Modeling disease spreading with adaptive behavior considering local and global information dissemination
MModeling disease spreading with adaptive behavior considering local and globalinformation disseminationXinwu Qian, Ph.D.
Lyles School of Civil Engineering, Purdue University550 Stadium Mall Dr, West Lafayette, IN, [email protected]
Jiawei Xue
Lyles School of Civil Engineering, Purdue University550 Stadium Mall Dr, West Lafayette, IN, [email protected]
Satish V. Ukkusuri, Ph.D.
Lyles School of Civil Engineering, Purdue University550 Stadium Mall Dr, West Lafayette, IN, [email protected](Corresponding Author)Word Count: 5870 words + 8 figures × ×
250 = 6120 wordsSubmission Date: August 26, 2020 a r X i v : . [ phy s i c s . s o c - ph ] A ug ian, Xue, Ukkusuri 2 ABSTRACT
The study proposes a modeling framework for investigating the disease dynamics with adaptivehuman behavior during a disease outbreak, considering the impacts of both local observationsand global information. One important application scenario is that commuters may adjust theirbehavior upon observing the symptoms and countermeasures from their physical contacts duringtravel, thus altering the trajectories of a disease outbreak. We introduce the heterogeneous mean-field (HMF) approach in a multiplex network setting to jointly model the spreading dynamics ofthe infectious disease in the contact network and the dissemination dynamics of information in theobservation network. The disease spreading is captured using the classic susceptible-infectious-susceptible (SIS) process, while an SIS-alike process models the spread of awareness termed asunaware-aware-unaware (UAU). And the use of multiplex network helps capture the interplaybetween disease spreading and information dissemination, and how the dynamics of one may affectthe other. Theoretical analyses suggest that there are three potential equilibrium states, dependingon the percolation strength of diseases and information. The dissemination of information mayhelp shape herd immunity among the population, thus suppressing and eradicating the diseaseoutbreak. Finally, numerical experiments using the contact networks among metro travelers areprovided to shed light on the disease and information dynamics in the real-world scenarios andgain insights on the resilience of transportation system against the risk of infectious diseases.
Keywords : Disease spreading, information dissemination, multiplex network, adaptive behavior,transportation contact networkian, Xue, Ukkusuri 3
INTRODUCTION
People are now engaged in more intensive daily activities and exposed to massive informationfrom various sources than ever before. However, one negative consequence of intensive activitiesis the accelerated outbreaks of infectious diseases through frequent travels and the exchange ofgoods, which turns small-scale local transmissions in the past into large-scale global pandemicssuch as the SARS, H1N1, MERS and more recently the COVID-19. On the other hand, the ex-change of vast local and global information offers the public a distinct opportunity to track andmonitor the state of ongoing disease outbreaks. One particular example is the global outbreak ofthe COVID-19, where we have witnessed the number of infections growing at an unprecedentedpace, and there are now over 16.3 million confirmed infections worldwide(
1, 2 ). Meanwhile, nu-merous COVID-19 monitoring dashboards are gaining popularity among the public ( ) and theepidemic has been the focus of the state media and social media platforms. As intensive activitiesare deemed to facilitate the spread of infectious diseases with frequent encounters, the exposureto mass information, however, may contribute to reshaping the activity patterns and how travel-ers are getting into contact and eventually altering the fate of disease outbreaks. This motivatesus to investigate the disease dynamics in the contact network with adaptive travelers’ behavior,considering the co-evolution of disease and information dynamics.Numerous studies investigated disease percolation over networks with individuals as thenodes and the connections between them (e.g., physical encounters) as the edges, and detailed re-views of related works can be found in (
4, 5 ). One important threshold value for network epidemi-ology is the disease threshold λ c , whose value determines the stable state of a disease outbreak.Studies have shown that λ c is a function of the degree distribution of the network (
6, 7 ) and isinversely proportional to the eigenvalue of the adjacency matrix of the network (
8, 9 ), which mo-tivates subsequent studies to investigate disease dynamics under different network layouts. Boguaet al. ( ) modeled spreading diseases using susceptible-infectious-susceptible (SIS) model in un-correlated and correlated networks and established a concise analytical expression for λ c in theuncorrelated network and used the eigenvalue of the network matrix to describe λ c when networkis correlated. Dickison et al. ( ) divided individuals into two inter-connected network layers andfocused on disease spreading within and between the two layers using simulations. Sanz et al. ( )separated the dynamics of two diseases into two networks, investigated the correlated states be-tween the two network layers, and established the analytical expression for the disease threshold.In addition, disease spreading in two networks with partially overlapped nodes was researched byBuono et al. ( ).In general, the co-evolution of disease dynamics and information dynamics may be inves-tigated following the same inter-correlated multiplex network setting, as in the above-mentionedstudies. Several recent studies made the initial attempts to address this issue. For instance, Wang etal. modeled information and disease outbreak in communication and contact layers, respectively.They assumed that one would be vaccinated if and only if they are aware of the disease ( ), andthey later extended their work by introducing the analytical threshold for disease-infected neigh-borhoods and improving the original vaccination strategy ( ). Kabir et al ( ) classified the wholepopulation into six categories based on two awareness