Where to cut to delay a pandemic with minimum disruption? Mathematical analysis based on the SIS model
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Where to cut to delay a pandemic with minimum disruption?Mathematical analysis based on the SIS model
Paolo Bartesaghi
Department of Statistics and Quantitative Methods,University of Milano - Bicocca, Via Bicocca degli Arcimboldi 8, 20126, Milano, [email protected]
Ernesto Estrada ∗ Institute of Mathematics and Applications, University of Zaragoza,Pedro Cerbuna 12, Zaragoza 50009, Spain;ARAID Foundation,Government of Aragón, Spain; Institute for Cross-Disciplinary Physics and Complex Systems(IFISC, UIB-CSIC),Campus Universitat de les Illes Balears E-07122, Palma de Mallorca, [email protected]
Received (Day Month Year)Revised (Day Month Year)Communicated by (xxxxxxxxxx)The problem of modifying a network topology in such a way as to delay thepropagation of a disease with minimal disruption of the network capacity to reroutegoods/items/passengers is considered here. We find an approximate solution to theSusceptible-Infected-Susceptible (SIS) model, which constitutes a tight upper bound toits exact solution. This upper bound allows direct structure-epidemic dynamic relationsvia the total communicability function. Using this approach we propose a strategy toremove edges in a network that significantly delays the propagation of a disease across thenetwork with minimal disruption of its capacity to deliver goods/items/passengers. Weapply this strategy to the analysis of the U.K. airport transportation network weightedby the number of passengers transported in the year 2003. We find that the removalof all flights connecting four origin-destination pairs in the U.K. delays the propagationof a disease by more than 300%, with a minimal deterioration of the transportationcapacity of this network. These time delays in the propagation of a disease represent animportant non-pharmaceutical intervention to confront an epidemics, allowing for betterpreparations of the health systems, while keeping the economy moving with minimaldisruptions.
Keywords : Networks Theory; SIS Model; CommunicabilityAMS Subject Classification: 92D39; 05C82, 37N25 ∗ Corresponding author: Ernesto Estrada, email: [email protected] anuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv Paolo Bartesaghi and Ernesto Estrada
1. Introduction
Epidemics propagate through networks . They include international trans-portation webs , nationwide and urban commuting systems , as well as face-to-facenetworks of human contacts . Temporarily disrupting these networks is the firstelection to avoid the propagation of an epidemic from local to global scales .The problem is that these networks also move our economy and society. Arguably anetwork exists to transport “something” among its nodes. Therefore, the disruptionof transportation networks affects the flow of raw materials needed for production,of goods needed for consumption and of people who directly or indirectly participatein the economic life of modern society. During the COVID-19 pandemic ,expanding from Wuhan in China to the rest of the globe since February 2020, bothhuman and economic damages have been catastrophic across the world .One lesson learned from this pandemic is that sometimes delaying thepropagation of the infection a few days allows for a better preparation of thehealth services which impact significantly in saving lives . In order to delaysuch propagation we have to act directly over the networks which the disease usesto expand and, in this sense, mathematical modeling has played an importantrole . In practice, we can cut some of the connections between the differentnodes of these networks with the hope of delaying the pandemic. Typically, weare talking about canceling or reducing international and national flights, isolatingregions of a country and/or neighborhoods of a city, and/or limiting the sizesof social groups allowed . The question is then: “Where to cut?”, thinkingsimultaneously in delaying the pandemic and not affecting dramatically the flowof goods/items/passengerss through the network. Let us illustrate this situationwith a toy example. If a disease is propagating across the network illustrated inFig. 1 it would be tempting to cut the edge between the nodes 1 and 12 to delaythe propagation. Indeed, the time at which the whole network is infected in aSusceptible-Infected-Susceptible (SIS) scenario is delayed by 9.45% respect tothe original network, but you have increased the average shortest path distance by36.3%, making the network much inefficient. If instead you decided to cut 5-6, youincrease the SIS time of global infection by 8.92% but also the average shortestpath by 25.2%. In contrast, removing the edge (2,3) increases the SIS time of globalinfection by 3.15% with an increase of only 0.59% in the average shortest path.anuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv Where to cut to delay a pandemic with minimum disruption? Fig. 1: A toy network illustrating two communities (squares) connected by two pathsof 2 and 4 edges, respectively. The toy model is used to illustrate the strategies ofdelaying disease propagation with minimum connectivity cost by cutting strategicedges in graphs.The concepts stated before are clarified in the next sections of this paper, butthey are used here to exemplify the lack of triviality of the problem in question.After the definition of all concepts and notation used in this work, we state andprove the main result, namely an approximate SIS model whose solution representsan upper bound to the exact solution of this model, which is always below thediverging solution of the linearized model. One of the main advantages of thisupper bound, apart from representing a worse case scenario for the propagationof a SIS disease, is that it directly connects the structure of the network with thedisease dynamics. That is, we show here that the upper bound found here for theSIS model can be expressed in terms of an exponential function of the adjacencymatrix, which is known to capture the contributions of subgraphs of a network toa property, e.g., node infectivity, via the use of walks in graphs. These functions,known nowadays as communicability functions, have found many applications acrossthe disciplines. Using our structural-transparent upper bound to the SIS model westudy the propagation of a disease across the network of commercial airports inthe U.K. We consider a network of airports with edges weighed by the number ofpassengers transported in year 2003. We devised here a strategy to remove edgesin a network which delays the disease propagation with minimum disruption of thecapacity of the network to reroute goods/items/passengers. For the U.K. airportnetwork we found that by removing 4 edges, i.e, removing all flights connecting4 pairs of origin-destination places, a disease can be delayed by more than 300%relative to the original network without disruption of the network efficiency todiffuse goods/items/passengers or to reroute them by shortest paths connections inthe network.anuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv Paolo Bartesaghi and Ernesto Estrada
