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1 Physia A 388 (2009) Contents lists available at SieneDiret Physia A journal homepage: wwwelsevierom/loate/physa Imperfet targeted immunization in sale-free networs Yubo Wang a, Gaoxi Xiao a,, Jie Hu a, Tee Hiang Cheng a, Limsoon Wang b, a Shool of Eletrial and Eletroni Engineering, Nanyang Tehnologial University, Singapore , Singapore b Shool of Computing, National University of Singapore, Singapore , Singapore Shool of Mediine, National University of Singapore, Singapore , Singapore a r t i l e i n f o a b s t r a t Artile history: Reeived 3 Otober 2008 Reeived in revised form 24 January 2009 Available online 5 Marh 2009 PACS: Fb 0570Ln Keywords: Sale-free networ Imperfet immunization Epidemi threshold Average outbrea size Degree-orrelated SIR Sale-free networs are prone to epidemi spreading To provide ost-effetive protetion for suh networs, targeted immunization was proposed to seletively immunize the hub nodes In many real-life appliations, however, the targeted immunization may not be perfet, either beause some hub nodes are hidden and onsequently not immunized, or beause the vaination simply annot provide perfet protetion We investigate the effets of imperfet targeted immunization in sale-free networs Analysis and simulation results show that there exists a linear relationship between the inverse of the epidemi threshold and the effetiveness of targeted immunization Therefore, the probability of epidemi outbrea annot be signifiantly lowered unless the protetion is reasonably strong On the other hand, even a relatively wea protetion over the hub nodes signifiantly dereases the number of networ nodes ever getting infeted and therefore enhanes networ robustness against virus We show that the above onlusions remain valid where there exists a negative orrelation between nodal degree and infetiousness 2009 Elsevier BV All rights reserved 1 Introdution Many real-life omplex systems an be desribed as networs [1,2] The well-nown examples inlude the Internet [3], World-Wide Web (WWW) [4], food web [5 7], and the human soiety [8] Studies on these systems as networs have spawned a new researh area alled omplex networs [2,9] Statistial studies on real-life omplex networs show that though different systems have different features [2], many of them share some nontrivial features One of the most notieable features is that a large number of them turn out to be sale-free networs of whih the nodal degrees follow a power-law distribution Speifially, the probability that a node is onneted to other nodes is P() r, where the exponent r usually ranges between 2 and 3 [2] In a sale-free networ, a large number of nodes have low degrees while a small number of nodes have very high degrees It is nown that sale-free networs enable effiient ommuniations but they are prone to disease/virus spreading [10,11] Studies on epidemi spreading are strongly motivated by the threats we are faing, eg, virus spreading in the Internet [11] and infetious diseases suh as HIV in human soiety [8] Within networs, virus lives and proliferates in networ nodes and propagates via lins Reent researh shows that epidemi spreading in an infinitely large sale-free networ with an exponent r 3 does not possess any epidemi threshold below whih the infetion annot produe a major epidemi outbrea or an endemi state [11 14] In other words, statistially speaing, a virus an easily survive and (1) Corresponding author Tel: ; fax: address: egxxiao@ntuedusg (G Xiao) /$ see front matter 2009 Elsevier BV All rights reserved doi:101016/jphysa

2 2536 Y Wang et al / Physia A 388 (2009) ause an outbrea in an infinitely large sale-free networ no matter how wea its spreading apability is Further studies on the finite-size sale-free networs show that the epidemi threshold remains to be low and dereases with an inreasing networ size [15] Suh analytial results help to explain our real-life experienes, eg, persistent virus spreading in the Internet Due to the peuliar onnetion pattern in sale-free networs, a small number of nodes have very high degrees The random immunization strategy is not effetive in preventing an epidemi outbrea or reduing the number of infetions in a sale-free networ though suh a strategy wors very well in homogeneous random networs [16,17] New immunization strategies have to be developed to reover the epidemi threshold One of the most effiient approahes is to immunize those nodes with the highest degrees, or, more speifially, to immunize those nodes (hereafter termed as hubs or hub nodes) with degrees higher than a preset ut-off value Suh a strategy is nown as targeted immunization [16] In existing studies on targeted immunization, it is generally assumed that all hub nodes are immunized; and one a hub is immunized, it will never be infeted We realize that in many real-life systems, this may not always be the ase, either beause the immunizations have missed some hubs or beause the vaination is not 100% effetive [18 21] In this paper, we onsider three different ases as follows: (i) Some hub nodes may be hidden and hene not immunized This is more liely to happen in large-sale systems We term this ase Partial Node