states (unaware, aware) and three infectionstates (susceptible, infected, recovered), and they introduced eight parameters to model the statetransition among the six categories. Similarly, Xia et al. modeled the state transitions between fivestates of awareness and disease infection ( ). They assumed that the infected but unaware statedid not exist, which contradicted the evidence of asymptomatic patients who were not aware of theian, Xue, Ukkusuri 4disease but was still infectious during the COVID-19 outbreak ( ). These studies established ageneral framework that is applicable for analyzing the interplay between disease and informationdynamics. Nevertheless, few studies consider the coexistence of both local and global informationdissemination, and global information such as mass media is often assumed to be independentof the network’s disease dynamics. In a real-world setting, individuals may obtain informationthrough their daily encounters (e.g., observing the symptoms and preventative measures from othertravelers) and the mass media. Moreover, the information released by the mass media should de-pend on the actual disease patterns that are observable from the population. Besides, these earlystudies only focus on the equilibrium states for the disease, namely the disease-free state and theendemic state, while ignoring the terminal state of information and how the state of informationmay drive the equilibrium state of the infectious diseases.In this study, we develop a multiplex disease model that is suitable for modeling the diseasedynamics with local observations and global information to bridge the gaps mentioned above. Themultiplex model consists of the disease layer with a susceptible-infected-susceptible process (SIS),the local observation layer with an unaware-aware process (UA), and a global information nodethat disseminates information to individual nodes. The strength of the information depends on thestates of the disease layer. We introduce the heterogeneous mean-field model (HMF) ( ) to capturethe collective disease and information dynamics of nodes with the same degree. By analyzing thetheoretical properties of the UA-SIS model, we identify three critical equilibrium states that mayemerge depending on the percolation strength of the disease and the information: the disease andawareness free equilibrium (DAFE), the disease-free equilibrium with awareness (DFE-A) and theendemic state. In particular, the DFE-A state corresponds to the case where the local information isstrong enough to shape a herd-immunity-alike pattern among the nodes and eventually suppressedthe spread of the diseases. Furthermore, we also demonstrate the disease and information dynamicsof the UA-SIS model in the realistic contact network built from mobility patterns of metro travelers.The rest of the study is organized as follows. Section two introduces mathematical no-tations. Section three describes the assumption and supporting evidence. Section four gives anoverview of the UA-SIS model. Section five presents the mathematical formulation of the UA-SISmodel, and section six delivers a theoretical analysis of the stable states and conditions for theUA-SIS model. Section seven shows the numerical experiments and insights obtained from themodel. Section eight concludes the study with a summary and key findings. NOTATION
The mathematical notation used in this study is summarized in Table 1.ian, Xue, Ukkusuri 5
TABLE 1 : Table of notation
Variables Descriptions k Nodes with degree kU k Percentage of nodes in state U with degree k . U=unaware. θ A , k Percentage of nodes in state A with degree k . A=aware. S k Percentage of nodes in state S with degree k . S=susceptible. θ I , k Percentage of nodes in state I with degree k . I=infectious. β U Transmission probability when a node is in U state. β A Transmission probability when a node is in A state. β A << β U . γ Recovery rate of the aware state, γ ≤ γ Recovery rate of the given disease, γ ≤ p Probability of a node changes from U to A by observing that one of itsneighbors is in A state. N ( i ) The set of neighbor nodes of node i . ASSUMPTION
The following assumptions are made to support the development of the UA-SIS model:1. It is assumed that nodes of the same degree have the same behavior. So that we can makeuse of the HMF model, also known as the degree-based mean-field model, to understandUA-SIS dynamics.2. The discussion is currently limited to HMF without degree correlation. Nevertheless,the framework discussed in this study can be easily adapted to account for correlateddegree sequences.3. It is considered that both UA and SIS processes have the same network structure forthe numerical experiments. This is to model the scenario where commuters obtain localinformation by observing the symptoms and preventative measures of their encountersduring travel. Various network structures in different layers will be considered in futurework.
UA-SIS MODEL
In the UA-SIS model, individuals are modeled as nodes, and their pairwise connections are cap-tured by the edges. The UA-SIS model is capable of capturing the dynamics of three processesthat take place simultaneously:1. The physical contagion process, where individuals get in contact with others and thedisease may spread upon a contagion.2. The observation process (local information), where individuals may observe the behav-ior of his/her neighbors, and accumulate knowledge of the diseases. Each valid obser-vation will strengthen his understanding and increase the likelihood of an individualmigrating from U state to A state. Meanwhile, individuals may move from A to U astime proceeds. This is known as the fading of memory.3. The information gathering and dissemination process (global information). Unlike manyother studies, where information dissemination is assumed to occur among neighborsonly, this study considers the existence of a central system that gathers the disease in-ian, Xue, Ukkusuri 6formation from the population and disseminates the compiled information back to thepopulation within reach.The interactions among these three processes can therefore be captured by a multiplex networkwith three levels, as shown in Figure 1.
