were obtained from the USA National
Notifiable Diseases Surveillance System as digitized by Project Tycho [58–60]. As California is adjacent to Nevada in west
coast of the United States, their climatic conditions are similar. Thus, they can be also considered as two coupled networks.
Three more these kinds of examples have been shown in Fig. 1 of SI where the coupled networks are based on states-level
influenza data and cities-level measles data, respectively. Therefore, the synchronized and mixed outbreak phases are general in
recurrent epidemics. We will explain their underlying mechanisms in the next subsection.
A two-layered network model of coupled recurrent epidemics. Two classical models of epidemic spreading are the
Susceptible-Infected-Susceptible (SIS) model and Susceptible-Infected-Refractory(SIR) model [18]. In the SIS model, a sus-
ceptible node will be infected by an infected neighbor with rate β. In the meantime, each infected node will recover with a
probability γat each time step. After the transient process, the system reaches a stationary state with a constant infected density
ρI. Similarly, in the SIR model, each node can be in one of the three states: Susceptible, Infected, and Refractory. At each
time step, a susceptible node will be infected by an infected neighbor with rate βand an infected node will become refractory
with probability γ. The infection process will be over when there is no infected I. These two models have been widely used
in a variety of situations. However, it was pointed out that both the SIS and SIR models are failed to explain the recurrence of
epidemics in real data [57, 61].
To reproduce the synchronized and mixed outbreak phases, we here present a two-layered network model of coupled recurrent
epidemics, shown in Fig. 2 where (a) represents its schematic figure of network topology and (b) denotes the epidemic model
at each node. In Fig. 2(a), two networks Aand Bare coupled through some interconnections between them, which form the
inter-network AB. For the sake of simplicity, we let the two networks Aand Bhave the same size Na=Nb. We let hkai,hkbi,
and hkabirepresent the average degrees of the networks A,BandAB, respectively, see
Methods
for details. In Fig. 2(b), the
epidemic model is adopted from our previous work [57] by two steps. In step one, we extend the SIR model to a Susceptible-
Infected-Refractory-Susceptible (SIRS) model where a refractory node will become susceptible again with probability δ. In
step two, we let each susceptible node have a small probability p0to be infected, which represents the natural infection from
environment. Moreover, we let the infectious rate β(t)be time dependent, representing its annual and seasonal variations etc.
To distinguish the function of the interconnections from that of those links in Aand B, we let βab be the inter-layer infectious
rate. In this way, the interaction between Aand Bcan be described by the tunable parameter βab .
In numerical simulations, we let both the networks Aand Bbe the Eros-R´enyi (ER) random networks [62]. To guarantee a
recurrent epidemics in each of Aand B, we follow Ref. [57] to let β(t)be the Gaussian distribution N(0.1,0.12)and choose
p0= 0.01. Fig. 3(a) and (b) show the evolutions of the infected density ρIin Aand B, respectively, where the parameters are
taken as hkai= 6.5,hkbi= 1.5,hkabi= 1.0, and βab = 0.09. It is easy to observe that some peaks of ρIappear simultaneously
in Aand B, indicating the synchronized outbreak phase. We also notice that some peaks of ρIin Ado not have corresponding
peaks in B(see the light yellow shadowed areas in Fig. 3(a) and (b)), indicating the mixed outbreak phase. Moreover, we do
4
FIG. 3: (color online). Reproduced time series of recurrent epidemics in the two-layered network model. (a) and (b) represent the
evolutions of the infected density ρIin the two-layered networks Aand B, respectively, where the parameters are taken as hkai= 6.5,
hkbi= 1.5,hkabi= 1.0,β(t) N (0.1,0.12),βab = 0.09,γ= 0.2,δ= 0.02,p0= 0.01, and Na=Nb= 1000.
not find the contrary case where there are peaks of ρIin Bbut no corresponding peaks in A, which is also consistent with the
empirical observations in Fig. 1(a)-(d).
Mechanism of the synchronized and mixed outbreak phases. To understand the phenomenon of synchronized and mixed
outbreak phases better, we here study their underlying mechanisms. A key quantity for the phenomenon is the outbreak of
epidemic, i.e. the peaks in the time series of Fig. 1. Notice that a peak is usually much higher than its background oscillations.
To pick out a peak, we need to define its background first. As the distribution of real data in Fig. 1 satisfies the normal distribution
(see Fig. 2 SI), we define the background as µ+ 3σwith µand σbeing the mean and standard deviation, respectively, which
contains about 99.7% data in normal distribution [63]. Then, we divide the time series into different segments by a unit of time
interval T. If there are one or multiple data larger than the threshold µ+ 3σin the time duration T, we consider it as an outbreak.
In this way, we can count the number of outbreaks in a total measured time t. Let nbe the average number of outbreaks in
realizations of the same evolution t. Larger nimplies more frequent outbreaks. Let n=|n(A)n(B)|be the difference of
outbreak number between the networks Aand B. Larger nimplies more frequent emergence of the mixed outbreak phase. In
particular, the mixed outbreak phase will disappear when n= 0.
We are interesting how the average degrees and coupling influence the numbers nand n. Figure 4(a) and (b) show the
dependence of non the average degree hkbiof network Bfor βab = 0.09 and 0.3, respectively, where the average degree
of network Ais fixed as hkai= 6.5and Tis set as T= 52. It is easy to see from Fig. 4(a) and (b) that the number n(B)of
network Bwill gradually increase with the increase of hkbi, while the number n(A)of network Akeeps approximately constant,
indicating that a larger hkbiis in favor of the recurrent outbreaks. Specifically, n(B)will reach n(A)when hkbiis increased to
the value of hkbi=hkai= 6.5, see the insets in Fig. 4(a) and (b) for the minimum of n. For details, Fig. 3 in SI shows the
evolution of infected densities ρIfor the cases of hkai=hkbi, confirming the result of n= 0 in Fig. 4(a) and (b). On the
other hand, comparing the two insets in Fig. 4 (a) and (b), we find that nin Fig. 4(a) is greater than the corresponding one in
Fig. 4(b), indicating that a larger βab is in favor of suppressing the mixed outbreak phase. These results can be also theoretically
explained by the microscopic Markov-chain approach, see
Methods
for details. The solid lines in Fig. 4 (a) and (b) represent the
theoretical results from Eqs. (6) and (7). It is easy to see that the theoretical results are consistent with the numerical simulations
very well.
Fig. 4(c) and (d) show the influences of the coupling parameters such as βab and kab for the outbreak number nand the
difference of outbreak number n, respectively, where hkai= 6.5and hkbi= 1.5. From Fig. 4(c) we see that for the case of
kab = 0.6,n(A)is an approximate constant and n(B)gradually increase with βab . While for the case of kab = 1.2, both n(A)
and n(B)change with βab, indicating that both kab and βab take important roles in the synchronized and mixed outbreak phases.
