On a Network SIS Epidemic Model with Cooperative and Antagonistic Opinion Dynamics
11 On a Network SIS Epidemic Model with Cooperative and AntagonisticOpinion Dynamics
Baike She, Ji Liu, Shreyas Sundaram, and Philip E. Paré*
Abstract — We propose a mathematical model to study cou-pled epidemic and opinion dynamics in a network of com-munities. Our model captures SIS epidemic dynamics whoseevolution is dependent on the opinions of the communities to-ward the epidemic, and vice versa. In particular, we allow bothcooperative and antagonistic interactions, representing similarand opposing perspectives on the severity of the epidemic,respectively. We propose an Opinion-Dependent ReproductionNumber to characterize the mutual influence between epi-demic spreading and opinion dissemination over the networks.Through stability analysis of the equilibria, we explore theimpact of opinions on both epidemic outbreak and eradication,characterized by bounds on the Opinion-Dependent Reproduc-tion Number. We also show how to eradicate epidemics byreshaping the opinions, offering researchers an approach fordesigning control strategies to reach target audiences to ensureeffective epidemic suppression.
I. I
NTRODUCTION
A. Motivation
Epidemiological models have been extensively studied forthe purpose of understanding the spread of infectious dis-eases through societies [1]–[4]. A key epidemiologic metricin these models is the basic reproduction number ( R ) ,which describes the expected number of cases directly gen-erated by one case in an infection-free population; similarly,the effective reproduction number ( R t ) characterizes theaverage number of new infections caused by a single infectedindividual at time t in the partially susceptible population [5].These reproduction numbers can be affected by a varietyof factors, including biological properties of the epidemic,environmental conditions, and the opinions and behaviors ofthe population [6]. Communities believing that an outbreakis severe will react quickly to suppress the epidemic, in termsof policy-making, rule-following, etc [7]. Instead, misinfor-mation or disbelief in an epidemic could potentially leadcommunities to react incorrectly, perpetuating outbreaks [8].Further, social media significantly increases the rate at whichopinions spread through communities, enabling shifts in theirreactions towards an epidemic in different ways. During theSARS-CoV-2 pandemic, both cooperative and antagonisticinteractions over social networks affected different commu-nities’ beliefs in the pandemic, influencing the transmissions *Baike She, Shreyas Sundaram, and Philip E. Paré are with the School ofElectrical and Computer Engineering, Purdue University. Ji Liu is with theDepartment of Electrical and Computer Engineering, Stony Brook Univer-sity. E-mails: {bshe, sundara2, philpare}@purdue.edu. Research supportedin part by the C3.ai Digital Transformation Institute sponsored by C3.aiInc. and the Microsoft Corporation, and in part by the National ScienceFoundation, grants NSF-CMMI of the virus [9]. Motivated by the critical role that opiniondynamics play in the epidemic spread (and vice versa), in thiswork we study a networked epidemic model coupled with anetworked opinion model possessing both cooperative andantagonistic interactions to understand the mutual influencebetween epidemic spreading and opinion dissemination overcommunities. Below, we discuss related literature on thistopic, and then describe our contributions. B. Literature Review
Network epidemic models have attracted considerableattention in the controls community [2]–[4], [10], as thenetwork SIS model and its variants can model the spreadof many types of objects, such as malware in computernetworks [11] and attacks in cyber-physical systems [12]. Tocapture the co-existence of rival opinions among differentpopulations, we use networks with both positive and neg-ative edges to characterize the cooperative and antagonis-tic interactions, respectively, of opinion exchange betweencommunities, as studied in [13]–[15]. Recent work has com-bined disease spread models with human awareness models[16]–[18], where the infection rates scale with the humanawareness towards the epidemics. However, these modelslack an explicit dynamical model to represent the change ofthe population’s perception on the severity of the epidemicsover time [19]–[21].To couple the networked SIS model with opinion dy-namics, we employ the health belief model, which is thebest known and most widely used theory in health behaviorresearch [22]. The health belief model proposes that peo-ple’s beliefs about health problems, perceived benefits ofactions, and/or perceived barriers to actions can explain theirengagement, or lack thereof, in health-promoting behavior.Therefore, people’s beliefs in their perceived susceptibilityand/or in their perceived severity of the illness affect howsusceptible they are and/or how effective they will be athealing from these epidemics. Our previous work leveragedthe health belief model to couple the SIS network model withcooperative opinion dynamics [21]. In this article, we modelthe population’s beliefs of the severity of the epidemic usingopinion dynamics with both cooperative and antagonisticinteractions. C. Contribution
The contributions of this work are the following.1) We develop a network SIS model coupled with bothcooperative and antagonistic opinion dynamics. The In this article, beliefs, attitudes, and opinions are used interchangeably. a r X i v : . [ ee ss . S Y ] F e b opinion dynamics evolve on an opinion-dependent signswitching topology, which characterizes the change ofthe opinions of the communities towards the epidemicover time.2) We define an Opinion-Dependent Reproduction Num-ber ( R ot ) of the epidemic model. We use R ot tocapture the severity of the epidemic (mild, moderate,severe), equilibria, and stability, which reveal the mu-tual influence between epidemic spreading and opiniondissemination over the communities.3) We interpret our results in the context of real-worldphenomena under the SARS-CoV-2 pandemic. Wepropose ways to guide control design and select targetcommunities to shift the opinions of the communitiesin order to better control the epidemic. D. Outline of the article
This work is organized as follows. In Section II, we statethe motivation and present the Epidemic-Opinion networkmodel. Section III introduces the preliminaries that are usedthroughout this work. In Section IV, we study the equilibriaof the Epidemic-Opinion network model. Section V definesthe Opinion-Dependent Reproduction Number, which char-acterizes the behavior of the epidemic spreading process.Section V also explores how to influence the opinions inorder to suppress the outbreak. Section VI and Section VIIpresent simulation and conclusion/future works, respectively.
E. Notations
For any positive integer n , we use [ n ] to denote the indexset { , , . . . , n } . We view vectors as column vectors andwrite x T to denote the transpose of a column vector x . Weuse x n to denote the vector, of the same size as x , whoseeach entry equals the n th power of the corresponding entryof x . For a vector x , we use x i to denote the i th entry.For any matrix M ∈ IR n × n , we use M i, : , M : ,j , M ij , todenote its i th row, j th column and ij th entry, respectively.We use M = diag { m , . . . , m n } to represent a diagonalmatrix M ∈ IR n × n with M ii = m i , ∀ i ∈ [ n ] . We use and e to denote the vectors whose entries all equal 0 and 1,respectively, and I to denote the identity matrix. The dimen-sions of the vectors and matrices are to be understood fromthe context. Let ∂ [ c, d ] n and Int [ c, d ] n denote the boundaryand interior of the cube [ c, d ] n , c, d ∈ R , respectively.For a real square matrix M , we use ρ ( M ) and s ( M ) todenote its spectral radius and the largest real part among itseigenvalues, respectively. For any two vectors v, w ∈ IR n ,we write v ≥ w if v i ≥ w i , and v (cid:29) w if v i > w i , ∀ i ∈ [ n ] .The comparison notations between vectors are feasible formatrices as well, for instance, for A, B ∈ IR n × n , A (cid:29) B indicates that A ij > B ij , ∀ i, j ∈ [ n ] . For any two sets A and B , we use A \ B to denote the set of elements in A but notin B . We also employ a modified signum function: sgnm ( x ) = (cid:40) , if x ≥ − , if x < . The Dirac delta function, which is the first derivative of sgnm ( · ) , is represented by θ ( · ) . Consider a directed graph G = ( V , E ) , with the node set V = { v , . . . , v n } and the edge set E ⊆ V × V . Let matrix A = [ a ij ] ∈ IR n × n denote the adjacency matrix of G =( V , E ) , where a ij ∈ IR if ( v j , v i ) ∈ E and a ij = 0 otherwise.Graph G does not allow self-loops, i.e., a ii = 0 , ∀ i ∈ [ n ] . Let k i = (cid:80) j ∈N i | a ij | , where N i = { v j | ( v j , v i ) ∈ E} denotesthe neighbor set of v i and | a ij | denotes the absolute valueof a ij . The graph Laplacian of G is defined as L (cid:44) K − A ,where K (cid:44) diag { k , . . . , k n } is a diagonal matrix.II. M ODELING AND P ROBLEM F ORMULATION
In this section, we introduce an SIS model, coupled withan opinion dynamics model. In particular, we assume anepidemic is spreading over a group of communities, wherethe interactions among the communities facilitate the spread-ing. Furthermore, the opinion of each community about theepidemic evolves as a function of the community’s infectedproportion, the community’s own opinion, and the opinionsof other communities.
