Centrality-Based Traffic Restriction in Delayed Epidemic Networks
11 Centrality-Based Traffic Restriction in DelayedEpidemic Networks
Atefe Darabi and Milad Siami Abstract
During an epidemic, infectious individuals might not be detectable until some time after becominginfected. The studies show that carriers with mild or no symptoms are the main contributors to thetransmission of a virus within the population. The average time it takes to develop the symptoms causesa delay in the spread dynamics of the disease. When considering the influence of delay on the diseasepropagation in epidemic networks, depending on the value of the time-delay and the network topology,the peak of epidemic could be considerably different in time, duration, and intensity. Motivated bythe recent worldwide outbreak of the COVID-19 virus and the topological extent in which this virushas spread over the course of a few months, this study aims to highlight the effect of time-delayin the progress of such infectious diseases in the meta-population networks rather than individualsor a single population. In this regard, the notions of epidemic network centrality in terms of theunderlying interaction graph of the network, structure of the uncertainties, and symptom developmentduration are investigated to establish a centrality-based analysis of the disease evolution. A convex trafficvolume optimization method is then developed to control the outbreak. The control process is done byidentifying the sub-populations with the highest centrality and then isolating them while maintaining thesame overall traffic volume (motivated by economic considerations) in the meta-population level. Thenumerical results, along with the theoretical expectations, highlight the impact of time-delay as well asthe importance of considering the worst-case scenarios in investigating the most effective methods ofepidemic containment.
I. I
NTRODUCTION
The large-scale spread of an infectious disease occurs every few years and leads into seriouscrises before it eventually dies out [1]. The extent through which a high-speed epidemic continuesdepends mostly on first, the government interventions, and second, the existence of an effective Department of Electrical & Computer Engineering, Northeastern University, Boston, MA 02115 USA (e-mails: { darabi.a, m.siami } @northeastern.edu ). a r X i v : . [ ee ss . S Y ] N ov treatment against the disease. In this regard, the study of epidemic propagation by networkmodels has been theoretically proven to be useful in determining the most effective methods ofepidemic management and control as well as allocating treatment and immunization resources[2], [3].The macro-modeling (or meta-population) representation of the epidemic networks is studiedin [4], where the author introduces the general form of the spread dynamics in a community ofsub-populations. Many studies have been using meta-population network of Susceptible-Infected-Susceptible (SIS) [5], Susceptible-Infected-Removed (SIR) [6], [7], and Susceptible-Exposed-Infected-Removed (SEIR) [7] models to develop the dynamics of the epidemic in macro scale.The centrality analysis of a time-delayed consensus network has been studied in [8] and aclosed form of the node and link centralities in the presence of different noises in the network isoffered. In [9], the authors have studied the convergence analysis of the time-delayed networkswith linear dynamics. It is shown that the convergence rate of delayed networks with lineardynamics has a correlation with time-delay and the largest eigenvalue of the graph. The stochasticdelayed SIS model has been exploited in [10] to find the threshold behavior of the systems withvaccination and double diseases.Significant work has been done in the area of epidemic elimination, disease spread control,and designing public health control measures against the epidemics [11]–[13]. Some suggest thatan optimized alternation in the disease network structure could hinder the disease effectively.Node or edge removal is the direct way of modifying the network structure which is also calledspectral control, as it aims to minimize the largest eigenvalue of the network to locally stabilizeit around the disease-free equilibrium [5], [14]. However, the spectral radius minimization hasbeen proven to be an NP-hard problem [15]. Network metric-based node or edge removal isanother common approach in spectral control. A removal strategy based on the number of theclosed walks and assortativity effect has been proposed in [15].On the other hand, vaccination and traffic flow restriction modify the epidemic recovery andinfection rates, which in turn, affect the disease progress pattern. Various control strategies havebeen proposed in this area. A geometric programming-based approach has been presented in[5] for the optimal resource allocation in a meta-population network. In this study, the infectionand recovery rates are assumed to be the optimization variables. The intercity traffic restrictionof an SIS mate-population model has also been studied in [16]. Optimal and heuristic local solutions are proposed to completely eradicate the disease, where the cost of the second solutionis shown to be higher than the cost of the first one. In terms of social awareness effect on theepidemic evolution, [17] has proposed a semidefinite programming-based optimization to controla network that follows the Susceptible-Alert-Infected-Susceptible (SAIS) epidemic model.In the real world cases, node removal happens by social distancing, quarantining, vaccinatingindividuals, or complete shut down of certain areas in the city [18]. In this regard, [19] hasintroduced PageRank-based vaccination strategy for realistic social networks. The authors in[20] have also proposed a network attack strategy based on degree and betweenness centralityfor both passive and online node and edge removal approaches. A Pulse vaccination strategy hasbeen offered in [6] which can eradicate the disease in all the sub-populations.In 2020, many network scientists have established their study on the modeling and controlof COVID-19, as it has turned into a global crisis. In the recent research study [21], theoptimal intervention starting time and duration that will reduce the epidemic peak and lengthin an SIR epidemic model has been proposed. The effect of a complete suppression, mistimedinterventions, and sustained interventions have also been investigated in this study [21]. Theeffect of social distancing start time, period, and duration on the infection peak of COVID-19 inepidemic models has also been investigated in [22]. It is shown that a well-timed interventionimproves flattening the infection curve in various epidemic models, such as SIR, feedback-SIR(fSIR), Susceptible-Infected-Quarantined-Susceptible (SIQR), Susceptible-Infected-Diagnosed-Ailing-Recognized-Threatened-Healed-Extinct (SIDARTHE), and 6 compartment SIR [22].A more generalized SEIR model of COVID-19 nationwide spread in China including self-protection and quarantine compartments has been established to predict the disease propagationpattern and to find a possible extinction time for some of the provinces [23]. An SIRD modelhas been developed to estimate the COVID-19 epidemic parameters in Italy, using real-dataand a non-convex parameter identification approach [24]. An agent-based model of COVID-19dynamics is utilized to simulate the multi-wave behavior of the spread while applying differentpossible government interventions such as testing and tracing, and travel restriction on it [25]. Our contributions.
