Distributed Link Removal Strategy for Networked Meta-Population Epidemics and its Application to the Control of the COVID-19 Pandemic
Fangzhou Liu, Yuhong Chen, Tong Liu, Zibo Zhou, Dong Xue, Martin Buss
aa r X i v : . [ phy s i c s . s o c - ph ] J un Distributed Link Removal Strategy for NetworkedMeta-Population Epidemics and Its Application tothe Control of the COVID-19 Pandemic
Fangzhou Liu , Yuhong Chen , Tong Liu , Zibo Zhou , Dong Xue , and Martin Buss Abstract —In this paper, we investigate the distributed link re-moval strategy for networked meta-population epidemics. In par-ticular, a deterministic networked susceptible-infected-recovered(SIR) model is considered to describe the epidemic evolvingprocess. In order to curb the spread of epidemics, we presentthe spectrum-based optimization problem involving the Perron-Frobenius eigenvalue of the matrix constructed by the networktopology and transition rates. A modified distributed link removalstrategy is developed such that it can be applied to the SIRmodel with heterogeneous transition rates on weighted digraphs.The proposed approach is implemented to control the COVID-19pandemic by using the reported infected and recovered data ineach state of Germany. The numerical experiment shows thatthe infected percentage can be significantly reduced by using thedistributed link removal strategy.
Index Terms —distributed link removal strategy, networkedmeta-population epidemics, COVID-19 pandemic
I. I
NTRODUCTION
Various models have been proposed to mathematicallycharacterize the spread of epidemics [1], [2]. Among oth-ers, the compartmental models, e.g., the susceptible-infected-susceptible (SIS) model and the susceptible-infected-recovered(SIR) model, play the fundamental role. One important class ofthe compartmental models are the scalar deterministic models,which can be referred to in the survey [3]. These models havebeen widely investigated and qualitatively characterize themacroscopic behavior of the dynamics of infectious diseases,for example, the COVID-19 pandemic [4], [5]. However,the drawback of the scalar models is that they are basedon the hidden assumption that there exists a well-mixedpopulation, i.e., individuals have the same chances to interactwith each other. In fact, this assumption introduces not onlythe homogeneity in network structure but also in individualbehaviors, which does not generally hold in the globalizedworld with close connection via, for instance, face-to-facesocial networks and traffic networks. Both of these hetero-geneities, nonetheless, play significant roles in shaping theepidemic spreading process. This brings us to the networkepidemic models, where the nodal dynamics are considered. F. Liu, Y. Chen, T. Liu, Z. Zhou, and M. Buss are with theChair of Automatic Control Engineering (LSR), Department of Electricaland Computer Engineering, Technical University of Munich, Theresien-str. 90, 80333, Munich, Germany; { fangzhou.liu, yuhong.chen,tong.liu, ga84sih, mb } @tum.de D. Xue is with the Key Laboratory of Advanced Control and Optimizationfor Chemical Processes, East China University of Science and Technology,Shanghai 200237, China; [email protected]
There are two kinds of interpretations of the network epidemicmodels: (a) the disease spreads on a network where eachnode represents one individual and (b) the disease spreadson a network of interconnected sub-population (groups ofpopulation), i.e., meta-population. Clear, the meta-populationinterpretation provides an efficient and comprehensive way ofdepicting pandemics which breaks out would-wide and spreadsrapidly in communities. Thus, in this paper, we investigate thecontrol strategy for networked meta-population epidemics.In general, from the perspective of network systems, controlstrategies for network epidemics are categorized into node ma-nipulation and edge manipulation. Previous literatures mainlyfocus on node manipulation, especially solving the resourceallocation problem [6]–[8] by interacting with transition rates.Although they manage to control the disease spreading processwith respect to certain optimal criteria, how to explicitly imple-ment the control signal, which is associated with the modelingof the impact of the resources, remains to be explored. Instead,the edge manipulation has been widely applied in the world tocurb the epidemic spreading. For example, city lock-down canbe regarded as cutting off all the connections in the graph. Byusing link-removal strategy, the spectral radius of the graphcan be decreased such that it is below the epidemic threshold[9] resulting in the disease-free equilibrium of the networkepidemic model. However, it is a combinatorial and NP-hardproblem to optimally design the network topology on whichthe epidemic spreads [10]. Recently, Xue and Hirche proposean algorithm combining power iteration (PI), max-consensus,and event-trigger optimization to approximately solve thisproblem in a distributed manner [11]. Their method requiresthe network to be undirected. Nonetheless, real epidemicspreading network is generally directed or at least bi-directedbased on the fact of asymmetric traffic flow between locations.In addition, the epidemic threshold adopted in [10], [11] isrooted in an epidemic model with homogeneous transitionrates, i.e., the transition rates of each sub-population areidentical. Thus new criterion based on epidemic models withheterogeneous transition rates [12] needs to be introduced.The main contribution of this paper is to develop a dis-tributed link removal strategy to control networked SIR meta-population epidemics. We extend the algorithm in [11] suchthat it is applicable for meta-population epidemics with hetero-geneous transition rates on weighted digraphs. This distributedstrategy enjoys the advantage of locally retrievable informationand no centralized decision-maker. Besides, from practicalpoint of view, we implement the proposed algorithm to curb the spread of COVID-19 pandemic by using real data. Weidentify the infection matrix (infection rates and adjacency ma-trix of the network) and the curing rates by using the reportedinfection and recovered cases of each state of Germany. Builtupon the identified infection matrix and curing rates, we showthat the infected percentage of Germany can be significantlyreduced by applying the proposed algorithm.