B - Disease NetworkA - Observation Network(U) Unaware (A) Aware (S) Susceptible (I) Infectious
T1 T2
FIGURE 1 : Illustration of the UA-SIS modeling process. (In the figure, the system has two layers of networks: the disease network and the observation network. TheSIS process takes place on the disease network while the UA process is on the observation network. On theleft (time=T1), the system starts with one infectious node, and two out of four of its neighbors are aware ofhis illness. On the right (time=T2), the system evolves over time, and one more node turns into A state. Andthe neighbor node who is still in U state is infected by the infectious node.)
In Figure 1, there are two correlated networks that represent disease dynamics and ob-servation dynamics. In the observation network, nodes observe their neighbors and obtain localinformation about the disease information. If one of the neighbors is in A, then the node has acertain probability that it will evolve into A as well. As for the disease network, the classic SISprocess is considered where the node is either in susceptible or infectious state. A susceptible nodemay turn into an infectious one upon physical contact with another infectious neighbor. However,since the two networks are inter-correlated and having the same topology, a susceptible node maynot be infected by its infectious neighbors in the disease network if it is in the aware state in theobservation network. This illustrates why using the SIS model itself may overestimate the outbreakscale of the disease and the necessity for including the information layer (observation as the casein our study).Up to now, only local information dissemination upon physical contact is considered. How-ever, one major source for the general public to obtain information is through online resources suchas social media and news agencies. People may value this information differently, among whichian, Xue, Ukkusuri 7the most reliable source is the official data and news. However, it is in general difficult for anyofficial media to understand the whole picture of the disease pattern. Consequently, the otherimportant research question is to understand how different levels of information may affect thedisease spreading dynamics over the disease network. This process is described in Figure 2.
FIGURE 2 : Illustration of the UA-SIS information dissemination process. (During the process, the central system (node G in yellow) obtains the disease information from its neighborsin the disease layer at time T =
1, and the disseminates this information to the same set of nodes in theobservation layer. This in combination with the observed information turns one of the nodes from U state toA state.)
FORMULATION
The HMF method is used which assumes that nodes of same degree are homogeneous. Mathemat-ically, the interplay among the three layers can be written as following.
Observation layer
The observation layer describes the dynamics where each individual observes from his or herphysical contacts and accumulating awareness of the disease states: dU k dt = − [ − ( − p ) k Θ A ( t ) + λ g ( t ) + β U k Θ I ( t )] U k ( t )+ γ θ A , k ( t ) (1) d θ A , k dt =[ − ( − p ) k Θ A ( t ) + λ g ( t ) + β U k Θ I ( t )] U k ( t ) − γ θ A , k ( t ) (2)where Θ A ( t ) is the probability that an arbitrary neighbor of a node is in state A at time t . 1 − p denotes the probability that the node will remain in U , so that ( − p ) k Θ A ( t ) gives the probabilitythat node i will remain in U after observing all his neighbors in A . And λ ( g ) captures the impactian, Xue, Ukkusuri 8on the behavior due the information from central information node such as mass media, which willbe explained in details in the following sections. As a consequence, the first term in equation 1refers to the proportion of nodes of degree k that migrates from state U to state A . Such a migrationmay take place upon observing the neighbors in A state, if the node itself is infected (captured by β U k Θ I ( t ) ) or if the node is exposed to central information (e.g., the total number infections in thesystem) that motivates the change of behavior. And the second term describes the decreasing levelof awareness over time so that people move from A back to U at the rate of γ . Disease layer
In disease networks, individuals are considered to be in one of the two states: susceptible (S) andinfectious (I). In particular, for individual in S, the chance of being infected depends on if they arein A or U states, with different transmission coefficient β A and β U respectively. And the contagiondynamics between S and I states can be mathematically expressed as: dS k dt = − β A k Θ I ( t ) S k ( t ) θ A , k ( t ) − β U k Θ I ( t ) S k ( t )( − θ A , k ( t ))+ γ θ I , k ( t ) (3) d θ I , k dt = β A k Θ I ( t ) S k ( t ) θ A , k ( t ) + β U k Θ I ( t ) S k ( t )( − θ A , k ( t )) − γ θ I , k ( t ) (4)In the equations, Θ I ( t ) , similar to the notion of Θ A ( t ) , characterizes the probability thatan arbitrary neighbor of a node is in I state at time t . The first two terms in equations 3 capturesthe how susceptible individuals are infected under U and A states respectively, with θ A , k ( t ) and1 − θ A , k ( t ) referring to the proportion of nodes in U and A states. And the third term in equation 3suggests the recovery of individuals in I state with the rate of γ . Information layer
