Asymmetrically interacting dynamics with mutual confirmation from multi-source on multiplex networks
aa r X i v : . [ phy s i c s . s o c - ph ] J a n Asymmetrically interacting dynamics with mutual confirmation frommulti-source on multiplex networks
Jiaxing Chen a,1 , Ying Liu a,c,1, ∗ , Ming Tang b,d, ∗ , Jing Yue a a School of Computer Science, Southwest Petroleum University, Chengdu 610500, P. R. China b School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China c Big Data Research Center, University of Electronic Science and Technology of China, Chengdu 610054, P. R. China d Shanghai Key Laboratory of Multidimensional Information Processing, East China Normal University, Shanghai 200241, China
Abstract
In the early stage of epidemics, individuals’ determination on adopting protective measures, which can reduce theirrisk of infection and suppress disease spreading, is likely to depend on multiple information sources and their mutualconfirmation due to inadequate exact information. Here we introduce the inter-layer mutual confirmation mechanisminto the information-disease interacting dynamics on multiplex networks. In our model, an individual increases theinformation transmission rate and willingness to adopt protective measures once he confirms the authenticity of newsand severity of disease from neighbors status in multiple layers. By using the microscopic Markov chain approach,we analytically calculate the epidemic threshold and the awareness and infected density in the stationary state, whichagree well with simulation results. We find that the increment of epidemic threshold when confirming the awareneighbors on communication layer is larger than that of the contact layer. On the contrary, the confirmation of neigh-bors’ awareness and infection from the contact layer leads to a lower final infection density and a higher awarenessdensity than that of the communication layer. The results imply that individuals’ explicit exposure of their infectionand awareness status to neighbors, especially those with real contacts, is helpful in suppressing epidemic spreading.
Keywords: multiplex network, epidemic spreading model, asymmetrically interacting dynamics, mutualconfirmation mechanism, microscopic Markov chain
1. Introduction
The human beings have been fighting with epidemic disease for a long history. The recent coronavirus disease2019 (COVID-19) has rapidly infected more than 25 millions of people and killed more than 800 thousands in justa few months [1]. Unraveling the mechanism that underlies the disease dynamics and adopting timely and accurateaction are very important to save lives of many people and reduce the impact on the society and economics [2–5].In the longrun through social and economic activities, people form kinds of networks, such as the contact network,communication network, trade network, et al [6]. The global spread of epidemics is thus a complex, network-drivenprocess [7]. The underlying networks through which the disease transmits play a critical role in determining thespatiotemporal patterns of the dynamic process [8]. Using the complex network approach, some key properties of theepidemic process, such as the epidemic threshold, the disease arrival time, the critical transmission path, the spatialorigin of spreading process, can be reliably predicted [9–11].Nowadays with the rapid development of information technology, people communicate and obtain informationmuch more easily and through multiple channels, e.g., kinds of online social networks, email, mobile phone and massmedia. When epidemics outbreaks, the information about disease spreads and stimulates risk awareness among peo-ple. The preventive measures people adopt help to reduce infection and thus suppress the spreading of disease [12]. ∗ Corresponding author
Email addresses: [email protected] (Ying Liu), [email protected] (Ming Tang) These authors contributed equally to this paper.
Preprint submitted to Elsevier January 6, 2021 he coevolving dynamics of information spreading and disease spreading can be modeled and analyzed as the asym-metrically interacting processes on the multiplex network, where the communication layer is formed by online socialnetwork and the physical contact layer is formed through persistent contacts in daily life [13–15]. The multiplexnetwork is a particular kind of multilayer network where the same set of individuals form di ff erent layers. This mul-tilayer approach has proved to be very successful in modeling the real complex systems where layers of networks areinterrelated with each other [16–18].In the e ff orts to understand the information-disease coevolving spreading dynamics on multiplex network, a num-ber of excellent models are proposed and some non-trivial phenomena are discovered which are substantially di ff erentfrom the independent spreading dynamics [13, 19–23]. For example, researchers demonstrated that there is a metacrit-ical point for the oneset of the epidemics which is determined by the topology of the virtual network and the dynamicsof information spreading [13]. The outbreak threshold of epidemic is not a ff ected by the information spreading, butan optimal information transmission rate may exist to markedly suppress the disease spreading [24]. The time scaleof information propagation doesn’t a ff ect the outbreak threshold, but an optimal relative timescale of information anddisease spreading may reduce the epidemics incidence [25, 26]. Some works define heterogenous disease or infor-mation transmission rate for nodes by taking the local or global infection or awareness density into consideration,where the awareness density is obtained from the communication layer and the infection density is obtained from thephysical contact layer [27–29]. In all the studies, the individuals receive information on one layer and transmit diseaseon the other layer.In the real-world, people usually receive information from multiple source and the mutual confirmation of theinformation is likely to strengthen their willingness to accept the information and adopt responding activities. Forexample, before a customer makes a decision to purchase, he or she may get information from websites, social media,digital advertisements, catalogs, mobile and face-to-face recommendation. A consistent and positive evaluation of thefocused product from multiple channels is probably to facilitate the customer’s decision to purchase. In multilayernetworks, the spreading processes on each layer are dynamically dependent on each other and their interactions areeither interdependent or competitive [30], which impact the activity of individual in di ff erent ways. In the information-disease coupling spreading dynamics, an individual can obtain the information on wether his neighboring nodes havebeen infected, either through the neighbors on the physical contact layer, e.g., from the neighbor’s daily symptomssuch as cough, fever, di ffi culty breathing, or through the neighbors on the communication layer, e. g., from theneighbor’s blog contents or online shared pictures. These mutual confirmation on the infection of disease is likely toenhance the individual to accept and transmit the information about the epidemics. On the other hand, an individualcan get to know wether his neighbors are aware of the disease either through the contact layer, from the neighbors’actions such as wearing mask, frequently cleaning hands or using hand sanitizers, intentionally maintaining at least onemeter distance from others [31], or through the communication layer, e.g., the neighbor’s attitude toward disease andsafeguard in communications. These mutual confirmation of the information on disease may enhance the individual’swillingness to adopt higher level of preventive measures, and thus further suppress the di ff usion of disease.In this manuscript, we introduce the mutual confirmation mechanism into the asymmetrically interacting spreadingprocess on multiplex networks. By using the microscopic Markov chain approach (MMCA), we are able to capture thekey quantities in epidemic spreading such as the epidemic threshold and the infection density in stationary state. Therest of the paper is organized as follows. In Section 2, the model with mutual confirmation mechanism is introduced.In Section 3, we use the Markov chain approach to analyze the epidemic threshold. In Section 4, we compare theMMCA predictions with the numerical simulation results and analyze the parameters. Finally in Section 5, we makeconclusions.