Notations:
Let R , N , and N ≥ be the set of real numbers,nonnegative integers, and positive integers, respectively. Givena matrix M ∈ R n × n , λ i ( M ) is the i th largest eigenvalue of M sorted in the decreasing order | λ ( M ) | ≥ | λ ( M ) | ≥ . . . ≥| λ n ( M ) | , ρ ( M ) is the spectral radius of M , i.e., ρ ( M ) =max i | λ i ( M ) | . Let Re ( λ ) be the real part of the eigenvalue λ . α ( M ) denotes the largest real part of M ’s eigenvalues, i.e., α ( M ) = max i Re ( λ i ( M )) . For a matrix M ∈ R n × r and avector a ∈ R n , M ij and a i denote the element in the i th rowand j th column and the i th entry, respectively. For any twovectors a, b ∈ R n , a ≫ ( ≪ ) b represents that a i > ( < ) b i ,for all i = 1 , . . . , n ; a > ( < ) b means that a i ≥ ( ≤ ) b i , forall i = 1 , . . . , n and a = b ; and a ≥ ( ≤ ) b means that a i ≥ ( ≤ ) b i , for all i = 1 , . . . , n or a = b . These component-wisecomparisons are also applicable for matrices with the samedimension. Vector ( ) represents the column vector of allones (zeros) with appropriate dimensions. I n stands for theidentity matrix of order n and e i is the i th column of I n .II. P ROBLEM F ORMULATION
In this section, we recall some necessary notions from graphtheory, introduce the meta-population SIR model, and providethe problem formulation.
A. Preliminaries
We consider a social network described by a weighteddirected graph G ( V , E , W ) with n ( n ≥ nodes, where V = { , , . . . , n } and E ⊆ V × V are the sets of nodes andedges, respectively. The adjacency matrix W = [ w ij ] ∈ R n × n is nonnegative and with zero diagonal entries. For two distinctnodes, w ij > if and only if there exists a link from node j to i , i.e., ( j, i ) ∈ E . For the convenience of further presentation,the in-neighborhood of node i is also introduced as N in i = { j : w ij > , j ∈ V} . (1)In this article, we confine ourselves that the graph G is stronglyconnected, i.e., W is irreducible. We then introduce thePerron-Frobenius Theorem for irreducible nonnegative matrix. Lemma 1. [13, Theorem 2.7] Given that a square matrix M is an irreducible nonnegative matrix. The following statementshold:(i) The largest eigenvalue of M , λ ( M ) , is a positive realeigenvalue equal to its spectral radius ρ ( M ) .(ii) ρ ( M ) is a simple eigenvalue of M .(iii) There exist a unique right eigenvector y ≫ and aunique left eigenvector z ⊤ ≫ ⊤ corresponding to ρ ( M ) .Since the adjacency matrix W of the strongly connectedgraph G is nonnegative irreducible, Lemma 1 can be directlyapplied. In the remaining of this article, we denote positive vectors y = [ y , y , . . . , y n ] ⊤ and z ⊤ = [ z , z , . . . , z n ] theright and left eigenvector corresponding to ρ ( W ) . B. Meta-Population Susceptible-Infected-Recovered Model
