On the Stability of the Endemic Equilibrium of A Discrete-Time Networked Epidemic Model
OOn the Stability of the EndemicEquilibrium of A Discrete-Time NetworkedEpidemic Model ⋆ Fangzhou Liu ∗ Shaoxuan Cui ∗ Xianwei Li ∗∗ Martin Buss ∗∗ Technical University of Munich, Munich, 80333 Germany(e-mail: { fangzhou.liu; shaoxuan.cui; mb } @tum.de). ∗∗ Shanghai Jiaotong University, Shanghai, 200240 China(e-mail: [email protected]).
Abstract:
Networked epidemic models have been widely adopted to describe propagationphenomena. The endemic equilibrium of these models is of great signi fi cance in the fi eld of viralmarketing, innovation dissemination, and information di ff usion. However, its stability conditionshave not been fully explored. In this paper we study the stability of the endemic equilibrium of anetworked Susceptible-Infected-Susceptible (SIS) epidemic model with heterogeneous transitionrates in a discrete-time manner. We show that the endemic equilibrium, if it exists, isasymptotically stable for any nontrivial initial condition. Under mild assumptions on initialconditions, we further prove that during the spreading process there exists no overshoot withrespect to the endemic equilibrium. Finally, we conduct numerical experiments on real-worldnetworks to demonstrate our results. Keywords:
Networked epidemic model, endemic equilibrium, stability, discrete-time1. INTRODUCTIONOriginating from epidemiology, the study of epidemicspreading processes has become an attractive area dueto its wide applications in propagation phenomena, suchas computer virus dissemination and information di ff u-sion Draief and Massouli (2010); Kephart and White(1991); Nowzari et al. (2016). In order to mathematicallymodel such a process, multitudes of epidemic models in-corporating di ff erent compartments have been proposed,e.g., susceptible-infected-susceptible (SIS), susceptible-infected-recovered-susceptible (SIRS), and susceptible-exposed-infected-susceptible (SEIR) Kermack and McK-endrick (1927); Li et al. (2017). In addition, taking intoconsideration the local in fl uence, networked epidemic mod-els which emphasize individual dynamics have been in-troduced. In this case, epidemic di ff usion process can begenerally described as a graph-based Markov chain. Partic-ularly, for the infection process of one agent, the transitionrate is dependent on the states of her neighbors Liu andBuss (2016); Qu and Wang (2017). However, the exactMarkov chain model faces di ffi culties due to the greatnumber of states especially in large-scale networks. Thusin the seminal work Mieghem et al. (2009), a continuous-time N -intertwined SIS model has been proposed by im-plementing mean- fi eld approximation (MFA). By reducingthe number of states from 2 N to N , this model not onlytackles the issue due to high dimensions but also exhibitsacceptable accuracy in the sense of approximation. In Par´e ⋆ This work was partially supported by the joint Sino-Germanproject “Control and Optimization for Event-triggered NetworkedAutonomous Multi-agent Systems” funded by the German ResearchFoundation (DFG) and the National Science Foundation of China(NSFC). et al. (2018), a discrete-time version of the N -intertwinedSIS model is introduced and further validated by realdata from the fi elds of epidemic spreading and informationdi ff usion. In this paper, we investigate this discrete-timenetworked SIS model and focus on the stability analysis ofits nonzero equilibrium. Epidemic models in an SIS manner have been widely stud-ied in both continuous-time and discrete-time Fall et al.(2007); Wang et al. (2003). The N -intertwined SIS modelon an undirected network with homogeneous transitionrates in Mieghem et al. (2009) is known to be the fi rstprobability-based networked epidemic model. It is thenextended in Khanafer et al. (2014); Mieghem and Omic(2014) by taking into consideration directed communica-tion topology and heterogeneous transition rates. Furtherextensions incorporate the SIS model on time-varyingnetworks and bi-virus competing SIS models, where thestability conditions on disease-free (trivial) equilibriumand endemic (nontrivial) equilibrium are highlighted Liuet al. (2019); Par´e et al. (2017).A discrete-time networked SIS model with heterogeneoustransition rates is introduced in Par´e et al. (2018) by ap-plying Euler method to the N -intertwined SIS model. Theauthors con fi rm the stability of the disease-free equilibriumunder the condition regarding the transition rates and net-work connections. In addition, the endemic equilibrium isproved to be existed under certain conditions and possessesall positive components. However, the stability analysis onthe endemic equilibrium has not been fully explored. Veryrecent work Prasse and Van Mieghem (2019) shows thathe endemic equilibrium is exponentially stable but limitedto special initial conditions. To the best our knowledge,there exist no results on the stability of the endemicequilibrium of the networked