2. Preliminaries
We consider here (weighted) graphs
Γ = (
V, E, W, ϕ ) where V is the set of vertices(nodes), E is the set of edges and W is a set of weights w ij ∈ W , such that w ij ∈ R + . The weights are assigned to the edges by the surjective mapping ϕ : E → W . In the case of unweighted graphs we consider that w ij = 1 for ( i, j ) ∈ E . We always consider V = n and E = m. All graphs here are undirected,therefore their adjacency matrices A are symmetric. Let k i denotes the (weighted)degree of the node i . Then, k = [ k , . . . , k n ] T is the degree vector. The eigenvaluesand eigenvectors of the adjacency matrix are designated, respectively, by λ i , i =1 , . . . , n , λ > λ ≥ · · · ≥ λ n and ψ i , i = 1 , . . . , n . To complete the notation we use u = [1 , . . . , T as the all ones vector, U = uu T , = [0 , . . . , T and I the identitymatrixIn the implementation of compartmental epidemiological models we consider x i ( t ) to be the probability that node i is infected at time t . For the wholenetwork, we define the vector x ( t ) = [ x ( t ) , . . . , x n ( t )] T . We designate by p theinitial probability of being infected, by q = 1 − p the initial probability of beinghealthy; β is the infection rate per link, γ the recovering rate, β e = β/γ the effectiveinfectivity rate and τ the epidemic threshold, that is the critical effective rate abovewhich the disease infects a non-zero fraction of the whole population.We consider here a Susceptible-Infected-Susceptible epidemiological model onthe graph Γ . In this case an infected node can infect any of its nearestsusceptible neighbors which then become infected with infection rate β > . Theinfected node can recover with recovery rate γ > and become susceptible again(see Fig. 2).Fig. 2: Diagram illustrating the flux of susceptible nodes (S) that becomeinfected/infectious (I) with rate β and which can cure without immunity and becomesusceptible again with rate γ .The evolution of the probability of getting infected for a node i is describedanuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv Where to cut to delay a pandemic with minimum disruption? by : dx i ( t ) dt = β (1 − x i ( t )) X j ∈N i A ij x j ( t ) − γx i ( t ) , t ≥ t , (2.1)where A ij are the entries of the adjacency matrix for the pair of nodes i and j , and N i is the set of nearest neighbors of i . In matrix-vector form, it becomes: ˙ x ( t ) = d x ( t ) dt = β [ I N − diag ( x ( t ))] A x ( t ) − γ x ( t ) , (2.2)with initial condition x (0) = x = p u . The following two results can be found in and characterize the behavior of network SIS below and over the epidemic threshold,which we adapt here to the case of undirected networks only: Theorem 2.1. If βλ /γ < we have the following results:(i) if x ∈ [0 , n then x ( t ) ∈ [0 , n for all t > ;(ii) there exists a unique equilibrium point x ⋆ = and it is exponentially stable;(iii) the linearization of the model around the point is given by ˙ x ( t ) = ( βA − γI ) x ( t ) . (2.3) Theorem 2.2. If βλ /γ > we have the following results:(i) if x ∈ [0 , n then x ( t ) ∈ [0 , n for all t > and if x > then x ( t ) > for all t > ;(ii) there exists an equilibrium point x ⋆ = , the epidemic outbreak, exponentiallyunstable;(iii) there exists an equilibrium point x ⋆ = , the endemic state, exponentiallystable such that: x ⋆ → a (cid:16) βγ λ − (cid:17) ψ if β → (cid:16) γλ (cid:17) + u − γβ ( diag ( k )) − if γ → , (2.4) where a = k ψ k ψ T diag ( ψ ) ψ . Lee-Tenneti-Eun approximation of the SI model
Using similar notations as before, the Susceptible-Infected (SI) model is writtenas : dx i ( t ) dt = β (1 − x i ( t )) X j ∈N i A ij x j ( t ) , t ≥ t , (2.5)which in matrix-vector form becomes:anuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv Paolo Bartesaghi and Ernesto Estrada d x ( t ) dt = β [ I N − diag ( x ( t ))] A x ( t ) , (2.6)with initial condition x (0) = x . It is well-known that the linearization of the modelaround the point is given by d x ( t ) dt = βA x ( t ) (2.7)which is exponentially unstable. Lee-Tenneti-Eun (LTE) rewrote the SI equationas − x i ( t ) dx i ( t ) dt = β X j ∈N i A ij (cid:16) − e − ( − log(1 − x j ( t ))) (cid:17) , (2.8)which is equivalent to dy i ( t ) dt = β X j ∈N i A ij f ( y j ( t )) , (2.9)where y i ( t ) := g ( x i ( t )) = − log (1 − x i ( t )) ∈ [0 , ∞ ] , f ( y ) := 1 − e − y = g − ( y ) .They then considered the following linearized version of the previous nonlinearequation d ˆ y ( t ) dt = βA diag (1 − x ( t )) ˆ y ( t ) + β h ( x ( t )) , (2.10)where ˆ x ( t ) = f (ˆ y ( t )) in which ˆ x ( t ) is the approximate solution to the SI model, ˆ y ( t ) = g ( x ( t )) and h ( x ) := x + (1 − x ) log ( u − x ) . They then proved the following result.
Theorem 2.3.
For any t ≥ t , x ( t ) (cid:22) ˆx ( t ) = f (ˆ y ( t )) (cid:22) ˜x ( t ) , (2.11) where ˆ y ( t ) = e β ( t − t ) A diag (1 − x ( t )) g ( x ( t ))+ ∞ X k =0 ( β ( t − t )) k +1 ( k + 1)! [ A diag (1 − x ( t ))] k A h ( x ( t )) , (2.12) is the solution of the approximate model of LTE and ˜x ( t ) = e β ( t − t ) A x ( t ) , (2.13)anuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv Where to cut to delay a pandemic with minimum disruption? is the solution of the linearized model. When t = 0 and x i (0) = p , ∀ i = 1 , , . . . , N the previous equation istransformed to ˆ y ( t ) = (1 /q − e qβtA u − (1 /q − q ) u . (2.14)
3. Mathematical results3.1.
Tight upper bound to the SIS model
We start here by rewriting the SIS model in Eq. (2.1) with the use of the variables y i ( t ) , such that x i ( t ) = 1 − e − y i ( t ) and ˙ x i ( t ) = e − y i ( t ) ˙ y i ( t ) . Then, we have ˙ y i ( t ) = β n X j =1 A ij (cid:16) − e − y j ( t ) (cid:17) − γ ( e y i − , (3.1)which can be transformed to ˙ y i ( t ) = n X j =1 h βA ij (cid:16) − e − y j ( t ) (cid:17) − γδ ij ( e y j − i , (3.2)using Kronecker δ ij function.Let us remark that f ( y ) = 1 − e − y is an increasing concave function. Then f ( y ) < f ( y ) + f ′ ( y ) ( y − y ) = e − y y + 1 − e − y ( y + 1) . (3.3)Also g ( y ) = e y − is an increasing convex function, such that g ( y ) > g ( y ) + g ′ ( y ) ( y − y ) = e y y − − e y ( y − . (3.4)We can now apply these conditions to Eq. (3.2) to obtain ˙ y i ( t ) < βe − y n X j =1 A ij y j ( t ) − γe y n X j =1 δ ij y j ( t )+ β (cid:2) − e − y ( y + 1) (cid:3) n X j =1 A ij + γ [1 + e y ( y − n X j =1 δ ij := ˆ y i ( t ) , (3.5)where we have called the upper bound ˆ y i . Using the notation settled for the initialconditions, this equation is written as ˙ˆ y i = βq n X j =1 A ij ˆ y j − γq n X j =1 δ ij ˆ y j + β ( p + q log q ) n X j =1 A ij − γ p + log qq , (3.6)anuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv Paolo Bartesaghi and Ernesto Estrada or in matrix-vector form as ˙ˆ y ( t ) = (cid:18) βqA − γq I (cid:19) ˆ y ( t ) + (cid:20) β ( p + q log q ) A − γq ( p + log q ) I (cid:21) u . (3.7)It is straightforward to realize that (3.7) has the form: ˙ˆ y ( t ) = B ˆ y ( t ) + b , (3.8)which can then be solved using the method of variation of parameters. Thereforewe have our main result. Theorem 3.1.