From Fig. 4(d) we see that the case of kab = 1.2decreases faster than the case of kab = 0.6, indicating that both the larger kab
5
FIG. 4: (color online). (a) and (b): Dependence of non the average degree hkbiof network Bfor βab = 0.09 and 0.3, respectively,
where the average degree of network Ais fixed as hkai= 6.5,Tis set as T= 52,hkabi= 1.0, and the insets show the dependence of
n=|n(A)n(B)|on hkbi. The solid lines represent the theoretical results from Eqs. (6) and (7). (c): Dependence of non βab with
hkai= 6.5and hkbi= 1.5where the “squares” and “circles” represent the case of hkabi= 0.6for the networks Aand B, respectively, and
the “up triangles” and “down triangles” represent the case of hkab i= 1.2for the networks Aand B, respectively. (d): Dependence of non
βab with hkai= 6.5and hkbi= 1.5where the “squares” and “circles” represent the cases of hkabi= 0.6and 1.2, respectively. The other
parameters are set as β(t) N (0.1,0.12),γ= 0.2,δ= 0.02,p0= 0.01, and Na=Nb= 1000. All the results are averaged over 1000
independent realizations with the time duration T= 52 and simulation steps t= 20000.
and larger βab are in favor of the synchronized outbreak phase. That is, the stronger coupling will suppress the mixed outbreak
phase but enhance the synchronized outbreak phase. On the contrary, the weaker coupling is in favor of the mixed epidemic
outbreak phase but suppress the synchronized outbreak phase. For details, Fig. 4 in SI shows the evolution of infected densities
ρIfor different βab, confirming the above results.
Correlation between the epidemic outbreaks of the two coupled networks. Except the framework of the synchronized and
mixed outbreak phases, the relationship between the epidemics of the two coupled layers can be also measured by the cross-
correlation defined in Eq. (19). Fig. 5(a) shows that the correlation coefficient rbetween the two time series of GOPC and
GP is about 0.66, indicating that these two data are highly correlated. To check the influence of coupling on the correlation,
Fig. 5(b) shows the dependence of ron βab for hkabi= 0.2,0.5,1.0and 2.0, respectively. We see that rincreases with βab for
a fixed hkabiand also increases with hkabifor a fixed βab, indicating the enhanced correlation by the coupling strength. Very
interesting, we find that for the case of hkab i= 1.0in Fig. 5(b), the point of r= 0.66 corresponds to hβab i= 0.09 (see the
purple “star”), implying that the coupling in Fig. 5(a) is equivalentto the case of hkab i= 1.0and hβabi= 0.09 in Fig. 5(b). In
6
FIG. 5: (color online.) Correlation between the two coupled layers. (a) Correlation between the two time series of GOPC and GP. (b)
Dependence of the correlation coefficient ron βab for hkabi= 0.2,0.5,1.0and 2.0, respectively, where the purple “star” and its related
dashed line represent the point of r= 0.66. Other parameters are the same as in Fig. 3(a) and (b).
this sense, we may draw a horizontal line passing the purple “star” in Fig. 5(b) (see the dashed line) and its crossing points with
all the curves there will also represent the equivalent coupling in Fig. 5(a). However, we notice that the horizontal line has no
crossing point with the curve of hkab i= 0.2, indicating that there is a threshold of hkabifor the appearance of r= 0.66 in Fig.
5(a). Figs. 5 and 6 in SI show more information on the cross-correlation.
Discussion
The dependence of the synchronizedand mixed outbreak phases on the average degrees of networks Aand Bmay be also un-
derstood from the aspect of their epidemic thresholds. By the theoretical analysis in Methods we obtain the epidemic thresholds
as βA
c=γ
hkaiand βB
c=γ
hkbiin Eq. (18), when the two networks are weakly coupled. For the case of hkai>hkbi, we have
βA
c< βB
c. When βsatisfies β < βA
c< βB
c, the epidemics cannot survive in each of the networks Aand B. Thus, the infected
fraction will be approximately zero, i.e. no epidemic outbreak in both Aand B. When βsatisfies β > β B
c> βA
c, the epidemics
will survive in both the networks Aand B, indicating an outbreak will definitely occur in both of them. These two cases are
trivial. However, when βis in between βA
cand βB
c, some interesting results may be induced by coupling. When coupling is
weak, it is possible for epidemic to outbreak only in network Abut not in network B. Once coupling is increased, it will be
also possible for epidemic to outbreak sometimes in network B, i.e. resulting a mixed outbreak phase. When coupling is further
increased to large enough, a synchronized outbreak phase will be resulted.
So far, the reported results are from the ER random networks Aand B. We are wondering whether it is possible to still observe
the phenomenon of the synchronized and mixed outbreak phases in other network topologies. For this purpose, we here take the
scale-free network [62] as an example. Very interesting, by repeating the above simulation process in scale-free networks we
have found the similar phenomenonas in ER random networks, see Figs. 7-9 in SI for details. Therefore, the synchronized and
mixed outbreak phases are a general phenomenon in multi-layered epidemic networks.
In sum, the epidemic spreading has been well studied in the past decades, mainly focused on both the single and multi-layered
networks. However, only a few works focused on the aspect of recurrent epidemics, including both the models in homogeneous
population [61] and our recent model in a single network [57]. We here report from the real data that the epidemics from different
networks are in fact not isolated but correlated, implying that they should be considered as a multi-layered network. Motivated
by this discovery,we present a two-layered network model to reproduce the correlated recurrent epidemics in coupled networks.
More importantly, we find that this model can reproduce both the synchronized and mixed outbreak phases in real data. The
two-layered network is preferred to show the synchronized phase when the average degrees of the two coupled networks have
a large difference and shows the mixed phase when their average degrees are very close. Except the degree difference between
the two networks, the coupling strength between the two layers has also significant influence to the synchronized and mixed
outbreak phases. We show that both the larger βab and larger hkab iare in favor of the synchronized phase but suppress the mixed
7
phase. This finding thus shows a new way to understand the epidemics in realistic multi-layered networks. Its further studies
and potential applications in controlling the recurrent epidemics may be an interesting topic in near future.
