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Reactive immunization on complex networks Eleonora Alfinitoa , Matteo Beccariab , Alberto Fachechib , and Guido Macorinib aDipartimento di Ingegneria dell’Innovazione, Università del Salento, Campus Ecotekne, 73100 Lecce, Italy bDipartimento di Matematica e Fisica Ennio De Giorgi, Università del Salento & INFN, Via Arnesano, 73100 Lecce, Italy E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: Epidemic spreading on complex networks depends on the topological structure as well as on the dynamical properties of the infection itself. Generally speaking, highly connected individuals play the role of hubs 7 andarecrucialtochannelinformationacrossthenetwork. Ontheotherhand,statictopologicalquantitiesmeasuring 1 the connectivity structure are independent on the dynamical mechanisms of the infection. A natural question is 0 2 therefore how to improve the topological analysis by some kind of dynamical information that may be extracted fromtheongoinginfectionitself. Inthisspirit,weproposeanovelvaccinationschemethatexploitsinformationfrom b e the details of the infection pattern at the moment when the vaccination strategy is applied. Numerical simulations F of the infection process show that the proposed immunization strategy is effective and robust on a wide class of 4 complex networks. 1 PACS number(s): 89.75.-k, 87.23.Ge, 87.10.Rt ] E P 1 Introduction to target nodes with high connectivity degree be- o. cause they act as hubs in the infection spreading. A i similar degree-based approach, but exploiting only b Epidemic diffusion on complex networks [1–3] is - local information, is the Acquaintance Immuniza- q a general paradigm to describe a large variety of [ tion (AI) [9]. Some variations and improvements real world outbreaks of infections, ranging from the 2 are discussed in [10, 11]. strictly biological case to malware diffusion as well v Instead, reactive immunization strategies start 3 as opinion propagation [4]. A central issue is the 4 with the network undergoing a propagating infec- design of efficient immunization strategies able to 9 tion and take into account dynamical aspects of the 3 prevent or control the epidemic spreading [2]. In 0 network and of the epidemic itself to identify which this context, numerical simulations are a flexible . 1 are the best sites to be vaccinated. Several scores and well-controlled framework to study epidemic 0 have been designed considering, for instance, per- 7 dynamics. In particular, they allow to understand 1 sonal awareness about the epidemics [12], message- the effectiveness of vaccination strategies that we : v passing interactions [13], dynamical reaction of the shall broadly classify as preventive vs. reactive i X schemes. Preventive immunization strategies aim networks [14, 15], information from previous infec- r tions [16], finite time for the vaccination to become to strengthen the network against epidemics using a effective [17], etc. A remarkably simple example of information about the healthy configuration, i.e. reactive protocols is the so-called High-Risk Immu- identifying the nodes to be immunized according to nization (HRI) [18], where the healthy neighbors of some score before the epidemic event. The score infected nodes are vaccinated. may require local or global knowledge about the network topological structure. An important ex- In this paper, we propose a modification of TI ample of the preventive approach is the Targeted scheme which exploits a refined score based on a Immunization scheme (TI) [5] (see also [6–8]), orig- local-global mixed strategy. Specifically, it intro- inally designed for scale-free networks. The idea is duces a modified score that is designed to consider bothhubsandindividualsatriskofcontagionasrel- the system always reaches a steady final state with- evant in the epidemic spreading. In other words, we out infected individuals. The density d of recov- R attempt to use the infection itself as a source of in- eredindividualsinthisstateisclearlyrelatedtothe formation and as a probe of how the network reacts spreading strength of the epidemic on the network. to the disease. On a regular network the infection A good immunization strategy would therefore re- may display a well defined propagating front, then, duce the final density d at the cost of a relatively R a good strategy is to vaccinate in a neighborhood of low vaccinated density d . The average values (cid:104)d (cid:105) V R it. It is not clear whether this strategy makes sense and (cid:104)d (cid:105) are computed by repeating the SIR evolu- V on a complex network and we precisely try to an- tion with vaccination a large number of times. swer this question. The effectiveness of our strategy Weproposeanovelstrategyofvaccinationwhich is tested by a Monte Carlo implementation of the interpolates between preventive and reactive immu- SIR model [19, 20] on a variety of complex theoret- nizations. In doing so, we take