Inspired by the recent COVID-19 outbreak, this study is dedicated to theepidemiological investigation of the infectious diseases with the following main contributions:i. Modeling the time-delayed epidemic dynamics of meta-populations using the network SISmodel (Section II). ii. Investigating the effect of time-delay and different sources of uncertainty on the modelednetwork dynamics (Sections II and III).iii. Developing the explicit centrality measure of sub-populations with respect to time-delayand transportation network structure (Section III).iv. Designing optimal and robust methods of traffic restriction based on the steady-stateperformance of the network (Section IV).The simulation results for a core-periphery network and also the network of United Statesbusiest airports are presented in Section V.II. D
ELAYED EPIDEMIC NETWORK OF META - POPULATION MODELS
The proportion of pre-symptomatic individuals in the population could play an important rulein the disease propagation pattern. In the case of COVID-19 for instance, studies indicate thata significant proportion of positive tests belongs to the pre-symptomatic individuals who arejust the carriers [26] and might not develop any symptoms up to 14 days [27]. The effect ofpre-symptomatic individuals in the spread of the disease can be modeled as a time-delay in thenetwork dynamics. In this study, the average time of symptom development for all the individualsis assumed to be identical and equal to a non-negative constant τ . A. Deterministic meta-population SIS model with time-delay
Let the undirected and weighted graph G = ( V , E , w ) represent a meta-population in theepidemic network. V = { , , . . . , n } shows the set of nodes in the graph or the group of cities,states, countries, or in general, sub-populations in the epidemic network. E denotes the edge setwhich shows the connection between every two member of V with the corresponding weightof w e = a i j for all e = { i , j } ∈ E for i (cid:54) = j . The counterpart of weighted edges in the epidemicnetwork would be the transportation capacity or traffic volume between every two sub-populationin V . In addition to its neighbors, the infection state of a sub-population depends on the progressof disease within the sub-population as well, which itself is a function of various factors suchas social distancing. Let us project the final effect of social distancing in the sub-population i on parameter a ii , where a ii = a ii increases, the social distancing rules become less strict. The adjacency matrix of the corresponding graph is then defined as, A = a a · · · a n a a · · · a n ... ... . . . ... a n a n · · · a nn . (1)The state of the epidemic network at time t ≥ p ( t ) = [ p ( t ) , . . . , p n ( t )] T ,where p i ∈ [ , ] is the marginal probability of sub-population i being infected at time t such that p i ( t ) = i is infected and p i ( t ) = τ due to the reasons explained earlier, the approximatedspread dynamics of sub-population i can then be described using the mean-field approximation model with time-delay as below, ˙ p i ( t ) = − δ i p i ( t − τ ) + β i n ∑ j = a i j p j ( t − τ ) (cid:0) − p i ( t − τ ) (cid:1) + b T i ξ ( t ) , (2) where β i is the infection rate at which sub-population j will contaminate sub-population i and also the rate in which the infected individuals in sub-population i will contaminate itssusceptible individuals. δ i is the recovery rate of sub-population i . ξ = [ ξ , . . . , ξ l ] T is the effectof an uncertainty in the disease spread dynamics modeled as the vector of independent Gaussianwhite noise with zero mean and b as the uncertainty coefficient vector. a i j is the i j th componentof the adjacency matrix of the coupling graph. The compact form of meta-population SIS modelcan be expressed as below,˙ p ( t ) = A p ( t − τ ) − P ( t − τ ) BA p ( t − τ ) + B ξ ( t ) , (3)where P ( t − τ ) = diag ( p ( t − τ )) , and infection and recovery matrices are B = diag ([ β , . . . , β n ]) , and ∆ = diag ([ δ , . . . , δ n ]) . (4) Matrix A = BA − ∆ determines the stability and performance of network dynamics. A = β a − δ β a · · · β a n β a β a − δ · · · β a n ... ... . . . ... β n a n β n a n · · · β n a nn − δ n . (5)In spread dynamics (2), B = [ b , . . . , b n ] T ∈ R n × l is the noise coefficient matrix and p i ( t ) isthe infected proportion of sub-population i at time t . For every sub-population in the selectedcommunity, it can be assumed that p i ( t ) is a value close to zero. For instance, this is a validassumption for the recent COVID-19 pandemic, because even though there were a substantialnumber of infected individuals from the beginning, the proportion of the infected populationbetween January 19 (The day that first case was reported) through February 28 within the USwas still close to zero . It therefore makes sense to linearize the epidemic model around thezero state. Assuming p i ( t ) (cid:28)