Consider epidemics spreading on a weighted directed graph G ( V , E , W ) . The dynamics of each group i ∈ V satisfies themeta-population SIR model as follows ˙ x i ( t ) = (1 − x i ( t ) − r i ( t )) n X j =1 β i w ij x j ( t ) − δ i x i ( t )˙ r i ( t ) = δ i x i ( t ) , (2)where x i ( t ) , r i ( t ) ∈ R represent the proportions of infected(I) and recovered (R) cases in group i at time instant t ,respectively. β i , δ i > are the infection and curing rate ofgroup i , respectively. Since the transition rates are generallydifferent for each group and the groups are not well-mixed, themodel in (2) characterizes the heterogeneity of the epidemicspreading process. From practical point of view, we adoptedthe SIR model described by x i and r i , because the infected andrecovered cases are regularly reported while the proportion ofsusceptible individuals in population can be hardly known. Inaddition, the dynamics of i th group’s proportion of susceptiblecases s i ( t ) can be omitted in the SIR model (2) in light of thefact that s i ( t ) + x i ( t ) + r i ( t ) ≡ for all i ∈ V and t ≥ .Furthermore, it is desirable that the states in the model (2)stay in the following simplex. ∆ = { ( a, b ) : a, b ≥ , a + b ≤ } . (3)Let x ( t ) = [ x ( t ) , x ( t ) , . . . , x N ( t )] ⊤ and r ( t ) =[ r ( t ) , r ( t ) , . . . , r N ( t )] ⊤ be the stacked infection propor-tions and recovering proportions, respectively. Let β =[ β , β , . . . , β N ] ⊤ and δ = [ δ , δ , . . . , δ N ] ⊤ . By denoting X ( t ) = diag ( x ( t )) , R ( t ) = diag ( r ( t )) , B = diag ( β ) , and D = diag ( δ ) , the compact form of the meta-population SIRmodel (2) reads ˙ x ( t ) = (1 − X ( t ) − R ( t )) BW x ( t ) − Dx ( t )˙ r ( t ) = − Dx ( t ) . (4)The SIR model is considered in this article due to its wideapplication in the describing epidemic spreading process.Nonetheless, the results and algorithms in the following sec-tions can be straightforwardly extended to other compartmen-tal models, e.g., SI, SIS, SIRS, SEIR.In the scenarios of epidemic curbing and rumor mitigation,the disease-free case, i.e., x = , is of great significance. Thefollowing lemma collects the behavior of the meta-populationSIR model (4). Lemma 2.
Consider the meta-population SIR model (4) withpositive transition rates on a strongly connected weighteddigraph G ( V , E , W ) . The following statement hold:(i) If ( x i (0) , r i (0)) ∈ ∆ for all i ∈ V , there holds ( x i ( t ) , r i ( t )) ∈ ∆ for all i ∈ V and t ≥ .(ii) The set of equilibrium points is the set of pairs ( N , r ∗ ) ,for any r ∗ ∈ [0 , n . (iii) If α ( BW − D ) ≤ − ǫ for some ǫ > , x ( t ) approaches exponentially fast, i.e., k x ( t ) k ≤ k x (0) k Ke − ǫt , forsome K > . Proof.
Consider the dynamics of each node in (2). Assumethat for some time instant τ ≥ , there hold ( r i ( τ ) , x i ( τ )) ∈ ∆ for all i ∈ V . We then inspect the following three cases: (a) if x i ( τ ) = 0 , then ˙ x i ( τ ) ≥ ; (b) if r i ( τ ) = 0 , then ˙ r i ( τ ) ≥ ;and (c) if x i ( τ )+ r i ( τ ) = 1 , then ˙ x i ( τ )+ ˙ r i ( τ ) = − δ i x i ( τ ) ≤ . By combining the above three cases, we can obtain thestatement (i).The statement (ii) has been proved in [1] and the proof issaved for triviality.We then prove the statement (iii). In light of the statement(i), we can obtain ˙ x < ( BW − D ) x. (5)Thus by comparison principle [14], we only need to prove theauxiliary system ˙ y = ( BW − D ) y converges to exponentiallyfast. It is straightforwardly true since there hold α ( BW − D ) ≤− ǫ . Thus we complete the proof. Remark 1.