Methods
A two-layered network model of recurrent epidemic spreading
We consider a two-layered network model with coupling between its two layers, i.e. the networks Aand Bin Fig. 2(a). We
let the two networks have the same size Na=Nb=Nand their degree distributions PA(k)and PB(k)be different. We may
image the network Aas a human contact network for one geographic region and the network Bfor a separated region. Each
node in the two-layered network has two kinds of links, i.e. those links in Aor Band the others between Aand B. The former
consists of the degree distributions PA(k)and PB(k)while the latter the interconnection network. We let hkai,hkbi, and hkab i
represent the average degrees of networks A,Band interconnection network AB, respectively. In details, we firstly generate
two separated networks Aand Bwith the same size Nand different degree distributions PA(k)and PB(k), respectively. Then,
we add links between Aand B. That is, we randomly choose two nodes from Aand Band then connect them if they are not
connected yet. Repeat this process until the steps we planned. In this way, we obtain an uncorrelated two-layered network.
To discuss epidemic spreading in the two-layered network, we let each node represent the SIRS model, see Fig. 2(b) for its
schematic illustration. In this model, a susceptible node has three ways to be infected. The first one is the natural infection
from environment or unknown reasons, represented by a small probability p0. The second one is the infection from contacting
with infected individuals in the network A(or B), represented by β(t). And the third one is the infection from another network,
represented by βab (see Fig. 2(a)). Thus, a susceptible node will become infected with a probability 1(1p0)(1β(t))kinf (1
βab)kinf
ab where kinf is the infected neighbors in the same network and kinf
ab is the infected neighbors in another network. At the
same time, an infected node will become refractory by a probability γand a refractory node will become susceptible again by a
probability δ.
In numerical simulations, the dependence of β(t)on time is implemented as follows [57]: we divide the total time tinto
multiple segments with length Tand let T= 52, corresponding to the 52 weeks in one year. We let β(t)be a constant in each
segment, which is randomly chosen from the Gaussian distribution N(0.1,0.12). Once a β(t)<0or β(t)>1is chosen, we
discard it and then choose a new one. At the same time, we fix γ= 0.2and δ= 0.02 and set βab as the tunable parameter.
A theoretical analysis based on microscopic Markov-chain approach
Let PS
i,A(t),PI
i,A(t),PR
i,A(t)be the probabilities for node iin network Ato be in one of the three states of S, I and Rat time
t, respectively. Similarly, we have PS
i,B (t),PI
i,B (t)and PR
i,B (t)in network B. They satisfy the conservation law
PS
i,A(t) + PI
i,A(t) + PR
i,A(t) = 1, P S
i,B (t) + PI
i,B (t) + PR
i,B (t) = 1.(1)
According to the Markov-chain approach [43, 57, 64–66], we introduce
ρA
S(t) = 1
N
N
X
i=1
PS
i,A(t), ρA
I(t) = 1
N
N
X
i=1
PI
i,A(t), ρA
R(t) = 1
N
N
X
i=1
PR
i,A(t),
ρB
S(t) = 1
N
N
X
i=1
PS
i,B (t), ρB
I(t) = 1
N
N
X
i=1
PI
i,B (t), ρB
R(t) = 1
N
N
X
i=1
PR
i,B (t),
where ρA
S(t),ρA
I(t)and ρA
R(t)represent the densities of susceptible, infected, and refractory individuals at time tin network A,
respectively. Similarly, we have ρB
S(t),ρB
I(t)and ρB
R(t)in network B.
Let qS,I
i,A (t),qI,R
i,A (t)and qR,S
i,A (t)be the transition probabilities from the state Sto I,Ito R, and Rto Sin network A,
respectively. By the Markov chain approach [64, 66] we have
qS,I
i,A (t) = 1 (1 p0)Y
lΛi,A
(1 β(t)PI
l,A(t)) Y
vΛi,B
(1 βabPI
v,B (t)),
qI,R
i,A (t) = γ, (2)
qR,S
i,A (t) = δ,
8
FIG. 6: (color online.) Comparison between the theoretical solutions and numerical simulations. The left and right panels are for the
networks Aand B, respectively. All the parameters are the same as in Fig. 3(a) and (b). (a) and (e) β(t)versus t; (b) and (f) ρSversus t;
(c) and (g) ρIversus t; (d) and (h) ρRversus t. In (b)-(d) and (f)-(h), the solid curves represent the theoretical solutions while the “circles”
represent the numerical simulations.
where Λi,A represents the neighboring set of node iin network A. The term (1 p0)in Eq. (2) represents the probability that
node iis not infected by the environment. The term QlΛi,A (1 β(t)PI
l,A(t)) is the probability that node iis not infected by
the infected neighbors in network A. While the term QvΛi,B (1 βabPI
v,B (t)) is the probability that node iis not infected by
the infected neighbors in another network. Thus, (1 p0)QlΛi,A (1 β(t)PI
l,A(t)) QvΛi,B (1 βab PI
v,B (t)) is the probability
for node ito be in the susceptible state. Similarly, for the node in network B, we have
qS,I
i,B (t) = 1 (1 p0)Y
lΛi,B
(1 β(t)PI
l,B (t)) Y
vΛi,A
(1 βabPI
v,A (t)),
qI,R
i,B (t) = γ, (3)
qR,S
i,B (t) = δ.
Based on these analysis, we formulate the following difference equations
PS
i,A(t+ 1) = PS
i,A(t)(1 qS,I
i,A (t)) + PR
i,A(t)qR,S
i,A (t),
PI
i,A(t+ 1) = PI
i,A(t)(1 qI ,R
i,A (t)) + PS
i,A(t)qS,I
i,A (t),(4)
9
PR
i,A(t+ 1) = PR
i,A(t)(1 qR,S
i,A (t)) + PI
i,A(t)qI ,R
i,A (t).
PS
i,B (t+ 1) = PS
i,B (t)(1 qS,I
i,B (t)) + PR
i,B (t)qR,S
i,B (t),
PI
i,B (t+ 1) = PI
i,B (t)(1 qI,R
i,B (t)) + PS
i,B (t)qS,I
i,B (t),(5)
PR
i,B (t+ 1) = PR
i,B (t)(1 qR,S
i,B (t)) + PI
i,B (t)qI,R
i,B (t).
The first term on the right-hand side of the first equation of Eq. (4) is the probability that node iis remained as susceptible state.
The second term stands for the probability that node iis changed from refractory to susceptible state. Similarly, we have the
same explanation for the other equations of Eqs. (4) and (5). Substituting Eqs. (2) and (3) into Eqs. (4) and (5), we obtain the
microscopic Markov dynamics as follows
PS
i,A(t+ 1) = PS
i,A(t)[(1 p0)Y
lΛi,A
(1 β(t)PI
l,A(t)) Y
vΛi,B
(1 βabPI
v,B (t))] + PR
i,A(t)δ,
PI
i,A(t+ 1) = PI
i,A(t)(1 γ) + PS
i,A(t)[(1 p0)Y
lΛi,A
(1 β(t)PI
l,A(t)) Y
vΛi,B
(1 βabPI
v,B (t))],(6)
PR
i,A(t+ 1) = PR
i,A(t)(1 δ) + PI
i,A(t)γ.
