Moving the epidemic tipping point through topologically targeted social distancing
Sara Ansari, Mehrnaz Anvari, Oskar Pfeffer, Nora Molkenthin, Frank Hellmann, Jobst Heitzig, Juergen Kurths
MMoving the epidemic tipping point through topologically targeted social distancing
Sara Ansari, Mehrnaz Anvari, Oskar Pfeffer,
2, 3
Nora Molkenthin, Frank Hellmann, Jobst Heitzig, and J¨urgen Kurths
2, 4 FutureLab on Game Theory and Networks of Interacting Agents,Complexity Science Department, Potsdam Institute for Climate Impact Research,Member of the Leibniz Association, PO Box 601203, 14412 Potsdam, Germany Potsdam Institute for Climate Impact Research, Potsdam 14473, Germany Institute of Theoretical Physics, Technische Universit¨at Berlin, Hardenbergstr. 36, D-10623 Berlin,Germany Institute of Physics, Humboldt University, Berlin 12489, German
The epidemic threshold of a social system is the ratio of infection and recovery rate above whicha disease spreading in it becomes an epidemic. In the absence of pharmaceutical interventions (i.e.vaccines), the only way to control a given disease is to move this threshold by non-pharmaceuticalinterventions like social distancing, past the epidemic threshold corresponding to the disease, therebytipping the system from epidemic into a non-epidemic regime. Modeling the disease as a spreadingprocess on a social graph, social distancing can be modeled by removing some of the graphs links.It has been conjectured that the largest eigenvalue of the adjacency matrix of the resulting graphcorresponds to the systems epidemic threshold. Here we use a Markov chain Monte Carlo (MCMC)method to study those link removals that do well at reducing the largest eigenvalue of the adjacencymatrix. The MCMC method generates samples from the relative canonical network ensemble witha defined expectation value of λ max . We call this the “well-controlling network ensemble” (WCNE)and compare its structure to randomly thinned networks with the same link density. We observe thatnetworks in the WCNE tend to be more homogeneous in the degree distribution and use this insightto define two ad-hoc removal strategies, which also substantially reduce the largest eigenvalue. Atargeted removal of 80% of links can be as effective as a random removal of 90%, leaving individualswith twice as many contacts. Finally, by simulating epidemic spreading via either an SIS or anSIR model on network ensembles created with different link removal strategies (random, WCNE, ordegree-homogenizing), we show that tipping from an epidemic to a non-epidemic state happens ata larger critical ratio between infection rate and recovery rate for WCNE and degree-homogenizednetworks than for those obtained by random removals. INTRODUCTION
In the absence of pharmaceutical interventions, theprevention of infection through a reduction of social con-tacts presents an effective method to slow or even haltthe spread of an epidemic [1, 2]. However, limiting socialcontacts comes at a significant psychological [3] and eco-nomical [4, 5] cost. To reduce those socio-economic ad-verse effects, it is therefore desirable to use targeted socialdistancing measures, in order to remove fewer links forachieving the same effect. If the infection patterns for aparticular disease are known, for example, measures canbe taken to remove the most common routes of infec-tion [6–8]. Similarly, methods have been introduced fortargeted link removal if the infection status of individualnodes is known [8–10] as well as systems where the entirenetwork structure is known in detail [10, 11]. This wouldbe the case for example in more aggregated settings, likethe transport network, which has been shown to directlyaffect disease spreading [12].Often, however, diseases without pharmaceutical inter-ventions are new and we lack detailed knowledge abouttheir infection patterns, and in the absence of functioningcontact tracing many infections may go unnoticed, espe-cially if the symptoms are mild or unspecific. Thereforesocial distancing as a behavioral change is the primaryintervention for disease prevention [13]. In the extreme case of reducing the number of contactsin the social graph to zero, by necessity the disease diesout and is fully controlled. If there are no links left inthe network, the disease can not spread. Intuitively weexpect that the more contacts are removed, the better adisease can be controlled and prevented from becomingan epidemic. This can be made precise given a modelof the disease spread. For an approximation of SIS dy-namics with recovery rate δ and infectiousness β it wasshown in [14] that the epidemic threshold, τ , is deter-mined by the inverse of the largest eigenvalue of the ad-jacency matrix of the underlying contact network, i.e., τ = λ − max . If τ > β/δ , the disease quickly dies out andremains confined to a small section of the network, if τ < β/δ , however, the disease