Probing Complexity with Epidemics: A New Reactive Immunization

Strategy

E. Alﬁnito

, M. Beccaria

, A. Fachechi

and G. Macorini

Dipartimento di Ingegneria dell’Innovazione, Universit

a del Salento, Campus Ecotekne, 73100 Lecce, Italy

2,3,4

Dipartimento di Matematica e Fisica E. De Giorgi, Universit

a del Salento & INFN, Via Arnesano, 73100 Lecce, Italy

Keywords:

Epidemic Models, Reactive Immunization, Complex Networks.

Abstract:

Epidemic evolution on complex networks strongly depends on their topology and the infection dynamical

properties, as highly connected nodes and individuals exposed to the contagion have competing roles in the

disease spreading. In this spirit, we propose a new immunization strategy exploiting the knowledge of network

geometry and dynamical information about the spreading infection. The ﬂexibility and effectiveness of the

proposed scheme are successfully tested with numerical simulations on a wide set of complex networks.

1 INTRODUCTION

Epidemic processes and the related immunization

problem is a very hot topic in statistical mechanics of

complex networks (Pastor-Satorras et al., 2015). The

central issue in the realization of an effective vacci-

nation program is in fact the identiﬁcation of the most

relevant features in the infection spreading. This is in

general a highly non-trivial problem, so that novel im-

munization programs, looking at some particular as-

pects of the complex system, have been realized. Ac-

cording to their application and the required knowl-

edge about the system, immunization programs can

be roughly classiﬁed in two opposite approaches. The

ﬁrst one is the preventive case, which aims to make

the healthy network stronger against future epidemic

events by using (global or local) information about

its topology. This is the case of Targeted Immuniza-

tion (TI) (Pastor-Satorras and Vespignani, 2002), Ac-

quaintance Immunization (AI) (Cohen et al., 2003)

as well as their variations and reﬁnements (Stauffer

and Barbosa, 2006; Hu and Tang, 2012). On the

other hand, the reactive approach is designed for deal-

ing with an already spreading infection. In this case,

topological knowledge and information about the epi-

demic state can be combined so that a dynamical reac-

tion can be carried out, see for example (Ruan et al.,

2012; Altarelli et al., 2014; Liu et al., 2014; Perra

et al., 2012; Yan et al., 2014; Pereira and Young,

2015). A paradigmatic example of this approach is

the so-called High-Risk Immunization (HR) (Nian

and Wang, 2010), in which only individuals interact-

ing with infected nodes are vaccinated. In some real

situations, only the reactive setting is possible, for ex-

ample to control unexpected disease outbreaks or, in

a technological context, to ﬁght malware diffusion.

In such situations, a good control of epidemic con-

tagion needs a fast intervention, so it is more conve-

nient to have an immunization program working well

on the largest possible set of networks. Unfortunately,

known immunization strategies have been usually de-

signed to work in very speciﬁc settings. We propose

a new immunization strategy which generalizes TI in

order to take into account the presence of an already

propagating epidemic. Our claim is that the most rel-

evant individuals in the epidemic diffusion have to

be identiﬁed according to a clever balance between

network topological features and the epidemic state

at the vaccination time. We test the effectiveness of

our protocol with a Monte Carlo implementation of

SIR model (Kermack and McKendrick, 1927; May

and Anderson, 1979) and extensively compare it with

standard immunization strategies on a variety of com-

plex (theoretical and real) networks.

2 THEORETICAL BACKGROUND

2.1 Implementation of SIR Model

In the SIR model (Kermack and McKendrick, 1927),

individuals are classiﬁed according to their condition

in susceptible (S), infected (I) and recovered (R). Epi-

116

Alﬁnito, E., Beccaria, M., Fachechi, A. and Macorini, G.

Probing Complexity with Epidemics: A New Reactive Immunization Strategy.

DOI: 10.5220/0006361301160123

In Proceedings of the 2nd International Conference on Complexity, Future Information Systems and Risk (COMPLEXIS 2017), pages 116-123