2. Model
The information-disease interacting model is implemented on a multiplex network, as illustrated in Fig. 1. Thenetwork consists of a virtual communication layer labeled as layer A and a physical contact layer labeled as layerB. The information spreading process is an unaware-aware-unaware (UAU) dynamics, where a node can be in eitherthe unaware (U) state or aware state (A). The unaware individuals become aware once they have communicated withthe aware individuals, and the aware individuals return to unaware state if they have forgotten or lost confidence on2 igure 1: The asymmetrically interacting spreading model with mutual confirmation. (a) The interacting spreading dynamics on a multiplexnetwork. Layer A corresponds to the virtual communication layer where the information on disease spreads, which is a UAU dynamics. Layer Bcorresponds to the physical contact layer where disease transmits, which is an SIS dynamics. An infected node in layer B is automatically awareof the disease and its counterpart in layer A is in aware state. An aware node in layer A will increase its preventive measure against disease and itscounterpart in layer B has a reduced rate of being infected as β A = γβ U . In the multiplex network, the nodes can be in one of the three states: US,AS and AI. Schematic diagram for calculating the information receiving rate λ mi (b) and the attenuation factor γ mi (c) under mutual confirmationmechanism. Solid and dotted lines represent connections in layer A and B respectively. For a U node noted light blue in (b), its proportion ofinfected neighbors is 2 / / ω Ai = / ω Bi = /
2. For an S node noted dark blue in (c), its proportion ofaware neighbors is 1 in layer A and 1 / ν Ai = ν Bi = / the news. The transmission rate of information is λ and the recovery rate is δ . In the physical contact layer, theclassical susceptible-infected-susceptible (SIS) model is used, where an infected node propagates disease to each ofits susceptible neighbor at disease transmission rate β , and infected nodes recover with rate µ . An infected node inlayer B is automatically aware of the disease and thus is in state A in layer A. An aware node is probably to adoptsome preventive measures against disease, thus its infected probability is reduced. We use β A and β U to representthe disease transmission rate for nodes with and without awareness respectively, where β A = γβ U and β U = β . Theattenuation factor γ represents the extent of reduction of disease transmission rate. So the nodes in the multiplexnetwork can be in one of the three states: unaware and susceptible (US), aware and susceptible (AS) and aware andinfected (AI). In the mutual confirmation mechanism, an individual can obtain the infection status of its neighbors, either fromthe communication layer or contact layer. A large proportion of infected neighbors from both layers mutually confirmthat the disease is spreading and dangerous, which increases people’s belief on the information and the informationtransmission rate is thus likely to be enhanced. Here we define the information receiving rate of node i in the mutualconfirmation mechanism as λ mi = λ (1 + θ A ω Ai )(1 + θ B ω Bi ) , (1)where λ is the basic information transmission rate, ω Ai = P j a ij I j k Ai is the proportion of infected neighbors in communica-tion layer and ω Bi = P j b ij I j k Bi is the proportion of infected neighbors in contact layer. I j = j is in infected states,otherwise I j =
0. The two parameters θ A and θ B quantify the confirmation strength of ω Ai and ω Bi in promoting theinformation transmission rate. Meanwhile, an individual is able to obtain the awareness status of his neighbors from3oth the communication layer and the contact layer. A large proportion of aware neighbors from both layers mutuallyconfirm that the information about disease are widely accepted, which enhances the individuals’ willingness to adopthigher level of preventive measures, therefore reduces the transmission rate of disease. Here we define the attenuationfactor for disease transmission rate of node i as γ mi = γ (1 − α A ν Ai )(1 − α B ν Bi ) , (2)where γ is the basic attenuation factor, ν Ai = P j a ij A j k Ai is the proportion of aware neighbors in communication layer and ν Bi = P j b ij A j k Bi is the proportion of aware neighbors in contact layer. A j = j is in aware status, otherwise A j = α A and α B quantify the confirmation strength of ν Ai and ν Bi in reducing the disease transmissionrate. When the four parameters θ A = θ B = α A = α B =