discrete-time SIS model inPar´e et al. (2018) for general initial conditions. Our contribution is twofold. Firstly, we prove that theendemic equilibrium of the model in Par´e et al. (2018) isasymptotically stable for any nonzero initial conditions.This fundamental result solves the open problem andcompletes the analysis of the model in Par´e et al. (2018).Since the model in Par´e et al. (2018) has been validatedby real-world data, its detailed analysis could deepenthe understanding of practical spreading processes. Thestudy of the endemic equilibrium, as an essential part,may provide more insights in the fi eld of viral marketingand information di ff usion where nonzero steady-states aredesired. Secondly, we consider two representative initialconditions, i.e., compared with the endemic equilibrium,all agents are in 1) lower infection level and 2) higherinfection level. We show that during the spreading processthere exists no overshoot with respect to the endemicequilibrium. Thus the results in Prasse and Van Mieghem(2018) for the model with homogeneous transition rates isextended to the model with heterogeneous transition rates.Based on this property, the domain of attraction in Prasseand Van Mieghem (2019) is enlarged.The remainder of this paper is organized as follows: InSection 2, preliminaries on graph theory and nonnegativematrices are provided. We then introduce the networkeddiscrete-time SIS model, as well as necessary propertiesand assumptions. The main results regarding the stabilityof the endemic equilibrium are presented in Section 3.Numerical experiments on real networks are conducted inSection 4. Notations:
Let R and N be the set of real numbers andnonnegative integers, respectively. Given a matrix M ∈ R n × n , ρ ( M ) is the spectral radius of M . For a matrix M ∈ R n × r and a vector a ∈ R n , M ij and a i denotethe element in the i th row and j th column and the i thentry, respectively. For any two vectors a, b ∈ R n , a ≫ ( ≪ ) b represents that a i > ( < ) b i , i = 1 , . . . , n ; a > ( < ) b means that a i ≥ ( ≤ ) b i , i = 1 , . . . , n and a (cid:6) = b ; and a ≥ ( ≤ ) b means that a i ≥ ( ≤ ) b i , i = 1 , . . . , n or a = b .These component-wise comparisons are also applicable formatrices with the same dimension. Vector ( ) representsthe column vector of all ones (zeros) with appropriatedimensions.2. PRELIMINARIES AND MODEL DESCRIPTIONIn this section, we brie fl y introduce the background knowl-edge of graph theory and properties of nonnegative matri-ces. Then the networked discrete-time SIS model and itsproperties are presented. Consider a weighted directed graph G = ( V , E , A ) where V and E ⊆ V × V are the set of vertices and the set of edges, respectively. A = [ a ij ] ∈ R N × N is the nonnegativeadjacency matrix. a ij > j to node i . In this paper, we con fi ne ourselvesthat G is strongly connected. The adjacency matrix A isirreducible if and only if the associated graph G is stronglyconnected. A matrix M is nonnegative if all the entries of M isnonnegative. We now introduce the Perron-Frobenius The-orem for irreducible nonnegative matrices. Lemma 1. (Varga, 2000, Theorem 2.7) Given that asquare matrix M is an irreducible nonnegative matrix. Thethe following statements hold i ) M has a positive real eigenvalue equal to its spectralradius ρ ( M ). ii ) ρ ( M ) is a simple eigenvalue of M . iii ) There exist a unique right eigenvector x ≫ y ⊤ ≫ ρ ( M ).The following lemma collects the properties of nonnegativematrices which are necessary for this paper. Lemma 2. (Horn and Johnson, 2013, Corollary 8.1.29 and8.1.30) Given a nonnegative matrix M ∈ R n × n , a positivevector x ∈ R n ( x ≫ ) and a nonnegative scalar µ , therehold: if µx ≪ M x , then µ < ρ ( M ); if µx ≫ M x , then µ > ρ ( M ); and if µx = M x , then µ = ρ ( M ). In this paper, we consider the networked discrete-timeSIS model in Par´e et al. (2018) on an N -node network G = ( V , E , A ) as follows. x i ( k +1) = x i ( k )+ h (1 − x i ( k )) N (cid:3) j =1 β ij x j ( k ) − δ i x i ( k ) , (1)where x i ( k ) represents the infection probability of agent i or the infection proportion of group i at time instance k ; β ij := β i a ij ; β i and δ i are the infection rate and the curingrate of node i , respectively; and h is the sampling period.The compact form of (1) reads x ( k + 1) = [ I − h (( I − diag ( x ( k ))) B − D )] x ( k ) , (2)where x ( k ) = [ x ( k ) , x ( k ) , . . . , x N ( k )] ⊤ , B = [ β ij ] N × N ,and D = diag ( δ , δ , . . . , δ N ). Evidently, is an equi-librium of the model in (2). It plays an signi fi cant rolein epidemic spreading processes, since it represents thedisease-free case. The non-trivial equilibrium ( x ∗ (cid:6) = ),on the contrary, is named the endemic equilibrium . Theendemic equilibrium is of great importance in informationdi ff usion and viral marketing where the nonzero steady-states are desired.The following assumptions are assumed to hold through-out this paper. Assumption 1.