We next introduce the equation for the information layer, where it is assumed that there is a centralsystem that collects the information over the network. Typical examples of the information layercan be that people share their states and thoughts of the disease dynamics over the social networks,or the state mass media distribute the updated disease information from Centers for Disease Controland Prevention.For the functionality of the information layer, we consider two types of information gath-ering schemes: the targeted information fetching and the random information fetching. It is con-sidered that the central node can obtain the information over the disease network from the nodesthat are adjacent to it, and then compile the information and send back to the same set of nodes inthe observation network. And the expressions for the target and random information fetching arewritten as: [ Target ] λ g ( t ) = κ ∑ i ∈ T Θ I , T ( t ) (5) [ Random ] λ g ( t ) = κα ∑ k θ I , k ( t ) P ( k ) (6)ian, Xue, Ukkusuri 9where Θ I , T ( t ) denotes the probability that an arbitrary selected node is infected within the targetset T . κ is a discount factor that converts the total number of infectious nodes into the level ofrisk of the disease, and α is the ratio that accounts for the proportion of nodes that the central nodehas connection to randomly. Probability of a node in infectious state
With the awareness, disease and information dynamics presented in the previous sections, we cannow define the probability that a randomly selected neighbor of a node with degree k is in A or I states, namely Θ A ( t ) and Θ I ( t ) .Since θ A , k ( t ) gives the probability that a node of degree k is in A state at time t , we have: Θ A ( t ) = ∑ k (cid:48) k (cid:48) P ( k (cid:48) ) θ A , k (cid:48) ( t ) < k > (7)where < k > is the average degree of the network. Equation 7 is the weighted expectation of nodeswith degree k where the degree distribution is characterized by P ( k ) . The underlying intuitionbehind this equation suggests that high degree node usually have a much higher chance of beingin the infected states as compared to low degree nodes, so that such heterogeneity of node degreeshould be taken into consideration with the incorporation of the node degree distribution ( ).Similarly, we can formally written the probability that a randomly selected neighbor of anode with degree k in I state as: Θ I ( t ) = ∑ k (cid:48) k (cid:48) P ( k (cid:48) ) θ I , k (cid:48) ( t ) < k > (8) P ( k ) here is the same as that in equation 7 as we assume the same network structure for the obser-vation layer due to the consideration of physical contact. This consideration can be easily relaxedfor the sake of other applications with different networks structure in the observation layer (e.g.,observation through social network rather than physical contact network), where one can replacethe P ( k ) with a proper degree distribution that captures the network structure of the observationlayer. SYSTEM EQUILIBRIUM AND STABILITY ANALYSIS
SIS model is a well-studied epidemic model with two equilibrium states. One is the disease freeequilibrium (DFE) where all individuals are susceptible. This can be analogous to that all indi-viduals are in U and S states in our UA-SIS model, where such state is named as the disease andawareness free equilibrium (DAFE). The other equilibrium is known as the endemic equilibrium,where there will always be a proportion of nodes in infectious state, and the size of infectiouspopulation is equal to the size of the giant component in the graph. Finally, there is a special equi-librium point for the UA-SIS model, where the state is free from disease invasion but the awarenessitself is permanent and strictly positive. This regime is named as the disease free equilibrium withawareness (DFE-A). Disease and awareness free equilibrium
We first discuss the first equilibrium state, the DAFE, so that when t → θ A , k ( t ) = θ I , k ( t ) = k . Equivalently, this gives U k ( t ) = S k ( t ) = d θ A , k dt = − ( − p ) k Θ A ( t ) + λ g ( t ) − γ θ A , k ( t ) (9) d θ I , k dt = ∑ k (cid:48) β U kk (cid:48) P ( k (cid:48) ) θ I , k (cid:48) ( t ) < k > − γ θ I , k ( t ) (10)In the neighborhood of DFE, we further know that k θ A ( t ) → ( − p ) k Θ A ( t ) ≈ + ln ( − p ) k Θ A ( t ) (11)so that we can rewrite equation 9 as: d θ A , k dt = − ln ( − p ) k Θ A ( t ) + λ g ( t ) − γ θ A , k ( t )= − ln ( − p ) k Θ A ( t ) + κα ∑ k θ I , k ( t ) P ( k ) − γ θ A , k ( t ) (12)Let C OD be the correlation matrix between observation layer and disease layer, and C OO and C DD be the matrices for observation layer and disease layer respectively,where. C OOk , k = − ln ( − p ) k k P ( k ) < k > (13) C ODk , k = κα P ( k ) (14) C DDk , k = β U k k P ( k ) < k > (15)We should therefore have C = (cid:20) C OO C OD C DD (cid:21) (16)We can therefore write equations 10 and 12 as the following linear system: d θ dt = C θ − γ . θ (17)where θ = [ θ A , , ..., θ A , k max , θ I , , ..., θ I , k max ] T .We should have the following proposition: Proposition 7.1.