0, then λ mi = λ , γ mi = γ , and the model reduces to the classicalmodel without mutual confirmation mechanism. Symbols used in this paper are list in Table 1. Table 1: Symbols used in the paper.Symbol Description A ( a ij ) adjacent matrix for communication layer A (element in matrix A ) B ( b ij ) adjacent matrix for contact layer B (element in matrix B ) k Ai ( k Bi ) degree of node i in layer A(B) β basic disease transmission rate µ disease recovery rate λ basic information transmission rate δ information recovery rate λ mi information receiving rate for node i under mutual confirmation β U disease transmission rate for unaware node, β U = ββ A disease transmission rate for aware node γ basic attenuation factor, β A = γβ U γ mi attenuation factor for node i under mutual confirmation ω Ai ( ω Bi ) proportion of infected neighbors in layer A(B) for node i ν Ai ( ν Bi ) proportion of aware neighbors in layer A(B) for node i θ A ( θ B ) confirmation strength of ω Ai ( ω Bi ) in λ mi α A ( α B ) confirmation strength of ν Ai ( ν Bi ) in γ mi p Aj ( t ) probability of node j in A state at time tr i ( t ) probability of node i not being informed by any neighbor at time tq Ui ( t ) probability of a U node not being infected by any neighbor at time tq Ai ( t ) probability of an A node not being infected by any neighbor at time t ρ I infection density in the stationary state ρ A awareness density in the stationary state
3. Microscopic Markov chain approach
We use the microscopic Markov chain approach (MMCA) [13] to describe the information-disease coupled spread-ing process in the proposed model. We denote the probability that node i is in one of the three states US, AS and AIat time t as p USi ( t ), p ASi ( t ) and p AIi ( t ) respectively. On the communication layer, the probability that a node is notinformed by any neighbor at time t is r i ( t ) = Π [1 − a i j p Aj ( t ) λ mi ] , (3)where λ mi = λ (1 + θ A ω Ai )(1 + θ B ω Bi ) is the information receiving rate of node i under mutual confirmation mechanism.On the contact layer, the probability of an unaware node or an aware node not being infected by any neighbor at time4 igure 2: Transition probability tree for the node states. There are three possible states for a node, US (unaware and susceptible), AS (aware andsusceptible) and AI (aware and infected). t are respectively q Ui ( t ) = Π [1 − b i j p AIj β U ] (4)and q Ai ( t ) = Π [1 − b i j p AIj β A ] , (5)where β U and β A = γ mi β U are the disease transmission rates for U and A nodes respectively, and γ mi = γ (1 − αν Ai )(1 − α B ν Bi ) is the attenuation factor for disease transmission rate under the mutual conformation mechanism. The transitionprobability tree for node states is demonstrated in Fig. 2. Then the evolution equations of the three states in the coupledspreading dynamics can be written as p USi ( t + = p AIi ( t ) δµ + p USi r i ( t ) q Ui ( t ) + p ASi ( t ) δ q Ui ( t ) , (6) p ASi ( t + = p AIi ( t )(1 − δ ) µ + p USi ( t )[1 − r i ( t )] q Ai ( t ) + p ASi ( t )(1 − δ ) q Ai ( t ) (7)and p AIi ( t + = p AIi ( t )(1 − µ ) + p USi { [1 − r i ( t )][1 − q Ai ( t )] + r i ( t )[1 − q Ui ( t )] } (8) + p ASi ( t ) { δ [1 − q Ui ( t )] + (1 − δ )[1 − q Ai ( t )] } . To obtain the epidemic threshold, we use the stationary solutions of the system by letting t → ∞ in Eqs. (6)-(8), whichsatisfy p AIi ( t + = p AIi ( t ) = p AIi , (9) p ASi ( t + = p ASi ( t ) = p ASi , (10) p USi ( t + = p USi ( t ) = p USi . (11)Around the epidemic threshold, the number of infected nodes is negligible compared to the total population, thus theprobability of nodes being infected can be assumed to be p AIi = ǫ i ≪
1. Eqs. (4) and (5) can be approximated as q Ui ( t ) = − β U X j b i j ǫ j , (12)and q Ai ( t ) = − β A X j b i j ǫ j . (13)Using the above conditions, we can simplify the stationary solution equations of Eqs. (6-8) as p USi = p USi r i + p ASi δ, (14)5 ASi = p USi [1 − r i ] + p ASi (1 − δ ) , (15) ǫ i = ǫ i (1 − µ ) + ( p ASi β A + p USi β U ) X j ǫ j b i j . (16)By substituting β A = γ mi β U into Eq. (16), we obtain µǫ i = β U ( p ASi γ mi + p USi ) X j ǫ j b i j . (17)Near the threshold p AI = ǫ i ≪
1, then p Ai = p AIi + p ASi ≈ p ASi . (18)In addition, there is US node in the network, but no UI node, so p USi = p Ui . As p AIi + p ASi + p USi =
1, then p Ui = − p Ai .Eq. (17) can be rewritten as µǫ i = β U ( γ ′ i p Ai + p Ui ) X j ǫ j b i j . (19)The Eq. (19) can be written in the format of matrix as X j { ( p Ui + γ ′ i p Ai ) b i j − µβ U σ i j } ǫ j = , (20)where σ i j is the element of identity matrix. When epidemic starts to pervade the network in stationary state, theepidemic threshold β c is the minimum value that satisfies Eq. (17). As a self-consistent equation, obtaining β c reducesto the eigenvalue problem [20]. Let Λ max ( H ) be the largest eigenvalue of matrix H , where the elements of H is h i j = ( p Ui + γ mi p Ai ) b i j = { [ γ (1 − α A ν Ai )(1 − α B ν Bi ) − p Ai + } b i j . (21)Then the epidemic threshold can be written as β c = µ Λ max ( H ) . (22)In calculating h i j , we iterate Eqs. (6) and (7) to get p Ai in the stationary state, and ν Ai = P j a ij p Aj k Ai , ν Bi = P j b ij p Aj k Bi .