A. Epidemic Dynamics
Consider an epidemic spreading over n connected com-munities represented by a directed graph G = ( V , E ) , wherethe node set V = { v , . . . , v n } and the edge set E ⊆ V × V represent the communities and the epidemic spreading in-teractions, respectively. The epidemic spreading interactionsover n communities are captured by an adjacency matrix A = [ a ij ] ∈ IR n × n , where a ij ∈ IR ≥ . A directed edge ( v j , v i ) indicates that community j can influence community i , in terms of infection.We use a networked SIS model to capture the epidemicdynamics of the n communities, such that x i ∈ [0 , represents the n communities captured the proportion ofthe infected population in community i , i ∈ [ n ] . Note that x i × ∈ [0% , . In this work, x and x ( t ) are usedinterchangeably. The epidemic dynamics for each of the n communities evolve as follows: ˙ x i ( t ) = − δ i x i ( t ) + (1 − x i ( t )) (cid:88) j ∈N i β i a ij x j ( t ) , (1)where δ i ∈ IR ≥ is the average curing rate of community i ,and β i a ij ∈ IR ≥ is the average infection rate of community j to community i . B. Opinion Dynamics
We let the opinions disseminate over a graph ¯ G = (cid:0) V , ¯ E (cid:1) ,with the adjacency matrix ¯ A = [¯ a ij ] ∈ IR n × n , ¯ a ij ∈ IR . The opinion graph ¯ G allows both positive and nega-tive edge weights, representing cooperative and antagonisticinteractions, respectively, on opinions exchanged betweenthe communities. Let ¯ A u ∈ IR n × n , with (cid:2) ¯ A u (cid:3) ij ∈ IR ≥ ,and (cid:2) ¯ A u (cid:3) ij = | ¯ a ij | . Therefore, ¯ A u captures the couplingstrength of the opinion dynamics without considering thesigns. Similar to graph G , we define neighbor set of graph ¯ G as ¯ N , and use ¯ K and ¯ L to denote the degree matrix andLaplacian matrix of ¯ G , respectively.For all t ≥ , o ( t ) ∈ IR n is the opinion vector of the n communities, o i ( t ) ∈ [ − . , . , i ∈ [ n ] . The range o i ( t ) ∈ [ − . , . denotes the belief of community i aboutthe severity of the epidemic at time t . Note that o ( t ) and o are used interchangeably in this work. The opinion o i ( t ) =0 . indicates that community i considers the epidemic tobe extremely serious, while o i ( t ) = − . implies thatcommunity i thinks the epidemic is not worth addressing.Assuming that communities with a neutral opinion o i ( t ) = 0 marginally lean towards treating the epidemic as a threat.It is natural to consider that communities with thesame attitude towards the epidemic exchange their opin-ions cooperatively, while communities with different atti-tudes exchange their opinions antagonistically. Therefore,based on the communities’ beliefs towards the epidemicat any given time, we allow the edge signs of the opin-ion graph ¯ G to switch. We achieve this behavior by par-titioning the node set of the communities V into twogroups, V ( o ( t )) = { v i ∈ V | sgnm ( o i ( t )) = 1 , i ∈ [ n ] } and V ( o ( t )) = { v i ∈ V | sgnm ( o i ( t )) = − , i ∈ [ n ] } .Then we construct the adjacency matrix ¯ A of graph ¯ G as ¯ A ( o ( t )) = Φ ( o ( t )) ¯ A u Φ ( o ( t )) , where Φ ( o ( t )) = diag { sgnm ( o ) , . . . , sgnm ( o n ) } is a gauge transformationmatrix. The entries of the gauge transformation matrix Φ ( o ( t )) are chosen as φ i ( o i ( t )) = 1 if i ∈ V p ( t ) and φ i ( o i ( t )) = − if i ∈ V q ( t ) , p (cid:54) = q , and p, q ∈ { , } , ∀ i ∈ [ n ] . Through the construction of ¯ A ( o ( t )) , | ¯ a ij | is fixedand sign | ¯ a ij | is switchable. In particular, for all non-zeroentries in ¯ A ( o ( t )) , sign | ¯ a ij | = 1 if signm ( o j ( t )) and signm ( o i ( t )) are the same, otherwise, sign | ¯ a ij | = − .Therefore, ¯ A ( o ( t )) captures the switch of the opinion inter-actions through the attitude changing towards the epidemic.Note that an opinion graph ¯ G constructed in this manner isalways structurally balanced by the following definition andlemma. Definition 1. [Structural Balance [13]] A signed graph ¯ G = (cid:0) V, ¯ E (cid:1) is structurally balanced if the node set V can bepartitioned into V and V with V ∪V = V and V ∩V = ∅ ,where ¯ a ij ≥ if v i , v j ∈ V q , q ∈ { , } , and ¯ a ij ≤ if v i ∈ V q and v j ∈ V r , q (cid:54) = r , and q, r ∈ { , } . Lemma 1. [13] A connected signed graph ¯ G is structurallybalanced if and only if there exists a gauge transformationmatrix Φ = diag { φ , . . . , φ n } ∈ IR n × n , with φ i ∈ {± } ,such that Φ ¯ A Φ ∈ IR n × n is non-negative. After defining the opinion interaction matrix ¯ A ( o ( t )) , avariant of the opinion dynamics model in [13] evolving overthe n communities with both cooperative and antagonisticinteractions is given by ˙ o i ( t ) = (cid:88) j ∈ ¯ N i | ¯ a ij ( o ( t )) | ( sign (¯ a ij ( o ( t ))) o j ( t ) − o i ( i )) , (2)with the compact form ˙ o ( t ) = − Φ ( o ( t )) ¯ L u Φ ( o ( t )) o ( t ) , (3)where ¯ L u represents the Laplacian matrix of ¯ A u . C. Coupled Epidemic-Opinion Dynamics
Having introducing the SIS network epidemic modelspreading over the n communities, and opinions spreadingover the same n communities, we now introduce networkdynamical models that coupled the epidemic dynamics withthe opinion dynamics.Assume that a community’s opinion/attitude towards theseverity of an epidemic will affect its actions, which leadsto the variation of the community’s average healing rateand infection rate. For instance, a community being verycautious about the epidemic will broadcast the influenceof the epidemic more frequently, and make policies tosuppress the epidemic, while the people in that communityare more likely to follow the instructions given by scientificinstitutions, and seek treatments in a timely manner. Theseactions will result in the community having a lower averageinfection rate and a higher average healing rate. To betterdescribe the situation, we use δ min and β min to denote thepossible minimum average healing rate and infection rate forall communities, respectively . To incorporate the opiniondynamics in (2) into the epidemic dynamics in (1), weconsider ˙ x i ( t ) = − [ δ min + ( δ i − δ min ) o (cid:48) i ( t )] x i ( t )+ (1 − x i ( t )) (cid:88) j ∈N i [ β ij − ( β ij − β min ) o (cid:48) i ( t )] x j ( t ) , (4)where o (cid:48) i ( t ) = o i ( t ) + 0 . shifts the opinion into the range [0 , . In particular, the term o (cid:48) i ( t ) scales the average healingand infection rates between their maximum and minimumvalues. In the case when o i ( t ) = − . , which implies thatcommunity i does not consider the epidemic a threat at time t , community i will take no action to protect itself and thusis maximally exposed to the infection. In the case when o i ( t ) = 0 . , which implies that community i believes theepidemic is extremely serious, it will implement policies andlimitations to decrease the infection rate and seek out all thepossible medical treatment options to improve its healingrate. Therefore, the model allows the communities’ opinionsto affect how susceptible they are and how effectively theyheal from the virus, capturing the health belief model [22],as explained in the Introduction.The proportion of the infected population of a com-munity i can also have an effect on its opinion/attitudetowards the epidemic. Consider the epidemic dynamics in(1) incorporated with (2) as follows: ˙ o i = ( x i ( t ) − o (cid:48) i ( t ))+ (cid:88) j ∈ ¯ N i | ¯ a ij ( o ( t )) | ( sign (¯ a ij ( o ( t ))) o j ( t ) − o i ( t )) . (5)The first term on the right hand side of (5) captures howthe proportion of infections of a community affects its ownopinion. If o i ( t ) is small but the community is heavilyinfected, i.e., x i ( t ) is large, o i ( t ) will increase. If o i ( t ) is We assume homogeneous minimum healing and infection rates forsimplicity. The results in this work can be extended to the heterogeneouscase. large but the community has few infections, i.e., x i ( t ) issmall, o i ( t ) will decrease. This behavior is sensible sincea community’s infection level should affect its belief in thesevereness of the virus, which is consistent with the healthbelief model in [22]. The second term on the right handside of (5) is from (2). The neighbors of community i affect its opinion cooperatively ( sign (¯ a ij ( o ( t ))) = 1 ) orantagonistically ( sign (¯ a ij ( o ( t ))) = − ). D. Problem Statements
Now that we have presented the Epidemic-Opinion modelin (4) and (5), we can state the problem of interest in thiswork. We are interested in exploring the mutual influence be-tween the epidemic spreading over n communities capturedby graph G in (4) and the opinions of the n communitiesabout the epidemic captured by graph ¯ G in (5). We willanalyze the equilibria of the system in (4)-(5) under differentsettings. In particular, we will define an Opinion-DependentReproduction Number to characterize the spreading of thevirus. We will study the stability of the equilibria of thesystem to infer the behaviors of the epidemic and opinionsspreading over the n communities. Finally, we will generatestrategies to eradicate the epidemic by affecting the opinionstates of the system, which could potentially guide the useof social media to broadcast the severity of the pandemic toappropriate communities to suppress the epidemic.III. P RELIMINARIES
We impose the following natural restrictions on the pa-rameters of the models throughout the article.