The DAFE of the UA-SIS system is asymptotically stable if the spectral radiusof ρ ( C ) < γ . In other words, if the maximum eigenvalue of C is smaller than γ , then the DFE isasymptotically stable. ian, Xue, Ukkusuri 11Since C is an upper-triangular matrix, we should have the maximum eigenvalue of C as λ = max ( λ ( C OO ) , λ ( C DD )) (18)When t → ∞ , according to ( ), the disease transmission thresholds for observation layerand disease layer can be written as: λ ( C OO ) = λ O = − ln ( − p ) < k >< k > (19) λ ( C DD ) = λ D = β U < k >< k > (20)We now describe the details to get 19 and 20. In equations 13 and 15, the constant itemsare respectively − ln ( − p ) / < k > and β U / < k > . Except for the constant items, we focus onthe matrix M = [ k i k j P ( k j )] i , j . M is a square matrix and can be expressed as the multiplication of acolumn vector and a row vector: M = k k . . . k l (cid:2) k P ( k ) k P ( k ) . . . k l P ( k l ) (cid:3) (21)We know for two matrices A , B , rank ( AB ) ≤ min { rank ( A ) , rank ( B ) } , so rank ( M ) ≤
1. Inaddition, all entries of M are positive so rank ( M ) ≥
1. It follows that rank ( M ) =
1. So 0 is aneigenvalue of M with the geometric multiplicity as l −
1. As geometric multiplicity of an eigenvalueis not smaller than its algebraic multiplicity, the algebraic multiplicity of 0 is l − l . We alsonotice that the sum of all eigenvalues of M is equal to the sum of all diagonal items in M , whichis ∑ li = k i k i p ( k i ) = < k >>
0. Therefore, the algebraic multiplicity of 0 is t − < k > . So the spectral radius of M is < k > and we have equations 19and 20.Following equation 19, we immediately observe that the spreading spread of awarenessdecays exponential with the strength of individual perceptions of the disease, as shown in Figure 3ian, Xue, Ukkusuri 12 FIGURE 3 : Spreading rate of awareness with the change of individual perception of diseaseThe second observation from the structure of C is that the availability of global informationdoes not affect the disease threshold in a network. That is, λ g does not determine the value of λ O and λ D , and hence λ C . This observation may sound counter-intuitive at first glance. However, whendisease is approaching DFE, the value of λ g is nearly zero as there are barely any infected people inthe network. As a consequence, it is always of lower order as compared to the personal awarenessof the disease, which plays a major role in the spreading process. The individual perception leveldirectly gives the duration that the awareness may persist, and therefore how likely that people maystay in a safer state. Disease free equilibrium with awareness
By observing the system equations, there are actually two different DFEs rather than one for basicSIS model: the DAFE, which is disease and awareness free equilibrium, and the DFE-A, which isthe disease free equilibrium with positive awareness population. The DAFE state is discussed inthe previous section. As for DFE-A, the equilibrium point of interest is θ I , k = θ A , k ≥ Θ I = Θ A >
0. By taking d θ A , k dt = [ − ( − p ) k Θ A ]( − θ A , k ) − γ θ A , k = θ A , k = − ( − p ) k Θ A − ( − p ) k Θ A + γ (23)ian, Xue, Ukkusuri 13Introducing this equation to equation 7 for Θ A ( t ) , we get a self-consistent equation for Θ A as Θ A = < k > ∑ k (cid:48) kP ( k (cid:48) ) − ( − p ) k (cid:48) Θ A − ( − p ) k (cid:48) Θ A + γ (24)where 0 is an trivial solution that corresponds to the DAFE equilibrium. The function f ( Θ A ) = < k > ∑ k (cid:48) kP ( k (cid:48) ) − ( − p ) k (cid:48) Θ A − ( − p ) k (cid:48) Θ A + γ − Θ A (25)is a concave function, where we have1 − ( − p ) k (cid:48) Θ A − ( − p ) k (cid:48) Θ A + γ < f ( ) < d f ( x ) dx | x = >
0. As a consequence, Θ A admits a positive solutionwithin the interval ( , ) . Without loss of generality, consider the DFE-A solution being ( µ , − µ , , ) where µ k = γ − ( − p ) k Θ ∗ A + γ (27)And the solution with positive awareness is stable as long as ρ ( C ) > γ , which can be derived basedon our analysis for the stability of the DAFE. Now we know that γ + γ ≤ µ k ≤
1. More importantly,for nodes with higher degree, µ k → γ + γ . Meanwhile, the higher the p is, the lower the µ k will be.And the value of µ k is sensitive to γ value, which is the fading rate of memory. If it takes longerfor people to forget the impacts of the diseases, then the γ value should be smaller and there willbe fewer people in U state.We now linearize the UA-SIS system at DFE-A as: d θ A , k dt = [ − ( − p ) k Θ A ( t ) + λ g ( t ) + β U k Θ I ( t )] µ k − γ θ A , k ( t ) (28) d θ I , k dt = β A k Θ I ( t )( − µ k ) + β U k Θ I ( t ) µ k − γ θ I , k ( t ) (29)Rearranging the right hand side gives: d θ I , k dt = β A k Θ I ( t ) + ( β U − β A ) µ k k Θ I ( t ) − γ θ I , k ( t ) (30)To ensure that the DFE-A is a.s.s, we just need to ensure that I = λ DFE − A = β A < k >< k > + ( β U − β A ) < µ k k >< k > < γ (31)ian, Xue, Ukkusuri 14If we consider that β A = β A << β U , which is equivalent to that those people who areaware of the disease will be totally vaccinated or quarantined, we have that λ DFE − AD is proportionalto ( β U − β A ) < µ k k >< k > . This indicates that the local information contributes significantly to loweringthe disease threshold as compared to the state when there is no information available. But themarginal gain will be considerably weaker as we keep increasing