The initial condition satis fi es x (0) ∈ [0 , N . Assumption 2.
For every i ∈ V , we have β i > δ i ≥ ssumption 3. The sampling period h is positive and forevery node i ∈ V , there holds h ( δ i + (cid:6) Nj =1 β ij ) ≤ Lemma 3.
Suppose that Assumptions 1, 2, and 3 hold.Given the networked discrete-time SIS model in (2), thereholds x ( k ) ∈ [0 , N for every k ∈ N . Proof.
We prove Lemma 3 by induction. If k = 0, we have x (0) ∈ [0 , N by Assumption 1. Suppose x ( k ) ∈ [0 , N holds for some k >
0. By using the dynamics in (1),Assumptions 2 and 3, we have x i ( k + 1) ≤ x i ( k )(1 − h δ i ) + (1 − x i ( k )) h N (cid:3) j =1 β ij ≤ − h δ i + h N (cid:3) j =1 β ij ≤ x i ( k + 1) ≥ x i ( k )(1 − h δ i ) ≥ . Thus we conclude that x ( k ) ∈ [0 , N for every k ∈ N .Lemma 3 implies that [0 , N is a positive invariant set of x ( k ). Bearing in mind that x i ( k ) is the infection probabil-ity or the infection proportion, it is natural to limit thevalue within the interval [0 , fi ned. Note that in order to obtain Lemma3, Assumption 3 can be relaxed to the assumption that forevery node i ∈ V , there hold h δ i ≤ h (cid:6) Nj =1 β ij ≤ h is short. In this paper, we accept Assumption 3 becauseit does lead to a well-de fi ned model and most of our resultsrely on this technical setting.Further properties of the equilibria of the model in (2) areconcluded in the following lemma. Lemma 4. (Par´e et al., 2018) Under Assumptions 1, 2, and3, the following statements hold for the dynamics in (2): i ) If ρ ( I − hD + hB ) ≤
1, then is the only equilibriumof the model in (2) and asymptotically stable withdomain of attraction [0 , N . ii ) If ρ ( I − hD + hB ) >
1, then the model in (2)possesses two equilibria, and x ∗ . Moreover, thereholds x ∗ ≫ .Lemma 4 provides the condition for the existence of an en-demic equilibrium x ∗ with all strictly positive components.However, It is still an open problem whether the endemicequilibrium is stable or not. In the remainder of this paper,we solve this problem and provide further properties of thenetwork discrete-time SIS model. To avoid ambiguity, wediscuss the endemic equilibrium only when it exists, i.e.,the following assumption holds Assumption 4.
For the dynamics (2), we have ρ ( I − hD + hB ) >
1. 3. MAIN RESULTSBefore embarking on the stability analysis, we presentfurther properties of the endemic equilibrium. The bound- aries of the endemic equilibrium are provided in the fol-lowing lemma.
Lemma 5.
Suppose that Assumptions 1-4 hold. Then thesteady-state x ∗ i of any node i ∈ V satis fi es1 − δ m (cid:6) Nj =1 β mj ≤ x ∗ i ≤ − δ i δ i + (cid:6) Nj =1 β ij , (3)where m = arg min i ∈ V { x ∗ i } . Proof.