PS
i,B (t+ 1) = PS
i,B (t)[(1 p0)Y
lΛi,B
(1 β(t)PI
l,B (t)) Y
vΛi,A
(1 βabPI
v,A(t))] + PR
i,B (t)δ,
PI
i,B (t+ 1) = PI
i,B (t)(1 γ) + PS
i,B (t)[(1 p0)Y
lΛi,B
(1 β(t)PI
l,B (t)) Y
vΛi,A
(1 βabPI
v,A(t))],(7)
PR
i,B (t+ 1) = PR
i,B (t)(1 δ) + PI
i,B (t)γ.
Instead of getting the analytic solutions of Eqs. (6) and (7), we solve them by numerical integration. We set the initial
conditions as PS
i,A(0) = 1.0,PI
i,A(0) = 0.0,PR
i,A(0) = 0.0,PS
i,B (0) = 1.0,PI
i,B (0) = 0.0, and PR
i,B (0) = 0.0. To conveniently
compare the solutions with the numerical simulations in the section Results, we use the same set of β(t)for both the integration
and numerical simulations. Fig. 6 shows the results where the left and right panels are for the networks Aand B, respectively.
In Fig. 6(b)-(d) and Fig. 6(f)-(h), the solid curves represent the theoretical solutions while the “circles” represent the numerical
simulations. It is easy to see that the theoretical solutions are consistent with the numerical simulations very well.
A theoretical analysis based on mean fieldtheory
To go deeper into the mechanism of the synchronized and mixed outbreak phases, we try another theoretical analysis on mean
field equations. Let sA(t),iA(t)and rA(t)represent the densities of susceptible, infected, and refractory individuals at time tin
network A, respectively. Similarly, we have sB(t),iB(t)and rB(t)in network B. Then, they satisfy
sA(t) + iA(t) + rA(t) = 1, sB(t) + iB(t) + rB(t) = 1.(8)
According to the mean-field theory, we have the following ordinary differential equations
dsA(t)
dt =p0sA(t)β(t)hkaisA(t)iA(t)βabhkabisA(t)iB(t) + δrA(t),(9)
diA(t)
dt =p0sA(t) + β(t)hkaisA(t)iA(t) + βabhkab isA(t)iB(t)γiA(t),(10)
drA(t)
dt =γiA(t)δrA(t),(11)
dsB(t)
dt =p0sB(t)β(t)hkbisB(t)iB(t)βabhkab isB(t)iA(t) + δrB(t),(12)
diB(t)
dt =p0sB(t) + β(t)hkbisB(t)iB(t) + βabhkab isB(t)iA(t)γiB(t),(13)
drB(t)
dt =γiB(t)δrB(t).(14)
10
Specifically, we consider a case of single epidemic outbreak with extremely weak coupling, i.e. p0= 0 and βab 0. In the
steady state, we have
dsA(t)
dt = 0,diA(t)
dt = 0,dsB(t)
dt = 0,diB(t)
dt = 0,(15)
which gives
sA(t) = γ
β(t)hkai, rA(t) = γ
δiA(t), sB(t) = γ
β(t)hkbi, rB(t) = γ
δiB(t).(16)
Substituting Eq. (16) into Eq. (8) we obtain
γ
β(t)hkai+ (γ
δ+ 1)iA(t) = 1,γ
β(t)hkbi+ (γ
δ+ 1)iB(t) = 1.(17)
The critical point can be given by letting iA(t)and iB(t)in Eq. (17) change from zero to nonzero, which gives
βA
c=γ
hkai, βB
c=γ
hkbi.(18)
For the considered networks with hkai>hkbi, we have βA
c< βB
c.
Cross-correlation measure
In statistics, the Pearson correlation coefficient is a measure of the linear correlation between two variables. If two time series
{Xt}and {Yt}have the mean values Xand Y, we can define the correlation coefficient ras follows
r=
n
P
t=1
(XtX)(YtY)
sn
P
t=1
(XtX)2·
n
P
t=1
(YtY)2
(19)
To analyze the correlations of the growth trends between the two time series, we investigates their cross-correlation r(t)in a
given window wt[67, 68], i.e. {Xt, Xt+1 , . . . , Xt+wt}and {Yt, Yt+1, ...,Yt+wt}.{Xt}and {Yt}will share the same trend
in the time interval wtwhen r(t)>0and the opposite growth trend when r(t)<0.
[1] Pastor-Satorras, R., & Vespignani, A., Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200 (2001).
[2] Bogu˜n´a, M., & Pastor-Satorras, R., Epidemic spreading in correlated complex networks. Phys. Rev. E 66, 047104 (2002).
[3] Ferreira, S. C., Castellano, C., & Pastor-Satorras, R., Epidemic thresholds of the susceptible-infected-susceptible model on networks: A
comparison of numerical and theoretical results. Phys. Rev. E 86, 041125 (2012).
[4] Bogu˜n´a, M., Castellano, C., & Pastor-Satorras, R., Nature of the epidemic threshold for the susceptible-infected-susceptible dynamics in
networks. Phys. Rev. Lett. 111, 068701 (2013).
[5] Parshani, R., Carmi, S., & Havlin, S., Epidemic threshold for the susceptible-infectious-susceptible model on random networks. Phys.
Rev. Lett. 104, 258701 (2010).
[6] Castellano, C., & Pastor-Satorras, R., Thresholds for epidemic spreading in networks. Phys. Rev. Lett. 105, 218701 (2010).
[7] Colizza, V., Pastor-Satorras, R., & Vespignani, A., ReactionCdiffusion processes and metapopulation models in heterogeneous networks.
Nature Phys. 3, 276-282 (2007).
[8] Colizza, V., & Vespignani, A., Invasion threshold in heterogeneous metapopulation networks. Phys. Rev. Lett. 99, 148701 (2007).
[9] Baronchelli, A., Catanzaro, M., & Pastor-Satorras, R., Bosonic reaction-diffusion processes on scale-free networks. Phys. Rev. E 78,
016111 (2008).
[10] Tang, M., Liu, L., & Liu, Z., Influence of dynamical condensation on epidemic spreading in scale-free networks. Phys. Rev. E 79, 016108
(2009).
[11] Vazquez, A., Racz, B., Lukacs, A., & Barabasi, A. L., Impact of non-Poissonian activity patterns on spreading processes. Phys. Rev. Lett.
98, 158702 (2007).
[12] Meloni, S., Arenas, A., & Moreno, Y., Traffic-driven epidemic spreading in finite-size scale-free networks. Proc. Natl. Acad. Sci. USA
106, 16897-16902 (2009).