1, equation (2) can be linearized as follows:˙ p i ( t ) = − δ i p i ( t − τ ) + β i n ∑ j = a i j p j ( t − τ ) + b T i ξ ( t ) . (6)The compact version of this network is in the following form.˙ p ( t ) = A p ( t − τ ) + B ξ ( t ) , (7)It should be noted that the following assumptions apply throughout the rest of this paper. Assumption 1:
The infection rate of all the sub-populations is equal. Therefore, β i = β for all i ∈ V . Assumption 2:
The network graph is assumed to be undirected, i.e., a i j = a ji . B. Stability analysis
For the undirected network (3), the initial infection p ( ) ∈ [ , ] n will exponentially die out if λ max ( A ) ≤ − ε . (8) where λ max ( . ) provides the maximum real eigenvalue and ε → + . In other words, an α > (cid:107) p i ( t ) (cid:107) ≤ α (cid:107) p i ( ) (cid:107) e − ε t and as a result, the disease-free equilibriumof the system is globally exponentially stable with rate ε [5].The minimum eigenvalue of A is also lower-bounded by the time-delay as below, λ min ( A ) ≥ − π τ (9)which is the direct result of frequency domain stability analysis of the delayed systems [9]. Thenetwork is globally asymptotically stable if and only if this condition is satisfied. Therefore, thestability of the network depends not only on the maximum but also the minimum eigenvalue of A .Combining conditions (8) and (9) results in the following matrix inequality with respect tothe positive semi-definite cone S n + , ε I n (cid:22) − A (cid:22) π τ I n . (10)The reproductive characteristics of a disease determine if it will result in an epidemic or not.The basic reproduction number, R M , of the epidemic meta-population (3) can be defined asbelow, R M : = βδ λ max ( A ) (11)The initial infection will converge to zero if R M < i on the other hand, determines the progress of disease within a single population, R S ( i ) : = β i a ii δ i (12)where for R S < C. Performance analysis
The steady-state performance, ρ ss , of network (7) can be expressed as, ρ ss ( A ; B ; τ ) = lim t → ∞ E (cid:2) y ( t ) T y ( t ) (cid:3) (13)where y ( t ) = [ p ( t ) , . . . , p n ( t )] T is the vector of infection probability at time t which is requiredto monitor and update the network performance measure. y ( t ) is a function of A ; therefore, the bounds on the eigenvalues of A determine the range in which the long-run performance of thenetwork will change. More details on the performance measure of a class of consensus networksunder the influence of exogenous white noises can be found in the reference papers [29]–[32].According to [33], the performance of a network can also be found by the frequency domaindefinition of its H -norm as below, ρ ss ( A ; B ; τ ) = π Tr (cid:20) (cid:90) + ∞ − ∞ G H ( j ω ) G ( j ω ) d ω (cid:21) , (14)where G ( j ω ) is the transfer function of the network and G H ( j ω ) corresponds to the complexconjugate transpose of G ( j ω ) . Lemma 1:
For the undirected network (7), the closed form solution of (14) is, ρ ss ( A ; B ; τ ) = n ∑ i = − Φ i λ i cos ( λ i τ ) + sin ( λ i τ ) , (15)in which Φ i is the i th diagonal element of the matrix Q T BB T Q , where Q = [ q , . . . , q n ] ∈ R n × n is the orthonormal matrix of eigenvectors of A . λ i ( A ) for i = , , . . . , n is the i th eigenvalueof matrix A . Proof 1:
The transfer matrix of (7) is, G ( s ) = (cid:0) sI n − e − τ s A (cid:1) − B = Q diag (cid:32)(cid:20) s − λ e − τ s , · · · , s − λ n e − τ s (cid:21) T (cid:33) Q T B . (16)For this transfer function matrix we have, Tr (cid:2) G H ( j ω ) G ( j ω ) (cid:3) = Tr (cid:34) Q T BB T Q diag (cid:32)(cid:20) − λ e j τω − j ω , · · · , − λ n e j τω − j ω (cid:21) T (cid:33)(cid:35)(cid:34) diag (cid:32)(cid:20) j ω − λ e − j τω , · · · , j ω − λ n e − j τω (cid:21) T (cid:33)(cid:35) (17) and by substituting (17) in (14), the performance will be, ρ ss ( A ; B ; τ ) = π n ∑ i = (cid:90) + ∞ − ∞ Φ i d ω ( j ω + λ i e j τω ) ( λ i e − j τω − j ω ) . (18)A proof follows by simplifying the above integral. It should be noted that the smaller values of ρ ss result in a better performance, therefore, alower value of performance is desired.III. C ENTRALITY INDICES