Let x ( t ) , ˜ x ( t ) and ˆx ( t ) be, respectively, the solution of the exact,linearized ˙˜ x ( t ) = ( βA − γI ) ˜ x ( t ) , and approximate (3.7) SIS model, with the sameinitial conditions: x (0) = ˆx (0) = ˜ x (0) = x = p u . Then, x ( t ) (cid:22) ˆx ( t ) (cid:22) ˜ x ( t ) . (3.9)We will prove this result by two parts using the following Lemmas. We shouldremark that this result indicates that the solution of the approximate (3.7) SISmodel represents an upper bound to the exact solution, which is always below thediverging solution of the linearized SIS model. Let us now prove the first part ofthis results using the following. Lemma 3.1.
Let y ( t ) be the transformed solution of the SIS model. Let ˆ y ( t ) bethe solution of the (3.7) model, then y ( t ) (cid:22) ˆ y ( t ) = e Bt (cid:2) B − b − log q u (cid:3) − B − b , (3.10) where B = βqA − γq I (3.11) and b = (cid:20) ( p + q log q ) βA − ( p + log q ) γq I (cid:21) u . (3.12) Remark 3.1.
Let’s make some remarks about solution (3.10):(i) if t = 0 : ˆ y (0) = − log q u and ˆ x (0) = p ;(ii) if γ = 0 , we have B = qβA and b = ( p + q log q ) βA u so that B − b = (cid:16) pq + log q (cid:17) u . Solution (3.10) reduces to ˆ y ( t ) = pq e qβAt u − (cid:18) pq + log q (cid:19) u (3.13)anuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv Where to cut to delay a pandemic with minimum disruption? which is equal to the solution for the SI Model by LTE in Eq. (2.14). Let us observethat for t → + ∞ , ˆ y ( t ) → + ∞ and ˆ x ( t ) → .(iii) if β = 0 : B = − γq I and b = − ( p + log q ) γq u so that B − b = ( p + log q ) u .Solution (3.10) becomes ˆ y ( t ) = p (cid:16) e − γq t − (cid:17) u − log q u (3.14)Let us observe that for t → + ∞ , ˆ y ( t ) → − p − log q and ˆ x ( t ) → − qe p . Thisbound doesn’t converge to as t → + ∞ but to x ⋆ = (1 − qe p ) u . Observe that < − qe p < p , as expected, and that − qe p → as p → . For instance, x ⋆ < . if p < . ; x ⋆ < . if p < . ; x ⋆ < . if p < . .(iv) if β = 0 and γ = 0 , the exponential term in equation (3.10) can be writtenas e Bt = e ( βqA − γq I ) t = e ( βqM Λ M T − γq I ) t = e M ( βq Λ − γq I ) M T t = M e ( βq Λ − γq I ) t M T (3.15)where Λ is the diagonal matrix of the eigenvalues of A and M is the orthogonalmatrix whose columns are the eigenvectors of A . As t grows to + ∞ , the diagonalexponential terms e ( βqλ i − γq ) t grows to + ∞ if ( βqλ i − γq ) > . In particular, if ( βqλ − γq ) < , no one of these terms grows to + ∞ and the epidemic decays. Thus,we can identify a threshold given by the following condition: β e = βγ < q λ = τ (3.16)such that τ = q λ is the threshold of this bound solution. Then, if β e < τ theepidemic decays; if β e > τ the epidemic grows. We should remark that as λ increases (and so does the average degree in the network), condition above becomestricter and the spread of epidemics is facilitated. Moreover, in general, τ is biggerthan /λ , which is the threshold in the exact solution of SIS Networked Model,and it approaches such a threshold as p decreases. Lemma 3.2.
Let ˜ x ( t ) be the solution of the linearized SIS problem ˙˜ x ( t ) =( βA − γI ) ˜ x ( t ) . Then ˆx ( t ) (cid:22) ˜ x ( t ) . (3.17) Proof.
We focus on the above-the-threshold behavior, i.e., we assume β e = βγ > q λ = τ. Then, because q < , we have that β e = βγ > λ . Following Lemma A.1 by LTE(see Eq. (29) in ), since the initial conditions are the same for the bound solutionand the linearized process, i.e., ˆx (0) = ˜ x (0) = x = p u , it is enough to prove thatanuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv Paolo Bartesaghi and Ernesto Estrada d ˆx ( t ) dt (cid:22) d ˜ x ( t ) dt (3.18)for all t ≥ . Let us remind that ˆx ( t ) = 1 − e − ˆy ; then we have d ˆx ( t ) dt = e − ˆy d ˆy ( t ) dt (cid:22) d ˆ y ( t ) dt , (3.19)for all t ≥ , where the inequality follows from e − ˆ y i < for all ˆ y i ∈ [0 , ∞ ] . By (3.10)we have d ˆy ( t ) dt = e Bt B (cid:2) B − b − log q u (cid:3) = e Bt [ b − log q B u ] , where B and b are given by Eq. (3.11) and Eq. (3.12), respectively. Since b − log q B u = p (cid:18) βA − γq I (cid:19) u , we have d ˆy ( t ) dt = e ( βqA − γq I ) t (cid:20) p (cid:18) βA − γq I (cid:19) u (cid:21) (cid:22) e ( βA − γI ) t [ p ( βA − γI ) u ] = d ˜ x ( t ) dt , where the last inequality is justified by the fact that e ( βqλ i − γq ) t < e ( βλ i − γI ) t and (cid:16) βA − γq I (cid:17) u (cid:22) ( βA − γI ) u . This finally proves the result. Remark 3.2.
It is well-known that the linearized SIS model approaches the exactsolution only when t → . This is the case particularly when β and γ are bothsmall, and β > γ as can be seen in Fig. 3(a), which refers to the toy network in Fig.1 with parameters β = 0 . , γ = 0 . and p = 1 / . Notice that we have usedlogarithmic scale in the time to specially highlight the short time behavior. In thiscase, it can be seen that the upper bound found here also coincides with the exactsolution for short times. For longer times, the linearized solution quickly diverges,while our upper bound behaves appropriately and converges to the steady statealmost as the same time as the exact solution. When, β < γ and both values arenot so small, the situation is pretty different from the previous one for the linearizedmodel. In this case, not even for very small times, the linearized solution coincideswith the exact one as can be seen in Fig. 3(b) for β = 0 . , γ = 0 . and p = 5 / . The upper bound found here coincides with the exact solution for a relatively longperiod of time and then converges to a steady state far from the 100% of contagionas expected for these given set of parameters.