into account both ical and real networks and systematically compar- static information (like the network geometry) and ing our proposal with some standard immunization dynamicalinformation(likethepatternofaspecific strategies [5, 9, 18]. infection). To this aim, we consider the score (cid:20) (cid:21) (cid:88) δj,I δj,S di−dj S = d + β +γ , (2.1) i i 2 Epidemics modeling and reactive im- (d )1/2 d d +d j i i j j∈Ni munizations: a new score where N denotes the set of neighbors of the i-th i node, d its degree (i.e. the number of links point- The SIR model is a simple compartmental model i ing to it), δ and δ are the Kronecker deltas of disease spreading [19]. Individuals are divided in j,I j,S which select only infected or suspectible neighbors three classes: susceptible (S), infected (I) and re- and β,γ are free parameters. We call our proposal covered (R). The epidemic evolution is then mod- Locally-Modified Targeted Immunization(LMTI ). eled by the transitions S → I and I → R. In more β,γ Forβ = γ = 0,thescorereducestothatofTargeted details, itstartswithasingle(patientzero)infected Immunization [5]. The β-term in the r.h.s. of (2.1) node. Then, at each step of the Monte Carlo pro- favors the immunization of individuals near the epi- cess, a randomly chosen infected individual can re- demic front. The damping factor (d )−1/2 selects cover with probability p . Otherwise, one of its j SIR neighbors with lower connectivity, which constitute first neighbors is randomly selected and, if suscepti- bottlenecksfortheepidemicdiffusion. Itistherefore ble, gets infected. The reactive immunization takes possible to reduce the contagion by cutting them place when a fraction f (the epidemic threshold) of the population is infected.1 The vaccination is a off. The γ-term is a further improvement involv- ing the so-called leverage centrality [21] restricted single-step process in which a fraction g of suscep- to the susceptible neighbors. It measures the recip- tibles individuals is immunized according to some rocal influence of the i-th node and its neighbors in score. The finite size of the network ensures that theepidemicdiffusion. Infact, leveragecentralityis 1Duetothestochasticnatureoftheprocess,epidemicmay a natural metric quantifying the local influence of a die out before reaching the threshold f and immunization node on its neighbors and therefore it gives comple- does not take place in these cases. The relation between mentary local information with respect to the com- the quantities g and (cid:104)d (cid:105) is (cid:104)d (cid:105) = P g, where P is the V V f f probabilitythattheinfectionreachesthethreshold(whichof mon (local) clustering coefficient. coursedependsonthenetworkandthethresholditself). We Wetesttheeffectivenessofthescore(2.1)against choose to take into account these events because they give the following benchmark immunization strategies an information about the exposure of a given network to a pandemic outbreak without vaccination. Given the value of • Targeted Immunization (TI). Our imple- p ,thenon-spreadingeventsarerelativelyrare,forexample SIR mentationofTIfollowstheoriginalidea: nodes for a BA[2] the probability to reach the lower threshold is roughly 90%. are vaccinated according to their degree. The 2 only modification is that the immunization is performed as a reactive process when the epidemic reaches the threshold f. Only nodes yet susceptible at the vaccination time are protected. • Acquaintance Immunization (AI). As in the previous case, AC immunization [9] is im- plemented as a reactive process. The choice of the nodes to be vaccinated follows the original proposal. Random first neighbors of randomly selected nodes are vaccinated (if susceptible) according to the desired immunized fraction g. • High Risk Immunization (HRI). Our implementation retains the idea of [18] to vaccinate neighbors of infected nodes, but the process is instantaneous and permanent. We test Figure 1. Some examples of randomly connected BA this strategy by immunizing up to the 99% of networks. The first line shows 5 BAs connected with the first neighbors of the infected nodes at the 100and500and2000randomextralinks. Inthesecond vaccination time. and third line the plots show examples of the networks obtained starting with 10 and 20 centers. 