The importance of every sub-population in the disease spread can be analysed by variousindices. In this study, the centrality index, η i , is the basis to rank the sub-populations.For network (7), let ξ i ( t ) ∼ N ( , σ i ) be the noise affecting the i th sub-population’s infectiondynamics, which might stem from modeling imperfections, testing error or inaccurate epidemicrates. The centrality of sub-population i is then defined as the rate of performance with respectto disturbance variance, η i : = ∂ ρ ss ∂ σ i . (19)The centrality index with respect to two important sources of disturbance will be establishedin the following sections. A. Modeling error
Model simplifications implemented on the epidemic dynamics affect the state of the infectedsub-populations as below,˙ p i ( t ) = − δ i p i ( t − τ ) + β i n ∑ j = a i j p j ( t − τ ) + ξ i ( t ) , (20)where ξ i ( t ) = σ i ˆ ξ i . The compact form of (20) would be,˙ p ( t ) = A p ( t − τ ) + B ˆ ξ ( t ) , (21)in which B = diag ([ σ , . . . , σ n ]) ∈ R n × n . Corollary 1:
For the network (21), the centrality index of the i th sub-population is, η i ( A ; τ ) = − (cid:104) A − cos ( τ A ) ( I n + sin ( τ A )) − (cid:105) ii , for all i ∈ V . Proof 2:
In the case of having modeling noise, the network dynamics is the same as (7) with B = B . Hence, the performance will be in the following matrix operator form, ρ ss ( A ; B ; τ ) = −
12 Tr (cid:104) diag (cid:16)(cid:2) σ , . . . , σ n (cid:3) T (cid:17) A − cos ( τ A ) ( I n + sin ( τ A )) − (cid:105) . (22) On the other hand, the value of centrality measure ρ ss is a linear function of variance of elementsof noise input. For the centrality index (19) the performance is defined as below, ρ ss = ∑ i ∈ V η i σ i . (23)Substituting equation (22) in the above definition, the centrality will be obtained. B. Testing error
In many cases of epidemic, especially when an infectious disease like COVID-19 first emerges,the testing methods are not completely accurate in terms of identifying the infected individuals.The incorrect results generate inaccurate statistics regarding the population of the confirmedcases which triggers impaired judgment and inappropriate containment methods.In theory, the testing error affects every sub-population’s state in the following way, ˙ p i ( t ) = − δ i ( p i ( t − τ ) + ξ i ( t )) + β i n ∑ j = a i j ( p j ( t − τ ) + ξ j ( t )) , (24) where ξ i ∼ N ( , σ i ) for i ∈ V . The state of the infected population in this case will be thesame as (7) with B = B = A diag ([ σ , . . . , σ n ]) ∈ R n × n as below,˙ p ( t ) = A p ( t − τ ) + B ˆ ξ ( t ) . (25) Corollary 2:
For the network (25), the centrality index is, η i ( A ; τ ) = − (cid:104) A cos ( τ A ) ( I n + sin ( τ A )) − (cid:105) ii , (26)for all i ∈ V . Proof 3:
With the testing error noise, the network dynamics is the same as (7) with B = B . Therefore, the compact form of ρ ss is, ρ ss ( A ; B ; τ ) =
12 Tr (cid:104) − diag (cid:16)(cid:2) σ , . . . , σ n (cid:3) T (cid:17) A cos ( τ A ) ( I n + sin ( τ A )) − (cid:105) . (27)A proof follows by using the definition (23) and above equation to find centrality.IV. E PIDEMIC CONTAINMENT BY TRAFFIC VOLUME OPTIMIZATION AT COMMUNITYLEVELS
A. Optimal traffic restriction
Monitoring and regulation of the traffic volume is one of the potential government interven-tions to mitigate the epidemic threat. Regarding the underlying epidemiological network, trafficrestriction between two sub-populations directly changes the corresponding edge weight, w e , ofthose sub-populations in the network. Therefore, the stability around disease-free state couldbe obtained by monitoring and management of the transportation network and restriction ofthe traffic volume between the highly infected and highly susceptible candidates. Although thecomplete isolation of the sub-populations seems to be the easiest and safest prevention method,especially in the case of COVID-19 which has now lasted for several months, it is not a permanentsolution mostly because of the economic considerations. Therefore, a proper balance must befound in the decision-making process. In this study, a convex optimization method is offeredto determine the proper volume of the transportation by minimizing the value of correspondingnetwork performance, and consequently improving the performance, with respect to the graphweights.This optimal traffic control problem for the network with modeling error noise can be expressedas below, minimize w e , ∀ e ∈ E ρ ss ( A ; B ; τ ) (28)subject to: w e ≤ w e ≤ w e ∑ e ∈ E w e = c A = B ∑ e ∈ E w e A e + B diag ([ a , . . . , a nn ]) − ∆ − π τ I n (cid:22) A (cid:22) − ε I n . Here, the first constraint determines lower and upper bounds for every the network weight. w e and w e are the minimum and maximum traffic flow in edge e which are arbitrary values selectedby economic considerations or traffic management methods to avoid decreasing traffic volumeless than a minimum threshold or increasing it more than the maximum capacity. The secondconstraint determines the total weight of the network edges or overall traffic volume, which couldacquire any desired value c depending on the intensity of isolation. The third constraint in which A e is the adjacency matrix of link e gives the definition of A with respect to edge weights. Thelast constraint imposes another limitation on the edge weights to respect the domain of stablesolutions. B and ∆ are the infection and recovery rate matrices defined in equation (4). Consideran optimization problem with the following general form,minimize x f ( x ) (29)subject to: f i ( x ) ≤ b i , i = , . . . , qh i ( x ) = , i = , . . . , r . This problem is considered a convex problem if all the functions f , . . . , f q are convex andall the equality constraints h , . . . , h r are affine [34]. According to this definition, the problem(28) does not fall into the category of convex problems. Hence, some modifications need tobe implemented on the original optimal problem (29). In this regard, an approximation of theperformance has been offered by [32] which converts the product of non-convex trigonometricfunctions to a linear function of A and its inverse. Using this approximation, the networkperformance will be, ρ ss ( A ; B ; τ ) (cid:39)
12 Tr (cid:20) − A o A − + τπ A o (cid:16) π I n + τ A (cid:17) − − c τ A o A + c τ A o (cid:105) , (30)where A o = B B T1 and the constant parameters c = . c = − .
01 are estimated tominimize the mean squared error of the approximated performance. This approximation is stillnot a convex function, as it includes the non-convex inverse functions A − and (cid:0) π I n + τ A (cid:1) − . Substituting the epigraph variables X = A − and X = (cid:0) π I n + τ A (cid:1) − will turn a convexfunction. The optimization problem (28) can now be cast as the following approximate form,which is a convex optimization problem as the objective function as well as all the inequalityconstraints are convex and the equality constraints are affine with respect to the only optimizationvariable w e . minimize X ; X ; w e , ∀ e ∈ E Tr (cid:20) A o X + τπ A o X − c τ A o A (cid:21) (31)subject to: w e ≤ w e ≤ w e ∑ e ∈ E w e = c A = B ∑ e ∈ E w e A e + B diag ([ a , . . . , a nn ]) − ∆ A + π τ I n (cid:23) − A − ε I n (cid:23) X I n I n − A (cid:23) X I n I n π I n + τ A (cid:23) . B. Robust traffic restriction
While the target of proposed optimal approach is to improve the overall network performancewith a uniform uncertainty distribution, i.e. σ i = i = , . . . , n , there are cases in whichthe sub-populations with highest centrality experience higher levels of disturbance. Worst casenoise distribution highlights the role of high centrality sub-populations in epidemic growth whichrequires us to design a robust containment approach.In this section, the traffic restriction problem is investigated as a robust design optimizationwhere the worst-case scenario is the optimization target. Such a case can be expressed as amin-max optimization presented in problem 32. The first constraint limits the sum of squarednoise variances and the rest of the constraints are the same as explained for problem (28). The inner optimization loop is to find the highest performance of the network with respect to theuncertainty σ i in order to improve the robustness against the disease spread.minimize w e , ∀ e ∈ E maximize σ i , ∀ i ∈ V ρ ss ( A ; B ; τ ) subject to: ∑ i ∈ V σ i = nw e ≤ w e ≤ w e ∑ e ∈ E w e = c A = B ∑ e ∈ E w e A e + B diag ([ a , . . . , a nn ]) − ∆ − π τ I n (cid:22) A (cid:22) − ε I n . (32)Using the performance definition in (23), the inner optimization problem in (32) can be rewrittenin the following form, minimize σ i , ∀ i ∈ V ∑ i ∈ V η i ( A ; τ ) σ i (33)subject to: ∑ i ∈ V σ i = n . Here, the cost function and constraint are only linear functions of the optimization variable σ i . Hence, the maximum performance occurs in the boundary, where for one variable we have σ i = n and η i ( A ; τ ) has its maximum value, and for the rest σ i =