4. Numerical simulations
To validate our MMCA method, we carry out extensive numerical simulations and find a good agreement betweenthe analytical and numerical results. In simulations, the uncorrelated configuration model (UCM) is used to generateeach layer of the multiplex network, which follows a power law degree distribution [32]. The number of nodes is setto be N = c = .
5, the minimal and maximal degree are k min = k max = N − ( c − respectively. The nodes of two layers are randomly connected thus each node in one layer has a counterpart in theother layer. Initially, 10% nodes are randomly selected as infected seeds and the remaining nodes are susceptible.Fig. 3 demonstrates the density of aware nodes ρ A and infected nodes ρ I in the stationary state as a function ofbasic disease transmission rate β under di ff erent parameters, where ρ I = N P i p AIi and ρ A = N P i ( p AIi + p ASi ). InFig. 3 (a) and (b), three cases are compared, which are (1) information-disease interacting dynamics with no mutualconfirmation, corresponding to θ A = θ B = α A = α B =
0, (2) only the infection of neighbors are mutually confirmedto increase the information receiving rate, corresponding to θ A = θ B = α A = α B =
0, and (3) only the awarenessof neighbors are mutually confirmed to reduce the attenuation factor of disease transmission rate, corresponding to θ A = θ B = α A = α B =
1. It can be seen that the when β is smaller than 0 .
6, the awareness density ρ A is thehighest when there is mutual confirmation from infected neighbors. This is because the information transmission rateis directly enhanced due to the confirmation from infected neighbors. The awareness density is the lowest when there6 A (a) A = B = A = B =0 A = B =1, A = B =0 A = B =0, A = B =1 I (b) A = B = A = B =0 A = B =1, A = B =0 A = B =0, A = B =1 A (c) A = B = A = B =0 A = A =1, B = B =0 A = A =0, B = B =1 I (d) A = B = A = B =0 A = A =1, B = B =0 A = A =0, B = B =1 Figure 3: The awareness density and the infection density obtained by the MMCA (lines) and Monte Carlo simulations (shapes) as a function ofbasic disease transmission rate β . (a) The awareness density under classical model(black circle), confirmation from infected neighbors (red triangle)and confirmation from aware neighbors (blue square). (b) The infection density under classical model, confirmation from infected neighbors andconfirmation from aware neighbors. (c) The awareness density under classical model (black circle), confirmation from layer A (red triangle) andconfirmation from layer B (blue square). (d) The infection density under classical model, confirmation from layer A and confirmation from layerB. Other dynamic parameters are set as λ = . δ = . γ = . µ = .
6. The simulation results are obtained by averaging over 100 independentruns. is mutual confirmation from the aware neighbors. This is because the disease transmission rate is reduced in this case,and the number of aware nodes informed by their infected counterparts decreases correspondingly. When β is greaterthan 0 .
6, the ρ A corresponding to α A = α B = ω Ai and ω Bi become large and thus significantlyenhance the information receiving rate of nodes, as defined in Eq. (1). As for the infection density ρ I , it can beseen that for both cases with mutual confirmation, the infection is suppressed compared with the case with no mutualconfirmation. The infection density is the lowest when the aware neighbors are mutually confirmed, corresponding tothe case of θ A = θ B = α A = α B =
1. This is because the disease transmission rate is reduced directly. When theinfected neighbors are mutually confirmed, the information receiving rate of nodes increases, resulting in the increaseof aware nodes in layer A and decrease of infection in layer B. So ρ I for the case of θ A = θ B = α A = α B = ff ects of confirmation from layer A an layer B are demonstrated. It can be seen thatneighbors of layer A (corresponding to θ A = α A = θ B = α B =
0) or layer B (corresponding to θ A = α A = θ B = α B =
1) have a similar impact on ρ A , which is very close to that of the classical model when β is small and abit higher than that of the classical model when β is greater than 0 .