3 Benchmark complex networks tween nodes belonging to different BAs. Here, we We test the effectiveness of our protocol on a vari- consider a starting network with N = 5000 nodes, etyofnetworksrangingfromtheoreticalmodelstoa equally distributed in m = 5,10,20 initial clusters, selection of real networks. In the first class, we con- andk = 100,500,2000. Thisvariantcanbethought sidertheclassicalexamplesofBarabàsi-Albert(BA) as a toy model for the epidemic spreading in clus- and Watts-Strogatz (WS) models. The first one is tered communities with relatively loose links. Some the prototype of scale-free networks [4, 22] and it is examplesoftheresultingnetworksareshowninFig. based on a growth algorithm with preferential at- 1. tachment. We denote with BA[Q] the network built Besides these theoretical models, we also con- adding Q new links at each step of the algorithm. sider the epidemic spreading in the following real The second one is the prototype of small-world net- networks: works [4, 22, 23]. WS graphs are built starting from regular ones with N nodes (each one connected to 1. Internet_AS, 11174 nodes, 23408 links. It de- 2Q consecutive sites) and then rewiring the links scribestheundirectedunweightedInternetNet- with probability θ. Here, we consider WS[Q] net- work2 [24] at the Autonomous System level. works with Q = 2,3 and θ = 0.1, 0.5. The data were collected by the Oregon Route We also propose two modifications of BA model. Views Project http://www.routeviews.org/ The first one is based on a partial randomization in May 2001. Nodes represent Internet service procedure. We start with a standard BA[Q] net- providers and edges connections among them. work with N nodes, then we randomly rewire R links. In our tests, we consider Q = 2, N = 1000 2. AA, 1057 nodes, 2502 links. It describes the and R = 100,500,1000,2000. The second variant interactions between the metabolites of E. coli is realized starting with m disconnected BA[2] centers, further connected adding k random links be- 2https://sites.google.com/site/cxnets/research222 3 in the course of the metabolic cycle3 [25]. We 4 Results consider the AA case. In this section, we report the main results of our 3. CA-HepTh-pruned, 8638 nodes, 24836 links. Monte Carlo simulations. In particular, we com- The Arxiv HEP-TH (High Energy Physics - pare the various immunization strategies according Theory) collaboration network4 from the e- to their ability in reducing the epidemic prevalence print arXiv. A paper generates a completely (cid:104)d (cid:105) by 50% and 75% (the horizontal dotted lines R connected subgraph in which nodes represent in the plots) and in reaching the epidemic threshold its authors. (red solid line in the plots). 4. p2p-Gnutella08, 6300 nodes, 20776 links. It is a sequence of snapshots of the Gnutella peer-to- peer file sharing network from August 2002.5 Nodes represent hosts in the Gnutella network and edges are connections among them. 5. ProteinYeast, 1870 nodes, 2350 links. It is the Protein Interaction Network6 [26]. Figure 2. The recovered mean final density (cid:104)d (cid:105) as R a function of the mean fraction of vaccinated (cid:104)d (cid:105), for V To provide some additional informations, in Tab. 1 BA[2] with N = 1000 nodes. The LMTI scheme is wereporttheglobalclusteringcoefficientsandmean compared to TI, AI and HR immunization strategies. distances among the nodes for the above real net- The horizontal red solid line is the epidemic threshold, works, and a comparison with the same quantities f = 0.05 (a), 0.15 (b), while the horizontal dotted lines computed for random networks. are 25% and 50% of the mean final density of recovered without any vaccination. For BA and WS models, we consider 50 different realizations for each network and perform 105 Fig. 2 collects the results for BA[2] for the two Monte Carlo runs with different initial conditions different epidemic thresholds. As it can be ex- for each of them. For the BA variants, we consider pected, degree-based schemes are the most efficient 20 different realizations of each graph and average in the pure BA setting. In particular, TI is the best 104 runs for each one. Finally, for real networks choice in reducing the epidemic prevalence (cid:104)d (cid:105) by R the statistics varies from 104 and 105 runs, depend- 50%. Our strategy (with the optimal choice β = 20 ing on their size. With such a choice, we keep the and γ = 10) performs very similarly at low (cid:104)d (cid:105) V statistical error on the final recovered density (cid:104)d (cid:105) R for both values of the epidemic threshold. How- undercontrol(forinstance, itisoftheorderof0.1% ever, if we want to reduce the prevalence to the in theoretical models).7 In all cases, we fix the re- 25%, a fast response to the outbreak is crucial, i.e. covering probability to p = 0.1 and consider two SIR f = 0.05. In this case, TI and LMTI are the most epidemic thresholds f = 0.05 or f = 0.15.8 indicated strategies as they requires a vaccinated 3http://www3.nd.edu/~networks/resources/ fraction around 10%. Moreover, LMTI can further metabolic/ 4http://snap.stanford.edu/data/ca-HepTh.html threshold is pc,SIR = 0.1765 [27], so pSIR = 0.1 would be 5http://snap.stanford.edu/data/p2p-Gnutella08. in the spreading phase. In this work, our main goal is a html comparison of the relative effectiveness of the various vacci- 6http://www3.nd.edu/~networks/resources/protein/ nationstrategies. AchangeinpSIR willsurelyaffectthefinal bo.dat.gz balance of the epidemic, but, from the point of view of the 7Statisticalfluctuationsaremainlydeterminedbythesim- comparison of the strategies, the dependence on pSIR is not ulation length, i.e. by the number of MC steps, while the crucial. ProvidedthatpSIR islowenoughtogiveaspreading dependence on the particular network realization is rather epidemic, a change of the value of the recovering probabil- weak due to self-averaging. ity results in an overall shift of all the curves, but does not 8By comparison, in a regular square lattice the epidemic change the relative performances. 