ISBN: 978-989-758-244-8

demic dynamics is therefore represented by the tran-

sitions S → I and I → R. In our implementation of

SIR model, the initial condition is a single infected

node (patient zero) in the network. At each step, an

infected individual is randomly selected and recov-

ers with probability p

SIR

. If this is not the case, one

of its susceptible neighbors is randomly chosen and

gets infected. The immunization takes place as a re-

action to the epidemic spreading when a given frac-

tion f (the epidemic threshold) of the total population

is infected. In the procedure, the fraction g of indi-

viduals to be vaccinated is selected according to some

score assigned to each node, which results in a prior-

ity list for the nodes to be vaccinated. The spreading

algorithm goes on as long as infected individuals are

present. Once the system reaches its ﬁnal state, we

consider the ﬁnal density d

of recovered individuals

as a measure of total number of individuals involved

in the epidemic. Clearly, an immunization strategy is

effective if it signiﬁcantly reduces the density d

immunizing a relatively small fraction d

of the pop-

ulation. Since the evolution of the system is stochas-

tic, we perform a large number of simulations and the

effectiveness of a given strategy is measured in terms

of the averages hd

i and hd

2.2 A New Immunization Strategy

Motivated by the issues discussed in the introduction,

we propose a local reﬁnement of TI. Speciﬁcally, our

score takes into account both static knowledge of the

network geometry and dynamical information about

the epidemics, and it is given by

= d

∑

j∈N



j,I

)

1/2

+ γ

j,S

− d

+ d



. (1)

Here, N

is the i-th node neighborhood, d

its con-

nectivity degree and δ

j,I

and δ

j,S

are Kronecker deltas

selecting respectively infected or susceptible nodes.

Finally, β and γ are tunable free parameters. Clearly,

the β-term favors the vaccination of nodes near the

epidemic front, where the factor (d

)

−1/2

is justiﬁed

by the fact that less connected neighbors constitute

bottlenecks for the epidemic spreading. The γ-term is

a further improvement based on the leverage central-

ity (Joyce et al., 2010).

2.3 Standard Immunization Strategies

In our numerical tests, we extensively compare our

proposal with the following standard immunization

strategies, suitably modiﬁed in the reactive setting:

• Targeted Immunization (TI). Originally de-

signed for scale-free networks, this scheme se-

lects nodes to be immunized among the suscep-

tible ones according to their degree.

• Acquaintance Immunization (AI). In this case,

random neighbors of randomly selected nodes are

vaccinated (if susceptible). This is a degree-based

vaccination strategy using only local information

about the network.

• High Risk Immunization (HR). Nodes to be im-

munized are chosen among the susceptible neigh-

bors of infected nodes.

3 COMPLEX NETWORKS

In this section, we brieﬂy describe the networks we

considered as background for the epidemic spreading.

They range from the standard theoretical cases to a

collection of real world systems.

3.1 Theoretical Networks

In the ﬁrst class, we consider the Barab

asi-Albert

(Albert and Barab

asi, 2002; Bornholdt and Schuster,

2006) and the Watts-Strogatz (Albert and Barab

asi,

2002; Bornholdt and Schuster, 2006; Watts and Stro-

gatz, 1998) models, which are respectively the proto-

types of scale-free and small-world graphs. We de-

note with BA[Q] the Barab

asi-Albert graph obtained

adding Q new links at each step of the growth algo-

rithm. On the other hand, we indicate as WS[Q] the

Watts-Strogatz graph obtained starting with a regu-

lar network with each node connected to 2Q adja-

cent sites. The rewiring probability in constructing

WS graphs is denoted by θ. In this paper, we con-

sider BA[2] and WS[2] networks with two different

rewiring probabilities θ = 0.1,0.5 for the WS case.

Since some real networks are believed to be approxi-

mately or locally scale-free, we also propose two vari-

ants of Barab

asi-Albert model. The ﬁrst one is re-

alized starting with a BA graph and then randomly

rewiring R of its links. This variant can be thought as

a model that interpolates, depending on R , between

the pure scale-free networks and the completely ran-

dom graphs. We start with a BA[2] network and con-

sider R = 500, 1000,2000 rewiring events. The sec-

ond variant is realized starting with a network with

N nodes equally distributed in m BA centers. The

centers are then further randomly connected adding

new links between individuals in different clusters. In

this case, we start with m = 5,10,20 BA clusters with

N = 5000 total nodes and add k = 100,500,2000 new

edges.

Probing Complexity with Epidemics: A New Reactive Immunization Strategy

117

Table 1: Some common graph metrics for the real networks considered in the MC simulations: in the four columns we report

the mean value, variance, maximum and minimum for the degree, vertex eccentricity, closeness centrality and betweenness

centrality.