Immunization (PNI); (ii) The vaination annot provide 100% protetion over the immunized nodes but an lower their hane of getting infeted We all this ase Partial Effetive Immunization (PEI); (iii) The vaination annot stop disease propagation through some networ lins In other words, an immunized hub may still be infeted by some, though not all, of its adjaent nodes This may happen between family members or trusted partners, et We all it Partial Lin Immunization (PLI) For all the three different ases, we introdue the immunization rate a Speifially, in PNI, it denotes the perentage of the hubs that are (perfetly) immunized; in PEI, it denotes the effetiveness of vaination In other words, eah immunized hub has a probability of 1 a to at as a suseptible node; and in PLI, it denotes the perentage of proteted lins onneted to the hub nodes Based on the Suseptible-Infeted-Removed (SIR) model [22,23], we analyze the epidemi threshold and the portion of networ nodes ever got infeted until the disease dies out (hereafter termed as Average Outbrea Size (AOS)) for these three different ases in sale-free networs We find that there is a linear relationship between the inverse of the epidemi threshold and the immunization rate; therefore, only a high immunization rate an signifiantly inrease the epidemi threshold However, even a relative low immunization rate an still greatly redue AOS; the networ robustness against epidemi spreading is therefore enhaned We show that suh onlusions remain valid where there exists a negative orrelation between nodal degree and infetiousness (ie, a higher-degree node has a lower hane to pass the disease to eah of its adjaent nodes) 2 Models 21 Suseptible-Infeted-Removed (SIR) model In this paper, due to page limit, we onsider only the SIR model Study results based on suseptible-infeted-suseptible (SIS) model [21,22] are briefly summarized in Setion 5, while further details have to be presented in a separate report The original SIR model was proposed (though never published) by Reed and Frost in 1920s In SIR, networ nodes are divided into three groups: Suseptible (S), Infeted (I), and Removed (R) Suseptible nodes are free of disease but an be infeted via diret ontats with infeted nodes Infeted nodes arry the disease and an pass it to suseptible nodes Removed nodes have either reovered from the disease or died; in either ase, they annot pass the disease to other nodes or be infeted again In the lassi SIR model, time is slotted In eah time slot, a suseptible node beomes infeted at a rate of λ (λ 1) if it is diretly onneted to at least one infeted node [10,13] The parameter λ is the mirosopi spreading rate (also nown as the infetion rate) Meanwhile, an infeted node beomes a removed node at the rate of δ (δ 1) Without loss of generality, throughout this paper we set δ 1 [10,13] Reent studies onsidered the ase where there exists a negative orrelation between nodal degree and infetiousness [24 26] In other words, a higher-degree node may have a lower hane to admit the disease from or pass the disease to eah of its adjaent nodes Suh models probably better resemble some real-life ases, eg, the transmission of sexual diseases [25] In this paper, we adopt the simple yet rather general model in Ref [24] that a -degree infeted node spreads the disease to eah of its adjaent nodes at a possibility of T(); and a -degree suseptible node admits an infetion at a rate of λ ( ) λa( ) if it is onneted to at least one infeted node T() and A() are the orrelation funtions To differentiate from the lassi SIR model, we term suh an extended model as degree-orrelated SIR model Unless otherwise speified, in this paper we study the effets of imperfet targeted immunizations in the lassi SIR model The main onlusions however remain valid for the degree-orrelated SIR model, as we shall see in Setions 34 and 43 As mentioned earlier, two metris will be adopted in the evaluations: (i) epidemi threshold λ, whih is a ritial spreading rate below whih infetions die out exponentially and above whih an epidemi outbrea will happen; and (ii) AOS

3 Y Wang et al / Physia A 388 (2009) Imperfet Targeted Immunization In the lassi targeted immunization, vainations are deployed to all hub nodes with degrees greater than while nonhub nodes are not immunized In PNI with an immunization rate of a, a fration a of the hub nodes are randomly seleted to be perfetly immunized while the others are not immunized In PEI, a hub node is infeted at a rate of (1 a)λ if it is diretly onneted to at least one infeted node In PLI, immunized hub nodes an be infeted at a rate of λ by a fration 1 a of its randomly seleted adjaent nodes but never by the rest of the adjaent nodes In all the three different ases, a non-hub node an be infeted at a rate of λ if it is diretly onneted to at least one infeted node; and the infeted nodes are removed at a unity rate Apparently a 1 orresponds to the ase of the lassi perfet targeted immunization and a 0 orresponds to the ase of no immunization 23 Networ models The simulation results presented in this paper are based on two different networ models: Barabási Albert (BA) model [2,27,28] and Autonomous System (AS) level Internet model [29] The