spreads as an epidemicover large parts of the network and becomes endemic.At τ c = ( λ cmax ) − = β/δ the system tips from epidemicto non-epidemic regime. Decreasing λ max thus repre-sents a topological way of controlling an epidemic thatis independent of the unknown or unchangeable diseaseparameters β and δ .By removing links from a network, one can generallyreduce the value of λ max . However, the size of this re-duction depends on the specific links removed (see theexample network in [14]). Here, we introduce a way toremove links based on λ max in order to sample from en-sembles with a given disease-controlling property usingMarkov chain Monte Carlo (MCMC) methods [15]. We a r X i v : . [ phy s i c s . s o c - ph ] F e b thereby study what characterizes contact reductions thatperform well at controlling the disease spread. We willsee that well-controlling networks on average have signif-icantly lower λ max even at relatively low ν . We find thatwell-controlling networks have a more strongly peakeddegree distribution, as well as more homogeneously sizedconnected components once the network is no longer con-nected. We therefore suggest and test two heuristics forachieving a similar effect by homogenizing the node de-grees in the network through link removal and find themto lead to a similarly strong control, suggesting that thestrongly peaked degree distribution characterizes well-controlling networks. Finally, by running both SIS andSIR models on networks obtained from different removalstrategies, we show that the tipping point in the criticalratio of β/δ occurs for larger ratios in the WCNE anddegree-homogenized networks than in generic removals.The results are robust across all initial network ensem-bles tested here, namely Barabasi–Albert (BA) networks[16], random geometric (RG) graphs [17] and a real-worldnetwork of friendships between high-school students [18]. BASIC NOTIONS AND MODEL DESCRIPTION
We start from an initial graph ensemble representingthe social network relevant for the spreading. The BA-and RG- network ensembles were chosen for their depic-tion of different types and applications of networks, onwhich epidemic spreading processes can happen. BA-Networks were introduced as a simplified model for on-line social networks, which are relevant for the spread-ing opinions, ideas or computer viruses. RG-networks onthe other hand capture the spatially embedded structureof most in-person social networks most relevant for thespread of diseases.We consider a generic reduction in which a certain per-centage of links are removed at random. Given a socialnetwork N soc with E edges this defines an ensemble of re-duced networks q ρ ( N soc ) that are equidistributed on thesubgraphs of N soc with (1 − ρ ) E edges, where 0 < ρ < λ max . That is, the ensem-ble of networks that is least distinguishable from randomcontact removals at a given expectation value of λ max .The canonical network ensemble is given by the proba-bilities p ( N ) = 1 Z e − νλ max ( N ) q ρ ( N ) . (1)Here Z is a normalization factor, ν is the inverse gener-icity as defined in [19], which interpolates between thegeneric ensemble (at ν = 0) and one peaked at the bestcontrolling networks at ν → ∞ (optimized). We callthe family of ensembles with reasonably high values of ν the well-controlling network ensembles (WCNE) and samples drawn from such an ensemble well-controllingnetworks (WCN). As in [19] we obtain well-controllingnetworks by employing an MCMC Metropolis–Hastingsmethod [20, 21].In both synthetic cases, the initial networks N soc have100 nodes and an average degree of (cid:104) k (cid:105) ≈
18, amount-ing to around 900 links. The precise number of linksfor the random geometric networks fluctuates slightlyaround that value due to the network construction al-gorithm used. B A - N e t w o r k R G - N e t w o r k Generic link removal High epidemic thresholda) b)c) d)
FIG. 1: Example networks from network ensemblesafter removing 80% of contacts from an initial BA andRG graph with N = 100 and E ≈ ν = 0 or, in the otherword, they show generic networks with random removedlinks. b) and d) present WCNE at ν = 1000. In allfigures nodes with the same color and size have thesame degree. To compare WCNE networks, i.e. b) andd), with the corresponding generic networks withrandom removed links in a) and c), it is visually clearthat WCNE networks have more homogeneous degreedistribution as well as more homogeneous componentsizes.Starting from a reduced network drawn randomly from q ρ ( N soc ) at a given contact reduction rate ρ , we keepthe number of links constant through all changes of thenetwork. We then vary the set of removed links usingMCMC. The probability of increasing/decreasing λ max in this step depends on the selected value of ν .Fig. 1 shows example networks from a generic removaland WCNE ensembles of BA and RG starting ensembleswith a removal rate of ρ = 0 .