0, i.e., σ i : = n if i = arg max j ∈ V η j ( A ; τ ) n (cid:104) − A − cos ( τ A ) ( I n + sin ( τ A )) − (cid:105) ii for all i ∈ V . Note that this solution is a non-convexfunction of w e ; therefore, to use it in the outer minimization loop, its approximated epigraphversion will be used. Problem (32) can now be cast in the following form,minimize x ; w e , ∀ e ∈ E x (35)subject to: w e ≤ w e ≤ w e ∑ e ∈ E w e = c A = B ∑ e ∈ E w e A e + B diag ([ a , . . . , a nn ]) − ∆ A + π τ I n (cid:23) − A − ε I n (cid:23) x (cid:23) n (cid:20) A o X + τπ A o X − c τ A o A (cid:21) ii , ∀ i ∈ V X I n I n − A (cid:23) X I n I n π I n + τ A (cid:23) . It is worth mentioning that the same methods in the preceding sections can be implementedon the networks with the described testing error noise as well.V. R
ESULTS
In this section, the established models and control methods will be implemented on a core-periphery network, as an academic example, and a US airport network, as a real world example.It should be noted that most of the presented results are based on the assumption that membersof every sub-population are not completely following social distancing rules, a ii (cid:54) = i = , , . . . , n . Therefore, they are free to join other sub-populations through air transportation.This will allow us to track the disease through the United States. A. Core-periphery network
To evaluate the performance of the proposed methods, a core-periphery network consisting ofthree communities has been simulated as the representative of a meta-population. The simulationsare based on the dynamics (3) over a tree graph with three connected star graphs consisting of20 nodes and 19 edges weighted in the range of [ , ] (see Fig. 1(a)). A combination of multiplestar graphs is a good candidate for a meta-population, as in reality some of the sub-populationsare considered as hubs while the others connect to the rest of the sub-populations through thesehubs. In this network, nodes 1, 2, and 15 are the hubs. In Fig. 1(a), all the sub-populations are (a) (b) (c)Fig. 1. (a) Meta-population network of 20 sub-populations and their normal traffic volume. All the sub-populations areexperiencing 5 days lag. The nodes are ranked based on their centrality index, η i , which is reflected through the size of theirindicating circles. The interconnections are ranked by their corresponding traffic volume which is specified by the color of edges.(b) Optimal meta-population network of Fig. 1(a) designed by the optimal approach (31), where a uniform noise distribution isapplied. (c) Robust meta-population network of Fig. 1(a) designed by the robust approach (35), where the worst case of applyingthe maximum noise input to the sub-population with highest centrality is considered. experiencing 5 days of delay and they are ranked based on their centrality index, η i , which isreflected through the size of their indicating circles. The interconnections are ranked by theircorresponding traffic volume which is specified by the color of edges. Let us assume that thenetwork shown in Fig. 1(a) is infected by a virus and experiencing 5 days of lag. Using theconvex optimization method (31) with c being the overall traffic volume in the absence of trafficrestrictions, the network structure changes into Fig. 1(b). The result of the robust optimizationapproach (35) for the case of 5 days of delay is shown in Fig. 1(c). The robust optimization tendsto consider the worst case scenario where the uneven noise distribution amplifies the effect ofhub node 2 with the highest centrality, which makes it a bigger threat requiring it to be even moreisolated. Hence, the robust optimizer is more conservative in manipulating the traffic volumesand reducing the centrality of hubs.The effect of time-delay on the epidemic evolution of the simulated network meta-populationwhen there is effective social distancing within every sub-population ( a ii = i = , , . . . , n )has been illustrated in Fig. 2, where the structure of the modeled network is shown in Fig. 1(a).Based on Fig. 2, it can be concluded that the higher the delays in identifying infected individuals,the higher the risk of experiencing a more extreme epidemic peak with multiple pulses. The time-delay also shows a correlation with the onset of epidemic peak, which is a decisive factor indesigning the proper government interventions. Fig. 2.
The average infection size, ¯ p ( t ) = n ∑ i ∈ V p i ( t ) , of the meta-population network shown in Fig. 1(a) withdifferent time-delays. 50 percent of the meta-population is initially infected and R = .