CA-HepTh-pruned Degree Eccentricity Clos. centr. Bet. centr.

mean 5.74809 12.5302 0.171894 21357.1

variance 41.7448 1.07741 0.000597495 3.32398×10

max 65. 18. 0.247167 1.15586×10

min 1. 10. 0.0813124 0.

p2p-Gnutella08 Degree Eccentricity Clos. centr. Bet. centr.

mean 6.5966 7.16003 0.218169 11471.8

variance 72.9622 0.423737 0.000621478 4.16777×10

max 97. 9. 0.295973 305191.

min 1. 6. 0.147952 0.

AA Degree Eccentricity Clos. centr. Bet. centr.

mean 4.74715 8.01616 0.227506 1836.35

variance 77.3613 0.857017 0.00110411 7.90946×10

max 145. 11. 0.35615 213598.

min 1. 6. 0.13556 0.

Internet AS Degree Eccentricity Clos. centr. Bet. centr.

mean 4.18991 7.15294 0.282169 14661.

variance 1095.36 0.359583 0.00163744 6.53432×10

max 2389. 10. 0.474276 2.03136×10

min 1. 5. 0.141305 0.

ProteinYeast Degree Eccentricity Clos. centr. Bet. centr.

mean 2.73388 13.275 0.150965 4234.31

variance 12.2956 1.68203 0.000581309 1.38406×10

max 56. 19. 0.234773 225922.

min 1. 11. 0.0791633 0.

3.2 Real Networks

Finally, the set of real networks we consider in this

paper is the following:

1. Internet AS, 11174 nodes, 23408 links. Undi-

rected unweighted Internet Network

(Colizza

et al., 2006) at the Autonomous System level.

Nodes represent Internet service providers and

links connections between them. Data were

collected by the Oregon Route Views Project

(http://www.routeviews.org/) in May 2001.

2. AA, 1057 nodes, 2502 links. Interactions between

metabolites of E. coli during the metabolic cycle

(Jeong et al., 2000). We consider the AA case.

3. CA-HepTh-pruned, 8638 nodes, 24836 links.

Arxiv HEP-TH (High Energy Physics - Theory)

collaboration network

from the e-print arXiv. A

paper is represented as a completely connected

subgraph in which nodes are its authors.

4. p2p-Gnutella08, 6300 nodes, 20776 links. Se-

quence of snapshots of the Gnutella ﬁle sharing

https://sites.google.com/site/cxnets/research222

http://www3.nd.edu/ networks/resources/metabolic/

http://snap.stanford.edu/data/ca-HepTh.html

network from August 2002.

Nodes are hosts of

Gnutella network and links connections between

them.

5. ProteinYeast, 1870 nodes, 2350 links. Protein In-

teraction Network

(Jeong et al., 2001).

Further information about the collection of real net-

works can be found in Table 1, in which some stan-

dard metrics are reported.

4 RESULTS

The discussion of our results is focused on the abil-

ity of the various immunization strategies to reduce

the epidemic prevalence hd

i as function of the vacci-

nated fraction of the population. In particular we use

the 50% and 25% of the original (i.e. without vac-

cination) prevalence as references (horizontal dotted

lines in the plots).

COMPLEXIS 2017 - 2nd International Conference on Complexity, Future Information Systems and Risk

118

Figure 1: Results for various immunization strategies for randomly rewired BA[2] networks with N = 1000 nodes and

R = 0 (pure BA case, a), 500 (b), 1000 (c), 2000 (d) rewiring events. The LMTI scheme is compared to TI, AI and HR strate-

gies for the pure BA case, while various choice of β and γ are compared with TI only for non zero R . The horizontal 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.

4.1 Barab

asi-Albert Setting

Let us ﬁrst consider the pure BA case, see panel (a) in

Fig. 1. Degree-based immunizations are clearly the

most effective ones in this setting. In particular, the

reduction of the epidemic prevalence by 50% is bet-

ter accomplished with TI. Besides, our strategy (with

the best choice β = 20 and γ = 10) has similar perfor-

mances at low hd

i. If we want to cut the prevalence

down to the 25%, TI and LMTI are almost identical,

but the latter is most advisable to further reduce the

epidemic strength. The results for partially random-

ized BAs are collected in panels (b), (c) and (d) of

Fig. 1. Here, we compare LMTI for different choices

of β and γ only with TI, as they are the best perform-

ers in the original setting. The ﬁrst visible feature is

the gradual worsening of the TI performances for in-

creasing R . This behavior has a simple explanation:

http://snap.stanford.edu/data/p2p-Gnutella08.html

http://www3.nd.edu/ networks/resources/protein/bo.dat.gz

the random rewiring procedure works in opposite di-

rection with respect to the preferential attachment of

the BA growth algorithm, and the role of big spread-

ers in the original setting is here gradually downsized.