8. At first glance, we seethat the WCNE examples have fewer disconnected nodesand a generally more homogeneous distribution of de-grees and component sizes. In the following, we describethe applied MCMC approach and its energy function indetail.The state of the system at a time step t is given bythe network structure and thus the adjacency matrix A t .The energy function for the process is given by the largesteigenvalue of the adjacency matrix A t , describing the re-maining network (i.e., the one obtained after removingthe current candidate set of edges), ε ( A ) = λ max ( A ) . (2)As mentioned in the introduction, it was shown in [14]that the epidemic threshold τ is proportional to the in-verse of λ max for an approximation of the SIS model. Inthe same paper it was also conjectured that this propor-tionality also holds in the SIR model. Thus, followingthat conjecture, we use λ max as our energy function.At each Monte Carlo step t → t + 1, a proposal A (cid:48) isgenerated to swap one edge between the remaining setof edges in the network and the set of currently removededges, thus keeping the number of removed edges con-stant. The proposal is then accepted with a probability P A → A (cid:48) = min (cid:16) , e ν ( ε ( A ) − ε ( A (cid:48) )) (cid:17) , (3)sampling from the canonical network ensemble at inversegenericity ν . Thus, a small ν results in almost every pro-posal being accepted and many random changes beingmade. The final ensemble is then not different from theone obtained by randomly removing the edges. In theother extreme of very large ν , almost only those propos-als are accepted that lower the energy, resulting in anensemble with a lower average λ max .This procedure is repeated until a steady state in λ max is reached, fluctuating around λ max ( t → ∞ ). We foundthis to be the case after 11000 steps. It should be notedthat for both the BA and RG initial network ensem-bles we repeat the explained strategy for n = 100 dif-ferent network configurations and, then, average over allfinal values of the largest eigenvalue to obtain (cid:104) λ max (cid:105) = n (cid:80) ni =1 i λ max (final). RESULTS
Examples for the evolution of λ max are shown in Figs. 2a) and c) for a range of ν and ρ = 0 . ν = 0)has largest eigenvalues fluctuating around λ max ≈ . ρ = 0 .
4, for theleast general ensemble with ν = 1000 this goes downto λ max ≈
11 in case of BA networks and λ max ≈ . ν > ν = 500 and ν = 1000 were alreadyminimal. Secondly our practical interest was in samplingthe WCNE, rather than finding the optimal link removal,because in practical scenarios, the underlying network isneither known, nor always perfectly controllable.In Fig. 2 b) and d) we see that the improvement fromMCMC persists in both network ensembles and over allremoval fractions. We see that the impact of removing90% of links randomly can alternatively be achieved byremoving only 80% in a targeted manner, leaving eachindividual with twice as many contacts on average. B A - E n s e m b l e R G - E n s e m b l e a) b)c) d) FIG. 2: MCMC substantially decreases λ max at higherinverse genericity ν . (a), (c) and (e) show respectivelythe decrease of λ max with MCMC for different initialnetworks for a range of values of the genericity ν . In allnetworks the removal rate is ρ = 0 .