6. As for ρ I , the e ff ects of confirmation from layerA or layer B are very obvious, leading to a reduced infection density than the classical model. This is because thedisease transmission rate is reduced and information transmission rate is enhanced at the same time, which togethersuppress the disease transmission. This result implies that if the neighbors in either layer can explicitly transmit theirattitude and infection status, it will help to reduce the infection of epidemic disease. In the later part, we will discussthe impact of confirmation from layer A and layer B in detail. ff ect of confirmation mechanism on the epidemic threshold According to the MMCA method, the epidemic threshold is strongly dependent on the matrix H , whose elementsare h i j = [ p Ui + γ (1 − α A ν Ai )(1 − α B ν Bi ) p Ai ] b i j . Here the ν Ai and ν Bi represent the proportion of aware neighbors inlayer A and B respectively, and the parameter α A and α B represent the strength of confirmation from neighbors. If theinformation spreading does not outbreak when the information transmission rate λ < λ c , then ν Ai = ν Bi = α A and α B will have no impact on the epidemic threshold. Only when the information spreading outbreakswhen λ > λ c , the information spreading can suppress the disease spreading [19]. In this case, ν Ai , ν Bi , F c (a) A (MC) A (MMAC) B (MC) B (MMAC) F < i m > (b) AB Figure 4: The impact of information confirmation on the epidemic threshold. (a) The epidemic threshold as a function of confirmation strength α F . F can be either A or B . The result obtained by the MMCA (lines) and Monte Carlo (MC) simulations of susceptibility method (shapes) aredemonstrated. (b) The average attenuation factor for disease transmission rate as a function of confirmation strength α F . Other dynamic parametersare set as: λ = . δ = . γ = . µ = . α A and α B work. On the other hand, as the infected individuals are very few near the epidemic threshold, ω Ai ≈ ω Bi ≈
0. Then as described in Eq.(1), the two parameters θ A and θ B representing the confirmation strength of ω Ai and ω Bi have no impact on λ mi and the epidemic threshold. So we focus on the impact of α A and α B on the epidemicthreshold β c . Fig. 4 shows the epidemic threshold as a function of α . The threshold obtained by our MMCA methodand numerical simulations are compared. The simulated epidemic threshold is obtained by using a method calledsusceptibility [33, 34], where a χ is defined as χ = N h ρ i − h ρ i h ρ i . (23)In the equation, ρ is the infection density in the stationary state in one run, and h ρ i is the average over several runs. Weimplement 100 runs under each of the infection rate to get a χ , and the infection rate that corresponds to the largest χ is identified as the simulated epidemic threshold. As shown in Fig. 4, the epidemic threshold increases with α F , where F is either A or B. The increment of β c is larger when α A is applied than that of α B , which is due to the lower averageattenuation factor for disease transmission h γ mi i of α A as shown in Fig. 4 (b).The reason why the average γ mi of α A is lower than that of α B can be explained by comparing the size of the averageproportion of aware neighbors h v Ai i in layer A and h v Bi i in layer B . According to Eq.(2), γ mi = γ (1 − α A ν Ai )(1 − α B ν Bi ).It can be seen from Fig. 5 (a) that around the epidemic threshold β c , h v Ai i > h v Bi i . Thus at a fixed value of α A or α B , h γ mi i is smaller when α A is applied than that of α B . When the disease transmission rate is under or around the epidemicthreshold, the epidemic does not breakout and there are very few infected nodes. The aware nodes are mostly generatedby the information spreading on layer A. As the spreading is from an aware node to its neighbors, the newly generatedaware nodes are locally around the previous aware nodes. Thus h v Ai i is relatively high for nodes with aware neighbors.While for nodes in layer B, as there are few infected nodes, their counterparts as aware nodes are correspondinglyfew. Meanwhile the aware nodes in layer A, if mapping to layer B, are randomly distributed in layer B because thetwo layers have no overlapping edges. Thus h v Bi i is relatively small. When the disease transmission rate is above theepidemic threshold, the disease outbreaks and there are more and more infected nodes. In such stage, the proportionof aware neighbors for nodes in layer B exceeds that of nodes in layer A. This is because the nodes which have apossibility of being infected in layer B has at least one infected neighbors and thus one aware neighbors. For nodes inlayer A, there is no such constraints on neighbors. So h v Bi i is a bit greater than h v Ai i when β becomes larger. The crosspoint at which h v Bi i exceeds h v Ai i is larger than β c . This means the local e ff ect of information spreading dominates atfirst and then the coevolution of information-spreading and disease-spreading makes the relative proportion of awareneighbors in the two layers changes. 8 < i * > (a) AB < i * > (b) Figure 5: The average proportion of aware or infected neighbors as as a function of basic disease transmission rate. (a) The average proportion ofaware neighbors in layer A and B. (b) The average proportion of infected neighbors in layer A and B. In calculating h γ mi i , we only take the ASnodes with at least one I neighbor into consideration, because only these nodes will be a ff ected by γ mi . Similarly, we only take U nodes with at leastone A neighbor into consideration when calculating h λ mi i , because only these nodes will be a ff ected by λ mi . The cross point at which h v Bi i exceeds h v Ai i is larger than β c , which is the result of local e ff ects in both information-spreading and disease-spreading. Other parameters are set as λ = . δ = . γ = . µ = . θ A = θ B = α A = α B =