11
[13] Balcan, D., Colizza, V., Gon?alves, B., Hu, H., Ramasco, J. J., & Vespignani, A., Multiscale mobility networks and the spatial spreading
of infectious diseases. Proc. Natl. Acad. Sci. USA 106, 21484-21489 (2009).
[14] Ruan, Z., Tang, M., & Liu, Z., Epidemic spreading with information-driven vaccination. Phys. Rev. E 86, 036117 (2012).
[15] Liu, S., Perra, N., Karsai, M., & Vespignani, A., Controlling contagion processes in activity driven networks. Phys. Rev. Lett. 112, 118702
(2014).
[16] Tang, M., Liu, Z., & Li, B., Epidemic spreading by objective traveling. Europhys. Lett. 87, 18005 (2009).
[17] Liu, Z., Effect of mobility in partially occupied complex networks. Phys. Rev. E 81, 016110 (2010).
[18] Pastor-Satorras, R., Castellano, C., Van Mieghem, P., & Vespignani, A., Epidemic processes in complex networks. Reviews of modern
physics,87(3), 925 (2015).
[19] A. Barrat, M. Barthelemy, & A. Vespignani, Dynamical Processes on Complex Networks. (Cambridge University Press, Cambridge,
England, 2008).
[20] Dorogovtsev, S. N., Goltsev, A. V., & Mendes, J. F., Critical phenomena in complex networks. Rev. Mod. Phys. 80, 1275 (2008).
[21] Boccaletti, S., Bianconi, G., Criado, R., del Genio, C. I., G´omez-Garde˜nes, J., Romance, M., Sendi˜na-Nadal, I., Wang, Z., & Zanin, M.,
The structure and dynamics of multilayer networks. Phys. Rep. 544, 1 (2014).
[22] Kivel¨a, M., Arenas, A., Barthelemy, M., Gleeson, J. P., Moreno, Y. & Porter, M. A., Multilayer networks. J. Complex Netw. 2, 203 (2014).
[23] Feng, L., Monterola, C. P., & Hu, Y., The simplified self-consistent probabilities method for percolation and its application to interdepen-
dent networks. New Journal of Physics,17(6), 063025 (2015).
[24] Sahneh, F. D., Scoglio, C., & Chowdhury, F. N. Effect of coupling on the epidemic threshold in interconnected complex networks: A
spectral analysis. In 2013 American Control Conference (pp. 2307-2312). IEEE (2013).
[25] Wang, H., Li, Q., D’Agostino, G., Havlin, S., Stanley, H. E., & Van Mieghem, P. Effect of the interconnected network structure on the
epidemic threshold. Phys. Rev. E 88(2) 022801 (2013).
[26] Yagan, O., Qian, D., Zhang, J., & Cochran, D. Conjoining speeds up information diffusion in overlaying social-physical networks. IEEE
J. Sel. Areas Commun. 31(6), 1038-1048 (2013).
[27] Newman, M. E. Threshold effects for two pathogens spreading on a network. Phys. Rev. Lett. 95(10) 108701 (2005).
[28] Marceau, V., No¨el, P. A., H´ebert-Dufresne, L., Allard, A., & Dub´e, L. J., Modeling the dynamical interaction between epidemics on
overlay networks. Phys. Rev. E 84 026105 (2011).
[29] Zhao, Y., Zheng, M., & Liu, Z., A unified framework of mutual influence between two pathogens in multiplex networks. Chaos 24,
043129 (2014).
[30] Buono, C., & Braunstein, L. A., Immunization strategy for epidemic spreading on multilayer networks. Europhy. Lett. 109(2), 26001
(2015).
[31] Buono, C., Alvarez-Zuzek, L. G., Macri, P. A., & Braunstein, L. A., Epidemics in partially overlapped multiplex networks. PloS one,
9(3), e92200 (2014).
[32] Little, R. G., Controlling cascading failure: Understanding the vulnerabilities of interconnected infrastructures. J. Urban Technology 9,
109-123 (2002).
[33] Rosato, V., Issacharoff, L., Tiriticco, F., Meloni, S., Porcellinis, S., & Setola, R., Modelling interdependent infrastructures using interact-
ing dynamical models. J. Critical Infrast. 463 (2008).
[34] Buldyrev, S. V., Parshani, R., Paul, G., Stanley, H. E., & Havlin, S., Catastrophic cascade of failures in interdependent networks. Nature
464, 1025 (2010).
[35] De Domenico, M., Sol´e-Ribalta, A., G´omez, S., & Arenas, A., Navigability of interconnected networks under random failures. Proc.
Natl. Acad. Sci. USA 111, 8351 (2014).
[36] Parshani, R., Rozenblat, C., Ietri, D., Ducruet, C., & Havlin, S., Inter-similarity between coupled networks. Europhy. Lett. 92, 68002
(2010).
[37] Reis, S. D., Hu, Y., Babino, A., Andrade Jr, J. S., Canals, S., Sigman, M., & Makse, H. A., Avoiding catastrophic failure in correlated
networks of networks. Nat. Phys. 10, 762 (2014).
[38] Vidal, M., Cusick, M. E., & Barab´asi, A. L., Interactome networks and human disease. Cell 144, 986 (2011).
[39] White, J. G., Southgate, E., Thomson, J. N., & Brenner, S., The structure of the nervous system of the nematode Caenorhabditis elegans.
Phil. Trans. Royal Soc. London B,314, 1 (1986).
[40] Min, B., Gwak, S. H., Lee, N., & Goh, K. I., Layer-switching cost and optimality in information spreading on multiplex networks. Sci.
Rep. 6, 21392 (2016).
[41] Dickison, M., Havlin, S., & Stanley, H. E., Epidemics on interconnected networks. Phys. Rev. E 85, 066109 (2012).
[42] Saumell-Mendiola, A., Serrano, M. A., & Bogun´a, M., Epidemic spreading on interconnected networks. Phys. Rev. E 86, 026106 (2012).
[43] Granell, C., G´omez, S., & Arenas, A., Dynamical interplay between awareness and epidemic spreading in multiplex networks. Phys. Rev.
Lett. 111, 128701 (2013).
[44] Cozzo, E., Ba˜nos, R. A., Meloni, S., & Moreno, Y., Contact-based social contagion in multiplex networks. Phys. Rev. E 88, 050801(R)
(2013).
[45] Funk, S., & Jansen, V. A., Interacting epidemics on overlay networks. Phys. Rev. E 81, 036118 (2010).
[46] Allard, A., No¨el, P. A., Dub´e, L. J., & Pourbohloul, B., Heterogeneous bond percolation on multitype networks with an application to
epidemic dynamics. Phys. Rev. E 79, 036113 (2009).
[47] Son, S. W., Bizhani, G., Christensen, C., Grassberger, P., & Paczuski, M., Percolation theory on interdependent networks based on
epidemic spreading. Europhy. Lett. 97, 16006 (2012).