By substituting the endemic equilibrium x ∗ i intothe model (1), we obtain(1 − x ∗ i ) N (cid:3) j =1 β ij x ∗ j = δ i x ∗ i . (4)It follows that x ∗ i = 1 − δ i δ i + (cid:6) Nj =1 β ij x ∗ j . (5)In light of x ∗ i ≤ , ∀ i ∈ V , it is clear that x ∗ i ≤ − δ i δ i + (cid:6) Nj =1 β ij . (6)Let m = arg min i ∈ V { x ∗ i } . Thus by (5), we have x ∗ m ≥ − δ m δ m + x ∗ m (cid:6) Nj =1 β mj = x ∗ m (cid:6) Nj =1 β mj δ m + x ∗ m (cid:6) Nj =1 β mj . (7)Bearing in mind that x ∗ i > , ∀ i ∈ V by Lemma 4, it yieldsthat x ∗ m ≥ − δ m (cid:6) Nj =1 β mj . (8)By combining (6) and (8), it is evident that the relation (3)holds.In Lemma 5, we obtain explicit boundaries of the endemicequilibrium compared with Lemma 4 ( x ∗ ≫ ), which isnecessary for our further analysis. Equipped with Lemma5, we present the following results on the stability of theendemic equilibrium of the discrete-time networked SISmodel in (2). Inspired by Prasse and Van Mieghem (2019), we start thestability analysis of the endemic equilibrium from specialcases. Speci fi cally, we consider two categories of repre-sentative initial conditions: compared with the endemicequilibrium, all agents are in 1) lower infection level and2) higher infection level, i.e., the initial conditions are inthe following two respective sets D l = { x (0) ∈ [0 , N : < x (0) ≤ x ∗ }D h = { x (0) ∈ [0 , N : x ∗ < x (0) ≤ } . (9)These initial conditions commonly exist in real-worldspreading processes. Speci fi cally, initial conditions in D l appear at early stage in the epidemic or informationdi ff usion processes when all the nodes are in a low infectionlevel. The initial conditions in D h , however, mimic thescenario when viruses have been widely spread and weshould take actions to reduce the infection level. Note that is excluded from D l , due to the fact that x ( k ) will stayat the origin if x (0) = .he following proposition con fi rms the stability of theendemic equilibrium given the two aforementioned initialconditions. Proposition 1.
Suppose that Assumptions 1-4 hold. Giventhe networked discrete-time SIS model in (2) with endemicequilibrium x ∗ , the endemic equilibrium is exponentiallystable with domain of attraction D = D l ∪ D h . Moreover,the following two statements hold i ) if x (0) ∈ D l , then x i ( k ) ≤ x ∗ i , ∀ i ∈ V , k ∈ N , ii ) if x (0) ∈ D h , then x i ( k ) ≥ x ∗ i , ∀ i ∈ V , k ∈ N . Proof.
The proof for the case when x (0) ∈ D l has beendetailed in Prasse and Van Mieghem (2019) and we saveit for triviality. Now we focus on the case when x (0) ∈ D h .Since x ∗ i is an equilibrium of (1), there holds x ∗ i = x ∗ i + h (1 − x ∗ i ) N (cid:3) j =1 β ij x ∗ j − δ i x ∗ i In conjugation with Assumption 2 and Lemma 4, we have x ∗ i < , ∀ i ∈ V . It follows that N (cid:3) j =1 β ij x ∗ j = δ i x ∗ i − x ∗ i . (10)Let z i ( k ) = x i ( k ) − x ∗ i . By using (10), the error dynamicsreads z i ( k + 1) = − h δ i − h N (cid:3) j =1 β ij x j ( k ) z i ( k )+ (1 − x ∗ i ) h N (cid:3) j =1 β ij z j ( k ) . (11)We fi rst prove statement ii) is true based on dynamics (11)by induction. If k = 0, it straightforward that x i (0) ≥ x ∗ i since x (0) ∈ D h . Suppose for some k >