0. When any of the parameters θ A , θ B , α A , α B is non-zero, λ mi increases or γ mi decreases, which donot change the local e ff ect in spreading. Similarly, we plot the average proportion of infected neighbors h ω Ai i in layer A and h ω Bi i in layer B as a functionof β respectively. As can be seen in Fig.5 (b) that h ω Bi i is greater than h ω Ai i under all disease transmission rate.This is because the infection spreads on layer B. As an infected node infects its neighbors, the newly infected nodesare locally around the infecting nodes. The infected nodes are locally clustered, resulting in a relatively large h ω Bi i .Mapping these infected nodes to their counterparts in layer A, the infected nodes are scattered randomly. Thus h ω Ai i is relatively small.These results imply that when the information on disease spreads only by relationships, it is local at the initialstage, which is not beneficial for suppressing the disease spreading. If at this stage the public health authorities canpropagate the information on disease through mass media, it will help suppress the spreading of epidemics.In Fig. 6 we compare the theoretical and numerical values of the density of aware individuals ρ A and infectedindividuals ρ I under di ff erent β and λ . It can be seen from Fig. 6 (a) and (b) that, with the increase of β or λ , ρ A increases. In Fig. 6 (c) and (d), ρ I increases with β and decreases with λ . When λ < λ c , the information spreadingdoes not break out thus has no e ff ect on disease spreading. This corresponds to the plateau of β c marked in Fig. 6 (c)and (d). When λ > λ c , the epidemic threshold β c increases with λ . This is because in h i j , ν Ai and ν Bi represent theproportion of aware neighbors. Increasing the information transmission rate λ can increase the value of ν Ai and ν Bi ,thus decrease the attenuation factor of disease transmission rate γ mi and increases the epidemic threshold. ff ect of di ff erent layers on suppressing the disease spreading In the proposed mutual confirmation mechanism, the infected and aware neighbors of communication layer andcontact layer can a ff ect the information receiving rate and disease transmission rate of nodes. Then the confirmationfrom which layer suppresses the disease spreading and promotes the information spreading more e ff ectively is thequestion we are interested in. Fig. 7 demonstrates the ρ A and ρ I in the stationary state when the confirmation fromlayer A or layer B is applied. It can be seen from Fig.7 (a) that when only the confirmation from neighbors in layerB is applied, the density of aware individuals is larger than that of layer A. This is because the average informationtransmission rate h λ mi i is higher when the confirmation from layer B is applied than that when only confirmation fromlayer A is applied, as can be seen in Fig.7 (c). The higher h λ mi i when the confirmation from layer B is applied is dueto the higher h ω Bi i as demonstrated in Fig.5 (b), which is a result of the locally clustered newly infected nodes in thedisease-spreading process. 9 igure 6: The density of aware individuals ρ A and infected individuals ρ I obtained from Monte Carlo simulations and MMCA. (a) Simulateddensity of aware individuals. (b) Theoretical density of aware individuals. (c) Simulated density of infected individuals. (d) Theoretical density ofinfected individuals. In Monte Carlo simulations, each result is obtained by averaging over 100 runs. Other parameters are set as δ = . γ = . µ = . θ A = θ B = α A = α B = . A (a) A = A = B = B = I (b) < i m > (c) < i m > (d) Figure 7: Comparison of confirmation e ff ect from layer A and layer B. (a) E ff ect of confirmation on ρ A . (b) E ff ect of confirmation on ρ I . (c) E ff ectof confirmation on average information transmission rate h λ mi i . (d) E ff ect of confirmation on average attenuation factor h γ mi i . In calculating h γ mi i ,we only take the AS nodes with at least one I neighbor into consideration, because only these nodes will be a ff ected by γ mi . Similarly, we only takeU nodes with at least one A neighbor into consideration when calculating h λ mi i , because only these nodes will be a ff ected by λ mi . Other parametersare set as δ = . δ = . β = . µ = . γ = . h γ mi i when the confirmation from layer B is applied is smaller than that of layer A, as can be seen from Fig.7 (d). So thelower ρ I of confirmation from layer B may due to the smaller γ mi and the wider spreading of awareness ρ A as shownin (a).The above results imply that the confirmation from contact layer can promote the di ff usion of awareness morewidely and suppress the disease more deeply than that of the communication layer. In real-world scenarios, for thepublic healthy authorities to control the di ff usion of disease, let the individuals timely and accurately reveal theirinfected status and aware status, especially for those having real contacts in daily life, is e ff ective to further suppressthe epidemics. In this part, we focus on the impact of each confirmation information, which are the ν Ai , ν Bi , ω Ai and ω Bi , onpromoting information di ff usion and suppressing infection. To do this, we vary the confirmation strength of eachconfirmation information, which are θ A , θ B , α A and α B respectively, and investigate the density of aware individuals ρ A and infected individuals ρ I . From Fig.8 (a) it can be seen that ρ A is the highest when θ B , the confirmation strengthof the proportion of infected neighbors ω Bi in layer B, is introduced. While θ A is introduced, which is the confirmationstrength of the proportion of infected neighbors ω Ai in layer A , ρ A is lower than that of θ B . This is because the averageinformation transmission rate h λ mi i is higher when θ B is applied than that of θ A as shown in Fig. 8 (c). The higher h λ mi i of θ B is due to the larger ω Bi shown in Fig.5 (b), which is the result of local e ff ect in disease spreading process.When α A and α B are applied respectively, ρ A is the lowest. This is because the non-zero α A and α B in the attenuationfactor for disease transmission γ mi reduce the disease transmission rate directly, thus leading to a decreased number ofinfected individuals as well as their counterparts as aware nodes.In Fig. 8 (b) when α A and α B are applied respectively, ρ I is lower than that of θ A and θ B . This is because the diseasetransmission rate is directly reduced when α A or α B is applied. The average attenuation factor for disease transmission h γ mi i of α B is lower than that of α A as shown in Fig. 8 (d), which leads to the lower ρ I of α B . The smaller h γ mi i when α B is introduced is the result of larger proportion of aware neighbors h ν Bi i in layer B as shown in Fig.5 (a) when thebasic disease transmission rate is far above the threshold, which is the result of the local e ff ect in disease spreadingprocess. The reason ρ I of θ A is higher than that of θ B is that θ A results in a lower ρ A than θ B , thus the suppressinge ff ect is smaller than that of θ B .The above results indicate that in suppressing disease spreading, the rank for the e ff ect of confirmation fromneighbors is α B > α A > θ B > θ A . This implies that directly reducing the disease transmission rate, other thanpromoting awareness spreading, can suppress the disease spreading more e ff ectively. In addition, the local e ff ect indisease spreading makes the average proportion of infected neighbors in the physical contact layer larger than that ofcommunication layer, which leads to the impact of θ B > θ A .