[48] Sanz, J., Xia, C. Y., Meloni, S., & Moreno, Y., Dynamics of interacting diseases. Phys. Rev. X 4, 041005 (2014).
[49] Leicht, E. A., & D’Souza, R. M., Percolation on interacting networks. arXiv:0907.0894 (2009).
[50] Hackett, A., Cellai, D., G´omez, S., Arenas, A., & Gleeson, J. P., Bond percolation on multiplex networks. Phys. Rev. X 6(2), 021002
(2016).
12
[51] Halu, A., Mukherjee, S., & Bianconi, G., Emergence of overlap in ensembles of spatial multiplexes and statistical mechanics of spatial
interacting network ensembles. Phys. Rev. E 89 012806 (2014).
[52] Barigozzi, M., Fagiolo, G., & Garlaschelli, D., Multinetwork of international trade: A commodity-specific analysis. Phys. Rev. E 81
046104 (2010).
[53] Barigozzi, M., Fagiolo, G., & Mangioni, G., Identifying the community structure of the international-trade multi-network. Physica A,
390 2051 (2011).
[54] Cardillo, A., G´omez-Garde˜nes, J., Zanin, M., Romance, M., Papo, D., del Pozo, F., & Boccaletti, S., Emergence of network features from
multiplexity. Sci. Rep. 31344 (2013).
[55] Kaluza, P., K¨olzsch, A., Gastner, M. T., & Blasius, B., The complex network of global cargo ship movements. Journal of the Royal
Society: Interface 71093 (2010).
[56] Department of Health, Hong Kong. Weekly consultation rates of influenza-like illness data.
http://www.chp.gov.hk/en/sentinel/26/44/292.html. Date of access: 15/06/2014.
[57] Zheng, M., Wang, C., Zhou, J., Zhao, M., Guan, S., Zou, Y., & Liu, Z., Non-periodic outbreaks of recurrent epidemics and its network
modelling. Sci. Rep. 5, 16010 (2015).
[58] The USA National Notifiable Diseases Surveillance System. Weekly measles infective cases. http://www.tycho.pitt.edu/. Date of access:
04/08/2016.
[59] Scarpino, S. V., Allard, A., & H´ebert-Dufresne, L. The effect of a prudent adaptive behaviour on disease transmission. Nat. Phys. 3832,
1745-2481 (2016).
[60] Van Panhuis,W. G.et al. Contagious diseases in the united states from 1888 to the present. N. Engl. J. Med. 369, 2152-2158 (2013).
[61] Stone L., Olinky R. & Huppert A., Seasonal dynamics of recurrent epidemics. Nature 446, 533-536 (2007).
[62] Albert, R. & Barabasi, A., Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47-97 (2002).
[63] Patel, J. K., & Read, C. B. Handbook of the normal distribution. CRC Press (Vol. 150) (1996).
[64] omez, S., Arenas, A., Borge-Holthoefer, J., Meloni, S., & Moreno, Y., Discrete-time Markov chain approach to contact-based disease
spreading in complex networks. Europhys. Lett. 89, 38009 (2010).
[65] omez, S., G´omez-Garde˜nes, J., Moreno, Y., & Arenas, A., Nonperturbative heterogeneous mean-field approach to epidemic spreading
in complex networks. Phys. Rev. E 84, 036105 (2011).
[66] Valdano, E., Ferreri, L., Poletto, C., & Colizza, V., Analytical computation of the epidemic threshold on temporal networks. Phys. Rev. X
5, 021005 (2015).
[67] Podobnik, B., & Stanley, H. E., Detrended Cross-Correlation Analysis: A New Method for Analyzing Two Nonstationary Time Series.
Phys. Rev. Lett. 100, 084102 (2008).
[68] Wang, W., Liu, Q. H., Cai, S. M., Tang, M., Braunstein, L. A., & Stanley, H. E., Suppressing disease spreading by using information
diffusion on multiplex networks. Sci. Rep. 6, 29259 (2016).
Acknowledgement
We acknowledge the Centre for Health Protection, Department of Health, the Government of the Hong Kong Special Admin-
istrative Region and the USA National Notifiable Diseases Surveillance System as digitized by Project Tycho for providing data.
We especially acknowledge Hern´an A. Makse for supporting the computer resource in the lab. This work was partially supported
by the NNSF of China under Grant Nos. 11135001, 11375066, 973 Program under Grant No. 2013CB834100, the Program for
Excellent Talents in Guangxi Higher Education Institutions and Guangxi Natural Science Foundation 2015jjGA10004.
Author contributions
M.Z. and Z.L. conceived the research project. M.Z., M.Z. B.M. and Z.L. performed research and analyzed the results. M.Z.
and Z.L. wrote the paper. All authors reviewed and approved the manuscript.
Additional information
Competing financial interests: The authors declare no competing financial interests. Correspondence and requests for materi-
als should be addressed to Z.L. (zhliu@phy.ecnu.edu.cn).

Supplementary resource (1)

... Then, differential equations are designed to describe the time evolution of the various compartments. The sub-populations can be chosen to represent (S)usceptible, (I)nfected and (R)ecovered individuals (SIR model), obeying the following differential equations: The system depends on three parameters, namely c, ε and N. Due to the conservation law (2), only two equations are independent, so that one can drop the equation for S. ...
... Empirical modifications of the basic SIR model exist and range from including new sub-populations to generalise the coefficients c, ε to be time-dependent in order to better reproduce the observed data (see Refs. [2][3][4][5] for recent examples). ...
Article
Full-text available
We generalise the epidemic Renormalization Group framework while connecting it to a SIR model with time-dependent coefficients. We then confront the model with COVID-19 in Denmark, Germany, Italy and France and show that the approach works rather well in reproducing the data. We also show that a better understanding of the time dependence of the recovery rate would require extending the model to take into account the number of deaths whenever these are over 15% of the cumulative number of infected cases.
... The spreading process is currently one of the hottest topics in the field of complex networks, such as the spread- ing of epidemics, opinions, rumors, new technologies and behaviors, and so on. So far, a great deal of signifi- cant progress has been achieved including the infinitesimal threshold, 1-6 reaction-diffusion model, 7-10 temporal and/or multilayer networks, [11][12][13][14][15][16][17][18][19][20][21][22][23][24] etc. (see the review Refs. 25-28 for details). ...
... The effects of multiple channels on the spreading process have been widely investigated based on a powerful analytical framework: multilayer or multiplex networks, [11][12][13][14][15][16][17][18][19][20][21][22] where the intra-links and inter-links represent the multiple social rela- tions (channels) among individuals. So far, the majority of research studies about multilayer networks are mainly focused on how the one-to-one interconnections influence the dynamic processes taking place on them. ...