0, there holds x ( k ) ∈ D h . It implies that z i ( k ) ≥ , ∀ i ∈ V . Taking intoaccount Assumption 3 and Lemma 4, it implies that thetwo summands in (11) are both nonnegative. It yields that z i ( k ) ≥ , ∀ i ∈ V . This is equivalent to statement ii).It remains to show x ∗ is exponentially stable if x (0) ∈ D h .Equivalently, we prove that the system (11) is exponen-tially stable. By using the relation (10), we can rewrite (11)as z i ( k +1)= (cid:7) − h δ i − x ∗ i (cid:8) z j ( k )+(1 − x ∗ i − z i ( k )) h N (cid:3) j =1 β ij z j ( k ) . (12)Denote z ( k ) = [ z ( k ) , z ( k ) , . . . , z N ( k )] ⊤ . The matrix formof (12) reads z ( k + 1) = Ξ z ( k ) − h diag ( z ( k )) Bz ( k ) , (13)where Ξ = I − diag (cid:7) h δ − x ∗ , h δ − x ∗ , . . . , h δ N − x ∗ N (cid:8) + h diag ( − x ∗ ) B. (14)We then show the exponential stability by comparisonprinciple. In accordance to statement ii) and (13), thereholds 0 ≤ z ( k + 1) i ≤ [ Ξ z ( k )] i , ∀ i ∈ V . (15)Thus there exists a dynamical system ¯ z ( k + 1) = Ξ ¯ z ( k ),with ¯ z (0) = z (0), such that ¯ z i ( k ) ≥ ¯ z i (0) , ∀ i ∈ V , k ∈ N . It follows that we only need to prove ρ ( Ξ ) <
1. For every i ∈ V , we have Ξ ii = 1 + h δ i x ∗ i − − x ∗ i ) h β ii ≥ − h δ i − x ∗ i − x ∗ i . In light of Assumption 3, Lemma 5, and the relation (10),we attain Ξ ii ≥ − h δ i / ( h δ i + h (cid:6) Nj =1 β ij ) − x ∗ i − x ∗ i ≥ . Notice that Ξ ij = h (1 − x ∗ i ) β ij ≥ , ∀ i, j ∈ V , i (cid:6) = j. It yields that Ξ ij ≥ , ∀ i, j ∈ V . Bearing in mind that G isirreducible and Assumption 3 holds, it implies that Ξ is anonnegative irreducible matrix. We then calculate the i thentry of ( Ξ − I ) x ∗ as follows[( Ξ − I ) x ∗ ] i = − h δ i x ∗ i − x ∗ i + h (1 − x ∗ i ) N (cid:3) j =1 β ij x ∗ j . According to the relation (10) and Lemma 4, we have[( Ξ − I ) x ∗ ] i = − h δ i x ∗ i − x ∗ i + h δ i x ∗ i < . It yields that Ξ x ∗ ≪ x ∗ . By Lemma 2, it implies that ρ ( Ξ ) <
1. Therefore the trajectories of ¯ z converge to theorigin exponentially. By comparison principle, the systemin (11) is exponentially stable. This completes our proof. Remark 1.
Proposition 1 incorporates two fundamentalresults under the initial condition x (0) ∈ D l ∪ D h . Firstly,the endemic equilibrium, if it exists, is exponentiallystable with domain of attraction D . Secondly, there existsno overshoot with respect to the endemic equilibrium.However, it does not lead to the monotonicity of thetrajectory of x ( k ), which will be illustrated in Section 4in detail. Note that the results under the initial condition x (0) ∈ D l has been reported in Prasse and Van Mieghem(2019). Statement ii) in Proposition 1 has been obtainedin Prasse and Van Mieghem (2018) for dynamics withhomogeneous transition rates. Here we not only extend theresults to heterogeneous epidemic models but also provethe exponential stability under initial condition x (0) ∈ D h .Based on the analysis on special initial conditions, we thengeneralize the results to any nontrivial initial condition. Remark 2.
In Ahn and Hassibi (2013), the authors showthat the endemic equilibrium is not stable by providinga counterexample where the system converges to a cycle.This result, however, is not applicable to the dynamics inof (13) because Assumption 3 is negated. Moreover, undertheir con fi gurations, [0 , N is no longer a positive invariantset which is inappropriate from practical perspective. Until now, we only consider the stability of the endemicequilibrium of the model (13) for certain initial conditions.It is of great importance to study the stability in the set[0 , N from both theoretical and practical perspectives.Thus we provide the following theorem. Theorem 1.
Suppose that Assumptions 1-4 hold. Giventhe networked discrete-time SIS model on graph G , thendemic equilibrium is asymptotically stable with domainof attraction [0 , N \ { } and the disease free equilibriumis stable if and only if x (0) = . Proof.