Remark 3.3.
We can summarize the strength points of this bound as follows: (i)it is an upper bound for any β , δ and p , (ii) it extends to SIS model the upperanuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv Where to cut to delay a pandemic with minimum disruption? Time P r opo r t i on o f i n f e c t ed node s (a) Time P r opo r t i on o f i n f e c t ed node s (b) Fig. 3: Time evolution of proportion of infected nodes in a SIS epidemic on thetoy network illustrated in Fig. 1. In blue (continuous) line we plot the exact SISsolution, in red (broken) line the solution provided by the upper bound found hereand in black (dotted) line the solution to the linearized model. The time (x-axis) isshown in logarithmic scale. In (a) we have β = 0 . , γ = 0 . and p = 1 / . In(b) β = 0 . , γ = 0 . and p = 5 / . bound for SI model by LTE, and so it generalizes it; (iii) it captures the presenceof a threshold τ consistent with classic models; (iv) for small initial probabilities p it gives a close approximation of the exact solution and a very accurate descriptionof the real spreading phenomenon, both above or under the threshold. At the sametime, it should be taken into account that, for big initial probabilities p , the solutionstill remains an upper bound, but the approximation gets worse as p → . Moreover,the bound predicts that ˆ x ( t ) → , below the threshold, only for small p . Epidemic spread and network communicability
One of the main goals of network theory is to understand dynamical processes interms of structural network properties. Here we use the upper bound found in thissection for the SIS model to understand how the communicability between nodesin a network captures the propagation of a disease through the nodes and edges ofthe network. We start by setting B = qβA − γq I = − γq (cid:0) I − q β e A (cid:1) =: − γq D . Thenwe can rewrite the vector B − b − log q u in the following way: B − b − log q u = pq (cid:2) I − p ( I − q β e A ) − (cid:3) u = pq (cid:2) I − pD − (cid:3) u , anuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv Paolo Bartesaghi and Ernesto Estrada where D = I − q β e A. In this way, solution (3.10) becomes ˆ y ( t ) = pq (cid:20) e − γq Dt − I (cid:21) (cid:2) I − pD − (cid:3) u − log q u . (3.20)Let v = (cid:2) I − pD − (cid:3) u with elements v i = P nj =1 (cid:2) I − pD − (cid:3) ij and let us set C = (cid:26) max i v i if β e > τ, min i v i if β e < τ. (3.21)We then have the following result. Lemma 3.3.
The probability that a node i ∈ V in a graph is infected at a giventime t is bounded by its total communicability R i as x i ( t ) ≤ − q exp − C pq R i ( ̺ ) − e γq t e γq t , (3.22) where R i ( ̺ ) = (cid:0) e ̺A u (cid:1) i and ̺ = qβt . Proof.
Based on the previous definitions we have that y ( t ) (cid:22) ˆ y ( t ) (cid:22) C pq (cid:20) e − γq Dt − I (cid:21) u − log q u =: ¯ y ( t ) . (3.23)For a given node i ∈ V we have ¯ y i ( t ) = C pq h(cid:16) e − γq Dt u (cid:17) i − i − log q = C pq (cid:0) e qβAt u (cid:1) i − e γq t e γq t − log q = C pq R i ( ̺ ) − e γq t e γq t − log q, (3.24)from which the solution immediately follows.We should notice that, if γ → then C → and ¯ x i ( t ) → − qe − pq ( R i − ,which equals the LTE solution of the SI model. We should also remark that, inthis approximation, all the structural information determining the dynamics of theSIS process is stored in the variable R i . We can interpret structurally this indexas follow. Let G ( ̺ ) = exp ( ̺A ) . Then, R i ( ̺ ) = P nj =1 G ij ( ̺ ) , where G ij ( ̺ ) =(exp ( ̺A )) ij = P ∞ k =0 h ( ̺A ) k i ij k ! . The important thing here is that (cid:0) A k (cid:1) ij counts the number of walks of length k between the nodes i and j . A walk of length k is any sequence of (not necessarily different) vertices v , v , . . . , v k , v k +1 such thatanuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv Where to cut to delay a pandemic with minimum disruption? for each i = 1 , , . . . , k there is an edge from v i to v i +1 . When i = j the walkis known as closed. In the expression of ¯ y i ( t ) , R i ( ̺ ) describes every trajectory ofthe infective particle starting at the node i (and ending elsewhere) at a given timeand under the fixed initial conditions. It is clear that with all the epidemiologicalparameters fixed, ¯ y i ( t ) depends linearly on R i ( ̺ ) . Network capacity to reroute goods/items/passengers
We consider that on the same graph Γ where the infective particle is diffusing,there are other desirable diffusive processes taking place, such as the diffusion ofgoods, items, and passengers, which move the economy. We consider that such goodsare diffusing through the graph by means of all available walks connecting a pair ofnodes i and j in exactly the same way as the infective particle is using them. Indeed,there are approaches to modeling the traffic on networks which use epidemiologicalmodels like SIS .Therefore, let us find what are the routes which are more probable forthe infective particle to travel through, which will be the same ones for thegoods/items/passengers moving in the network. For that we start by defining thefollowing difference: ξ ij ( ̺ ) := G ii ( ̺ ) + G jj ( ̺ ) − G ij ( ̺ ) , (3.25)which represents the difference between all those walks that start and end at thesame vertex to those walks which go from one node to another . The first two termsthen represent the circulability of a diffusive particle around a given node, whilethe last represents the transmissibility between two nodes . Before continuing weneed a clarification here. In the definition of ξ ij ( ̺ ) we are using the parameter ̺ used in SIS model of the disease propagation. However, we are expecting that thisparameter ξ ij ( ̺ ) captures the mobility of goods/items/passengers in the network,not of the disease. In the particular cases we are studying here we do not havean estimation for the parameter ̺ for the mobility of goods/items/passengers.Additionally, we have the problem that the measure ξ ij ( ̺ ) is dependent on ̺ .Therefore, to make comparable the results of the viral spreading and the mobilityof goods/items/passengers we use in the calculations of the last the same parameter ̺ as for the first. The theoretical justification for this assumption is that we need ̺ ≪ for the mobility of goods/items/passengers to avoid congestion problems atthe nodes and the values used for the SIS dynamics fulfill this requirement. In itwas proved the following result. Lemma 3.4.