4 Figure 3. The recovered mean final density (cid:104)d (cid:105) as Figure 5. The recovered mean final density (cid:104)d (cid:105) as R R a function of the mean fraction of vaccinated (cid:104)d (cid:105), for a function of the mean fraction of vaccinated (cid:104)d (cid:105), for V V WS[2]withN =1000nodesandtherewiringprobability randomly rewired BA[2] with N = 1000 nodes and θ = 0.1. The LMTI scheme is compared to TI, AI and R=100 (a), 2000 (b)rewiringevents. TheLMTIscheme HR immunization strategies. The horizontal red solid is compared to TI immunization strategy. The horizon- line is the epidemic threshold, f = 0.05 (a), 0.15 (b), talredsolidlineistheepidemicthresholdf =0.05,while whilethehorizontaldottedlinesare25%and50%ofthe thehorizontaldottedlinesare25%and50%ofthemean meanfinaldensityofrecoveredwithoutanyvaccination. final density of recovered without any vaccination. reduce the epidemic prevalence for lower (cid:104)d (cid:105) than V TI. On the other side, a late reaction to the epidemic (f = 0.15) causes the difficulty in controlling the spreading, so a massive vaccination process is needed. In fact, LMTI (which is the best choice in this eventuality) requires the vaccination of at least the 25% of the entire population. Instead, TI fails for (cid:104)d (cid:105) < 0.4. A similar behaviour holds also in Figure 6. The recovered mean final density (cid:104)d (cid:105) as a V R the BA[3] case, so we cease to give more details on functionofthemeanfractionofvaccinated(cid:104)dV(cid:105),forran- domlyconnectedBA[2]withN =5000totalnodes,m= this. 20 equally populated clusters and k =100 (a), 2000 (b) new links. The LMTI scheme is compared to TI immunization strategy. The horizontal red solid line is the epidemic threshold f = 0.05, while the horizontal dotted lines are 25% and 50% of the mean final density of recovered without any vaccination. However, both LMTI and HRI allow to reduce the Figure 4. The recovered mean final density (cid:104)dR(cid:105) as prevalence by 50% for a very small number of vac- a function of the mean fraction of vaccinated (cid:104)dV(cid:105), for cinations ((cid:104)d (cid:105) (cid:46) 0.05 for both values of the epi- V WS[2]withN =1000nodesandtherewiringprobability demic threshold). Most remarkably, our strategy θ = 0.5. The LMTI scheme is compared to TI, AI and can reduce it to 25% for both values of the epidemic HR immunization strategies. The horizontal red solid thresholdwithavaccinatedfractionlowerthan10% line is the epidemic threshold, f = 0.05 (a), 0.15 (b), of the entire population (for comparison, AI has whilethehorizontaldottedlinesare25%and50%ofthe meanfinaldensityofrecoveredwithoutanyvaccination. the same effect for (cid:104)dV(cid:105) = 0.2 ÷ 0.4). Therefore, a prompt reaction has the only effect of lowering In the WS setting, results are radically different, the vaccination coverage needed to reach the aim. see Fig. 3 for the WS[2] and θ = 0.1 case. Here, TI WS networks with different Q and θ present anal- immunization is a poor strategy when compared to ogous features, with the only difference that HRI LMTIandHRI.Thisisaconsequenceoftheabsence dramatically worsens as the rewiring probability in- of nodes acting as hubs for the epidemic spreading. creases, see Fig. 4. In both BA and WS cases, our 5 strategy allows to reach the epidemic threshold and equivalent. In particular, TI performs slightly bet- to effectively stop the epidemic. teronlyinCA-HepTh-prunedandAA.However,ifwe The importance of local terms in (2.1) can be wanttofurtherreducetheepidemicprevalenceupto better appreciated in the BA variants. Figs. 5 and 25%, LMTI is always the best choice. Moreover, it 6 collects the results for these models with the epi- allows to effectively stop the epidemics for a smaller demic threshold f = 0.05. In this case, we compare vaccinated fraction than TI. Remarkably, HRI is a only TI and LMTI, the best performers in the orig- rather inefficient choice also in ProteinYeast and In- inal BA setting. ternet_AS networks, which show a great structural For partially randomized BA[2] with R = 100, resistancetotheepidemics(evenwithoutanyvacci- the network keeps an approximate BA structure, so nation, the average size of an infection is relatively the results are very similar to the pure case. As small). When compared to HRI, AI seems to be the randomization increases, TI gradually becomes stronger,butitiscomparableinefficiencytoTIand inefficient(exceptforsmall(cid:104)d (cid:105)values),soitiscon- LMTIonlyinp2p-Gnutella08,inwhichitismoredif- V venient to vaccinate