By Lemma 2, it is straightforward that ∆ is aninvariant set for the infection and recovered proportions, givennonnegative transition rates. Note that distinct from the SISmodel, the SIR model always converge to a disease-free case ifthe curing rate δ i is positive. In this regard, what matters for theSIR model is not whether there will be healthy state, but howfast the disease dies out. By the statement (iii), the decay rateof the infection proportion is furnished by α ( BW − D ) . Thus,we control α ( BW − D ) to curb the spread of the epidemics. C. Link Removal Problem
Given a weighted digraph G = {V , E , W } , the link removalproblem is formally described as follows: for a fixed budget | ∆ E| = r ( r ∈ N + ) , select a set of edges ∆ E from E toconstruct a new graph G r = {V , E \ ∆ E , W r } , such that theexponential decay rate of the meta-population SIR model (4)with positive infection and curing rates is maximized, i.e., max ∆ E⊆E ǫ s . t . α ( BW r − D ) ≤ − ǫ |E| = r. (6)Note that since B and D are diagonal matrices with positivediagonal entries, the optimization problem (6) is equivalentto minimize α ( D − BW r ) , where the matrix D − BW r isirreducible nonnegative. For the convenience of presentation,we denote ∆ W = W − W r , A := D − BW,A r := D − BW r , ∆ A r := D − B ∆ W. (7)In light of Lemma 1, we can rewrite the optimization problemsas follows min ∆ E⊆E λ ( A r )s . t . |E| = r. (8) After labeling edge ( j, i ) ∈ E on graph G by l ij , theoptimization problem (8) can be reformulated as min m ∈{ , } |E| λ ( A − ∆ A r )s . t . ∆ A r = |E| X l ij =1 β i m l ij e i e ⊤ j w ij /δ i ⊤ m = r, (9)where m = [ m , m , . . . , m |E| ] ⊤ with m l ij = 1 if the edgelabeled as l ij is removed from E and m l ij = 0 , otherwise. Remark 2.
For epidemics with infection rate β and curingrate γ , we have the reproduction number R = βγ . If R < ,the smaller R is, the faster the epidemic dies out. For meta-population SIR model (4), the dominant eigenvalue of thematrix A can be considered as the reproduction number whichtakes into consideration the influence of the network topologyas well as the heterogeneous transition rates.III. D ISTRIBUTED L INK R EMOVAL S TRATEGY
In this section, we propose a distributed algorithm to solvethe link removal problem (9) for the meta-population SIRepidemic model on weighted digraphs. By using eigenvalue-sensitivity-based approximation, we introduce the dominantleft and right eigenvectors of the matrix A to solve the problemin question. Then we design a distributed algorithm based onpower iteration and max-consensus algorithm. A. Eigenvalue-Sensitivity-Based Approximation
Proposition 1.
The optimization problem (9) can be approx-imately solved by min m ∈{ , } |E| ∆ λ ( A, ∆ A r )s . t . ∆ λ ( A, ∆ A r ) = |E| X l ij =1 β i m l ij w ij z i y j /δ i ⊤ m = r. (10) Proof.
