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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 Erd˝os-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−(1−p0)(1−β(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

l∈Λi,A

(1 −β(t)PI

l,A(t)) Y

v∈Λi,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 Ql∈Λi,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 Qv∈Λi,B (1 −βabPI

v,B (t)) is the probability that node iis not infected by

the infected neighbors in another network. Thus, (1 −p0)Ql∈Λi,A (1 −β(t)PI

l,A(t)) Qv∈Λi,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

l∈Λi,B

(1 −β(t)PI

l,B (t)) Y

v∈Λi,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

l∈Λi,A

(1 −β(t)PI

l,A(t)) Y

v∈Λi,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

l∈Λi,A

(1 −β(t)PI

l,A(t)) Y

v∈Λi,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

l∈Λi,B

(1 −β(t)PI

l,B (t)) Y

v∈Λi,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

l∈Λi,B

(1 −β(t)PI

l,B (t)) Y

v∈Λi,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

(Xt−X)(Yt−Y)

sn

P

t=1

(Xt−X)2·

n

P

t=1

(Yt−Y)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.

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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).