nodes near the epidemic front. ficulttocontroltheepidemicspreading(withoutim- This is clear in the R = 2000 case. munization, the average size of an infection is about Now, we consider randomly connected BAs with the 65% of the entire population). This feature can an highly clustered structure (m = 20). If these be explained noting that this network is highly and clusters are poorly connected (k = 100), TI and uniformlyconnectedasitpresentsthehighestmean LMTI gives approximately the same performances, degree and lowest mean vertex eccentricity. with the only difference that our scheme allows C (cid:96) C (cid:96) to stop the epidemic with a much smaller vacci- R R CA-HepTh-pruned 0.28 5.9 0.0007 5.4 nated fraction ((cid:104)d (cid:105) ∼ 0.10) than TI. For a much V p2p-Gnutella08 0.020 4.6 0.0010 4.8 larger number of connections between the clusters AA 0. 4.4 0.0044 4.6 (k = 2000), the situation radically changes. In fact, Internet_AS 0.0096 3.6 0.00039 6.6 the reduction of the prevalence by 50% is better ac- ProteinYeast 0.079 6.8 0.0017 6.4 complishedwithTIscheme. ForLMTI,theincrease of local terms importance worsens the efficiency at Table 1. We report the global clustering coefficient C low (cid:104)dV(cid:105), but drastically improves the performance andthemeandistanceamongthenodes(cid:96)forthefivereal for a larger number of vaccinations. networks. The last two columns show, as comparison, This behaviour has a simple explanation. When the same quantities computed for a random graph with thenetworksortheirclustershaveanapproximately the same number of nodes and links. BA structure, nodes acting as hubs are still present. Therefore, in this case it is convenient to vaccinated 5 Conclusions nodes with higher degree. As the original structure is lost (increasing the randomization or the number In this work, we have proposed a new reactive im- of new links between the original clusters), the immunization strategy based on a local modification portance of hubs in the epidemic spreading is dras- of the Targeted Immunization protocol. The aim tically downsized. Once that highest degree nodes of the local term is to actively take into account areimmunized,itisbettertogivemuchmoreimpor- the presence of the epidemic outbreak and design tance to individuals near the epidemic front. This the reactive vaccination by exploiting the infection also explains the faster decaying of LMTI curves for itself as a probe of the complex network. Our pro- increasing β values. posalfitsintheframeworkofcommonlyveryappre- Finally, in Fig. 7 we report the results for real ciated techniques using local knowledge about com- networks. Inordertohalvetheepidemicprevalence, plex systems, see for instance the Hebbian learn- we note again that TI and LMTI are the most indi- ing rule [28] for an exemplary model for neural net- cated strategies and their performances are almost works and [29] for a detailed analysis. By means 6 pler models could help to settle this issue.9 Several extensions of our work can be foreseen. Onthetheoreticalside,onecanexploreotherclasses of ideal networks with good theoretical control, like weightedordirectedgraphs. Fromthepointofview of applications, it could be important to apply our scheme to actual specific diseases, e.g. Xylella fas- tidiosa,TBCandEbolaoutbreaks. Thiswillrequire a more realistic propagation model like the delayed SIR considered in [30], and a detailed cost bene- fit analysis taking into account the finite resources available for a real vaccination programme, see for instance [31]. Finally, we remark that our immunization scheme is clearly information-demanding, as it requires the full knowledge of the neighborhood of each node and the pattern of the epidemic at the vaccination time. This is rather unlikely in real situations and another natural evolution of the present work would be the study of an immunization strategy accounting the possibility of partial or Figure 7. The recovered mean final density (cid:104)d (cid:105) as a R corrupted information about the system. functionofthemeanfractionofvaccinated(cid:104)d (cid:105)foraset V of real networks (a-e). The LMTI scheme is compared to TI immunization strategy. The horizontal red solid line is the epidemic threshold f, while the horizontal References dotted lines are 25% and 50% of the mean final density of recovered without any vaccination. For Internet_AS [1] M. E. Newman, The structure and function of (a), the horizontal dotted line is the 50% of the mean complex networks, SIAM review 45 (2003), no. 2 final density of recovered without any vaccination. 167–256. [2] R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani, Epidemic of explicit simulations we have compared our im- processes in complex networks, Rev. Mod. Phys. 87 (2015) 925. munization scheme with other immunization strategies. We have shown that our protocol is a very [3] S. Boccaletti, V. Latora, Y. Moreno, M. 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