7. It is assumed that thereis effective social distancing within every sub-population, a ii = i = , , . . . , n . The changes in average infection size of network 1(a) when the optimal and robust controlsare applied is shown in Fig. 3. For a meta-population that is initially 10 percent infected, itis shown that 5 percent of meta-population will always be infected, while for the controllednetwork, the infection eventually dies out.Fig. 4(a) shows the logarithmic scale performance of the original, optimal, and robust networkswith respect to the desired traffic volume of the meta-population when all of the sub-populations,regardless of their centrality, experience an equal level of disturbance. As it was mentioned in thepreceding sections, the smaller the value of performance, the better the obtained performance.Hence, the optimal and robust controllers have been successful in improving the performanceof the original network, while the optimal network results in a slightly better performance dueto its less conservative weight distribution.Additionally, a comparison between the highest performance of both optimization methodsis presented in Fig. 4(b) to put more emphasis on the importance of considering worst caseswhile determining the intensity of traffic restriction during the epidemic. The direct correlation Fig. 3.
Average infection size of the three networks in Figs. 1(a), 1(b), and 1(c)with τ =
17 days, R (cid:39) .
12, andinitial infection of 50 percent in 4 nodes. (a) (b) (c)Fig. 4. (a) Performance comparison between the original network Fig. 1(a), the optimal network of Fig. 1(b) and the robustnetwork of Fig. 1(c) in the case of uniform noise distribution. (b) Performance comparison between the original network Fig. 1(a),the optimal network of Fig. 1(b) and the robust network of Fig. 1(c) in the case of extreme noise distribution. (c) Logarithmicperformance measure of networks Fig. 1 with respect to time-delay. between network performance measure, ρ ss , and time-delay can be found in Fig. 4(c). As wasexpected, increasing time-delay impairs the performance.To make a better comparison between the resulted networks, the bar diagram of the edgeweights, or traffic volumes, is represented in Fig. 5. It can be seen that between the connections Fig. 5.
Weight (traffic volume) comparison between the original network Fig. 1(a), the optimal network Fig. 1(b),and the robust network Fig. 1(c). with highest volume, ( , ) , ( , ) , ( , ) , ( , ) , and ( , ) , only the ones connecting twoof three hubs have considerable drop in volume. The centrality diagram of sub-populations canbe found in Fig. 6.The detailed results of the performance improvement through optimal and robust epidemiccontrols are represented in Table. I for Case 1 with uniform noise distribution, i.e. σ i = i = , . . . , n , and Case 2 is the worst case explained in (34). TABLE I. T HE VALUES AND PERCENTAGES OF PERFORMANCE ENHANCEMENT FOR THE NETWORKS SHOWNIN F IGS . 1( B ) AND C ) COMPARED WITH F IG . 1( A )Case 1 Case 2Original network (Fig. 1(a)) 283 (0%) 1317 (0%)Optimal network (Fig. 1(b)) 175 ( + + + + Fig. 6. Centrality comparison between the hubs of original network Fig. 1(a), optimal network Fig. 1(b), and robust networkFig. 1(c).
B. Network of United States busiest airports
Air transportation plays an important role in introducing a new disease to a meta-populationand spreading it within its sub-populations. Therefore, in this study a group of United Statesairports with the busiest airports is selected as the symbol of a real world epidemic network.The simulations are based on the dynamics (3) over the network of 15 hubs, see Fig. 7, and their104 weighted connections through air transportation. The United States air traffic data used inthis study can be found in [35].Let us assume the a virus first arrives to the United States by one of these airports andinfects this sub-population while spreading through other states by air transportation (Note thatthis scenario is just for the purpose of illustration and does not affect the generality of thestudied epidemic problem). The resulting epidemic network is represented in Fig. 7(a), wherethe hubs are experiencing 8 days of delay and ranked based on the centrality index, η i , of theirairport. The scale of circles is correlated with centrality index of its corresponding airport. Theinterconnection between every pair of hubs is ranked based on traffic volume between theirairports, which is specified by the color of links. Applying the proposed optimal traffic control (a) (b) (c)Fig. 7. (a) Meta-population network of 15 United States hubs and their normal traffic volume. All the sub-populations areexperiencing 8 days lag. The hubs are ranked based on their centrality index, η i , which is reflected through the size of theirindicating circles. The interconnections are ranked by their corresponding traffic volume which is specified by the color of edges.(b) Optimal meta-population network of Fig. 7(a) designed by the optimal approach (31), where a uniform noise distribution isapplied. (c) Robust meta-population network of Fig. 7(a) designed by the robust approach (35), where the worst case of applyingthe maximum noise input to the hub with highest centrality, New York, is considered. method (31) on this network will result in network Fig. 7(b) with a lower range of centrality forall the hubs. Additionally, implementing the robust optimization approach (35) on the originalnetwork will change the network structure to Fig. 7(c).Fig. 8 shows a comparison between the infection size of the simulated airport network when50 percent of the meta-population is initially infected and R = .