In this setup, the importance of local terms in (1) can

be better appreciated and immunization of the indi-

viduals close to the epidemic front becomes more and

more important in controlling the outbreak.

Another interesting result holds for the randomly

connected BAs, see Fig. 2. In this setting, the reduc-

tion to the 50% of the epidemic prevalence is generi-

cally best suited with TI, while the cutting to the 25%

is roughly equivalent for the two strategies (consid-

ering the best choice β = 20 and γ = 10 for LMTI).

However, LMTI always allows to stop the epidemic

for a sufﬁciently large vaccinated fraction (which in-

crease for higher k). TI generally appears more effec-

tive than LMTI for low values of hd

i. This can be

explained noticing that for increasing k with ﬁxed m

or increasing m with ﬁxed k, the connection of each

node with individuals belonging to other BA centers

Probing Complexity with Epidemics: A New Reactive Immunization Strategy

119

Figure 2: Results for various immunization strategies for randomly connected BA[2] clusters with N = 5000 total

nodes. The number of BA centers is m = 5 (a,b,c), 10 (d,e,f), 20 (g,h,i), each with k = 100, 500,2000 new links. and

R = 0 (pure BA case, a), 500 (b), 1000 (c), 2000 (d) rewiring events. The LMTI scheme is compared to TI only. The hor-

izontal 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.

acquires more and more importance. Thus, it is more

likely that the epidemic spreads on the whole network

rather than to be localized in the initial cluster, there-

fore acquiring an extended front. In this case it is

convenient to adopt a non-local immunization strat-

egy, unless there is a sufﬁcient vaccinations availabil-

ity to surround the infection. Finally, an interesting

aspect is that choices with higher β are more effec-

tive in stopping the epidemic as the network becomes

highly connected and clustered. This is clear from

Fig. 2 for k = 2000, where LMTI with the choice

β = 100 and γ = 10 allows to stop the epidemic for

i = 0.25 in networks with m = 5 centers and for

i = 0.20 for m = 20 initial clusters.

4.2 Watts-Strogatz Setting

Fig. 3 collects the results for Watts-Strogatz net-

works. They are radically different than the BA set-

ting. As a consequence of absence of hubs, a pure TI

is indeed a poor strategy in WS graphs. On the other

hand, our score shows a much better adaptability. The

presence of the local terms is fundamental in reducing

the epidemic prevalence, giving to the LMTI curves a

proﬁle with a rapid fall-off. The only strategy giving

comparable results for small θ is HR, which was the

worst performer for BA. Moreover, HR performances

are highly sensitive to the θ value and dramatically

worsen for θ = 0.5. An interesting aspect of our strat-

egy is the role of the leverage term. In fact, for WS

network keeping a residual regular structure (θ = 0.1),

the γ-term is totally ineffective. Moreover, it becomes

even deleterious for highly randomized WS (θ = 0.5).

The choice β = 20 and γ = 0 is by far the best per-

former in all WS settings, always allowing to stop

the epidemic with a relatively low vaccinations (note

that all the other immunization strategies fail in doing

this).

COMPLEXIS 2017 - 2nd International Conference on Complexity, Future Information Systems and Risk

120

Figure 3: Results for various immunization strate-

gies for WS[2] networks with N = 1000 nodes and

θ = 0.1 (a), 0.5 (b). The LMTI scheme is compared to TI,

AI and HR strategies. The horizontal solid line is the epi-

demic threshold f = 0.05, while the horizontal dotted lines

are 25% and 50% of the mean ﬁnal density of recovered

without any vaccination.

4.3 Real Networks

Fig. 4 collects the results for real networks. As a gen-

eral result, TI and LMTI are the most indicated strate-

gies for reducing the epidemic prevalence by 50%

(LMTI can cut it to the 25% with a slightly larger

vaccinated fraction). Moreover, HR generally works

worse than AI in all real networks we considered.