5. Conclusion and Discussion
In the early stage of epidemics, people are not certain about the credibility of information, thus may seek forconfirmation from multiple sources. The infection of neighbors from both the communication layer and the con-tact layer will convince people the prevalence of disease, thus making the transmission of information on epidemicsmore easily. Meanwhile if there is a large proportion of acquaintances that have accepted the information, people aremore likely to adopt protective measures to reduce the risk of infection. Based on this assumption, we articulate theinformation-disease interacting spreading model with inter-layer mutual confirmation mechanism on multiplex net-works. By using the microscopic Markov chain method, we analytically predict the epidemic threshold and infectiondensity in stationary state, which agree well with the simulation results.We find that the confirmation from aware neighbors in layer A can enhance the epidemic threshold more than theconfirmation from aware neighbors in layer B. This is because around the epidemic threshold there are few infectednodes and the spreading of information dominates the di ff usion of awareness. The local e ff ect of information spreadingmakes the proportion of aware neighbors in layer A larger than that of layer B. On the other hand, the infectedneighbors of either layer A or B has no impact on the epidemic threshold, because around the epidemic threshold thenumber of infected neighbors is approaching zero. 11 A (a) ABAB I (b) < i m > (c) < i m > (d) Figure 8: E ff ect of the confirmation information. (a) Comparison of ρ A for each of the four confirmation information. (b) Comparison of ρ I foreach of the four confirmation information. When we study on one confirmation strength, other three confirmation strength parameters are set to be0. E.g., in subgraph (a) the curve with blue triangle represents ρ A as a function of θ A , while other three confirmation strength parameters are set tobe θ B = α A = α B =
0. (c) Comparison of average information transmission rate λ mi for θ . (d) Comparison of average attenuation factor γ mi for α .In calculating h γ mi i , we only take the AS nodes with at least one I neighbor into consideration, because only these nodes will be a ff ected by γ mi .Similarly, we only take U nodes with at least one A neighbor into consideration when calculating h λ mi i , because only these nodes will be a ff ectedby λ mi . Other parameters are set as δ = . δ = . β = . µ = . γ = . As for the e ff ect of suppressing disease, when independently apply one of the four confirmation information, thereduction of ρ I can be ordered as ∆ α B > ∆ α A > ∆ θ B > ∆ θ A . There are two reasons for this ranking. Firstly, reducingthe disease transmission rate directly suppresses the epidemic more than enhancing the awareness spreading does.Secondly, the local e ff ect in disease spreading makes the proportion of infected neighbors in layer B, as well as theproportion of aware neighbors of layer B, larger than that of layer A, thus leading to the larger inhibitory e ff ect ofconfirmation from layer B. From the perspective of layers, confirmation from the aware and infected neighbors incontact layer can result in a lower infection density in the stationary state than that of the confirmation from thecommunication layer, which is also due to the local e ff ect in disease-spreading and information-spreading.The results in this work imply that when epidemic outbreaks, encouraging people to explicitly express their in-fected status and aware attitude is helpful to reduce the infection in the whole population. While the spreading ofdisease and information has a local e ff ect, the heath authority’s announcement of the information on epidemics to thepublic is important to suppress the disease. When the anti-epidemic resource is limited, investing the limited resourceinto real-world activities, such as distributing protective goods or medicines, cutting o ff disease transmission path, ismore e ff ective than propagating online. Acknowledgement
This work is supported by the National Natural Science Foundation of China (No. 61802321, 11975099), theSichuan Science and Technology Program (No. 2020YJ0125), the Natural Science Foundation of Shanghai (No.18ZR1412200) and the Science and Technology Commission of Shanghai Municipality (Grant No. 14DZ2260800).
ReferencesReferences [1] World health organization website (September 2020). URL https: // / [2] J. Zhang, M. Litvinova, Y. Liang, Y. Wang, W. Wang, S. Zhao, Q. Wu, S. Merler, C. Viboud, A. Vespignani, et al.,Changes in contactpatterns shape the dynamics of the covid-19 outbreak in china, Science 368 (6498) (2020) 1481-1486.