4. For highgenericity, such as ν → λ max fluctuates just aroundits initial value after link removal (yellow line), while itreaches a low steady state for the smaller genericity(black line). (b), (d) and (f) show the average of λ max for a range of removal fractions ρ = { . , . , . , . , . , . , . , . } and genericities.As it is clear in these figures, in the larger ν , the curveis shifted more downwards, indicating a significantincrease in the epidemic threshold λ − max . Network analysis
We now analyze the resulting ensembles and thus com-pare several network measures across random and WCNEremovals in Fig. 3. We find several indicators pointing toa homogenization of the networks through the reductionof λ max .Across all initial network ensembles we observe a morestrongly peaked degree distribution for WCNE (blue di-amonds), compared to the generic ensemble (red circles),as shown in Fig. 3 a) and b).Furthermore the number of connected components forthe WCNE remains at 1 until a removal ratio of ρ = 0 . ρ = 0 . ρ = 0 .
9, we find in Figs. 3(e)–(f) that thelargest component is smaller in the WCNE ensemble thanit is in the generic one. This means that it is not singlenodes or small sub-graphs that are disconnected but thenetwork splits into several networks of similar size, in linewith our ad-hoc observation from Fig. 1.While the MCMC procedure is effective at increasingthe epidemic threshold of networks compared to simi-larly dense networks with randomly removed edges, it isoften not practically feasible to use. The procedure re-quires knowledge of the original social network, which isnot typically accessible when suggesting social distancingmeasures. Moreover, there is not one fixed real networkthat can be optimized, but rather an ever-changing, un-known interaction structure. We need general character-istics of well-performing removal sets, that can be appliedto a time-dependent, unknown network. By understand-ing general characteristics of removals that reduce thespreading we can formulate policies for targeted reduc-tions. Our above observations suggest that we need tohomogenize the degree distribution.We therefore now introduce two ad-hoc methods forlink removal, with this aim, one operating with only localand one with global information.i) The first one we call the degree product method .Starting from the original network, we iteratively removeone of the links with the highest degree product at ran-dom, which we define as the product of the degrees ofthe two end nodes, until the fraction ρ of removed linksis reached. An illustrative example is shown in Fig. 5(top).ii) The second one we call the degree cap . In order ofascending node number, each node with degree > k max BA-Ensemble RG-Ensemble a) b) c) d)e) f) FIG. 3: Network measures in the ensembles. For bothinitial network ensembles (BA and RG), the ensemblesafter link removal are analyzed. (a) and (b) show thedegree distributions after removal of 80% of links.(d)and (e) show the average number of components afterthe removal of ρ links. (g) and (h) demonstrate the sizeof the giant component after removal of 90% links. has all but k max of its links removed at random. Thisis a random order in case of RG networks. In the BAensemble, low node numbers tend to be correlated withhigh degrees, thus high-degree nodes are reduced first.However this discrepancy results in similar effects on thenetwork structure as shown in Fig. 3. An illustrativeexample of the algorithm is shown inWe have included networks generated with bothdegree-homogenizing methods in the analyses of Fig. 3and show their reduction of the largest eigenvalues inFig. 4. Subfigure c) also shows a similar level of reduc-tion effect for a real world high school contact network.Both methods result in networks with a more peaked de-gree distribution than for random removals. The degree-product method (black stars) reaches the most peakedone and correlates very well with the network ensem-ble resulting from MCMC simulations in most networkmeasures. The degree cap (yellow crosses) results in aless pronounced peak. Both methods mimic the delay ofthe onset of network fracturing, again with the degreeproduct method having a more pronounced effect. In-terestingly for the size of the giant component shown inFig. 3(e)–(f), the results differ for the different initial en-sembles. While the degree product method consistentlyresults in smaller giant components at a similar rangeof λ max than the WCNE, the degree cap method resultsin component sizes comparable to the generic ensemblefor BA and RG networks. In the High-school example itproduces the same component size as the degree productmethod, which is lower than even those of WCNE. Simulation and real world networks