Let ξ ij ( ̺ ) for ̺ ∈ R be the difference between the circulabilities ofa diffusive particle around the nodes i and j , and the transmissibility between bothnodes. Then, ξ ij ( ̺ ) is a Euclidean distance between the corresponding nodes. We should recall that both terms, circulability and transmissibility, contributepositively to the infection propagation through: R i ( ̺ ) = G ii ( ̺ ) + G ij ( ̺ ) + P nk = i = j G ik ( ̺ ) . Therefore, we aim here at the following task:anuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv Paolo Bartesaghi and Ernesto Estrada • How to decrease significantly R i ( ̺ ) , and consequently ¯ x i ( t ) , withoutincreasing significantly ξ ij ( ̺ ) , and consequently minimally affecting thenetwork capacity to diffuse goods/items/passengers?Although ξ ij ( ̺ ) could be a good proxy for the network capacity of transportinggoods/items/passengers we should be aware that all the transport occurring in anetwork occurs through the paths connecting two nodes. That is, ξ ij ( ̺ ) does notnecessarily indicate the route followed by an item from the node i to the node j . For finding such routes we need a geometrization of the graph. This is carriedout by defining a length space on it . Let us consider e = ( i, j ) as a compact1-dimensional manifold with boundary ∂e = i ∪ j . Let the edge e = ( i, j ) be giventhe ξ ij ( ̺ ) metric such that e ij ∼ = isom (cid:26) [0 , ξ ij ( ̺ )] ( i, j ) ∈ E i, j ) / ∈ E. (3.26)We now extend the metric on the edges of Γ via infima of lengths of curvesin the geometrization of Γ . Then, the network becomes a metrically length space,which is locally compact, complete and geodetic . Now define the “shortest diffusivepath length” as: C ij ( Γ, ̺ ) := min
P,ij X ( i,j )= e ∈ Ee ∈ P ξ ij ( ̺ ) , (3.27)where P is a path in Γ , i.e., a walk with repetition neither of vertices nor ofedges, and the minimum is taken among all paths connecting the correspondingpairs of vertices. We can now define the capacity of a network to reroutegoods/items/passengers after the removal of an edge e by: ∆ ¯ C ( Γ − e, ̺ ) = ¯ C ( Γ − e, ̺ ) − ¯ C ( Γ, ̺ )¯ C ( Γ, ̺ ) , (3.28)where ¯ C ( Γ, ̺ ) is the mean shortest communicability path (SCP), that is the averageof C ij ( Γ, ̺ ) over all shortest diffusive paths connecting pairs of nodes in Γ , and Γ − e is the graph from which the edge e has been removed. Implementation of an edge-removal strategy
We are always interested in nontrivial edge removals here, i.e., those that donot disconnect the graph. Then, to respond to the main query formulated inthe previous subsection we will consider edge-removal strategies that decreasesignificantly R i ( ̺ ) , but not increase significantly ∆ ¯ C ( Γ − e, ̺ ) . It is obviousthat any strategy that increases the relative communicability between two vertices G ij ( ̺ ) will necessarily drops ξ ij ( ̺ ) . Unfortunately, it will also increase R i ( ̺ ) . The obvious strategy seems to drop G ii ( ̺ ) so that both R i ( ̺ ) and ∆ ¯ C ( Γ − e, ̺ ) diminish their values. However, very frequently dropping G ii ( ̺ ) also decreases G ij ( ̺ ) . Therefore, we cannot foresee at first hand a strategy that fulfill bothanuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv
Where to cut to delay a pandemic with minimum disruption? requirements and we then implement a computational approach to investigate theproblem. First, we start by defining the following term: O ( e, t ) := max e ∈ E ∆t ⋆ ∆ ¯ C ( Γ − e, ̺ ) , (3.29)where ∆t ⋆ = ( t ⋆ ( Γ − e ) − t ⋆ ( Γ )) /t ⋆ ( Γ ) and t ⋆ is the time at which every node inthe network is infected, i.e., the steady state of the SIS process, where the maximumis obtained among all the edges of the graph.We must be aware of an important characteristic of this process. The term ∆ ¯ C ( Γ − e, ̺ ) depends on t , which means that O ( e, t ) is different for different times.This means that the process of edge-removal is time-dependent, and we should goremoving edges as the time of the evolution of the epidemic goes on. This is a veryrealistic scenario and reflect some of the difficulties found in the current COVID-19 pandemics, where the measures taken at a given time are not necessarily theoptimal ones at another. In the Supplementary Information we provide a detailedlist of steps for the implementation of the edge-removal process and an R code forthe same. In Algorithm 1 we give the pseudo-code of the current implementation.3.4.1. Toy network example
It is time now to give some numbers and we will start by analyzing the toy modelillustrated in Fig. 1. The process evolves as follow. We consider the time evolutionof the SIS model in which we observe the evolution of the ratio of infected nodes bytime. At a given time, we make the following plot. For every potential edge-removalof interest, here made for every of the 16 edges of the graph, we plot ∆ ¯ C ( Γ − e, ̺ ) vs. ∆t ⋆ in a square unit as the one illustrated in Fig. 4(a) for t = 150 . The pointsin the plot correspond to the effects produced by removing the corresponding edge.The radii and color of these points are proportional to the values of O ( e, t = 150) .It can be clearly seen that there are two groups of edges. In the upper-right cornerwe have all the edges whose removal change very much the capacity of the networkto reroute goods/items/passengers. In the opposite corner we have all those edgeswhose removal increase the time for infecting the whole population with minimumdisruption of network operational capacity. We can select here a given number ofedges to be removed in dependence of other factors, of economic or logistical nature.We remove here one edge at a time. In this case we select the edge with the largest O ( e, t = 150) which is the edge (2 , . This single removal increases the time atwhich the SIS dynamics infects the of the whole population by 17.1% with aminimum change in the network capacity to operate, i.e., ∆ ¯ C ( Γ − e, ̺ ) ≈ . .This edge removal produces a change in the trajectory of the infection as observedin the plot Fig. 4(d), which is marked by the point (2 , , which represents the edgeremoved.We now continue observing the evolution of the epidemic until we decide thenext intervention. In this case we decided to do it at t = 300 (we simply use similarperiods of time here to make the interventions, in a real-life situation this can beanuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv Paolo Bartesaghi and Ernesto Estrada
Algorithm 1:
Optimal Downdating
Input:
The original network:
Γ = (
V, E ) , v i ∈ V and ( i, j ) ∈ E ;The downdating times: T k = k · a, k ∈ [0 , m ] , a fixed time step;The epidemic level ε . Output:
The downdated network after m steps: Γ m = ( V, E m ) set Γ = Γ , A = A , X ( t ) = X ( t ) and t = min ( t : X ( t ) ≥ − ε ) for k ∈ [1 , m ] do A ( i,j ) k ← A k − − A k − · U ij U Tji , ∀ i, j ∈ E k − generate Γ ( ij ) k with adjacency matrix A ( ij ) k , ∀ i, j ∈ E k − if count . components (cid:0) Γ ( ij ) k (cid:1) = 1 : A k ← A k − and stop; else (NULL) X ( ij ) k ( t ) ← X k − ( t ) , ∀ i, j ∈ E k − for t ≥ T k compute X ( ij ) k ( t ) on network Γ ( ij ) k with X ( ij ) k ( T k ) = X k − ( T k ) , ∀ i, j ∈ E k − compute t ( ij ) k = min (cid:16) t : X ( ij ) k ( t ) ≥ − ε (cid:17) , ∀ i, j ∈ E k − compute ∆t ( ij ) k = t ( ij ) k − t k − compute ∆ ¯ C ( ij ) k = ¯ C ( ij ) k ( a ) − ¯ C k − ( a )¯ C k − ( a ) where ¯ C ( ij ) k ( a ) is the mean SCP onnetwork Γ ( ij ) k at time T k = ka and ¯ C k − ( a ) is the mean SCP onnetwork Γ k − select ( i k , j k ) ∈ E k − corresponding to max i,j ∆t ( ij ) k ∆ ¯ C ( ij ) k remove edge ( i k , j k ) ∈ E k − from Γ k − and generate Γ k return Γ m and X m ( t ) done at irregular intervals). Notice that the value of ∆ ¯ C ( Γ − e, ̺ ) is dependenton the time at which we decide to make the plot. The new situation is observedin the plot Fig. 4(b) where the model inform us that the next best cut should bemade at edge (2 , . The combined interventions of cutting edges (2 , and (2 , increases the time to reach 90% of infected population by 53.8%. If we translatethis into days, for instance, it means to gain almost 54 days out of 100, which isvery significant. Now the capacity of the network to operate has drop by 3.0%. Thetrajectory of the infection changes again at the point (2 , of the plot in Fig. 4(d).In Fig. 4(c) we illustrate the result of the third intervention at t = 450 , whichindicates that the next cut should be made to the edge (3 , . The combinedinterventions which have removed three edges out of 16 in this toy network increasesthe time for infecting the of the whole network by 160.4%(!). That is, byremoving only 18.75% of edges, which does not disconnect the graph, we have morethan duplicated the time that we now have to take actions during the epidemic, from381 time units to 992 ones. All this by dropping only in 5.34% the total capacity ofthe network to operate in relation to normal conditions. The epidemic now followsanuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv
Where to cut to delay a pandemic with minimum disruption? the trajectory from the point (3 , in the plot in Fig. 4(d). (a) (b)(c) (d) Fig. 4: Plots ∆ ¯ C ( Γ − e, ̺ ) vs. ∆t ⋆ : the radii and color of the points are proportionalto the values of O ( e, t ) at different time units, specifically at a) t = 150 , b) t = 300 ,c) t = 450 ; figure d) shows the trajectory of the infection curve as a function oftime: each marked point refers to the removal of the corresponding edge.
4. Analysis of a real-life situation: the UK airports network
Here we consider the network of domestic flights between 44 commercial airportsin the United Kingdom in the year 2003. This was the year in which the SARSepidemic was spreading across the world. The network consists of 220 weighted edgesrepresenting air internal routes between these 44 airports. The weights correspondto the number of passengers transported during that year between the correspondinganuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv Paolo Bartesaghi and Ernesto Estrada airports. In Fig. 5 we show a representation of this network where the nodes aredrawn with size and colors proportional to their weighted degrees–total number ofpassengers arriving/departing to/from that airport in 2003. The weighted degree w i and the “standard” degree k i , i.e., number of edges incident to the node i ,are related to each other by means of a power-law relation: w ≈ e . k . with Pearson correlation coefficient r = 0 . . More importantly, rank correlationindicates that both indices rank the airports in very different ways. For instance,the Kendall τ coefficient between both indices is only τ ≈ . . According to w themost “important” airport was Heathrow, which was visited this year by 6,176,092passengers, representing 13.8% of all passengers traveling this year across the U.K.However, the degree of this node is only 9, possibly indicating its major role asan international hub and connecting only to those relevant national airports fromwhich passengers can easily move to other places. On the other side of the coinwe have the airports of Jersey and Aberdeen with degree 24 and 23, respectively.Each of them moves less than 3% of the total number of passengers in the U.K. thisyear. However, Jersey is a major touristic destination in the U.K. and Aberdeenhas become an important work hub in the U.K. due to the oil industry. Thus, theyreceive flights from many different cities across the U.K., although the number ofpassengers is relatively small.Fig. 5: Network of air transport of passengers in the U.K. The size and color of thenodes is proportional to the number of passengers transported from/to that airport.Two airports are connected if there is at least one flight between the two.anuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv Where to cut to delay a pandemic with minimum disruption? SIS models have been previously used to study the propagation of diseasesthrough airport networks, mainly due to the mathematical advantage of using thismodel over other compartmental ones . For the case of modeling diseaseslike SARS among a population it is more appropriate to use models based on theSIR methodology (see and references therein). Therefore, we explain here why dowe use our SIS approach to this particular case. First, we should remark that ournodes represent the airports in the U.K. Thus, we consider the cases in which anairport is either susceptible or infected with the disease. An airport is susceptible ifnone of the passengers in that airport at a given time is infected. This airport canbecome infected due to the fact that infected passenger(s) come from other nearestneighbor airports (see Fig. 6(a)). (a)(b) Fig. 6: Illustration of the S → I (a) and I → S (b) transformations of an airportlabeled as i in the airport network.On the other hand, it is clear that an airport cannot be considered “recovered”in the sense of creating immunity, at least in the absence of quarantines in thisairport (which are not considered here). Therefore, an infected airport can becomesusceptible again if the infected passenger(s) that were located in that airport moveaway from it (see Fig. 6(b)).We have seen before that there are significant differences in the degree profilesof the airports considering the weighted (number of passengers) and unweightedversions of the networks. Therefore, we consider here the analysis of both versionsof the U.K. airport transportation network for the propagation of a SIS disease andanuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv Paolo Bartesaghi and Ernesto Estrada the implementation of edge-removal strategies. We apply the edge-removal strategydescribed in this work for removals at t = 200 , , , , . In the case ofthe unweighted network, these removals correspond to the connections between thefollowing pairs of airports (in order): Glasgow-Manchester; Belfast City-Manchester;Stansted-Edinburgh; Isle of Man-Manchester and Bristol-Guernsey. In total, theremoval of these 5 connections increases the cumulative time for infecting thewhole network - precisely to overcome the epidemic level of 90% of infected nodes- by 161.9% with a decrease of 0.25% in the capacity of the network to reroutegoods/items/passengers.time Route ∆t ⋆ (%) ∆ ¯ C (%) ∆ ¯ l (%)200 Glasgow-Manchester 9.52 0.026 0.048400 Belfast C.