the value of p .Based on these analysis, there will be two conditions for the disease to reach DFE-A state.The first is the trivial case, where the disease system itself will stay in the DFE state and the awarepopulation is positive: λ O > γ , λ D < γ (32)The second case is more interesting where the disease itself may initially land a local out-break without significant level of awareness among the nodes. Then the awareness is built amongthe nodes and the strength of the disease is weakened by the spread of awareness so that the previ-ously disease outbreaks will eventually be suppressed and eventually eliminated. Mathematically,this requires the following condition to be satisfied: λ DFE − A < γ , λ O > γ , λ D > γ (33)where both local outbreaks for disease and awareness are possible due to the threshold valuesfor the disease and observation layers separately. But the joint eigenvalue is less than the criticalthreshold so that the disease will eventually be eliminated. Size of endemic state
We are not only interested in DFE of the diseases, but also would like to explore how local ob-servation and global information may affect the speed of the infectious diseases, and consequentlythe size of the outbreak (endemic state). This motivates us to conduct further analysis.When λ DFE − AD > γ and λ O > γ , the disease will eventually reach the endemic state. Fol-lowing ( ), we first calculate the size of the endemic disease. At endemic, we should have theequilibrium point being ( µ , − µ , s , − s ) with 1 ≥ µ > ≥ s >
0. As a consequence, weshould have d θ I , k dt =( β A k Θ I ( t ) + ( β U − β A ) µ k k Θ I ( t ))( − θ I , k ( t )) − γ θ I , k ( t ) (34)At endemic state, we should have d θ I , k dt =
0, so that γ θ I , k = [ β A + ( β U − β A ) µ k ]( − θ I , k ) k Θ I (35) θ I , k = α k k Θ I γ + α k k Θ I (36)where α k = β A + ( β U − β A ) µ k . And Θ I = ∑ k (cid:48) k (cid:48) P ( k (cid:48) ) θ I , k (cid:48) < k > (37)ian, Xue, Ukkusuri 15 Θ I = < k > ∑ k (cid:48) k (cid:48) P ( k (cid:48) ) α (cid:48) k k (cid:48) Θ I γ + α (cid:48) k k (cid:48) Θ I (38)Following the same analysis as for Θ A for DFE-A, we know that there will be a positive Θ I in theinterval ( , ) which satisfies above equation.Meanwhile, for θ A , k , by setting [ − ( − p ) k Θ A ( t ) + λ g ( t ) + β U k Θ I ( t )]( − θ A , k ( t )) = γ θ A , k ( t ) (39) θ A , k = − γ − ( − p ) k Θ A + λ g + β U k Θ I + γ (40)Let G ( k ) = − ( − p ) k Θ A + λ g , we have θ A , k = − γ G ( k ) + γ + β U k Θ I (41) µ k = γ G ( k ) + γ + β U k Θ I (42)Finally, we have Θ A = < k > ∑ k (cid:48) k (cid:48) P ( k (cid:48) )( − µ k (cid:48) ) (43) Θ I = < k > ∑ k (cid:48) k (cid:48) P ( k (cid:48) )( − γ γ + ( β A + ( β U − β A ) µ k (cid:48) ) k (cid:48) Θ I ) (44)Based on the equations, we observe that, for high degree nodes, the value of u k is dominatedby β U Θ I rather than the strength of the personal awareness. But for nodes with low degree, thestrength of information dictates the value of µ k . This suggests that at endemic state, high degreenodes are less prone to infections, while low degree nodes are more vulnerable to the risk ofinfectious diseases. NUMERICAL EXPERIMENTSExperiment setting
The ODE45 solver in MATLAB is used to simulate the UA-SIS model described in this study. Andwe present the dynamics of the UA-SIS model over two types of networks. The first network is thescale-free network, which is commonly observed network structure in the real-world such as theWorld Wide Web network and the social network ( ). The degree distribution for the scale-freenetwork follows: P ( k ) ∝ k − γ (45)ian, Xue, Ukkusuri 16The other type of network we consider here is the realistic contact network observed inthe transportation system. In particular, Qian et al. ( ) developed the degree distribution for thecontact networks in the metro system based on the smart card transaction data from three cities.The metro contact network (MCN) serves as an ideal candidate network to understand the changeof behavior upon observing the states of the neighbors during travel. The degree distribution of theMCN follows: P ( k ) = N ∑ i = ( Mw i ) k e − Mw i k ! P ( t i ) (46)where M = ∑ Ni α t γ t i ( N − ) measures the number of contacts among N travelers and P ( t i ) is thedistribution of travel time in metro systems. The shape of MCN is controlled by two parameters: γ t which measures the similarity of mobility pattern among travelers and α captures the structuralproperty of the metro networks. We set γ t = .
652 and α = .
004 following the parameters for theShanghai metro network in ( ). For both type of networks, the results are obtained by simulatingthe disease dynamics over the network with 10 nodes. And the scale-free network is generated sothat the average node degree < k > is the same as that of MCN for fair comparison.Finally, for all the experiments, we set γ = γ = . κ = α = .
05 unless otherwisespecified.ian, Xue, Ukkusuri 17
Results I -3 p -3 A -3 p -3 (a) Results for scale-free network I -3 p -3 A -3 p -3 (b) Results for metro MCN FIGURE 4 : Infectious population (left) and aware population (right) with varying β and p with t → ∞ .Figure 4 shows how varying β and p values may affect the final infectious and awareness popu-lation. The figure depicts the state transition among the three equilibrium states with different β and p values. In particular, when β < . β < . p < .