Article
Full-text available
A great deal of significant progress has been seen in the study of information spreading on populations of networked individuals. A common point in many of the past studies is that there is only one transition in the phase diagram of the final accepted size versus the transmission probability. However, whether other factors alter this phenomenology is still under debate, especially for the case of information spreading through many channels and platforms. In the present study, we adopt a two-layered network to represent the interactions of multiple channels and propose a Susceptible-Accepted-Recovered information spreading model. Interestingly, our model shows a novel double transition including a continuous transition and a following discontinuous transition in the phase diagram, which originates from two outbreaks between the two layers of the network. Furthermore, we reveal that the key factors are a weak coupling condition between the two layers, a large adoption threshold, and the difference of the degree distributions between the two layers. Moreover, we also test the model in the coupled empirical social networks and find similar results as in the synthetic networks. Then, an edge-based compartmental theory is developed which fully explains all numerical results. Our findings may be of significance for understanding the secondary outbreaks of information in real life.
... For instance, it is well known that, together with primary cascading failures, power grids may also suffer secondary cascading failures in local parts of the network, i.e., secondary disasters [48]. Moreover, similar phenomena are known to occur in epidemic spreading, when recurrent epidemics may break out from time to time [49][50][51][52]. ...
Article
Full-text available
We give evidence that consecutive explosive transitions may occur in two-layered networks, when a dynamical layer made of an ensemble of networking phase oscillators interacts with an environmental layer of oscillators which are in a state of approximate synchronization. Under these conditions, the interlayer coupling induces two consecutive explosive transitions in the dynamical layer, each one associated with a hysteresis loop. We also show that the same phenomenon can be observed when the environmental layer is simplified into a single node with phase lag. Theoretical arguments unveil that the mechanisms at the basis of the two transitions are in fact different, with the former originating from a coupling-amplified disorder and the latter originating from a coupling-induced synchronization. We discuss the relevance of the observed state in brain dynamics and show how it may emerge in a real brain network.
... In [26] , the scenario of recurrent epidemics is studied which is an extension from a single community. Furthermore, in [27] , the epidemic spreading process is investigated to reproduce the synchronized and mixed outbreak patterns in two communities. ...
Article
In practice, an epidemic might be spreading among multi-communities; while the communities are usually intra-connected. In this manuscript, each community is modeled as a multiplex network (i.e., virtual layer and physical one). The connections inside certain community are referred as inter-contacts while the intra-contacts denote the connections among communities. For the epidemic spreading process, the traditional susceptible-infected-recovered (SIR) model is adopted. Then, corresponding state transition trees are determined and simulations are conducted to study the epidemic spreading process in multi-communities. Here, the effect of incorporating virtual layer on the range of individual affected by the epidemic is pursued. As illustrated, multi-summits are incurred if the spreading in multi-communities is considered; furthermore, the disparity between summits varies. This is affected by various factors. As indicated, the incorporation of virtual layer is capable of reducing the proportion of individuals being affected; moreover, disparity of different summits is likely to be increased regarding with scenarios of excluding virtual layer. Furthermore, the summit is likely to be postponed if virtual layer is incorporated.
... Multiple channels can be represented by a multilayered or multiplex network [24][25][26][27], where the intralinks within the same layer and inter-links across distinct layers represent the multiple social channels among individuals. So far, studies about multi-layered networks have mainly focused on the influence of one-to-one interconnections between two layers on the dynamic processes [28][29][30][31][32]. ...
Article
Full-text available
Information spreading has been studied for decades, but the underlying mechanism why the information can be accepted by a large number of people overnight is still under debate, especially in the aspects of two-channel effects for information transmission and theoretical analysis. In this study, based on a susceptible-accepted-recovered (SAR) model, we examine the effects of two channels represented by a two-layered network, in which one channel is the intra-links within the same layer and the other is inter-links between layers. Different with the case of one single channel on a one-layered network, in the case of two channels, the spreading can be speeded up by the increase in the coupling strength, i.e., average node degree and transmission probability between the two layers. Strikingly, even if the parameter (social reinforcement) is small, the strongly coupling strength can induce explosive transition in the information spreading process. Additionally, a big gap closed to the critical point for the explosive transition was found in the phase space of theoretical analysis, which indicates the emergency of a global large-scope outbreak. These findings may be of significance on the understanding and controlling explosive information spreading in modern society.
... 这一模型可以用来解释季节性的病毒传播 [49,50] . : ...
Article
In modern society, new communication channels and social platforms remarkably change the way of people receiving and sharing information, but the influences of these channels on information spreading dynamics have not been fully explored, especially in the aspects of outbreak patterns. To this end, based on a susceptible–accepted–recovered model, we examined the outbreak patterns of information spreading in a two-layered network with two coexisting channels: the intra-links within a layer and the inter-links across layers. Depending on the inter-layer coupling strength, i.e., average node degree and transmission probability between the two layers, we observed three different spreading patterns: (i) a localized outbreak with weak inter-layer coupling, (ii) two peaks with a time-delay outbreak appear for an intermediate coupling, and (iii) a synchronized outbreak for a strong coupling. Moreover, we showed that even though the average degree between the two layers is small, a large transmission probability still can compensate and promote the information spread from one layer to another, indicating by that the critical average degree decreases as a power law with transmission probability between the two layers. Additionally, we found that a large gap closed to the critical inter-layer average degree appears in the phase space of theoretical analysis, which indicates the emergence of a global large-scope outbreak. Our findings may, therefore, be of significance for understanding the outbreak behaviors of information spreading in real world.
Book
Full-text available
The World Health Organization (WHO) has declared the 2019 novel coronavirus outbreak (COVID-19) as a pandemic on March 11th. As of the end of April 2020, more than 3 million COVID-19 cases and 200 thousands death have been reported from more than 200 countries. It is therefore important to know what to expect in terms of the growth of the number of cases, and to understand what is needed to arrest the very worrying trends. In this disruptive period of the COVID-19 pandemic, scientists are investing an unprecedented effort to try to forecast and suggest measures to mitigate the ill-fated effects of the pandemic. Although recent literature indicates that travel control and restrictions of public activities are effective in delaying the spreading of the COVID-19 epidemic in China (Heory et al. 2020; Chinazzi et al. 2020), there is still an urgent need for greater understanding of the intrinsic dynamics and effective control methods which can offer in emergency and pandemic management. This Research Topic aims to extend the angles and collect articles which propose data driven mathematical or statistical models of the spread of the COVID-19, and/or of its foreseen consequences on public health, society, industry, economics and technology. It also focuses on collecting the real-time big data of COVID-19 spreading, and further helps the scientists to establish the efficient databases for the risk management. Furthermore, we also want to understand the impact of the pandemic on the economy and society of the whole world, and provide efficient suggestions for economic recovery and social order maintenance. The editors and reviewers of this special issue will guarantee a fast, but fair, peer-to-peer review procedure, in order to provide to society a reliable injection of scientific insights. The scopes and topics include but are not limited to: • nonlinear dynamics and non-equilibrium processes of COVID-19; • complex system and complex networks modeling of COVID-19; • computational epidemiology, biophysics, systems biology and computational biology aspects of COVID-19; • artificial intelligence, machine learning and big data analytics of COVID-19; • self-organization and emergent phenomena of social organization with COVID-19 pandemics; • applications to social science, Public health, economics, engineering and other aspects related to COVID-19 pandemics.