Assumption 1.
Let x i (0) ∈ [0 , , o i (0) ∈ [ − . , . , δ i ≥ δ min > , and β ij ≥ β min > , ∀ i ∈ [ n ] and ∀ j ∈ N i .The epidemic spreading graph G and opinion disseminationgraph ¯ G are strongly connected. Note that the adjacency matrix of a strongly connectedgraph is irreducible. Therefore, the adjacency matrix A of thegraph G is a nonnegative irreducible matrix. A real squarematrix M is called a Metzler matrix if M ij ≥ , ∀ i, j ∈ [ n ] and i (cid:54) = j , which implies that the adjacency matrix A isalso an irreducible Metzler matrix. Some of our results relyon properties of Metzler matrices and nonnegative matrices,which we briefly recall below. Lemma 2. [23, Prop. 2] For a Metzler matrix M ∈ IR n × n ,the following statements are equivalent: The matrix M is Hurwitz; There exists a vector v (cid:29) such that M v (cid:28) ; There exists a vector u (cid:29) such that u T M (cid:28) ; There is a positive diagonal matrix Q such that M T Q + QM is negative definite. Lemma 3. [10, Lemma A.1] For an irreducible Metzlermatrix M ∈ IR n × n , if s ( M ) = 0 , there exists a positivediagonal matrix Q such that M T Q + QM is negativesemidefinite. Lemma 4. [24, Thm. 2.7, and Lemma 2.4] Suppose thatM is an irreducible nonnegative matrix. Then, the followingstatements hold: M has a simple positive real eigenvalue equal to itsspectral radius, ρ ( M ) ; There is a unique (up to scalar multiple) eigenvector v (cid:29) corresponding to ρ ( M ) ; ρ ( M ) increases when any entry of M increases; If N is also an irreducible nonnegative matrix and M ≥ N , then ρ ( M ) ≥ ρ ( N ) . Lemma 5. [24, Sec. 2.1 and Lemma 2.3] Suppose that M is an irreducible Metzler matrix. Then, s ( M ) is a simpleeigenvalue of M and there exists a unique (up to scalarmultiple) vector x (cid:29) such that M x = s ( M ) x . Let z > be a vector in IR n . If M z < λz , then s ( M ) < λ . If M z = λz , then s ( M ) = λ . If M z > λz , then s ( M ) > λ . Lemma 6. [25, Prop. 1] Suppose that Λ is a nega-tive diagonal matrix in IR n × n and N is an irreduciblenonnegative matrix in IR n × n . Let M = Λ + N . Then, s ( M ) < if and only if ρ ( − Λ − N ) < , s ( M ) = 0 ifand only if ρ ( − Λ − N ) = 0 , and s ( M ) > if and only if ρ ( − Λ − N ) > . IV. E
QUILIBRIA OF E PIDEMIC -O PINION D YNAMICS
This section considers the mutual influence between theepidemic dynamics in (4) and opinion dynamics in (5). Inparticular, we introduce a compact form of the incorporatedsystem to explore the equilibria of the Epidemic-Opinionmodel. This section lays a foundation for analyzing thestability and convergence to the equilibria under differentconditions in the next section.We write (4) and (5) in a compact form as follows: (cid:20) ˙ x ( t )˙ o ( t ) (cid:21) = (cid:20) W ( o ( t )) I − Φ ( o ( t )) ¯ L u Φ ( o ( t )) (cid:21) (cid:20) x ( t ) o ( t ) (cid:21) − (cid:20) . e (cid:21) , (6)where W ( o ( t )) = − ( D min + ( D − D min ) ( O ( t ) + 0 . I ))+ ( I − X ( t )) ( B − ( O ( t ) + 0 . I ) ( B − B min )) and O ( t ) = diag { o ( t ) , . . . , o n ( t ) } , X = diag { x , . . . , x n } , D = diag { δ , . . . , δ n } , D min = δ min I , B = [ β ij ] ∈ IR n × n and B min = β min ˜ A , with ˜ A ∈ IR n × n being the unweighted adjacency matrix of graph G (with ˜ A ij ∈ { , } , ∀ i, j ∈ [ n ] ). Note that the Epidemic-Opinionmodel in (6) is a nonlinear system, due to the nonlinearitybrought by W ( o ( t )) and − Φ ( o ( t )) ¯ L u Φ ( o ( t )) . Thesystem in (6) follows: ˙ x ( t ) = − D ( o ( t )) x ( t ) + ( I − X ( t )) B ( o ( t )) x ( t ) (7) ˙ o ( t ) = Ix ( t ) − (cid:0) Φ ( o ( t )) ¯ L u Φ ( o ( t )) + I (cid:1) o ( t ) − . e , (8)where D ( o ( t )) = D min + ( D − D min ) ( O ( t ) + 0 . I ) , and B ( o ( t )) = B − ( O ( t ) + 0 . I ) ( B − B min ) capture theopinion-dependent healing and infection matrix, respectively. Remark 1.
Based on Assumption 1, B ( o ( t )) is an ir-reducible non-negative matrix and D ( o ( t )) is a positivedefinite diagonal matrix, ∀ t . It can be verified that thematrix W is a Metzler matrix. Since graph ¯ G is stronglyconnected and structurally balanced, the Laplacian matrix Φ ( o ( t )) ¯ L u Φ ( o ( t )) of graph ¯ G has only one zero eigen-value, and the rest of the eigenvalues of ¯ G have positivereal parts [13, Lemma 2]. Note that the gauge transformationdefined in Lemma 1 does not change the spectra of a matrix[13], and thus, Φ ( o ( t )) ¯ L u Φ ( o ( t )) and ¯ L u share the samespectra. Further, the matrix (cid:0) Φ ( o ( t )) ¯ L u Φ ( o ( t )) + I (cid:1) hasone eigenvalue located at one, and the rest of its eigenvalueshave positive real parts larger than one, which implies thatthe matrix − (cid:0) Φ ( o ( t )) ¯ L u Φ ( o ( t )) + I (cid:1) is Hurwitz, ∀ t . Notethat the opinion dynamics in (8) is a state-based switchingsystem. For each subsystem, the matrix Φ ( o ( t )) ¯ L u Φ ( o ( t )) is the standard signed Laplacian matrix. Now that we have introduced the Epidemic-Opinionmodel, to analyze the behavior of the system in (6), we needto show that the model is well-defined.
Lemma 7.
For the system defined in (6) , if x i ( t ) ∈ [0 , and o i ( t ) ∈ [ − . , . , ∀ i ∈ [ n ] , then x i ( t + τ ) ∈ [0 , and o i ( t + τ ) ∈ [ − . , . , ∀ i ∈ [ n ] , and ∀ τ ≥ .Proof. See Appendix.To explore the equilibria of (6), let z ( t ) = (cid:2) x T o T (cid:3) T denote the states of the system in (6), z ( t ) ∈ IR n , and z ∗ = (cid:104) ( x ∗ ) T ( o ∗ ) T (cid:105) T denote a equilibrium of (6), we say x ∗ and o ∗ are the equilibria of (7) and (8), respectively. Definition 2.