-Manchester 23.81 0.068 0.101600 Stansted-Edinburgh 47.62 0.085 0.152800 Isle of Man-Manchester 89.95 0.189 0.2031000 Bristol-Guernsey 161.90 0.247 0.254Table 1: Quantitative results of the edge-removal strategy using the unweightedversion of the U.K. air transportation network.In contrast, the consideration of the number of passengers between the differentairports produces a completely different picture. First, we need a normalizationof the weighted adjacency matrix to make the edge weights comparable to thoseof the unweighted version. This is carried out by dividing the weighted adjacencymatrix with the mean value of the edge weights in the network. In this way, boththe unweighted and the normalized weighted adjacency matrices have the samemean. Using this strategy the order of removals is as follows: Heathrow-Edinburgh;Heathrow-Manchester; Heathrow-Glasgow and Heathrow-Belfast City. The fifthremoval is not carried out as it is not necessary to drop to probability of infecting thewhole network below 90%, which was the target of the experiment. The evolutionof the mean probability that an airport gets infected at three different times isillustrated in Fig. 7.By removing all the flights between Heathrow and Edinburgh we delay the timefor infecting the whole network by 38.9%. The second removal increases this time to88.9% and the third one increases it up to 194.4%. Finally, the cumulative removalof 4 connections increases the time to infect the whole network by 333.3%. Howthe capacity of the airport network to reroute goods/items/passengers has changedafter these removals? The response is surprising! The remove of all flights betweenHeathrow and Edinburgh does not drop the capacity of the global network to diffusegoods/items/passengers and passengers through its nodes. In contrast, it increasesthis capacity by 9.9%. This is, of course, a consequence of considering that suchanuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv Where to cut to delay a pandemic with minimum disruption? Fig. 7: Illustration of the evolution of the probability of getting infected in theU.K. airports network at an initial (no edge removal), intermediate (after removalsof Heathrow-Edinburgh and Heathrow-Manchester) and advanced (after all fouredge removals) times. The radii and colors of the nodes are proportional to theprobability of getting infected which is illustrated in colorbars.goods/items/passengers move in the network in a completely diffusive way. If weconsider that they move using the shortest paths, then we observe a drop in thecapacity of the network equal to ∆ ¯ l = 0 . , i.e., the increase in the average shortestpath length in the network after the removal. After the four removals previouslydescribed the network has increased its diffusive capacity by 3.6% with a drop inits capacity to deliver goods/items/passengers via shortest paths of 0.4%. In eitherway, the removal of these four inter-airport connections produces a remarkable delayon the propagation of the SIS disease in comparison with a very small affection ofthe network operative capacity.time Route ∆t ⋆ (%) ∆ ¯ C (%) ∆ ¯ l (%)200 Heathrow-Edinburgh 38.89 -9.93 0.049400 Heathrow-Manchester 88.89 -9.34 0.152600 Heathrow-Glasgow 194.44 -5.61 0.254800 Heathrow-Belfast C. 333.33 -3.60 0.407Table 2: Quantitative results of the edge-removal strategy using the passengers-weighed version of the U.K. air transportation network with adjacency matrixnormalized by the mean number of passengers in the network.anuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv Paolo Bartesaghi and Ernesto Estrada (a) (b)
Fig. 8: Time evolution of the proportion of infected airports subjected to edge-removal strategies in (a) the unweighted version of the network, and (b) thepassengers-weighed network with adjacency matrix normalized by the mean numberof passengers.We have also conducted experiments to show whether the previous results aredependent on the normalization scheme used for the weighted adjacency matrix. Inthis case we use two other normalization schemes, namely by dividing the weightedadjacency matrix by the maximum edge weight or by dividing it by the total sumof edge weights. In both cases the edges identified to be removed are the same asfor the case of normalizing by the mean weight with the addition of a fifth edgeto be removed, which corresponds to Belfast Int.-Liverpool. The time to infect thewhole network increases by 286.4% and by 309.5% after the fifth removal using thetwo additional schemes of normalization, respectively. All in all, these experimentsshow that the results previously described using the mean-weight normalization ofthe adjacency matrix are not specific of this kind of normalization and stressed theimportance of using passengers-weighted version of the airport networks.
5. Conclusions
We have developed an approximate solution to the SIS epidemiological model whichrepresents an upper bound to the exact solution of that model. This upper boundhas several important features: (i) it does not diverge as the linearized SIS model;(ii) it represents a worse-case scenario for the propagation of a SIS disease; (iii)its solution can be expressed in terms of the communicability function, allowingclear structure-dynamic relations. Using this model and its connection with thecommunicability function, we proposed here a general strategy for mitigating theeffects of a disease propagation on a network with minimum disruption of networkanuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv
REFERENCES capacities to reroute goods/items/passengers. This strategy consists in removingsome connections which are found to delay the propagation of a disease on thenetwork but minimally altering the capacity of the network to diffuse items amongits nodes or to reroute them by alternative shortest paths. As a proof of concept, wehave studied the airport transportation network of U.K. in 2003, where the nodesrepresent airports and the edges represent the flight connections, weighed by thenumber of passengers transported this year, between them. We have shown thatusing the strategy proposed in this work, the removal of only 4 origin-destinationpairs in a time-dependent way delays the propagation of an epidemic by more than330% relative to the original network. This delay represent a very significant gainin time for preparations of health systems and non-pharmaceutical interventions toconfront such epidemic. In addition, these removals alter minimally the capacityof the U.K. airport system to transport goods/items/passengers either in diffusiveways or via shortest-paths routing.The main emphasis of the current work has been on the mathematical,methodological side. We expect that the extension of this approach to otherepidemiological models allow more realistic implementations to tackle thisimportant kind of non-pharmaceutical interventions that mitigate the effects ofepidemics in the future. Acknowledgments
The author thanks financial support from Ministerio de Ciencia, Innovaciony Universidades, Spain for the grant PID2019-107603GB-I00 ”Hubs-repelling/attracting Laplacian operators and related dynamics on graphs/networks”.