Article
Full-text available
Although there is always an interplay between the dynamics of information diffusion and disease spreading, the empirical research on the systemic coevolution mechanisms connecting these two spreading dynamics is still lacking. Here we investigate the coevolution mechanisms and dynamics between information and disease spreading by utilizing real data and a proposed model on multiplex network. Our empirical analysis finds asymmetrical interactions between the information and disease spreading dynamics. Our results obtained from both the theoretical framework and intensive stochastic numerical simulations suggest that an information outbreak can be triggered on a communication network by its own spreading dynamics or the disease outbreak on a contact network, but that the disease threshold is not affected by information spreading. Our key finding is that there is an optimal information transmission rate that markedly suppresses the disease speading. We find that the time evolution of the dynamics in the proposed model qualitatively agrees with the real-world spreading processes at the optimal information transmission rate.
Article
Full-text available
We study a model of information spreading on multiplex networks, in which agents interact through multiple interaction channels (layers), say online vs. offline communication layers, subject to layer-switching cost for transmissions across different interaction layers. The model is characterized by the layer-wise path-dependent transmissibility over a contact, that is dynamically determined dependently on both incoming and outgoing transmission layers. We formulate an analytical framework to deal with such path-dependent transmissibility and demonstrate the nontrivial interplay between the multiplexity and spreading dynamics, including optimality. It is shown that the epidemic threshold and prevalence respond to the layer-switching cost non-monotonically and that the optimal conditions can change in abrupt non-analytic ways, depending also on the densities of network layers and the type of seed infections. Our results elucidate the essential role of multiplexity that its explicit consideration should be crucial for realistic modeling and prediction of spreading phenomena on multiplex social networks in an era of ever-diversifying social interaction layers.
Article
Full-text available
The study of recurrent epidemic outbreaks has been attracting great attention for decades, but its underlying mechanism is still under debate. Based on a large number of real data from different cities, we find that besides the seasonal periodic outbreaks of influenza, there are also non-periodic outbreaks, i.e. non-seasonal or non-annual behaviors. To understand how the non-periodicity shows up, we present a network model of SIRS epidemic with both time-dependent infection rate and a small possibility of persistent epidemic seeds, representing the influences from the larger annual variation of environment and the infection generated spontaneously in nature, respectively. Our numerical simulations reveal that the model can reproduce the non-periodic outbreaks of recurrent epidemics with the main features of real influenza data. Further, we find that the recurrent outbreaks of epidemic depend not only on the infection rate but also on the density of susceptible agents, indicating that they are both the necessary conditions for the recurrent epidemic patterns with non-periodicity. A theoretical analysis based on Markov dynamics is presented to explain the numerical results. This finding may be of significance to the control of recurrent epidemics.
Article
Full-text available
An interconnected network features a structural transition between two regimes [F. Radicchi and A. Arenas, Nat. Phys. 9, 717 (2013)1745-247310.1038/nphys2761]: one where the network components are structurally distinguishable and one where the interconnected network functions as a whole. Our exact solution for the coupling threshold uncovers network topologies with unexpected behaviors. Specifically, we show conditions that superdiffusion, introduced by Gómez et al. [Phys. Rev. Lett. 110, 028701 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.028701], can occur despite the network components functioning distinctly. Moreover, we find that components of certain interconnected network topologies are indistinguishable despite very weak coupling between them.
Article
Full-text available
We present an analytical approach for bond percolation on multiplex networks and use it to determine the expected size of the giant connected component and the value of the critical bond occupation probability in these networks. We advocate the relevance of these tools to the modeling of multilayer robustness and contribute to the debate on whether any benefit is to be yielded from studying a full multiplex structure as opposed to its monoplex projection, especially in the seemingly irrelevant case of a bond occupation probability that does not depend on the layer. Although we find that in many cases the predictions of our theory for multiplex networks coincide with previously derived results for monoplex networks, we also uncover the remarkable result that for a certain class of multiplex networks, well described by our theory, new critical phenomena occur as multiple percolation phase transitions are present. We provide an instance of this phenomenon in a multipex network constructed from London rail and European air transportation datasets.
Article
Full-text available
Interdependent networks in areas ranging from infrastructure to economics are ubiquitous in our society, and the study of their cascading behaviors using percolation theory has attracted much attention in recent years. To analyze the percolation phenomena of these systems, different mathematical frameworks have been proposed, including generating functions and eigenvalues, and others. These different frameworks approach phase transition behaviors from different angles and have been very successful in shaping the different quantities of interest, including critical threshold, size of the giant component, order of phase transition, and the dynamics of cascading. These methods also vary in their mathematical complexity in dealing with interdependent networks that have additional complexity in terms of the correlation among different layers of networks or links. In this work, we review a particular approach of simple, self-consistent probability equations, and we illustrate that this approach can greatly simplify the mathematical analysis for systems ranging from single-layer network to various different interdependent networks. We give an overview of the detailed framework to study the nature of the critical phase transition, the value of the critical threshold, and the size of the giant component for these different systems.
Article
The spread of disease can be slowed by certain aspects of real-world social networks, such as clustering and community structure, and of human behaviour, including social distancing and increased hygiene, many of which have already been studied. Here, we consider a model in which individuals with essential societal roles—be they teachers, first responders or health-care workers—fall ill, and are replaced with healthy individuals. We refer to this process as relational exchange, and incorporate it into a dynamic network model to demonstrate that replacing individuals can accelerate disease transmission. We find that the effects of this process are trivial in the context of a standard mass-action model, but dramatic when considering network structure, featuring accelerating spread, discontinuous transitions and hysteresis loops. This result highlights the inability of mass-action models to account for many behavioural processes. Using empirical data, we find that this mechanism parsimoniously explains observed patterns across 17 influenza outbreaks in the USA at a national level, 25 years of influenza data at the state level, and 19 years of dengue virus data from Puerto Rico. We anticipate that our findings will advance the emerging field of disease forecasting and better inform public health decision making during outbreaks.