3] M. U. Kraemer, C.-H. Yang, B. Gutierrez, C.-H. Wu, B. Klein, D. M. Pigott, L. Du Plessis, N. R. Faria, R. Li, W. P. Hanage, et al., Thee ff ect of human mobility and control measures on the covid-19 epidemic in china, Science 368 (6490) (2020) 493-497.[4] S. M. Kissler, C. Tedijanto, E. Goldstein, Y. H. Grad, M. Lipsitch, Projecting the transmission dynamics of sarscov-2 through the postpan-demic period, Science 368 (6493) (2020) 860-868.[5] Z.-M. Zhai, Y.-S. Long, M. Tang, Z. Liu, Y.-C. Lai, When did covid-19 start? -optimal inference of time zero, Research Square (2020) 1-14.[6] M. E. J. Newman, The structure and function of complex networks, SIAM Review 45 (2) (2003) 167-256.[7] R. M. Anderson, B. Anderson, R. M. May, Infectious diseases of humans: dynamics and control, Oxford university press, 1992.[8] L. Hufnagel, D. Brockmann, T. Geisel, Forecast and control of epidemics in a globalized world, Proceedings of the National Academy ofSciences 101 (42) (2004) 15124-15129.[9] C. Castellano, R. Pastor-Satorras, Thresholds for epidemic spreading in networks, Physical Review Letters 105 (21) (2010) 218701.[10] D. Brockmann, D. Helbing, The hidden geometry of complex, network-driven contagion phenomena, Science 342 (6164) (2013) 1337-1342.[11] R. Pastor-Satorras, C. Castellano, P. Van Mieghem, A. Vespignani, Epidemic processes in complex networks, Reviews of Modern Physics87 (3) (2015) 925.[12] S. Funk, E. Gilad, C. Watkins, V. A. Jansen, The spread of awareness and its impact on epidemic outbreaks, Proceedings of the NationalAcademy of Sciences 106 (16) (2009) 6872-6877.[13] C. Granell, S. Gomez, A. Arenas, Dynamical interplay between awareness and epidemic spreading in multiplex networks, Physical ReviewLetters 111 (12) (2013) 128701.[14] C. Granell, S. Gomez, A. Arenas, Competing spreading processes on multiplex networks: awareness and epidemics, Physical Review E 90(1) (2014) 012808.[15] M. De Domenico, C. Granell, M. A. Porter, A. Arenas, The physics of spreading processes in multilayer networks, Nature Physics 12 (10)(2016) 901-906.[16] M. Kivela, A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno, M. A. Porter, Multilayer networks, Journal of Complex Networks 2 (3)(2014) 203-271.[17] S. Boccaletti, G. Bianconi, R. Criado, C. D. Genio, J. G´omez-Garde˜nes, M. Romance, I. Sendi˜na-Nadal, Z. Wang, M. Zanin, The structureand dynamics of multilayer networks, Physics Reports 544 (1) (2014) 1-122.[18] G. F. de Arruda, F. A. Rodrigues, Y. Moreno, Fundamentals of spreading processes in single and multilayer complex networks, PhysicsReports 756 (2018) 1-59.[19] W. Wang, M. Tang, H. Yang, Y. Do, Y.-C. Lai, G. Lee, Asymmetrically interacting spreading dynamics on complex layered networks,Scientific Reports 4 (2014) 5097.[20] Q. Guo, X. Jiang, Y. Lei, M. Li, Y. Ma, Z. Zheng, Two-stage e ff ects of awareness cascade on epidemic spreading in multiplex networks,Physical Review E 91 (1) (2015) 012822.[21] V. Nicosia, P. S. Skardal, A. Arenas, V. Latora, Collective phenomena emerging from the interactions between dynamical processes inmultiplex networks, Physical Review Letters 118 (13) (2017) 138302.[22] A. Moinet, R. Pastor-Satorras, A. Barrat, E ff ect of risk perception on epidemic spreading in temporal networks, Physical Review E 97 (1)(2018) 012313.[23] H. Yang, C. Gu, M. Tang, S.-M. Cai, Y.-C. Lai, Suppression of epidemic spreading in time-varying multiplex networks, Applied Mathemat-ical Modelling 75 (2019) 806-818.[24] W. Wang, Q.-H. Liu, S.-M. Cai, M. Tang, L. A. Braunstein, H. E. Stanley, Suppressing disease spreading by using information di ff usion onmultiplex networks, Scientific Reports 6 (2016) 29259.[25] H. Wang, C. Chen, B. Qu, D. Li, S. Havlin, Epidemic mitigation via awareness propagation in communication networks: the role of timescales, New Journal of Physics 19 (7) (2017) 073039.[26] P. C. V. da Silva, F. Velasquez-Rojas, C. Connaughton, F. Vazquez, Y. Moreno, F. A. Rodrigues, Epidemic spreading with awareness anddi ff erent timescales in multiplex networks, Physical Review E 100 (3) (2019) 032313.[27] H.-F. Zhang, J.-R. Xie, M. Tang, Y.-C. Lai. Suppression of epidemic spreading in complex networks by local information based behavioralresponses, Chaos: An Interdisciplinary Journal of Nonlinear Science 24 (4) (2014) 043106.[28] Y. Pan, Z. Yan, The impact of individual heterogeneity on the coupled awareness-epidemic dynamics in multiplex networks, Chaos: AnInterdisciplinary Journal of Nonlinear Science 28(6) (2018) 063123[29] V. Sagar, Y. Zhao, A. Sen, E ff ect of time varying transmission rates on the coupled dynamics of epidemic and awareness over a multiplexnetwork, Chaos: An Interdisciplinary Journal of Nonlinear Science 28 (2018) 113125.[30] M. M. Danziger, I. Bonamassa, S. Boccaletti, S. Havlin, Dynamic interdependence and competition in multilayer networks, Nature Physics15 (2) (2019) 178-185.[31] Coronavirus disease (covid-19) advice for the public from the who website (September 2020). URLhttps: // / emergencies / diseases / novel-coronavirus-2019 / advice-for-public[32] M. E. Newman, S. H. Strogatz, D. J. Watts, Random graphs with arbitrary degree distributions and their applications, Physical Review E 64(2) (2001) 026118.[33] S. C. Ferreira, C. Castellano, R. Pastor-Satorras, Epidemic thresholds of the susceptible-infected-susceptible model on networks: A com-parison of numerical and theoretical results, Physical Review E 86 (4) (2012) 041125.[34] P. Shu, W. Wang, M. Tang, Y. Do, Numerical identification of epidemic thresholds for susceptible-infectedrecovered model on finite-sizenetworks, Chaos: An Interdisciplinary Journal of Nonlinear Science 25 (6) (2015) 063104.advice-for-public[32] M. E. Newman, S. H. Strogatz, D. J. Watts, Random graphs with arbitrary degree distributions and their applications, Physical Review E 64(2) (2001) 026118.[33] S. C. Ferreira, C. Castellano, R. Pastor-Satorras, Epidemic thresholds of the susceptible-infected-susceptible model on networks: A com-parison of numerical and theoretical results, Physical Review E 86 (4) (2012) 041125.[34] P. Shu, W. Wang, M. Tang, Y. Do, Numerical identification of epidemic thresholds for susceptible-infectedrecovered model on finite-sizenetworks, Chaos: An Interdisciplinary Journal of Nonlinear Science 25 (6) (2015) 063104.