Let the state ( x ∗ , o ∗ ) denote an equilibrium of (6) , where ( x ∗ , o ∗ ) is a consensus-healthy state if ( x ∗ , o ∗ ) = ( , o ∗ ) , and o ∗ i = o ∗ j , ∀ i, j ∈ [ n ] ; a dissensus-healthy state if ( x ∗ , o ∗ ) = ( , o ∗ ) , and ∃ i, j ∈ [ n ] , s.t. o ∗ i (cid:54) = o ∗ j ; a consensus-endemic state if x ∗ ≥ , x ∗ (cid:54) = , and o ∗ i = o ∗ j , ∀ i, j ∈ [ n ] ; a dissensus-endemic state if x ∗ ≥ , x ∗ (cid:54) = , and ∃ i, j ∈ [ n ] , s.t. o ∗ i (cid:54) = o ∗ j . In this work, we use the term healthy state to describe bothcase 1) and case 2) in Definition 2, and the term endemicstate for both case 3) and case 4). It is obvious that theEpidemic-Opinion model in (6) has a consensus-healthy state ( , − . e ) as its trivial equilibrium. Further, from (8), when o ∗ = − . e , = x ∗ + (cid:0) ¯ L u + I (cid:1) × . e − . e = x ∗ + ¯ L u × . e + 0 . e − . e = x ∗ , which indicates x ∗ = . Therefore, the consensus state o ∗ = − . e must pair with the healthy state x ∗ = . The followingtheorem summarizes the healthy equilibria of (6). Theorem 1.
For the Epidemic-Opinion model in (6) , ahealthy equilibrium z ∗ is either the unique consensus-healthy state ( x ∗ = , o ∗ = − . e ) , or a dissensus-healthy state ( x ∗ = , o ∗ ) , with o ∗ = (cid:0) Φ ( o ∗ ) ¯ L u Φ ( o ∗ ) + I (cid:1) − ( − . e ) with both positive and negative entries.Proof. See Appendix.
Corollary 1.
For the Epidemic-Opinion model in (6) , theconsensus-healthy state ( x ∗ = , o ∗ = − . e ) is the uniqueequilibrium with consensus in opinions, that is, o = α e , α ∈ [ − . , . .Proof. See Appendix.Note that if ∃ i ∈ [ n ] , s.t. o ∗ i = 0 , from (2), it must be truethat o ∗ i = 0 , ∀ i ∈ [ n ] . However, Corollary 1 states that, z ∗ does not include the case that o ∗ = . Therefore, we havethe next corollary. Corollary 2.
For the Epidemic-Opinion model in (6) , theequilibrium z ∗ cannot include o ∗ i = 0 , ∀ i ∈ [ n ] . Remark 2.
Theorem 1 indicates the possible equilibriawhen the epidemic dies out. The consensus-healthy state ( x ∗ = , o ∗ = − . e ) implies that, at the stage that theepidemic disappears, all communities agree that the epi-demic is not a threat. However, the existence of dissensus-healthy states ( x ∗ = , o ∗ ) , with o ∗ having both positive andnegative entries, describes the scenario when communitieshold different beliefs towards the epidemic at the time whenthe epidemic is about to disappear, and thus the epidemicwill still cause possible contention between different com-munities. Corollary 1 states that the only way to ensure allcommunities reach agreement is that they all agree that theepidemic is not a threat at the moment the epidemic dies out.In other words, it is impossible for all communities to agreethat the epidemic is not worth treating seriously while theepidemic is still spreading. After analyzing the healthy equilibria, the followinglemma further explores the endemic equilibria of (6).
Lemma 8. If ( x ∗ , o ∗ ) is an endemic equilibrium of thesystem in (6) , then (cid:28) x ∗ (cid:28) e , − . e (cid:28) o ∗ (cid:28) . e .Proof. See Appendix.
Remark 3.
Lemma 8 states that if an endemic state exists,no community can be completely infection free or completelyinfected. Further, the equilibrium of a community can neverbe equal to the extreme scenario of its beliefs, which meansthe epidemic has an influence on the opinions of all commu-nities.
V. S
TABILITY A NALYSIS OF E PIDEMIC -O PINION D YNAMICS
In this section, we analyze the properties of the equilibriaof the Epidemic-Opinion model in (6), to reveal real-worldphenomena on the mutual influence between disease spreadand opinion formation during an epidemic.As mentioned in [5], the reproduction number R of anepidemic is critical in determining the spreading of the epidemic. In line with the expression on the reproductionnumber R , we define an Opinion-Dependent ReproductionNumber R ot to characterize the performance of the Epidemic-Opinion model in (6). Definition 3. [Opinion-Dependent Reproduction Number]Let R ot = ρ (cid:16) D ( o ( t )) − B ( o ( t )) (cid:17) denote the Opinion-Dependent Reproduction Number, where D ( o ( t )) and B ( o ( t )) are defined in (7) . Note that the Opinion-Dependent Reproduction Number R ot depends on the variation of the opinion states o ( t ) . Whenall communities think the epidemic is extremely serious, bydefining o max = 0 . e , we have R min = ρ (cid:16) D ( o max ) − B ( o max ) (cid:17) = ρ (cid:0) D − B (cid:1) . Instead, when all communities believe that the epidemic isnot real, by defining o min = − . e , we have R max = ρ (cid:16) D ( o min ) − B ( o min ) (cid:17) = ρ (cid:0) D − B min (cid:1) . Proposition 1.
The Opinion-Dependent Reproduction Num-ber R ot has the following properties: If o ( t ) ≤ o ( t ) , then R ot ≥ R ot , and vice versa; R min ≤ R ot ≤ R max .Proof. See Appendix.Proposition 1 indicates that the opinions towards theepidemic provide bounds on the Opinion-Dependent Repro-duction Number R ot . The more seriously community i treatsthe epidemic, i.e., with a higher o ( t ) , the lower R ot is, andvice versa. In the following sections, we interpret R ot asthe severity of an epidemic, and explore the behavior of (6)through bounds on R ot . A. Mild Viruses
In this section we explore the behavior of viruses that areonly slightly contagious, that is, where R ot ≤ R max ≤ .First we analyze the equilibrium of the system in (6) underthe condition that R max ≤ . Proposition 2. If R max ≤ , then any equilibrium of (6) isa healthy state.Proof. See Appendix.Proposition 2 claims that when R max ≤ , i.e., the severityof the epidemic is fairly mild, the Epidemic-Opinion modelin (6) has only healthy equilibria, despite the evolution of theopinions. To further explore the behavior of all the healthyequilibria of (6), we derive the Jacobian matrix df x,o of (6)evaluated at ( x, o ) as follows: (cid:20) W − ˜ V ( x, o ) − ( D − D min ) X − ( I − X ) ˜ BI − (cid:0) Φ ( o ) ¯ L u Φ ( o ) + I (cid:1) − ˜∆ (cid:21) , where ˜ V ( x, o ) and ˜ B are diagonal matrices with ( B − ( O + 0 . I ) ( B − B min )) x and ( B − B min ) x ,on their diagonals, respectively, and ˜∆ = (cid:0) ∆ (cid:0) ¯ L u + I (cid:1) Φ ( o ) o + Φ ( o ) (cid:0) ¯ L u + I (cid:1) ∆ o (cid:1) , with the Diracdelta function θ ( · ) and ∆ = diag { θ ( o ) , . . . , θ ( o n ) } .From Corollary 2, we have o ∗ i (cid:54) = 0 , ∀ i ∈ [ n ] , for allequilibria of (6). Therefore, ˜∆ = when evaluating at allthe equilibria, due to θ ( o i ) = 0 , when o i (cid:54) = 0 , ∀ i ∈ [ n ] .We evaluate the Jacobian matrix at all healthy equilibria, ( x = , o ∗ ) , df ,o ∗ = (cid:20) W I − (cid:0) Φ ( o ∗ ) ¯ L u Φ ( o ∗ ) + I (cid:1) (cid:21) . (9)Note that for each equilibrium with its opinion formation o ∗ , when all of the opinion states are evolving closely enoughto o ∗ , the gauge transformation matrix Φ ( o ∗ ) is fixed.From Remark 1, the spectrum of − (cid:0) Φ ( o ∗ ) ¯ L u Φ ( o ∗ ) + I (cid:1) is the same as − (cid:0) ¯ L u + I (cid:1) . Further, for any opinions o ∗ ,the matrices are Hurwitz. Hence, the stability of the systemdepends on the spectrum of W ( o ( t )) . From Lemma 6, s ( − ( D min + ( D − D min ) ( O ∗ + 0 . I ))+ ( B − ( O ∗ + 0 . I ) ( B − B min ))) < if and only if ρ (( D min + ( D − D min ) ( O ∗ + 0 . I )) − × ( B − ( O ∗ + 0 . I ) ( B − B min ))) < , Further, by Proposition 1, ρ (( D min + ( D − D min ) ( O ∗ + 0 . I )) − × ( B − ( O ∗ + 0 . I ) ( B − B min ))) ≤ R max ≤ , we have s ( W ( o ( t ))) ≤ . Hence, the Jacobian matricesevaluated at healthy equilibria are Hurwitz if R max < ,leading to the results, by Lyapunov’s indirect method. Proposition 3. If R max < , then all the healthy equilibria ( , o ∗ ) of (6) are locally exponentially stable. Theorem 2. If R max ≤ , then for any initial condition,the system in (6) will asymptotically converge to a healthyequilibrium. If R max < , the convergence is exponentiallyfast.Proof. See Appendix.