Since the graph G = {V , E , W } is strongly connected, λ ( A ) is positive and simple by Lemma 1. In addition, theright and left (normalized) eigenvectors, y and z ⊤ , are strictlypositive, i.e., y, z ≫ . According to the perturbation theoryin [15, p.183], the following expansion holds λ ( A − ∆ A r ) = λ ( A ) − z ⊤ ∆ A r yz ⊤ y + O ( k ∆ A r k ) . (11)For graphs G with a large spectral gap between λ ( W ) and λ ( A ) , the higher order items can be neglected and the first-order approximation equals λ ( A − ∆ A r ) . Since y and z ⊤ are normalized, i.e., z ⊤ y = 1 , we can obtain the expressionof ∆ λ ( A, ∆ A r ) by rewritten z ⊤ ∆ A r y in a component-wisemanner. B. Distributed Algorithm Design
To implement the algorithm in a distributed way, we firstlyintroduced distributed estimation of the eigenvectors. Basedon the estimated eigenvectors, we carried out the removalalgorithm. Power iteration (PI) is a common method to es-timate dominant eigenvalue.The eigenvector corresponding tothe dominant eigenvalue of matrix C is given by ˆ ξ ( t + 1) = C ˆ ξ ( t ) k C ˆ ξ ( t ) k , (12)where ˆ ξ ( t ) is estimation at step t . It worth noting that, poweriteration demand a primitive matrix C , which may not be thecase for defined matrix A . So we set C = I + A to acquirethe eigenvector.A problem for the distributed complement would be thenormalization in (12) at each iteration step. To get k C ˆ ξ ( t ) k ,global information is needed. Therefore, we use a max -consensus protocol to help getting the ˆ ξ ( t ) converged, notnormalized though.For ˆ y ( t ) = [ˆ y ( t ) , ..., ˆ y n ( t )] ⊤ , each node i has access to itsown value ˆ y i ( t ) and its neighbours’ value ˆ y j ( t ) , j ∈ N i . Thenthe PI in (12) can be modified as ˆ y i ( t + 1) = k i ( t ) ˆ y i ( t ) + n X j =1 w ij ˆ y j ( t ) , (13)where k i ( t ) helps in the convergence of ˆ y i ( t + 1) and canbe achieved in the following max -consensus way. Firstly, wecalculate a candidate locally, as h i ( t + 1) = 1ˆ y i ( t ) ˆ y i ( t ) + n X j =1 w ij ˆ y j ( t ) . (14)Then, every node shares this value with its neighbors andchoose the max value from the values of its neighbors’ anditself’s: p i ( t + t s + 1) = max j ∈N in i p j ( t + t s ) , p i ( t ) = h i ( t ) , (15)where t s ∈ N ≥ . Terminated at a mixing time T d , which meansat time t + T d a max -consensus is reached that p ( t + T d ) = · · · = p n ( t + T d ) = max j ∈V h j ( t ) . (16)Setting k i ( t ) = p i ( t ) , k i ( t ) can be formulated as k i ( t ) = 1max j ∈V h j ( t − T d ) . (17) Theorem 1.
Given a connected graph G , the eigenvector y corresponding to λ ( A ( G )) can be computed distributively byrepeating steps (13)-(17). Proof.
According to [16], if irreducible C ≥ is primitive, lim t → + ∞ max i h i ( t ) = ρ ( C ) . (18)Meantime, the power iteration (12) guarantee a compact formconverge to the true eigenvector corresponding to the largesteigenvalue of adjacency matrix. In this way, lim t →∞ ˆ y ( t + 1) = lim t →∞ C ˆ y ( t ) λ ( C ) = y ( t ) (19) can be achieved. Remark 3.
In the proposed link removal algorithm, the usingof Lemma 1 calls for an irreducible non negative A r , whichequals to a strongly connected W r , with the definition in (7).That can be a strong assumption for a matrix. However, asglobalized we are these days, it is nearly impossible for anycity or sub-population to stay cut off physically from theoutside world. Especially in the severe pandemic situation,necessary medical and living materials must be sent by people.Therefore, no vertex is supposed to be isolated, and theassumption of a strongly connected adjacency matrix afterremoval is reasonable and necessary.IV. S IMULATIONS
A. Parameter Learning via COVID-19 data in Germany
We identified the propagation network of the COVID-19virus consisting of 16 nodes. Each node represents a federalstate in Germany. The reasons to use the data in Germany aretwo folds. Firstly, Germany closed the borders with neighborson March th [17]. Considering that the incubation periodof COVID-19 is up to 14 days, the network of federal statesin Germany in April and May can be viewed as isolated, i.e.,infection from external nodes (other countries) is excluded.Secondly, Germany guaranteed sufficient testing capacity andnumerous intensive beds, which is far below the upper limit ofthe healthcare resource they can provide. Therefore, the dataof Germany can well interpret the infection characteristic ofthe virus.We use the infection data of the 16 federal states ofGermany from [18]. The identified network is a weightedasymmetric one consisting of 16 notes. From the geographicalpoint of view, not all the federal states are adjacent to eachother. However, the network can be treated as nearly fully-connected due to the logistic, business/personal trips, etc..Because the data from [18] is published once a day, thepropagation network is identified based on the followingdiscrete-time model. x [ k + 1] − x [ k ] = ( I − X [ k ] − R [ k ]) BW x [ k ] − X [ k ] δr [ k + 1] − r [ k ] = X [ k ] δ, (20)which can be further simplified in linear parameterization form ξ [ k ] = φ [ k ] ⊤ θ ∗ (21)with φ [ k ] ⊤ = (cid:20) x [ k ] ⊤ ⊗ ( I − X [ k ] − R [ k ]) − X [ k ]0 X [ k ] (cid:21) , (22)and θ ∗ = (cid:20) vec ( BW ) d (cid:21) , ξ [ k ] = (cid:20) x [ k + 1] − x [ k ] r [ k + 1] − r [ k ] (cid:21) , (23)where ⊗ represents the Kronecker product, vec ( BW ) denotesthe vectorization of matrix BW and I ∈ R n × n is an identitymatrix. We see that the network structure BW and thecuring rates d are stacked into the parameter vector θ ∗ . The identification of the propagation network of the virus can beformulated as a constrained optimization problem as follows θ ∗ = arg min θ (cid:13)(cid:13) Φ ⊤ θ − Ξ (cid:13)(cid:13) s.t. ≤ θ i ≤ , i = 1 , · · · , n ( n + 1) (24)with Φ = [ φ [ k ] , φ [ k + 1] , · · · , φ [ k + N ]] being the regressormatrix and Ξ = [ ξ [ k ] , ξ [ k + 1] , · · · , ξ [ k + N ]] being the vectorcontaining the data of daily increase in infected cases. N ∈ N is the number of utilized data. Since the network consists of16 nodes, the data of at least 17 days is required to ensure thefull rank of the regressor matrix. In our simulation, we utilizethe data of 25 days from May 24th to April 18th, i.e., N =25 . The optimal parameter θ ∗ is obtained by adopting lsqlin function of Matlab with interior-point algorithm. In reality,the recorded data is not perfect because of various reasonssuch as the delay of reporting cases and uncertain incubationperiods. The identified network is the nearest solution to thereal network, which satisfies the constrain in (24). Bayern Baden-Wuerttemberg
Berlin
Brandenburg
Bremen
Hamburg
Hessen
Mecklenburg-Vorpommern
Niedersachsen Nordrhein-Westfalen
Rheinland-Pfalz
Saarland
Sachsen
Sachsen-Anhalt
Schleswig-Holstein
Thueringen
Figure 1: The approximation of daily increment of infectioncases in each state of Germany. The red line and blue linecorrespond to the real data and approximation, respectively.
B. Implementation of the Link Removal Strategy
We slightly abuse the network obtained in Section IV.A bysetting the edges weight less than 0.001 to 0. This results in astrongly connected digraph and better validates our algorithm.Based on this processed graph, we initially test the estimationof the principal eigenvector, which turns out to converge wellas shown in Fig. 2. The colorful ˆ y i grows for several iterationsand finally converge to the corresponding component of theprincipal eigenvector represented by the black dash line.To verify the effectiveness of the proposed algorithm onsolving problem (9), we compare the dominant eigenvalue λ ( A r ) of the proposed removal algorithm with a randomremoval strategy. It worth noticing that the estimation of thedominant eigenvalue after every removal is not necessary. Toremove more than one edge after an estimation is possible andcan conserve computational efforts. With this concern, we setdifferent step-size in our simulation. The Fig. 3 shows that,regardless of the step-size, the proposed algorithm significantlyreduces the dominant eigenvalue of A , so that much betterperformance is achieved. As for the results of different step-sizes, similar effects are achieved by 1 edge per step and 2edges per step. Things are the same for 5 edges per step and 10edges per step, which show little worse performance than thatof 1 and 2 edges per step. In all, to minimize the step enhancesthe performance, but the difference is not significant if the gapbetween step-size settings is not huge. The links directed toBerlin and Mecklenburg-Vorpommern are more likely to beremoved. That would possibly because of the high weights oflinks pointing to them, as the removal of fast-spreading linkmay helps in containing the pandemic.We carry an experiment in one-step scenario and furthertest the connectivity by abandoning the proposal that harmconnectivity. In Fig. 3, the result is marked by the red cross,the line of which is only slightly different with that of bluespots representing removal without connectivity guarantee. Itis possibly because the graph is well connected. Despite ofthe removal we have made, the connectivity is not damaged.With the edges-removed graph, we make an estimation ofthe COVID-19 epidemic and compare the result with thatof the approximated graph and the real data. In Fig. 4, thedecreasing rate of infection number shows obvious differenceamong the three lines. The real infection number drops fasterthan the estimation with identified parameters, which mayattribute to the more experienced treatment and improvingsocial distancing awareness. Moreover, the estimation withgraph processed by the proposed algorithm shows a steeperdecline than real data, which verifies that our network-basedmethod is able to control the pandemic in an effective way.V. C ONCLUSION
In this paper, we propose a distributed link removal strategyfor the network meta-population SIR epidemics. Comparedwith the previous work, this approach enjoys a more generalsetting where the investigated network is relaxed to be aweighted digraph. From practical point of view, the proposedapproach is applied to the scenario of curbing the COVID-19spreading by using infection and recovered cases in each state
Iteration Step D o m i nan t E i gen v e c t o r Figure 2: Distributed estimation with power iteration.