002 in MCN, the strength of the information percolation is not strong enough to promotea local awareness outbreak over the observation network. And similar observation can be madefor the scale-free network. Under such circumstances, the system is in DAFE state with diseaseand awareness population (I and A) being 0. When β increases beyond 0 . p value. Such a statecorresponds to the DF E − A equilibrium, where the awareness is strictly positive, but the diseaseian, Xue, Ukkusuri 18is eradicated. And the contour plot suggests the relationship between β and p to achieve DFE-Afollows a convex function. As such, the increase in the pace of local awareness dissemination isexpected to be greater than the increase in disease intensity so that a DFE can be achieved. Finally,when the disease strength exceeds λ DFE − A , the system will transit into an endemic state, and bothdisease and awareness are permanent in the population. This corresponds to the areas with both Iand A population are greater than zero in the contour plots. By examining the differences of thedisease dynamics between the scale-free network and MCN as in Figure 4, we can tell that the areaof the regions for DAFE and DFE-A are both greater for MCN than those for the scale-free net-work. This indicates that the MCN is more resilient to the threat of infectious diseases as comparedto the scale-free network. This can be explained from their degree distribution where MCN decaysfaster than the scale-free network and presents hubs. These translate into lower < k > for MCNso that the disease and awareness thresholds are shown in equations 19 and 20 are smaller sinceall other parameters are the same. Consequently, the real-world contact networks in the transporta-tion system are less vulnerable than the scale-free network given its structural properties, wherethe latter is the network of interests in most network epidemiology studies. However, cautionsstill need to be exercised as the level of resilience of MCN is not significantly higher than that ofthe scale-free network, since < k > will still diverge with increasing network sizes as discussedin ( ). On seeing the major differences between the MCN and the scale-free network, we focusexclusively on MCN for the following analyses.Figure 5 reveals the effect of local information transmission on spreading dynamics. Theline corresponding to I=0 in Figure 5(a) presents that local information efficiently suppresses dis-ease outbreak, for the threshold increases (with increasing β ) as p increases. This conclusion canalso be drawn via equation 31, where µ k is less than 1. For reaching a higher infectious population( I = . . β lowers the threshold for awareness outbreak, asthe higher p value is required to retain the same level of terminal A population. An interestingobservation is the existence of â ˘AIJthreshold dropâ ˘A˙I, which happens at around p = . p = . . p increases from a lower value to above the observed threshold, the local informationthreshold drops rapidly. After the â ˘AIJthreshold dropâ ˘A˙I, even a small percent of infection willlead to the local awareness spreading very quickly. This finding demonstrates the essential roleof local information transmission related to an infectious disease outbreak. On the other hand,Figures 5(c)-(d) show the slices of the corresponding parts in Figure 4 and present that local infor-mation transmission reduces the limiting size of the infectious population and increases the limitingsize of the aware population. This asymmetrical phenomenon is consistent with the results shownin Figures 5(a)-(b). Finally, as β increases, p is found to have diminishing impacts on the finalinfectious and aware population which can be seen in Figures 5(c)-(d). In these cases, β dominatesthe system dynamics for both infectious and aware population, and this observation is consistentwith the convex relationship between β and p for I population as can be seen in Figure 4ian, Xue, Ukkusuri 19 p -3 infectious population=0infectious population=0.10infectious population=0.20 (a) Terminal I population with different β and p p -3 -3 awareness population=0.3awareness population=0.50awareness population=0.70 (b) Terminal A population with different β and p p -3 I n f e c t i ou s popu l a t i on I Beta=2.06e-03Beta=3.06e-03Beta=4.54e-03Beta=6.74e-03 (c) Change of terminal I population with respect to p p -3 A w a r ene ss popu l a t i on A Beta=2.06e-03Beta=3.06e-03Beta=4.54e-03Beta=6.74e-03 (d) Change of terminal A population with respect to p FIGURE 5 : The impact of local information on disease spreading dynamicsian, Xue, Ukkusuri 20 -3 A w a r ene ss popu l a t i on A p=0 -3 A w a r ene ss popu l a t i on A p=1.86e-03 m=m m=10m m=100m m=1000m m=10000m -3 I n f e c t i ou s popu l a t i on I p=0 -3 I n f e c t i ou s popu l a t i on I p=1.86e-03 a bc d FIGURE 6 : The impact of global information (random) on disease spreading dynamicsFigure 6 reveals the role of global information by fixing the value of p . It can be directlyverified from Figures 6(b) that global information has no impact on the threshold for disease-freeequilibrium, as the infectious population will only be greater than 0 once a certain β value isreached. Similarly, the global information will also not change the awareness threshold if there isno infectious population since the awareness population starts to increase as the infectious popula-tion is greater than 0 as demonstrated in Figure 6(d). These two results illustrate the dependency ofglobal information on the disease states, as media agencies release global information by monitor-ing the progress of the disease dynamics in a reactive manner. After an infectious disease breaksout, only very strong global information is able to restrain disease efficaciously (e.g., when globalinformation strength increased by 10 and 10 times). Similar observations can also be identifiedfrom Figures 6(a) and (c). And the key difference is that the combinations of p with initial β valuesresults in endemic states in Figures 6(a) and (c) and DFE-A states in Figures 6(b) and (d). In con-clusion, we observe both local and global information are powerful tools