To validate our removal strategies’ disease-slowingproperty, we run both SIS and SIR simulations on a realsocial network of high-school students. As a measure forthe disease becoming endemic, we compute the cumu-lative number of infected nodes. This is plotted versusdifferent disease parameter ratios β/δ in Figs. 6(a) and(b) for SIR and SIS, respectively. The size of the networkis N = 1062 and the figure averages over 640 realizationsof SIR and SIS simulations with 10 nodes initially in-fected at random. They were simulated for 1000 stepsfor SIS and until equilibrium was reached for SIR. Weconsistently find that tipping into the epidemic state oc-curs at lower values of β/δ in the generic networks thanin the WCNE across all removal ratios for both SIR andSIS. However, the figure also indicates that there maybe a kind of trade-off here. While the tipping point isshifted, the transition also becomes steeper, leading tolarger numbers of total infected people at very large β/δ .Finally, Fig. 6c) shows the smallest value of β/δ foreach removal rate ρ , for which the cumulative number ofinfected individuals exceeds 1% using the SIS model. Itclearly indicates the shift and verifies that it exists forboth ad-hoc strategies as well as for the WCNE, sup-porting our conjecture that the sharper peak in the de- RG-networks High-school a) b) c)
FIG. 4: The degree-based strategies reduce λ max almost as well as MCMC. The reduction is greater for smallerremoval ratios but persists even if 90 % of edges are removed. degree productdegree cap FIG. 5: Top: Successively removing the link with thehighest degree product results in networks with a verynarrow degree distribution. Bottom: Go through nodesin order of increasing node number and remove anylinks exceeding the cap k max = 3.gree distribution may be the cause of the reduction inspreading. DISCUSSION AND CONCLUSION
We have studied the effects of targeted link removal onthe epidemic threshold in a network by comparing ran-dom removal, removal based on the largest eigenvalue asan energy function and two types of degree-homogenizingremoval. We have found that at sufficiently low gener-icities (i.e., high ν ), all three types of targeted removalhave a significantly higher epidemic threshold (i.e., lower λ max ) than the random removal. This also results in ashift of the tipping point to the epidemic state in SIS and SIR simulations, such that more infectious diseases canbe controlled with fewer link removals necessary.We have found this to coincide with a more sharplypeaked degree distribution as well as fewer and more ho-mogeneously sized connected components, which we haveinterpreted as an overall homogenization. Consequently,we have proposed the two degree-homogenizing methods,which are also effective at decreasing λ max , as well asshift the onset of epidemic spreading.While running a full MCMC may be infeasible in prac-tice, where the social network is fluctuating and un-known, the two ad-hoc methods provide a simple topo-logical way of targeting links. This shows that a cap inthe number of permitted contacts per person, as alreadypractised in many countries as part of social distancingmeasures against Covid-19, is actually close to optimalin terms of topological link targeting in static networks,at least in rather simple model simulations such as ours.Realistically however, measures may also include non-binary changes (such as wearing masks, shortening con-tacts or keeping a distance) as well as temporal changesin infectiousness and fluctuating interaction patterns.While we have here exemplified the method using simplestatic SIR and SIS models, a more complex energy func-tion would in principle make such an analysis possiblealso in the case of inhomogeneous (weighted) or tempo-rally fluctuating transmission probabilities β , as well asnode, rather than edge removal (vaccinations). Changesof connection strengths or edge weights could be used in-stead of complete removals of edges as a model of trans-mission reductions through measures, such as mask wear-ing, meeting outdoors, improved ventilation and keepinga distance. [1] Michael Greenstone and Vishan Nigam. Does social dis-tancing matter? University of Chicago, Becker FriedmanInstitute for Economics Working Paper , (2020-26), 2020. [2] Savi Maharaj and Adam Kleczkowski. Controlling epi-demic spread by social distancing: Do it well or not atall.
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