Let us rewrite the dynamical system (13) in thefollowing form. z ( k + 1) = Φ ( k ) z ( k ) , (16)where Φ ( k ) = I − diag (cid:7) h δ − x ∗ , h δ − x ∗ , ..., h δ N − x ∗ N (cid:8) + diag ( − x ( k )) hB (17)Construct a square matrix F as F = I − diag (cid:7) h δ − x ∗ , h δ − x ∗ , ..., h δ N − x ∗ N (cid:8) + hB. (18)Evidently, we have Φ ( k ) ≤ F and equality is valid if andonly if x ( k ) = . By using the relation (10), matrix F canbe rewritten as F = − (cid:3) j (cid:4) =1 h β j x ∗ j x ∗ h β · · · h β N h β − (cid:3) j (cid:4) =2 h β ij x ∗ j x ∗ · · · h β N ... ... . . . ... h β N h β N · · · − (cid:3) j (cid:4) = N h β ij x ∗ j x ∗ N Let µ = [1 , x ∗ x ∗ , . . . , x ∗ N x ∗ ] ⊤ . Notice that F µ = µ. (19)Since F is an irreducible nonnegative matrix and µ ≫ ,there holds ρ ( F ) = 1 by Lemma 2. By Lemma 1, F possesses a positive left eigenvector v ⊤ corresponding toits spectral radius, i.e., v ⊤ F = v ⊤ . Then we discuss thefollowing two cases. Case 1: If x (0) = , by using the dynamics (2), it followsthat x ( k ) = , ∀ k ∈ N . Thus if x (0) = , the trajectories of x ( k ) stay at the origin and do not converge to the endemicequilibrium. In addition, if x ( k ) = and k ≥
1, by thedynamical system (1), we have(1 − h δ i ) x i ( k −
1) + h (1 − x i ( k − N (cid:3) j =1 β ij x j ( k ) = . (20)It is clear that both of the summands are nonnegative. Inlight of Assumptions 2 and 3, it follows that x ( k −
1) = .Thus x ( m ) = , ∀ m ∈ N , if there exists certain k such that x ( k ) = . Therefore, is stable if and only if x (0) = . Case 2: If x (0) (cid:6) = , we fi rst show that there mustexists a time instance s such that x ( s ) ≫ . Since G isstrongly connected, (cid:6) Nj =1 β ij > , ∀ i ∈ V by Assumption2. It follows that 1 − h δ i > x l (0) > l ∈ V . In light of the dynamicalsystem (1), we have x l ( k ) > , ∀ k ∈ N . We then considerthe worst case. Without loss of generality, we suppose only x p (0) > , p ∈ V and other entries of x (0) are 0. Basedon the aforementioned derivation, x p ( k ) stays positive forany k ∈ N . Since G is strongly connected, we can always fi nd a node q such that there exists an edge from q to p ,i.e., a pq >
0. By utilizing the dynamical system (1), it isapparent that x q (1) = h β pq x q (0) >
0. Similarly, all other entries turn to be positive in fi nite time steps thanks tothe strong connectivity of the N -node graph G .Construct the following auxiliary system for any k ≥ s . y ( k + 1) = Φ ( k ) y ( k ) , (21)with initial condition y i ( s ) = | z i ( s ) | , ∀ i ∈ V . Since Φ ( k )is nonnegative, it is straightforward that − y ( k ) ≤ z ( k ) ≤ y ( k ) , ∀ k ∈ N . Thus z ( k ) is asymptotically stable, if theorigin of the system (21) is asymptotically stable. Considerthe Lyapunov function V ( k ) = v ⊤ y ( k ). We can obtain itsincrement as follows. ∆ V ( k ) = V ( k + 1) − V ( k ) = v ⊤ ( Φ ( k ) − I ) y = v ⊤ ( Φ ( k ) − F ) y = − hv ⊤ diag ( x ( k )) By ≤ . (22)Since x, v ≫ and the sum of each row in B is positive,it implies that ∆ V = 0 if and only if y = . Thus byLaSalle’s invariance principle, the origin of the model (21)is asymptotically stable. Based on comparison principle,we conclude that the model (16) is asymptotically stablefor any non-zero initial condition. This is equivalent to thestatement in Theorem 1.Theorem 1 con fi rms that the endemic equilibrium of thesystem (2), if it exists, is asymptotically stable for anynontrivial initial condition. Thus the open problem on thestability of the endemic equilibrium is solved. Althoughmild assumptions are adopted, this theorem is applicablefor most of the cases where the sampling period is short.Stability analysis based on relaxed assumptions will beinvestigated in the future work.4. NUMERICAL EXAMPLESIn this section, we demonstrate our main results by numer-ical experiments. The simulations are conducted on a real-world network in Coleman (1964) with slight modi fi cation.This network describes the friendships between boys in asmall highschool in Illinois. The largest strongly connectedsubgraph containing 67 nodes is utilized. For the conve-nience of simulation, we normalize the weights accordingto the number of in-neighbors, i.e., (cid:6) Nj =1 a ij = 1 , ∀ i ∈ V .We take into account two types of parameters listed in thefollowing table.Table 1. Parameters in Model (1) β i δ i h Parameters I (0 . , .