It is interesting to note that Internet AS and Pro-

teinYeast networks show a great structural resistance

to epidemic outbreaks. Even without any vaccina-

tion, the average size of an epidemic remains rela-

tively small (less than the 30% of the total popula-

tion). According to the k-shell decomposition (Seid-

man, 1983), Internet AS and ProteinYeast networks

have a relatively small center and a large periphery,

as the ﬁrst two shells k = 1, 2 account for a very large

fraction of the nodes (respectively the 78% and the

86% of the entire population). ProteinYeast is also

the network with smallest mean degree and highest

vertex eccentricity. In these networks, the eradication

of the epidemic is particularly simple (with LMTI,

this can be done respectively with hd

i ∼ 0.15 and

i ∼ 0.05). The other networks are much more ex-

posed to the risk of a pandemic outbreak. In fact,

without vaccination the typical size of the epidemic

involves more than the 60% of the population. In

p2p-Gnutella08 and CA-HepTh-pruned networks,

the results for the TI and LMTI with β = 20 are

almost the same, while in both cases the HR strat-

egy fails even in signiﬁcantly reducing the epidemic

prevalence. It is interesting to note that in the p2p-

Gnutella08 network the AI is particularly efﬁcient,

with performances similar to TI or LMTI for small

values of hd

i. With reference to Table 1, this net-

work presents the higher mean degree and the lowest

mean vertex eccentricity, suggesting that it is highly

and uniformly connected. As a consequence, in this

network it is very difﬁcult to control the epidemic

spreading. Finally, results for AA network show again

that TI and LMTI are almost equivalent for small

values of hd

i, with the second one doing better for

slightly larger values of the vaccinated fraction.

5 CONCLUSION AND

DISCUSSION

In this work, we proposed a novel reactive immuniza-

tion strategy. It is based on a local modiﬁcation of

TI protocol which aims to actively take into account

the presence of a propagating epidemic and use it as

a source of information to trigger an optimized vacci-

nation response. We numerically compared our pro-

posal with other standard immunization strategies and

proved that it is a very efﬁcient choice in every case

we considered. Moreover, it always allows to stop the

epidemic with a relatively small vaccinated fraction.

Our proposal naturally ﬁts in the class of techniques

using local knowledge about complex systems (see

for example the Hebbian learning rule (Hebb, 1949)

in the case of neural networks).

There are two important aspects about our novel

immunization strategy we would like to stress. First,

our scheme, so as many other standard immunization

strategies, is strongly information-demanding. Of

course, a full knowledge about the complex system

is rather unlikely in real situations, then sampling and

interpolation (Ferguson et al., 2014), so as data assim-

ilative strategies (Rhodes and Hollingsworth, 2009)

have been recently developed. The second aspect

concerns the role of the tunable parameters in our

strategy. In fact, we compared our LMTI with other

benchmark strategies for an optimal choice of β and γ.

Such a choice consists in the evaluation of the com-

Probing Complexity with Epidemics: A New Reactive Immunization Strategy

121

Figure 4: Results for various immunization strategies on real networks (a-e). The LMTI scheme is compared to TI, AI and

HR strategies. The horizontal 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. For Internet AS (a), the horizontal dotted line is the 50%

of the mean ﬁnal density of recovered without any vaccination.

peting roles of global and local features in the epi-

demic spreading and, in general, it depends on the

network model and its size, as well as on the epidemic

threshold. The study of such a dependence is gener-

ally a non-trivial theoretical task and goes beyond the

goal of the present paper. Instead, in our purely phe-

nomenological approach, we tuned the local param-

eters choosing the best pair (β,γ) by looking at the

performances of LMTI in reducing the epidemics. We

therefore did not identify a method to a priori ﬁx the

best values of these parameters. Hopefully, this issue

can be addressed and theoretically better understood

in future works. In doing so, a more detailed exper-

imental analysis of the dependence of LMTI perfor-

mances on the free parameters is necessary.

In conclusion, several extensions of our work are

possible. For example, it can be generalized to other

classes of ideal networks with good theoretical con-

trol, like weighted or directed graphs. It can also be

applied to actual speciﬁc diseases, e.g. TBC, Xylella

fastidiosa (Alﬁnito et al., 2016) or Ebola infections.

In doing this, more realistic propagation models, like

the delayed SIR considered in (Agliari et al., 2013)

and a detailed cost beneﬁt analysis are needed.

REFERENCES

Agliari, E., Asti, L., Barra, A., Scrivo, R., Valesini, G.,

and Wallis, R. S. (2013). Application of a stochastic

modeling to assess the evolution of tuberculous and

non-tuberculous mycobacterial infection in patients

treated with tumor necrosis factor inhibitors. PloS

one, 8(1):e55017.

Albert, R. and Barab

asi, A.-L. (2002). Statistical mechan-

ics of complex networks. Reviews of modern physics,

74(1):47.

Alﬁnito, E., Beccaria, M., and Macorini, G. (2016). Criti-

cal behavior in a stochastic model of vector mediated

epidemics. Scientiﬁc reports, 6.