3. For the network with nodelay, τ = τ =
12 days and initial infectionof 5 percent is presented in Fig. 9, which indicates that the proposed methods are successfullydecreasing the infection size.The logarithmic scale of performance measure and its maximum for the networks in Fig. 7are presented in Figs. 10(a) and 10(b). In the case of uniform disturbance distribution, bothoptimal and robust controllers result in equal levels of performance for lower traffic percentages,while for high overall traffic volumes, over 70 percent of normal volume, the optimal controlperformance decreases with a lower rate than that of robust control. However, in the worst caseof noise distribution, robust controller shows a better performance, as it was expected. Fig. 8.
The average infection size, ¯ p ( t ) = n ∑ i ∈ V p i ( t ) , of the meta-population network shown in Fig. 7(a) withdifferent time-delays. 50 percent of the meta-population is initially infected and R = .
3. It is assumed that thereis effective social distancing within every sub-population, a ii = i = , , . . . , n .. Fig. 9.
Average infection size of the three networks in Figs. 7 with τ =
12 days, R (cid:39) .
03, and initial infectionof 5 percent. (a) (b) (c)Fig. 10. (a) Performance comparison between the original network Fig. 7(a), the optimal network of Fig. 7(b) and the robustnetwork of Fig. 7(c) in the case of uniform noise distribution. (b) Performance comparison between the original network Fig. 7(a),the optimal network of Fig. 7(b) and the robust network of Fig. 7(c) in the case of extreme noise distribution. (c) Logarithmicperformance measure of networks Fig. 7 with respect to time-delay.TABLE II. T HE VALUES AND PERCENTAGES OF PERFORMANCE ENHANCEMENT FOR THE NETWORKS SHOWN IN F IGS .7( B ) AND C ) COMPARED WITH F IG . 7( A ) Case 1 Case 2Original network (Fig. 7(a)) 567 (0%) 966 (0%)Optimal network (Fig. 7(b)) 413 ( + + + + Figs. 11 and 12 present detailed information about the changes in traffic volume and centralityindex of hubs, respectively. Unlike the core-periphery network, here the difference betweenoptimal and robust network outputs is visible. The robust control tends to isolate more of thehubs with high centrality, which results in a better performance in worst cases, as shown in10(b).The detailed results of the performance improvement through optimal and robust epidemiccontrols are represented in Table. II for Case 1 with uniform noise distribution, i.e. σ i = i = , . . . , n , and Case 2 is the worst case explained in (34).VI. D ISCUSSIONS
In this study, the nonlinear and linear dynamics of an SIS network model in epidemic networkshas been investigated. The studied meta-population is assumed to be experiencing delays due tothe considerable proportion of pre-symptomatic population. The explicit centrality indices in thepresence of model simplifications and testing errors have been derived and then used to developoptimal and robust traffic restriction methods for epidemic containment purposes. The proposed Fig. 11. Weight (traffic volume) comparison between the original network Fig. 7(a), optimal network Fig. 7(b), and robustnetwork Fig. 7(c).Fig. 12. Centrality comparison between the hubs of original network Fig. 7(a), optimal network Fig. 7(b), and robust networkFig. 7(c). methods are implemented on a core-periphery network and a network of United States busiestairports. The simulation results indicate that the unavoidable delays in symptom developmentand infection identification can result in a significant difference in epidemic evolution for bothcases, which requires more attention while designing potential government interventions. Theproposed optimal and robust approaches, both based on the convex control method, shownto be capable of enhancing the delayed network’s performance, and therefore, decreasing theinfection rate considerably. Although adding more compartments to the network model increasesthe complexity of deriving explicit centrality indices, it can provide more realistic results.Implementing the proposed methods on the directed graphs with time-varying weights is alsoan interesting direction for improving the results of the current study.Another aspect of an epidemic outbreak that requires more attention is vaccine tourism . Whenvaccination becomes available for the first time, it motivates traveling to the centers offering itwhich results in the vaccine tourism phenomenon. In the case of COVID-19 for instance, it ispredicted that the first successful country in developing the vaccination will attract many tourists,at least until the vaccination resources are well distributed. With the vaccine distribution facingseveral challenges such as keeping vaccines at subzero temperatures, having low efficacy rate,producing limited dozes, etc, air transportation to the country offering the vaccine will probablybe continued for several months. Therefore, most probably, with the vaccine tourism comes anew wave of infection raise which requires a near-optimal and fast traffic volume control tomitigate its adverse consequences. The offered policies in this study can be generalized to meetthe requirements for a safe and robust traffic control in the case of releasing a vaccination.R EFERENCES [1] W. Qiu, S. Rutherford, A. Mao, and C. Chu, “The pandemic and its impacts,”
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