References
1. D. Balcan, V. Colizza, B. Gonçalves, H. Hu, J. J. Ramasco and A. Vespignani,Multiscale mobility networks and the spatial spreading of infectious diseases,
Proceedings of the National Academy of Sciences (2009) 21484–21489.2. P. Bartesaghi, M. Benzi, G. P. Clemente, R. Grassi and E. Estrada, Risk-dependent centrality in economic and financial networks,
SIAM Journal onFinancial Mathematics (2020) 526–565.3. B. Baspinar and E. Koyuncu, A data-driven air transportation delaypropagation model using epidemic process models, International Journal ofAerospace Engineering (2016) 1–11.4. M. Benzi and P. Boito, Matrix functions in network analysis,
GAMM-Mitteilungen (2020) e202000012.5. M. Benzi and C. Klymko, Total communicability as a centrality measure, Journal of Complex Networks (2013) 124–149.6. M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature , volume319 (Springer Science & Business Media, 2013).anuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv REFERENCES
7. V. Colizza, A. Barrat, M. Barthélemy and A. Vespignani, The role of the airlinetransportation network in the prediction and predictability of global epidemics,
Proceedings of the National Academy of Sciences (2006) 2015–2020.8. D. M. Cutler and L. H. Summers, The covid-19 pandemic and the $16 trillionvirus,
JAMA (2020) 1495–1496.9. N. G. Davies, A. J. Kucharski, R. M. Eggo, A. Gimma, W. J. Edmunds,T. Jombart, K. O’Reilly, A. Endo, J. Hellewell, E. S. Nightingale et al., Effectsof non-pharmaceutical interventions on covid-19 cases, deaths, and demand forhospital services in the uk: a modelling study,
The Lancet Public Health (2020)e375 – e385.10. M. Enserink and K. Kupferschmidt, With covid-19, modeling takes on life anddeath importance, Science (2020) 1414–1415.11. E. Estrada, The communicability distance in graphs,
Linear Algebra and itsApplications (2012) 4317–4328.12. E. Estrada,
The structure of complex networks: theory and applications (OxfordUniversity Press, 2012).13. E. Estrada, Covid-19 and sars-cov-2. modeling the present, looking at the future,
Physics Reports (2020) 1 – 51.14. E. Estrada and N. Hatano, Communicability in complex networks,
Phys. Rev.E (2008) 036111.15. E. Estrada and D. J. Higham, Network properties revealed through matrixfunctions, SIAM review (2010) 696–714.16. E. Estrada and J. A. Rodriguez-Velazquez, Subgraph centrality in complexnetworks, Physical Review E (2005) 056103.17. S. Flaxman, S. Mishra, A. Gandy, H. J. T. Unwin, T. A. Mellan, H. Coupland,C. Whittaker, H. Zhu, T. Berah, J. W. Eaton et al., Estimating the effectsof non-pharmaceutical interventions on covid-19 in europe, Nature (2020)257–261.18. S. Gómez, A. Arenas, J. Borge-Holthoefer, S. Meloni and Y. Moreno, Discrete-time markov chain approach to contact-based disease spreading in complexnetworks,
EPL (Europhysics Letters) (2010) 38009.19. M. J. Keeling and K. T. Eames, Networks and epidemic models, Journal of theRoyal Society Interface (2005) 295–307.20. I. Z. Kiss, J. C. Miller, P. L. Simon et al., Mathematics of epidemics on networks:from exact to approximate models. (Springer, 2017).21. D. Koch, R. Illner and J. Ma, Edge removal in random contact networks and thebasic reproduction number,
Journal of mathematical biology (2013) 217–238.22. C. J. Kuhlman, G. Tuli, S. Swarup, M. V. Marathe and S. Ravi, Blockingsimple and complex contagion by edge removal, in (IEEE, 2013), pp. 399–408.23. C.-H. Lee, S. Tenneti and D. Y. Eun, Transient dynamics of epidemicspreading and its mitigation on large networks, in Proceedings of the TwentiethACM International Symposium on Mobile Ad Hoc Networking and Computing anuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv
REFERENCES (Association for Computing Machinery, New York, NY, USA, 2019), Mobihoc’19, p. 191–200.24. Y. Li, L. Zhao, Z. Yu and S. Wang, Traffic flow prediction with bigdata: A learning approach based on sis-complex networks, in (IEEE, 2017), pp. 550–554.25. F. Liljeros, C. R. Edling and L. A. N. Amaral, Sexual networks: implicationsfor the transmission of sexually transmitted infections, Microbes and infection (2003) 189–196.26. S. Markvorsen, Minimal webs in riemannian manifolds, Geometriae Dedicata (2008) 7.27. M. McKee and D. Stuckler, If the world fails to protect the economy, covid-19will damage health not just now but also in the future,
Nature Medicine (2020) 640–642.28. W. Mei, S. Mohagheghi, S. Zampieri and F. Bullo, On the dynamics ofdeterministic epidemic propagation over networks, Annual Reviews in Control (2017) 116–128.29. F. Ndairou, I. Area, J. J. Nieto and D. F. Torres, Mathematical modeling ofcovid-19 transmission dynamics with a case study of wuhan, Chaos, Solitons &Fractals (2020) 109846.30. M. E. J. Newman,
Networks: an introduction (Oxford university press, 2010).31. S. H. H. Nourzad and A. Pradhan, Network-wide assessment of transportationsystems using an epidemic spreading methodology, in
Computing in CivilEngineering (2013) (2013), pp. 387–394.32. M. Polyakova, G. Kocks, V. Udalova and A. Finkelstein, Initial economicdamage from the covid-19 pandemic in the united states is more widespreadacross ages and geographies than initial mortality impacts,
Proceedings of theNational Academy of Sciences (2020) 27934–27939.33. B. Qu and H. Wang, Sis epidemic spreading with heterogeneous infection rates,
IEEE Transactions on Network Science and Engineering (2017) 177–186.34. R. C. Reiner, R. M. Barber, J. K. Collins, P. Zheng, C. Adolph, J. Albright,C. M. Antony, A. Y. Aravkin, S. D. Bachmeier, B. Bang-Jensen et al., Modelingcovid-19 scenarios for the united states, Nature Medicine .35. Z. Ruan, C. Wang, P. M. Hui and Z. Liu, Integrated travel network model forstudying epidemics: Interplay between journeys and epidemic,
Scientific reports (2015) 11401.36. L. P. Sanders, B. Söderberg, D. Brockmann and T. Ambjörnsson, Perturbativesolution to susceptible-infected-susceptible epidemics on networks, PhysicalReview E (2013) 032713.37. O. S. Sukharev, Economic crisis as a consequence covid-19 virus attack: risk anddamage assessment, Quantitative Finance and Economics (2020) 274–293.38. T. Takaguchi and R. Lambiotte, Sufficient conditions of endemic threshold onanuary 5, 2021 2:3 WSPC/INSTRUCTION FILE Manuscript_ArXiv REFERENCES metapopulation networks,
Journal of theoretical biology (2015) 134–143.39. F. Wu, S. Zhao, B. Yu, Y.-M. Chen, W. Wang, Z.-G. Song, Y. Hu, Z.-W. Tao, J.-H. Tian, Y.-Y. Pei et al., A new coronavirus associated with human respiratorydisease in china,
Nature (2020) 265–269.40. H.-X. Yang, Z.-X. Wu and B.-H. Wang, Suppressing traffic-driven epidemicspreading by edge-removal strategies,
Physical Review E (2013) 064801.41. M. Zamir, Z. Shah, F. Nadeem, A. Memood, H. Alrabaiah and P. Kumam, Nonpharmaceutical interventions for optimal control of covid-19, Computer methodsand programs in biomedicine (2020) 105642.42. P. Zhou, X.-L. Yang, X.-G. Wang, B. Hu, L. Zhang, W. Zhang, H.-R. Si,Y. Zhu, B. Li, C.-L. Huang et al., A pneumonia outbreak associated with anew coronavirus of probable bat origin,
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