to suppress the spreadingof infectious diseases. Local information is found to be more effective than the global informationian, Xue, Ukkusuri 21as it may percolate across the networks even without the presence of the disease outbreak. Thiscan contribute to building a barrier against infectious diseases in a proactive manner. On the otherhand, global information reacts upon the state of the infectious population, and can only help mit-igate the scale of an outbreak after the disease invades the population. This can be understood asthe case where mass media may not catch up on the progress of the diseases if it has not become amajor threat to the general public. Time -8-6-4-2024 I n c r ea s i ng r a t e -3 IA Time P opu l a t i on IA (a) DAFE Time -0.0100.010.020.030.04 I n c r ea s i ng r a t e IA Time P opu l a t i on IA (b) DFE-A Time -0.0100.010.020.030.040.05 I n c r ea s i ng r a t e IA Time P opu l a t i on IA (c) Endemic FIGURE 7 : The rate of change for S and I in three different equilibrium statesFigure 7 presents the relative growth rates and dynamics of I and A population under dif-ferent equilibrium states. For both DFE-A and DAFE, the information population will reach itspeak shortly after the increase rate of the infectious population reaches its minimum. Note thatthe synchronizations and disease dynamics during DFE-A and DAFE are different even though thedisease gets eliminated in both cases. Under DAFE, the increasing rate of disease is, in general,faster than that of awareness. When the infectious population reaches zero, we observe that theamount of information also starts to drop. This implies that the growth of awareness populationis primarily driven by the disease spreading itself, and the strength of the information awarenessis not strong enough to persist. On the other hand, under DFE-A, we observe that the growthof awareness population is faster than the infectious population, and the relative growth rate ofawareness population is always positive. Moreover, the increase rate also drops with decreasingnumber of infectious population. Nevertheless, we observe that the growth rates of awareness andthe infectious population are positively correlated, and there exists a time lag for the two growthrates to be positively correlated. This time lag is found to be shorter under DAFE, and much longerian, Xue, Ukkusuri 22under DFE-A. Finally, the synchronizations and dynamics when the disease is endemic also differfrom both DAFE and DFE-A. In particular, the growth of awareness and the growth of infectiouspopulations are almost perfectly synchronized, but with awareness always spreading faster thanthe disease. This finding is found to be consistent with the real world observation reported in ( ),where the growth rate of patient visit data (corresponding to infectious population) is perfectlysynchronized with the trend of Google Flu index (corresponding to awareness population). Basedon these findings, we conclude that, when there are many people aware of the risk of the disease,the disease either should be either of minimum risk or it has almost reached its endemic state. N nodes D i s ea s e t h r e s ho l d SISUA-SIS
FIGURE 8 : Decay of disease threshold in MCN with increasing number of nodesFinally, Figure 8 presents how disease threshold changes with the growing size of the net-work, for both SIS and UA-SIS models. An important and well-known finding for scale-freenetworks is that there is a lack of disease threshold if the power term of the degree distribution2 ≤ γ <
3. The reason behind this is that the moment of the variance of degree distribution di-verges with the growing size of the network. As discussed earlier, < k > also diverges for MCNas the network size grows (equivalent to the increase in the number of travelers). And such adivergence leads to the decay of the disease threshold to 0 as shown in Figure 8, so that the sys-tem may become extremely vulnerable to the risk of infectious diseases. Nevertheless, with thehelp of information dissemination and local awareness through observing other travelers, even asmall amount of local information will significantly improve the resilience of the network and sig-nificantly delay the decay of the threshold. We observe that the disease threshold under UA-SISmodel may be several magnitudes higher than that of the SIS model, especially for very large-sizednetworks. This observation highlights the importance of incorporating the change of behavior oftravelers when we model the dynamics of the infectious diseases and supports the value of theproposed UA-SIS model in better understanding the actual trajectories of an infectious disease.This is a particularly important observation and articulates the effectiveness of local informationin preventing target attacks.ian, Xue, Ukkusuri 23 CONCLUSION
In this study, the multiplex network model for modeling the co-evolution of information and dis-ease dynamics over the networks is presented. In particular, travelers are assumed to change theirbehavior based on their observations of the states of their neighbors and by obtaining informationfrom global sources such as news agencies and social media. This percolation of information willhave a direct impact on the disease dynamics over the disease network. Meanwhile, the state of thedisease spreading also affects the level of information released by global sources and the state andbehavior of each individual travelers. The HMF method is used to model the co-evolution of thetwo dynamics, and obtained three possible stable states. Based on these findings, threshold valuesfor disease and information percolation that may result in one of the three stable states are alsodiscussed and validated by the numerical experiments.
AUTHOR CONTRIBUTIONS
The authors confirm contribution to the paper as follows: study conception and design: X. Qian,S.V. Ukkusuri; data collection: X.Qian; analysis and interpretation of results: X. Qian, J. Xue;draft manuscript preparation: X. Qian, J. Xue, S.V. Ukkusuri. All authors reviewed the results andapproved the final version of the manuscript.
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