Remark 4.
Proposition 3 and Theorem 2 reveal that, underthe condition that R max ≤ , the initial conditions of theepidemic (i.e., the level of the infection in each community)and/or the opinion states (i.e., how much the communitiesunderestimate the severity of the epidemic), will not hinderthe epidemic from disappearing quickly. The phenomena ismainly because R max ≤ , indicating that the epidemic is nota threat. Hence, it is unnecessary to broadcast the severity ofthe epidemic publicly, since the epidemic will die out quickly.The initial condition may affect the opinions after theepidemic disappears. Imagine the case where few infectionsappear in each community and no community believes theepidemic is serious at the beginning. Then, the epidemicwill disappear and all communities will reach a consensusthat the epidemic is not a threat, captured by the uniqueconsensus-healthy equilibrium. Alternatively, consider the case where some communities are heavily infected at thebeginning, hence they believe the epidemic is a threat.Even after the epidemic dies out quickly, disagreement willlinger between communities, corresponding to the dissensus-healthy equilibria.B. Severe Viruses In this section we explore the behavior of viruses thatare very contagious, that is, where R min > . Note thatTheorem 2 demonstrates that the healthy states ( , o ∗ ) , areequilibria for (6) under any R ot , s.t. R max ≤ .¯ Therefore,we consider the stability properties of the healthy equilibria. Lemma 9.
Under the condition that R min > , all thehealthy equilibria ( , o ∗ ) are unstable.Proof. See Appendix.
Remark 5.
Lemma 9 states that, when the Opinion-Dependent Reproduction Number is too large, R min > , i.e.,the epidemic is highly contagious, even if all communitieshave zero infection, one infected person appearing in anycommunity will result in an outbreak, leading to a pandemic.Further, with all communities being extremely cautious aboutthe epidemic ( o ∗ = 0 . e ), taking every action suggestedby the scientific institutions, public health officials, and themedia to protect themselves, the epidemic will continuespreading. Therefore, relying only on non-pharmaceuticalInterventions (NPIs) through social media, it would beimpossible to eradicate the epidemic After studying the healthy equilibria in Lemma 9, weexplore the existence of the endemic equilibria under thecondition that R min > . Lemma 8 claims that, if it exists,the endemic state ( x ∗ , o ∗ ) must satisfy e (cid:29) x ∗ (cid:29) , . e (cid:29) o ∗ (cid:29) − . e . Since ( − D + B min ) is an irreducible Metzlermatrix, ρ ( D − B min ) > implies s ( − D + B min ) > . FromLemma 5, let φ (cid:44) s ( − D + B min ) be the eigenvalue of ( − D + B min ) with an associated right eigenvector y (cid:29) .Without loss of generality, assume max i y i = 1 . Now, definefor any (cid:15) ∈ [0 , , a convex and compact subset of χ as Ξ (cid:15) (cid:44) { z ∈ χ : x i ≥ (cid:15)y i ∀ i ∈ [ n ] } , (10)where χ = { z ∈ IR n | z i ∈ [0 , , i = 1 , . . . , n ; z i ∈ [ − . , . , i = n + 1 , . . . , n } . Note that Ξ = χ and ∀ (cid:15) > , Ξ (cid:15) ⊂ χ . From the proof ofLemma 7 and the piece-wise continuity of the system in (6),we have the following results. Lemma 10.
Consider the system in (6) . If z ( t ) ∈ ∂χ \ ( , o ∗ ) ,where ( , o ∗ ) are the healthy equilibria of (6) , then z ( t + τ ) ∈ Int χ , ∀ τ ≥ . Theorem 3.
Suppose that R min > . Then, there exists asufficiently small ¯ (cid:15) such that Ξ (cid:15) defined in (10) for every (cid:15) ∈ (0 , ¯ (cid:15) ] is a positive invariant set for the system in (6) .Moreover, (6) has at least one endemic equilibrium in χ .Proof. Given the result of Lemma , it follows that thepositive invariance of Ξ (cid:15) is established if we can prove that, for all i ∈ [ n ] , ˙ x i > whenever x i = (cid:15)y i and x j ∈ [ (cid:15)y j , for j (cid:54) = i . Toward that end, observe from (4) that ˙ x i = − ( δ min (0 . − o i ) + δ i ( o i + 0 . (cid:15)y i + (1 − (cid:15)y i ) × (cid:88) j ∈N i ( β ij (0 . − o i ) + β min ( o i + 0 . x j − (cid:15)y j + (cid:15)y j ) , (11)and note that we have dropped the argument t for brevity.Since x j − (cid:15)y j ≥ and > (cid:15)y i > by hypothesis, (1 − (cid:15)y i ) (cid:88) j ∈N i ( β ij (0 . − o i )+ β min ( o i + 0 . x j − (cid:15)y j ) ≥ . This implies that (11) obeys the following inequality: ˙ x i ≥ − ( δ min (0 . − o i ) + δ i ( o i + 0 . (cid:15)y i + (cid:88) j ∈N i ( β ij (0 . − o i ) + β min ( o i + 0 . (cid:15)y j − (cid:15) y i (cid:88) j ∈N i ( β ij (0 . − o i ) + β min ( o i + 0 . y j . (12)Note that φy = ( − D + B min ) y implies that δ i y i + (cid:80) j ∈N i β min y j = φ i y i . Based on Assumption 1, δ i ≥ δ min and β ij ≥ β min for j ∈ N i . Therefore, we obtain − δ min y i + (cid:88) j ∈N i β ij y j ≥ − δ i y i + (cid:88) j ∈N i β min y j = φ i y i . (13)Since o i ∈ [ − . , . , it follows from (13) that ( z i + 0 .
5) ( − δ i y i + (cid:88) j ∈N i β min y j )+ (0 . − z i )( − δ min y i + (cid:88) j ∈N i β ij y j ) ≥ φ i y i > . (14)Using (10), the right-hand side of (12) can then be furtherbounded as ˙ x i ≥ (cid:15)φy i − (cid:15) y i (cid:88) j ∈N i ( β ij (0 . − z i ) + β min ( z i + 0 . y j . Obviously, for some sufficiently small (cid:15) i > , we then have ˙ x i ≥ (cid:15) i φy i − (cid:15) i y i (cid:88) j ∈N i ( β ij (0 . − z i )+ β min ( z i + 0 . y j > . By setting ¯ (cid:15) = min i (cid:15) i , we conclude that Ξ (cid:15) for every (cid:15) ∈ (0 , ¯ (cid:15) ] is a positive invariant set of (6). Since Ξ (cid:15) , for (cid:15) ∈ (0 , ¯ (cid:15) ] ,is compact and convex, the system in (6) is Lipschitz smoothin Ξ (cid:15) under each switching subsystem in (8). Therefore, theresult in [26, Lemma 4.1] immediately establishes that anysystem in (6) paired with one switching subsystem of (8)has at least one equilibrium in Ξ (cid:15) . Taking (cid:15) to be arbitrarilysmall, and Lemma 10 establishes that the system in (6) hasat least one equilibrium in Int χ . Therefore, the system in (6)can have more than one endemic equilibrium.Combined with Theorem 1, Theorem 3 states that, when R min > , the system in (6) has both healthy and endemicequilibria. Lemma 9 shows the healthy equilibria are un-stable. Complete analysis of the stability of the endemicequilibria is a subject of our future work. C. Moderate Viruses
Previous sections show that the opinion states can providelittle help when the epidemic is either highly contagious( R min > ) or very mild ( R max < ). In this sectionwe explore the behavior of viruses that are only slightlycontagious, that is, where R min < and R max > , andshow that the influence of stubborn communities on opinionstates can impact the behavior of the epidemic.Based on Proposition 1, R min ≤ R ot ≤ R max . If R min < and R max > , there must be an R ot ≈ . Therefore, thesystem in (6) may contain properties that both cases R min > and R max < have. Since the range of R ot depends on theopinion state o ( t ) , for healthy equilibria, we evaluate R o ∗ t regarding the opinion state o ∗ at the healthy equilibrium.Thus, we have the following results. Theorem 4.