Figure 3: Performance of the proposed removal algorithm withdifferent step sizes.of Germany. The simulation illustrates the effectiveness ofthe proposed approach. Future work will focus on developingdata-driven distributed topology manipulation strategies tocontrol network epidemic spreading processes.R
EFERENCES[1] W. Mei, S. Mohagheghi, S. Zampieri, and F. Bullo, “On the dynamicsof deterministic epidemic propagation over networks,”
Annual Reviewsin Control , vol. 44, pp. 116–128, 2017.[2] C. Nowzari, V. M. Preciado, and G. J. Pappas, “Analysis and controlof epidemics: A survey of spreading processes on complex networks,”
IEEE Control Systems , vol. 36, no. 1, pp. 26–46, 2016.[3] H. W. Hethcote, “The mathematics of infectious diseases,”
SIAM Review ,vol. 42, no. 4, pp. 599–653, 2000.[4] P. Shi, S. Cao, and P. Feng, “SEIR transmission dynamics model of2019 ncov coronavirus with considering the weak infectious ability andchanges in latency duration,” medRxiv , 2020.[5] J. Wu, K. Leung, and G. M. Leung, “Nowcasting and forecasting thepotential domestic and international spread of the 2019-nCoV outbreakoriginating in wuhan, china: a modelling study,”
The Lancet , vol. 395,no. 10225, pp. 689–697, 2020.[6] F. Liu and M. Buss, “Optimal control for heterogeneous node-basedinformation epidemics over social networks,”
IEEE Transactions onControl of Network Systems , p. Online, 2020.
Date I n f e c t i on N u m be r Estination with identified parametersEstination after Link-removalReal
Figure 4: The performance of the link-removal algorithm onthe control of the COVID-19 spreading process. [7] F. Liu, Z. Zhang, and M. Buss, “Robust optimal control of deterministicinformation epidemics with noisy transition rates,”
Physica A: StatisticalMechanics and its Applications , vol. 517, pp. 577–587, 2019.[8] C. Nowzari, V. M. Preciado, and G. J. Papas, “Optimal resource alloca-tion for control of networked epidemic models,”
IEEE Transactions onControl of Network Systems , vol. 4, no. 2, pp. 159–169, 2017.[9] P. Van Mieghem, J. Omic, and R. Kooij, “Virus spread in networks,”
IEEE/ACM Transactions on Networking , vol. 17, no. 1, pp. 1–14, 2009.[10] P. Van Mieghem, D. Stevanovi´c, F. Kuipers, C. Li, R. van de Bovenkamp,D. Liu, and H. Wang, “Decreasing the spectral radius of a graph by linkremovals,”
Pysical Review E , vol. 84, p. 016101, 2011.[11] D. Xue and S. Hirche, “Distributed topology manipulation to controlepidemic spreading over networks,”
IEEE Transactions on Signal Pro-cessing , vol. 67, no. 5, pp. 1163–1174, 2019.[12] J. Liu, P. Par´e, A. Nedi´c, C. Y. Tang, C. L. Beck, and T. Basar, “Analysisand control of a continuous-time bi-virus model,”
IEEE Transactions onAutomatic Control , vol. 64, no. 12, pp. 4891–4906, 2019.[13] R. Varga,
Matrix Iterative Analysis . Springer-Verlag, 2000.[14] H. K. Khalil,
Nonlinear systems; 3rd ed.
Upper Saddle River, NJ:Prentice-Hall, 2002.[15] G. Stewart and J. Sun,
Matrix Perturbation Theory . Boston, MA: USA:Academic, 1990.[16] R. Wood and M. O’Neill, “An always convergent method for finding thespectral radius of an irreducible non-negative matrix,”