Altarelli, F., Braunstein, A., Dall’Asta, L., Wakeling, J. R.,

and Zecchina, R. (2014). Containing epidemic out-

breaks by message-passing techniques. Physical Re-

view X, 4(2):021024.

Bornholdt, S. and Schuster, H. G. (2006). Handbook of

graphs and networks: from the genome to the internet.

John Wiley & Sons.

Cohen, R., Havlin, S., and Ben-Avraham, D. (2003).

Efﬁcient immunization strategies for computer net-

works and populations. Physical review letters,

91(24):247901.

Colizza, V., Flammini, A., Serrano, M. A., and Vespignani,

A. (2006). Detecting rich-club ordering in complex

networks. Nature physics, 2(2):110–115.

Ferguson, J. M., Langebrake, J. B., Cannataro, V. L., Gar-

cia, A. J., Hamman, E. A., Martcheva, M., and Osen-

berg, C. W. (2014). Optimal sampling strategies for

detecting zoonotic disease epidemics. PLOS Compu-

tational Biology, 10(6):1–13.

Hebb, D. O. (1949). The Organization of Behavior: A Neu-

ropsychological Theory. Wiley, New York.

Hu, K. and Tang, Y. (2012). Immunization for complex net-

work based on the effective degree of vertex. Interna-

tional Journal of Modern Physics B, 26(06):1250052.

Jeong, H., Mason, S. P., Barab

asi, A.-L., and Oltvai, Z. N.

COMPLEXIS 2017 - 2nd International Conference on Complexity, Future Information Systems and Risk

122

(2001). Lethality and centrality in protein networks.

Nature, 411(6833):41–42.

Jeong, H., Tombor, B., Albert, R., Oltvai, Z. N., and

Barab

asi, A.-L. (2000). The large-scale organization

of metabolic networks. Nature, 407(6804):651–654.

Joyce, K. E., Laurienti, P. J., Burdette, J. H., and Hayasaka,

S. (2010). A new measure of centrality for brain net-

works. PLoS One, 5(8):e12200.

Kermack, W. O. and McKendrick, A. G. (1927). A con-

tribution to the mathematical theory of epidemics. In

Proceedings of the Royal Society of London A: Math-

ematical, Physical and Engineering Sciences, volume

115-772, pages 700–721. The Royal Society.

Liu, S., Perra, N., Karsai, M., and Vespignani, A. (2014).

Controlling contagion processes in activity driven net-

works. Physical review letters, 112(11):118702.

May, R. M. and Anderson, R. M. (1979). Population

biology of infectious diseases: Part ii. Nature,

280(5722):455–461.

Nian, F. and Wang, X. (2010). Efﬁcient immunization

strategies on complex networks. Journal of theoret-

ical biology, 264(1):77–83.

Pastor-Satorras, R., Castellano, C., Van Mieghem, P., and

Vespignani, A. (2015). Epidemic processes in com-

plex networks. Rev. Mod. Phys., 87:925.

Pastor-Satorras, R. and Vespignani, A. (2002). Immu-

nization of complex networks. Physical Review E,

65(3):036104.

Pereira, T. and Young, L.-S. (2015). Control of epidemics

on complex networks: Effectiveness of delayed isola-

tion. Physical Review E, 92(2):022822.

Perra, N., Gonc¸alves, B., Pastor-Satorras, R., and Vespig-

nani, A. (2012). Activity driven modeling of time

varying networks. Scientiﬁc reports, 2.

Rhodes, C. and Hollingsworth, T. (2009). Variational data

assimilation with epidemic models. Journal of Theo-

retical Biology, 258(4):591 – 602.

Ruan, Z., Tang, M., and Liu, Z. (2012). Epidemic spreading

with information-driven vaccination. Physical Review

E, 86(3):036117.

Seidman, S. (1983). Network structure and minimum de-

gree. Social Networks, 5(3):269–287.

Stauffer, A. O. and Barbosa, V. C. (2006). Dissemination

strategy for immunizing scale-free networks. Physical

Review E, 74(5):056105.

Watts, D. J. and Strogatz, S. H. (1998). Collective dynamics

of ‘small-world’ networks. Nature, 393(6684):440–

442.

Yan, S., Tang, S., Pei, S., Jiang, S., and Zheng, Z. (2014).

Dynamical immunization strategy for seasonal epi-

demics. Physical Review E, 90(2):022808.

Probing Complexity with Epidemics: A New Reactive Immunization Strategy

123