Reactive immunization on complex networks

1.5 MB·

by  Eleonora Alfinito

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 epi-
demic reaches the threshold f. Only nodes
yet susceptible at the vaccination time are pro-
tected.
• 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 im-
plementation retains the idea of [18] to vacci-
nate neighbors of infected nodes, but the pro-
cess 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] cen-
ters, 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 differ-
ent 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 epi-
demic (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 immu-
nization strategy. The horizontal red solid line is the
epidemic threshold f = 0.05, while the horizontal dot-
ted 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 im- munization 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 immu-
nization scheme is clearly information-demanding,
as it requires the full knowledge of the neighbor-
hood 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 immuniza-
tion 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 strate-
gies. We have shown that our protocol is a very [3] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez,
efficient choice in all the considered cases, allowing and D.-U. Hwang, Complex networks: Structure
and dynamics, Physics reports 424 (2006), no. 4
to stop the epidemic with a relatively small vacci-
175–308.
nated fraction. The addition of a local term sensing
theinfectionwasmotivatedbyanaivepictureofthe [4] S. Bornholdt and H. G. Schuster, Handbook of
infection diffusion valid for a regular network. Nev- graphs and networks: from the genome to the
internet. John Wiley & Sons, 2006.
ertheless,itisrelevantalsoonabroadsetofcomplex
network totally far from being regular. We did not [5] R. Pastor-Satorras and A. Vespignani,
find a way to predict a priori the best choice for the Immunization of complex networks, Physical
parameters of our score. In a purely phenomenolog- Review E 65 (2002), no. 3 036104.
ical approach, the best free parameters are chosen
9Sincetheoutputofthescoreisanorderingofthenodes
empirically by looking at the performance of our
whichgivesthepriorityfortheimmunization,theresultturns
scheme as β and γ are changed. Hopefully, a deeper
out to be robust with respect small changes of the two free
investigation or the application of our score in sim- parameters.
7
[6] Y. Chen, G. Paul, S. Havlin, F. Liljeros, and H. E. [19] W. O. Kermack and A. G. McKendrick, A
Stanley, Finding a better immunization strategy, contribution to the mathematical theory of
Physical review letters 101 (2008), no. 5 058701. epidemics, in Proceedings of the Royal Society of
London A: Mathematical, Physical and
[7] L. Hébert-Dufresne, A. Allard, J.-G. Young, and
Engineering Sciences, vol. 115-772, pp. 700–721,
L. J. Dubé, Global efficiency of local immunization
The Royal Society, 1927.
on complex networks, Scientific reports 3 (2013).
[20] R. M. May and R. M. Anderson, Population
[8] S. Yan, S. Tang, W. Fang, S. Pei, and Z. Zheng,
biology of infectious diseases: Part ii, Nature 280
Global and local targeted immunization in
(1979), no. 5722 455–461.
networks with community structure, Journal of
Statistical Mechanics: Theory and Experiment [21] K. E. Joyce, P. J. Laurienti, J. H. Burdette, and
2015 (2015), no. 8 P08010. S. Hayasaka, A new measure of centrality for brain
networks, PLoS One 5 (2010), no. 8 e12200.
[9] R. Cohen, S. Havlin, and D. Ben-Avraham,
Efficient immunization strategies for computer [22] R. Albert and A.-L. Barabási, Statistical
networks and populations, Physical review letters mechanics of complex networks, Reviews of
91 (2003), no. 24 247901. modern physics 74 (2002), no. 1 47.
[10] A. O. Stauffer and V. C. Barbosa, Dissemination [23] D. J. Watts and S. H. Strogatz, Collective
strategy for immunizing scale-free networks, dynamics of ‘small-world’ networks, Nature 393
Physical Review E 74 (2006), no. 5 056105. (jun, 1998) 440–442.
[11] K. Hu and Y. Tang, Immunization for complex [24] V. Colizza, A. Flammini, M. A. Serrano, and
network based on the effective degree of vertex, A. Vespignani, Detecting rich-club ordering in
International Journal of Modern Physics B 26 complex networks, Nature physics 2 (2006), no. 2
(2012), no. 06 1250052. 110–115.
[12] Z. Ruan, M. Tang, and Z. Liu, Epidemic spreading [25] H. Jeong, B. Tombor, R. Albert, Z. N. Oltvai, and
with information-driven vaccination, Physical A.-L. Barabási, The large-scale organization of
Review E 86 (2012), no. 3 036117. metabolic networks, Nature 407 (2000), no. 6804
651–654.
[13] F. Altarelli, A. Braunstein, L. Dall’Asta, J. R.
Wakeling, and R. Zecchina, Containing epidemic [26] H. Jeong, S. P. Mason, A.-L. Barabási, and Z. N.
outbreaks by message-passing techniques, Physical Oltvai, Lethality and centrality in protein
Review X 4 (2014), no. 2 021024. networks, Nature 411 (2001), no. 6833 41–42.
[14] S. Liu, N. Perra, M. Karsai, and A. Vespignani, [27] T. Tomé and R. M. Ziff, Critical behavior of the
Controlling contagion processes in activity driven susceptible-infected-recovered model on a square
networks, Physical review letters 112 (2014), lattice, Phys. Rev. E 82 (2010), no. 5 051921.
no. 11 118702.
[28] D. O. Hebb, The Organization of Behavior: A
[15] N. Perra, B. Gonçalves, R. Pastor-Satorras, and Neuropsychological Theory. Wiley, New York,
A. Vespignani, Activity driven modeling of time June, 1949.
varying networks, Scientific reports 2 (2012).
[29] E. Agliari and A. Barra, A hebbian approach to
[16] S. Yan, S. Tang, S. Pei, S. Jiang, and Z. Zheng, complex-network generation, EPL (Europhysics
Dynamical immunization strategy for seasonal Letters) 94 (2011), no. 1 10002.
epidemics, Physical Review E 90 (2014), no. 2
[30] E. Agliari, L. Asti, A. Barra, R. Scrivo,
022808.
G. Valesini, and R. S. Wallis, Application of a
[17] T. Pereira and L.-S. Young, Control of epidemics stochastic modeling to assess the evolution of
on complex networks: Effectiveness of delayed tuberculous and non-tuberculous mycobacterial
isolation, Physical Review E 92 (2015), no. 2 infection in patients treated with tumor necrosis
022822. factor inhibitors, PloS one 8 (2013), no. 1 e55017.
[18] F. Nian and X. Wang, Efficient immunization [31] M. Bellingeri, E. Agliari, and D. Cassi,
strategies on complex networks, Journal of Optimization strategies with resource scarcity:
theoretical biology 264 (2010), no. 1 77–83. From immunization of networks to the traveling
8
salesman problem, Modern Physics Letters B 29
(2015), no. 29 1550180.
9