For the system in (6) , if R min < , and R max > , the following statements hold: All the dissensus-healthy equilibria ( , o ∗ ) satisfying R o ∗ t < , ∀ t are locally exponentially stable; All the dissensus-healthy equilibria ( , o ∗ ) satisfying R o ∗ t > , ∀ t are unstable; The consensus-healthy equilibrium ( , − . e ) is un-stable.Proof. See AppendixTheorem 4 implies that the local stability of the healthyequilibria depends on R o ∗ t . Moreover, one might never finda locally stable healthy equilibria, if no opinion state ofthe dissensus-equilibria satisfies Case 1) in Theorem 4. Toanalyze the stability of the healthy equilibria through theopinion states, we introduce the threshold opinion states. Corollary 3. If R min < and R max > , there mustexist one threshold opinion vector ¯ o such that R ¯ ot ≤ , andanother threshold opinion vector ˆ o , such that R ˆ ot > . Corollary 3 is a direct result of Proposition 1; note thatthere could be more than one ˆ o and/or ¯ o satisfying thethreshold condition. Further, if o ( t ) > ¯ o , ∀ t , then R ot < , ∀ t . Instead, if o ( t ) < ˆ o , ∀ t , then R ot > , ∀ t . From Lemma 3,we can capture the stability of healthy equilibria in Theorem4 through ˆ o and ¯ o . Corollary 4.
For the system in (6) , if R min < and R max > , the dissensus-healthy equilibrium ( , o ∗ ) satisfying o ∗ (cid:29) ¯ o are locally exponentially stable, while equilibria ( , o ∗ ) satisfying o ∗ (cid:28) ˆ o are unstable. Remark 6.
Case 3) of Theorem 4 and Corollary 4 implythat, when the epidemic is moderate, the epidemic cannotbe eradicated if all communities ignore it ( o ∗ = − . e ) ordo not treat it seriously enough ( o ∗ (cid:28) ˆ o ). Further, Case 1)of Theorem 4 and Corollary 4 imply that the epidemic willdisappear when all communities believe that the epidemic issevere past a certain degree ( o ∗ (cid:29) ¯ o ). The opinion thresholds ˆ o and ¯ o connect the stability of thehealthy equilibria to o ( t ) . Note that the system in (6) mighthave no stable healthy equilibrium. Assuming the system in (6) has at least one locally stable healthy equilibrium, fromLemma 3, there must exist an ¯ o , s.t. o ∗ ≥ ¯ o . Therefore, toeradicate the epidemic, we can employ external influenceon the communities to drive o ( t ) above ¯ o . Consider theexistence of stubborn communities in (8), i.e., the opinionsof the communities are not influenced by their neighbors [27,Eq. (7)]. The following theorem captures the behavior of (6)with stubborn communities. Theorem 5.
For the system in (6) , if R min < and R max > , the stubborn communities driving o ( t ) (cid:29) ¯ o , ∀ t ≥ , willensure that the system in (6) converges to the set of healthyequilibria. The proof of Theorem 5 is similar to the proof of Theo-rem 2, except that the opinion states are maintained abovethe threshold ¯ o instead of converging to an endemic state. Remark 7.
Theorem 5 states, when the epidemic is moder-ate, we can eradicate it by selecting stubborn communities todrive the opinions of all communities above the threshold ¯ o .The situation implies that, by broadcasting the severity of theepidemic to some target communities, we can influence all ofthe communities’ beliefs towards the epidemic. In particular,when all communities’ opinions are driven above a threshold,i.e., all the communities consider the epidemic somewhatserious, they will take the proper actions to end the epidemic. Theorem 5 shows the role of stubborn communitiesin epidemic suppression. However, optimally selecting theproper stubborn communities and designing external controlsignals to influence the stubborn communities is challenging.Hence, by Proposition 1, we explore a particular method thatconsiders stubborn communities with fixed opinions statesequaling to . , ∀ t . The following result provides a way ofselecting extreme stubborn communities for a particular case. Corollary 5.
Consider an opinion vector (cid:126)o with both positiveand negative entries, (cid:126)o i ∈ {− . , . } , ∀ i ∈ [ n ] . If ∃ (cid:126)o , s.t. R (cid:126)ot < , ∀ t ≥ , the system in (6) can reach a healthy stateby setting o i ( t ) = 0 . , ∀ t ≤ , ∀ i satisfying (cid:126)o i = 0 . .Proof. By selecting stubborn communities with opinionstates fixed at . by Corollary (5), from Proposition 1,the system in (6) satisfies R (cid:126)ot < , ∀ t ≥ . Since all non-stubborn communities will have their opinion states greateror equal than − . , from Proposition 1, R t < , ∀ t ≥ .Thus, the system in (6) converges to a healthy state.Corollary 5 offers a way of selecting communities tomake them stubborn in order to suppress the epidemic. Theelements of the opinion vector (cid:126)o i ∈ {− . , . } can beadjusted to generate different combinations of stubborn com-munities and opinions, e.g., exploring stubborn communitiesthrough Corollary 5 with the condition (cid:126)o i ∈ {− . , o } , with o = 0 . α , α ∈ ( − , being the stubborn state. Remark 8.
Corollary 5 reveals the role of stubborn com-munities in determining the behavior of the epidemic. Whenan epidemic is moderate ( R min < and R max > ),an appropriate choice of the stubborn communities withthe most serious attitudes towards the epidemic could help Fig. 1: The graph for the epidemic and opinion interactions. the whole network eradicate the epidemic. Further, we canadd external stubborn nodes to the opinion graph ¯ G , tomodel and study the impact of external media on epidemicspreading. Moreover, the method proposed by Corollary can also capture the influence of stubborn communities andmedia with negative attitudes, which may insist the epidemicis a hoax and treat it less seriously. VI. S
IMULATIONS
In this section, we illustrate the main results through thefollowing examples. Consider an epidemic spreading overten communities, with the epidemic and opinion spreadingthrough the same network satisfying Assumption 1, capturedby the graph G in Fig. 1. Note that we use the same graphstructure in G to capture the epidemic and opinion graphs tosimplify the simulation, and our results still apply to com-munities with different epidemic and opinion interactions.First, we consider the case that the epidemic is mild,which indicates R max ≤ . By generating parameters ofthe infection and healing rates randomly, we obtain R min =0 . , R max = 0 . . Consistent with Proposition 2 andTheorem 2, when R max ≤ , i.e., the epidemic is mild, allof the communities reach healthy states with zero infections,illustrated in Fig. 2 (a) and (c). Further, Theorem 2 statesthat the system either reaches the consensus-healthy state ( , − . e ) , where all of the communities agree that theepidemic is not serious, illustrated in Fig. 2 (a) and (b),or a dissensus-healthy state ( , o ∗ ) , with the communitiesholding both positive and negative opinion states towardsthe epidemic, illustrated in Fig. 2 (c) and (d).Fig. 3 illustrates the situation where the epidemic is severe,characterized by R min > . Through randomly generatingparameters satisfying the condition, we have R min = 1 . , R max = 2 . . Consistent with Lemma 9, none of thecommunities can ever reach a healthy state. Fig. 3 (a) alsoimplies the existence of an endemic equilibrium, illustratingTheorem 3. Fig. 3 (b) shows that all the ten communitiesreach dissensus. Note that the continuity at the opinion-switching points and the Lipschitz continuity between theopinion-switching points can be observed from the example. Fig. 2: Under the condition R min = 0 . , R max = 0 . ,with the graph of ten communities from Fig. 1, the commu-nities reach the healthy state: (a) epidemic states converge tozero, (b) opinion