25) (0 . , .
35) 1Parameters II (0 . , .
55) (0 . , .
35) 1
The infection rates and curing rates are randomly selectedfrom the corresponding intervals such that we end upwith an SIS model with heterogeneous transition rates.Note that we have ρ ( I − hD + hB ) < ρ ( I − hD + hB ) > fi ed.By using the same initial conditions x (0) ∈ [0 , . N inthe model (1) with Parameters I and II, we obtain thetrajectories of x ( k ) which are shown in Figures 1 and 2,respectively. Thus the results are consistent with Lemma4 and con fi rm the existence of the endemic equilibriumif ρ ( I − hD + hB ) >
1. Furthermore, it is clear that theendemic equilibrium is stable when using Parameters II. ✶✵ ✷✵ ✸✵ ✹✵ ✺✵❚(cid:0)✁✂ ✄☎✂✆✝✵✵✞✶✵✞✷
Fig. 1. The model in (2) with Parameters I and initialcondition x (0) ∈ [0 , . N converges to the disease-free equilibrium. ✵ ✶✵ ✷✵ ✸✵ ✹✵ ✺✵❚(cid:0)✁✂ ✄☎✂✆✝✵✵✞✶✵✞✷✵✞✸✵✞✹✵✞✺ Fig. 2. The model in (2) with Parameters II and initialcondition x (0) ∈ [0 , . N converges to the endemicequilibrium. There exist no overshoots in the case x i (0) ≤ x ∗ i , ∀ i ∈ V .Apart from the stability of the endemic equilibrium, Figure2 also manifests that, there exists no overshoot withrespect to the endemic equilibrium if the initial infectionprobabilities are small. In addition, we can observe thatnot all the trajectories are monotonically increasing. As acomparison, we conduct the simulation with large initialinfection probabilities ( p (0) ∈ (0 . , . N ) in model (1)with Parameters II. As is presented in Figure 3, x ( k )converges to the endemic equilibrium and x i ( k ) ≥ x ∗ i , ∀ i ∈ V , k ∈ N . These results support the two statements inProposition 1. We further conduct the simulation withrandom initial condition within the set [0 , N . Apparently,the trajectories in Figure 4 converge to the endemicequilibrium, which demonstrate the statement in Theorem1. ✟ ✠✟ ✡✟ ☛✟ ☞✟✌✍✎✏ ✑✒✏✓✔✟✕☛✟✕✖✟✕✗✟✕✘ Fig. 3. The model in (2) with Parameters II and initialcondition x (0) ∈ (0 . , . N converges to the endemicequilibrium. There exist no overshoots in the case x i (0) ≥ x ∗ i , ∀ i ∈ V . ✙ ✚✙ ✛✙ ✜✙ ✢✙✣✤✥✦ ✧★✦✩✪✙✙✫✛✙✫✢✙✫✬✙✫✭✚ Fig. 4. The model in (2) with Parameters II and initialcondition x (0) ∈ [0 , N converges to the endemicequilibrium. 5. CONCLUSIONIn this paper we investigate the networked discrete-timeSIS model in Par´e et al. (2018). The main focus is the sta-bility of the endemic equilibrium. We solve this open prob-lem by providing rigorous proof and we con fi rm that theendemic equilibrium, if it exists, is asymptotically stablefor any nontrivial initial condition under mild assumptionson the parameters. Additionally, for two kinds of initialconditions, i.e., high initial infection level and low initialinfection level, we show that no overshoots with respectto the endemic equilibrium occur in the trajectories of allindividual states.uture work focuses on relaxing Assumption 3 such thatthe stability of the endemic equilibrium can be analyzedin more general situations.REFERENCESAhn, H.J. and Hassibi, B. (2013). Global dynamicsof epidemic spread over complex networks. In the52nd IEEE Conference on Decision and Control (CDC) ,4579–4585.Coleman, J.S. (1964). Introduction to mathematical soci-ology. London Free Press Glencoe .Draief, M. and Massouli, L. (2010).
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