states reach consensus, (c) epidemic statesconverge to zero, (d) opinion states reach dissensus.Fig. 3: The communities under the condition R min = 1 . , R max = 2 . , with the graph of ten communities from Fig.1, reach a dissensus-endemic state: (a) epidemic states reachendemic equilibrium, (b) opinion states reach dissensus.Lastly, we consider the case where the epidemic is severebut still under control. Thus, we generate parameters leadingto R min = 0 . < , R max = 2 . > to characterizethis situation. As described in Theorem 4, the stabilityof the healthy equilibria depends on the opinion states.More importantly, from Theorem 5 and Corollary 5, we canappropriately select stubborn communities to eradicate theepidemic. Fig. 4 illustrates the role of stubborn communitiesin the epidemic suppression, showing the same system undertwo different settings. The system captured by Fig. 4 (a) and(b) reaches a dissensus-endemic state. With the exact sameconditions, we fix the opinion states of the communities 1,6, and 9 as [ o ( t )] = [ o ( t )] = [ o ( t )] = 0 . , ∀ t ≥ ,which means community 1, 6, and 9 always believe thatthe epidemic is extremely severe, and the opinions of othercommunities will not impact their beliefs in the severity ofthe epidemic. Meanwhile, communities 1, 6, and 9 keepbroadcasting the information that the epidemic is very severeto their neighbors. Through this setting, compared to the Fig. 4: All of the communities under the condition R min =0 . , R max = 2 . , with the graph of ten communities fromFig. 1: (a) epidemic states reach endemic equilibrium, (b)opinion states without stubborn opinions, (c) epidemic statesconverge to zero, (d) opinion states with stubborn opinions.same system captured by Fig. 4 (a) and (b), Fig. 4 (c)and (d) show that all the communities reach the healthystate, illustrating the results derived in Theorem 4, Theorem5, and Corollary 5, that appropriate selection of stubborncommunities with their cautious opinions broadcasting toother communities can suppress the epidemic.VII. C ONCLUSION
This work studies the mutual influence between epidemicspreading and opinion dissemination over connected com-munities. By defining an Opinion-Dependent ReproductionNumber, our work reveal the behavior of the Epidemic-Opinion model in (6). Our work also illustrates the role ofstubborn communities in epidemic eradication. The resultsof this work pave the way for a more detailed analysis ofthe extended models. In this work, the stability analysis ofthe endemic states under the condition that R min > stillneeds to be further explored. The study of the endemic statescould reveal the impact of opinions on the most seriousepidemics. Next, except for using stubborn opinions, onecould consider control design to shape the opinions of thecommunities to eradicate the epidemic. Further extensionsof the results in this work consist of validating the opinion-dependent sign switching network model with real-worlddata and extending the ideas to couple opinion dynamicswith the SIR (susceptible-infected-recovered) model.R EFERENCES[1] W. O. Kermack and A. G. McKendrick, “A contribution to themathematical theory of epidemics,”
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Nonlinear Systems . Prentice Hall, 2002. A PPENDIX I Proof of Lemma 7:
Consider the system captured by (7)and (8). Note that the system in (7) is a group of polynomialODEs over the compact set [0 , n . Between the switchingpoints, each subsystem in (8) is a group of polynomialODEs over the compact set [ − . , . n . Therefore, foreach subsystem of (8) paired with (7), the system (7) isLipschitz on [0 , n and each subsystem of (8) is Lipschitzon [ − . , . n . It can be verified that the solutions at theswitching points of (8) are continuous. Hence, the solutions x i ( t ) and o i ( t ) of (7) and (8) are continuous, ∀ i ∈ [ n ] ,respectively.Suppose there is an index i ∈ [ n ] such that x i ( t ) is thefirst state to reach zero at t , while the rest of the states x j ∈ Int [0 , n and o j ∈ Int [ − . , . n , ∀ j ∈ [ n ] , i (cid:54) = j .Based on (4) and Assumption , ˙ x i ( t ) = (cid:88) j ∈N i [ β ij − ( β ij − β min ) ( o i ( t ) + 0 . x j ( t ) ≥ . Hence, ˙ x i ( t ) ≥ indicates x i ( t ) cannot drop below zerowhen being the first to reach zero. The same statementshold for the situations where more than one of the epidemicstates reach ∂ [0 , n , simultaneously. Following the sameprocedure, we can verify that x ( t ) ≤ e , ∀ t ≥ . Considerthe opinion dynamics in (8). From the same analysis, it canbe verified that o i ∈ [ − . , . , ∀ t ≥ , ∀ i ∈ [ n ] . Proof of Theorem 1:
We first show that the healthy state x ∗ = can only pair with the unique consensus state o ∗ = − . e . Recall the definition of Φ ( o ( t )) in SectionII-B; if o ∗ i = o ∗ j , ∀ i, j ∈ [ n ] , then Φ ( o ∗ ) = I . From (8), theequilibria of the opinion dynamics satisfy − (cid:0) ¯ L u + I (cid:1) o ∗ = 0 . e . (15)Thus we have that − . e is an eigenvector of the matrix − (cid:0) ¯ L u + I (cid:1) paired with the largest eigenvalue − . From Re-mark 1, − (cid:0) ¯ L u + I (cid:1) is nonsingular, and thus o ∗ = − . e isthe unique solution of (15). Therefore, ( x ∗ = , o ∗ = − . e ) is the unique consensus-healthy equilibrium of (6).For dissensus-healthy states, ( x ∗ = , o ∗ ) , if o ∗ (cid:29) or o ∗ (cid:28) , which implies Φ ( o ∗ ) = I , the equilibriumof the opinion dynamics in (8) becomes − (cid:0) ¯ L u + I (cid:1) o ∗ =0 . e , which has only − . e , the consensus state, as itssolution. Therefore, o ∗ in ( x ∗ = , o ∗ ) must have bothpositive and negative entries. Based on the fact that (cid:0) Φ ( o ( t )) ¯ L u Φ ( o ( t )) (cid:1) is a nonsingular matrix, the equation (cid:0) Φ ( o ∗ ) ¯ L u Φ ( o ∗ ) + I (cid:1) o ∗ = x ∗ − . e , (16)has a unique solution for each (Φ ( o ∗ )) , given by o ∗ = (cid:0) Φ ( o ∗ ) ¯ L u Φ ( o ∗ ) + I (cid:1) − ( − . e ) , when x ∗ = . Now weshow that each solution must satisfy . e ≥ o ∗ ≥ − . e .Assume that [Φ ( o ∗ )] ii = 1 , ∀ i ∈ { , . . . , m } and [Φ ( o ∗ )] jj = − , ∀ j ∈ { m + 1 , . . . , n } . Let ¯ L =Φ ( o ∗ ) ¯ L u Φ ( o ∗ ) . Suppose to the contrary (without loss ofgenerality) that o ∗ < − . . Based on the assumption, o ∗ i < , ∀ i ∈ { , . . . , m } , while o ∗ j ≥ , ∀ j ∈ { m + 1 , . . . , n } .Considering the first row of (16), we have − (cid:12)(cid:12)(cid:2) ¯ L (cid:3) (cid:12)(cid:12) o ∗ − o ∗ + m (cid:88) i =2 (cid:12)(cid:12)(cid:2) ¯ L (cid:3) i (cid:12)(cid:12) o ∗ i − n (cid:88) j = m +1 (cid:12)(cid:12)(cid:12)(cid:2) ¯ L (cid:3) j (cid:12)(cid:12)(cid:12) o ∗ j = 0 . . For (cid:12)(cid:12)(cid:2) ¯ L (cid:3) (cid:12)(cid:12) = (cid:80) nk =2 (cid:12)(cid:12)(cid:2) ¯ L (cid:3) k (cid:12)(cid:12) , we have − m (cid:88) i =2 (cid:12)(cid:12)(cid:2) ¯ L (cid:3) i (cid:12)(cid:12) o ∗ − n (cid:88) j = m +1 (cid:12)(cid:12)(cid:2) ¯ L j (cid:3)(cid:12)(cid:12) o ∗ + m (cid:88) i =2 (cid:12)(cid:12)(cid:2) ¯ L (cid:3) i (cid:12)(cid:12) o ∗ i − n (cid:88) j = m +1 (cid:12)(cid:12)(cid:2) ¯ L j (cid:3)(cid:12)(cid:12) o ∗ j − o ∗ = 0 . . Note that if o ∗ i ≥ o ∗ , ∀ i ∈ { , . . . , m } , and o ∗ j ≤ − o ∗ , ∀ j ∈ { m + 1 , . . . , n } , the left side of the equation abovemust be greater than . . Therefore, there must exist at leastone o ∗ i , i ∈ { , . . . , m } , satisfying o ∗ i < o ∗ and/or at leastone o ∗ j , j ∈ { m + 1 , . . . , n } , satisfying o ∗ j > − o ∗ . Supposethat o ∗ < o ∗ , then for the second row of (16), the samestatement holds, that there must exist at least one o ∗ i , i ∈{ , . . . , m } , satisfying o ∗ i < o ∗ < o ∗ , and/or at least one o ∗ j , j ∈ { m + 1 , . . . , n } , satisfying o ∗ j > − o ∗ > − o ∗ . Followingthis procedure, for the last row corresponding to the lastentry in o ∗ , we can no longer find any entries satisfyingthe condition. Therefore, no solution of (16) can be smallerthan − . . A similar process can be applied to show thatno solution of the equation (16) can be an element largerthan . .Therefore, for each sign pattern of Φ ( o ∗ ) , the sys-tem in (6) has one unique dissensus-healthy state o ∗ = (cid:0) Φ ( o ∗ ) ¯ L u Φ ( o ∗ ) + I (cid:1) − ( − . e ) . Proof of Corollary 1:
Theorem 1 shows that the consensus-healthy state ( x ∗ = , o ∗ = − . e ) , is the unique equilib-rium when x ∗ = . Therefore, we need to show that thereexists no consensus-endemic state that is an equilibrium.Suppose to the contrary that ∃ o , s.t. o = α e , α ∈ ( − . , . .Based on (16), x = α e + 0 . e . Substituting ( x, o ) in (8), wehave ˙ o = α e + 0 . e − (cid:0) I ¯ L u I + I (cid:1) × . e − . e = α e − . e ≤ . The inequality becomes an equality only under the con-dition that ( x = e , o = 0 . e ) . However, by substituting ( x = e , o = 0 . e ) into (7), we have ˙ x < . Thus, ( x, o ) can-not be an equilibrium of (6). Therefore, by contradiction, thehealthy-consensus state ( x ∗ = , o ∗ = − . e ) is the uniqueequilibrium with consensus in opinions. Proof of Lemma 8:
First we show that x ∗ (cid:29) and o ∗ (cid:29)− . e . Suppose to the contrary that ∃ i ∈ [ n ] , s.t. x ∗ i = 0 , o ∗ i = − . , while x ∗ j (cid:54) = 0 , o ∗ j (cid:54) = − . , for all other j ∈ [ n ] , ∀ i (cid:54) = j . Based on the proof of Lemma 7, ˙ x ∗ i = 0 at x ∗ i = 0 ,and ˙ o ∗ i = 0 at o ∗ i = − . , unless x ∗ j = 0 , o ∗ j = − . , forall other j ∈ [ n ] . Therefore, ( x ∗ , o ∗ ) must satisfy x ∗ (cid:29) and o ∗ (cid:29) − . e . Similar method can be applied to showthe case that x ∗ (cid:28) e and o ∗ (cid:28) . e . Proof of Proposition 1:
1) Based on Assumption1, D ( o ( t )) is a positive definite diagonal matrix and B ( o ( t )) is an irreducible nonnegative matrix, ∀ t . Hence, D ( o ( t )) − B ( o ( t )) is an irreducible nonnegative matrix.Consider the case where, ∃ o i ( t ) < o i ( t ) , i ∈ [ n ] , t > t > , while o j ( t ) = o j ( t ) , ∀ i, j ∈ [ n ] , i (cid:54) = j .Recall that D ( o ( t )) = D min + ( D − δ min I ) ( O ( t ) + 0 . I ) and B ( o ( t )) = B − ( O ( t ) + 0 . I ) ( B − B min ) . Basedon o i ( t ) < o i ( t ) , we have D ii ( o ( t )) < D ii ( o ( t )) ,leading to D − ii ( o ( t )) > D − ii ( o ( t )) , and B i, : ( o ( t )) >B i, : ( o ( t )) , while the rest of D − ( o ( t )) and B ( o ( t )) areequal to D − ( o ( t )) and B ( o ( t )) . Hence, o i ( t ) < o i ( t ) leads to (cid:104) D ( o ( t )) − B ( o ( t )) (cid:105) i, : > (cid:104) D ( o ( t )) − B ( o ( t )) (cid:105) i, : . From the third statement of Lemma 5, we have ρ (cid:104) D ( o ( t )) − B ( o ( t )) (cid:105) > ρ (cid:104) D ( o ( t )) − B ( o ( t )) (cid:105) , which means R ot > R ot . The same method can verify thecase that o ( t ) ≥ o ( t ) , then R ot ≤ R ot .2) This statement is two special cases of 1). Since − . e ≤ o ( t ) ≤ . e , when o ( t ) = o min = − . e , ∀ i ∈ [ n ] , based on the first statement of Proposition 1, ρ (cid:16) D ( o ( t )) − B ( o ( t )) (cid:17) ≤ ρ (cid:104) D ( o min ) − B ( o min ) (cid:105) = ρ (cid:0) D − B (cid:1) = R max . When o ( t ) = o max = 0 . e , ∀ i ∈ [ n ] , ρ (cid:16) D ( o ( t )) − B ( o ( t )) (cid:17) ≥ ρ (cid:16) D ( o max ) − B ( o max ) (cid:17) = ρ (cid:0) D − B min (cid:1) = R min . Proof of Proposition 2:
Suppose to the contrary that thereis an endemic state ( x, o ) as the equilibrium of (6) under thecondition that R max ≤ . By Lemma 8, it must be true that x (cid:29) . Since ( x, o ) is an equilibrium, from (7), ( − D min + B ) x = XBx + ( O + 0 . I ) ( D − D min ) x + ( I − X ) (( O + 0 . I ) ( B − B min )) x. By Assumption 1, both ( B − B min ) and ( I − X ) (( O + 0 . I ) ( B − B min )) are nonnegative andirreducible, and ( O + 0 . I ) ( D − D min ) is a positivedefinite diagonal matrix. Hence, since x (cid:29) , wehave XBx (cid:29) , ( O + 0 . I ) ( D − D min ) x (cid:29) , ( I − X ) (( O + 0 . I ) ( B − B min )) x (cid:29) . Therefore, ( − D min + B ) x (cid:29) .Recall that ( − D min + B ) is an irreducible nonnegativematrix; from Lemma 5, s ( − D min + B ) > . However, byLemma 6, s ( − D min + B ) > leads to ρ ( D − B ) = R max > , which contradicts the assumption of the proposition that R max ≤ . Therefore, an endemic state ( x, o ) cannot be anequilibrium of (6) if R max ≤ . Proof of Theorem 2:
Note that ˙ x ( t ) = − [ D min + ( D − D min ) ( O ( t ) + 0 . I )] x ( t )+ ( I − X ( t )) [ B − ( O ( t ) + 0 . I ) ( B − B min )] x ( t )= − D min x ( t ) + ( I − X ( t )) Bx ( t ) − ( D − D min ) ( O ( t ) + 0 . I ) x ( t ) − ( I − X ( t )) ( O ( t ) + 0 . I ) ( B − B min ) x ( t ) ≤ − D min x ( t ) + ( I − X ( t )) Bx ( t ) . (17) The inequality implies further that ˙ x ≤ ˙ y = − D min y ( t ) + By ( t ) , since I − X ( t ) is a diagonal matrix and [ I − X ( t )] ii ∈ [0 , , ∀ i ∈ [ n ] . From Lemma 6, R max = ρ ( D − B ) ≤ implies s ( − D min + B ) ≤ . Initializing y (0) = x (0) ,and from the fact that ˙ y = − D min y ( t ) + By ( t ) convergesto y = exponentially when s ( − D min + B ) < , and x = is the unique equilibrium when R max < , weconclude that x ( t ) → ∞ exponentially fast. Additionally,since ( − D min + B ) is an irreducible Metzler matrix, if s ( − D min + B ) = 0 , by Lemma 3, there exists a positive diag-onal matrix P such that ( − D min + B ) (cid:62) P + P ( − D min + B ) is negative semidefinite. Therefore, consider the Lyapunovfunction V ( x ( t )) = x ( t ) (cid:62) P x ( t ) , from (7) and (17), when x ( t ) (cid:54) = , we have ˙ V ( x ( t )) = x ( t ) (cid:62) P ˙ x ( t ) ≤ x ( t ) (cid:62) P ( − D min + B − X ( t ) B ) x ( t ) . Using the proof of [25, Prop. 2], one can show that x ( t ) = is asymptotically stable with the domain of attraction [0 , n .Since − ( ¯ L u + I ) is Hurwitz, based on Remark 1 andTheorem 1, the convergence of each subsystem ˙ o ( t ) = − (Φ ( o ∗ ) ¯ L u Φ ( o ∗ ) + I ) o ( t ) − . e to o ∗ is exponentiallyfast under the fixed (Φ ( o ∗ )) . Further, each subsystem ofthe opinion dynamics in (8) is input-to-state stable [28,Lemma 4.6]. Therefore, the switching system in (8) withinput x ( t ) vanishing to asymptotically (or exponentially)fast, is asymptotically (or exponentially) stable with domainof attraction [ − . , . n . Proof of Lemma 9:
Through the Jacobian matrix evaluatedat ( , o ∗ ) in (9), where the spectrum of W ( o ( t )) determinethe stability of the Jacobian matrix, it can be shown that R min > leads to, s ( − ( D min + ( D − D min ) ( O ∗ + 0 . I ))+ ( B − ( O ∗ + 0 . I ) ( B − B min ))) > , following the same processing for Proposition 3. Therefore,the Jacobian matrix evaluated at ( , o ∗ ) in (9) has at least onepositive eigenvalue. Hence, all the healthy equilibria ( , o ∗ ) ,under the condition that R min > , are unstable. Proof of Theorem 4:
Proof of the local stability of all thehealthy equilibria in Theorem 4 is similar to the proof oflocal stability of the healthy equilibria in Theorem 2. Byswitching the condition R max < to R o ∗ t < , the Jacobianmatrix in (9) evaluated at ( , o ∗ ) is Hurwitz, which completesthe proof of Case 1) of the theorem. Case 2) follows the sameprocedure; showing that R o ∗ t > implies that the Jacobianmatrix in (9) evaluated at ( , o ∗ ) is not Hurwitz. Therefore,the equilibria are unstable. For Case 3), R o ∗ t at ( , − . e ) is ρ (cid:16) D ( o min ) − B ( o min ) (cid:17) = R max . Since R max >1