BA model inorporates two important general onepts: growth and preferential attahment It is an unorrelated networ for whih the probability that any two nodes are diretly onneted is proportional to the produt of their nodal degrees The AS-level Internet model omes from the real-life data olleted in the National Laboratory for Applied Networ Researh (NLANR) projet on January 2, 2000, whih ontains 6474 nodes onneted by 12,572 lins It has been verified that this model losely resembles a sale-free networ It is a disassortative networ [30] in whih high-degree nodes tend to onnet to low-degree nodes 3 Theoretial analysis We perform theoretial analysis on the unorrelated sale-free networ model (eg, the BA model) Analysis on orrelated networ models is notoriously ompliated [12 16] and thus has to be left out for a separate report We argue that the analysis on the unorrelated networ model, simple as it is, an nevertheless reveal some properties that are valid in many other networs, as has been observed in a few existing studies [10,13,15,16,31,32] In this paper, our theoretial analyses fous on studying the epidemi threshold and AOS 31 Partial Node Immunization (PNI) Denote the densities of infeted, suseptible, removed and immunized nodes with degree at time t as S (t), ρ (t), R (t) and I (t), respetively We have S (t) + ρ (t) + R (t) + I (t) 1 Following Moreno, Pastor-Satorras and Vespignani et al [13,16], by applying the dynamial mean-field (MF) theory [33], PNI an be desribed by the following differential equations: dρ (t) ρ (t) + λs (t)θ(t); ds (t) λs (t)θ(t); dr (t) ρ (t), where θ(t) denotes the probability that a randomly seleted lin is onneted to an infeted node at time t The term λs (t)θ(t) in Eqs (3) and (4) indiates the perentages of -degree nodes whih are newly infeted It is a reasonable approximation to the general expression λs (t)1 [1 θ(t)] } at a low spreading rate: where a small number of nodes are infeted at a lose-to-threshold spreading rate, the density of the infeted nodes ρ (t) approahes zero; hene θ(t) 1 For any unorrelated networ, the probability that a randomly seleted lin points to a -degree node is proportional to P() [10,30] Therefore in PNI, P()ρ (t) P()ρ (t) θ(t), (6) sp(s) s where denotes the average nodal degree Assume that at the beginning of epidemi spreading only a very small fration of nodes are infeted and randomly distributed within the networ; or in other words, ρ (0) 0 Sine in PNI the density (2) (3) (4) (5)

4 2538 Y Wang et al / Physia A 388 (2009) of the immunized nodes is I (t) a for > and I (t) 0 otherwise, the initial onditions an be stated as: ρ (0) 0, R (0) 0, S (0) 1 a, I (t) a if >, ρ (0) 0, R (0) 0, S (0) 1, I (t) 0 if (7) Considering Eq (4) with the initial onditions as in Eq (7), we derive the density of the suseptible nodes as e S (t) λ t 0 θ(t ) +ln(1 a) if >, e λ t 0 θ(t ) if Defining an auxiliary funtion φ(t) as representing the value of integration in the exponent in Eq (8) and replaing θ(t) by Eq (6), we have t t P()ρ (t ) P() t ρ φ(t) θ(t ) 0 (t ) t P()R (t) (9) From Eq (9), we see that φ(t) atually represents the probability that a randomly seleted lin points to a removed node at time t Taing differentiation of φ(t), we have a self-onsistent equation of φ(t) that dφ(t) P()ρ (t) 1 u +1 P()I (t) φ(t) 1 P()[1 R (t) S (t) I (t)] P()S (t) (10) Here t and u denote the minimum and maximum nodal degrees of the networ respetively, and At the dφ(t) steady state of epidemi spreading where t and lim t 0, we an derive from Eq (10) that φ( ) F(φ( )) u +1 P()I ( ) 1 u +1 P()I ( ) 1 P()S ( ) [ t P()e λφ( ) + u +1 P()e λφ( )+b Obviously φ( ) 0 is a solution of Eq (11), orresponding to the ase where there is no epidemi outbrea To have an epidemi outbrea, there must be a non-zero solution of φ( ) in the interval (0, 1 1 u +1 P()I (t)) satisfying df(φ( )) dφ( ) > 1, (12) φ( )0 whih defines the epidemis threshold λ for PNI as [11,14] [ ] λ 1 1 u u 2 P() a 2 p() (13) +1 t Eq (13) applies to any unorrelated networ with degree distribution P() For an N-node BA model where P() 2m 2 / 3, 2m and u mn 1/2 [27,34], applying the ontinuum theory to Eq (13), we have the expression of epidemi threshold that λ 1 1 1/2 mn m ln N am ln 2 +1 Eqs (13) and (14) show that there is a linear relationship between the inverse epidemi threshold λ 1 and the immunization rate a Hene the epidemi threshold annot be signifiantly inreased unless the immunization protetion is reasonably strong Fig 1 shows the dependene of epidemi threshold on the immunization rate and the ut-off It is also lear that in infinitely large sale-free networs, the epidemi threshold would remain lose to zero even when only a small portion of hub nodes are not very well proteted ] (8) (11) (14)

5 Y Wang et al / Physia A 388 (2009) Fig 1 Dependene of epidemi threshold on immunization rate and immunization ut-off in the BA model Note that for the speial ases where a 0 and 1, Eqs (13) and (14) orrespond to the well-nown ases with no immunization [13] and perfet targeted immunization [16] respetively By setting t, Eqs (13) and (14) desribe the ase with global random immunization [16,17] Denote R as the networ AOS By adopting the ontinuum theory, we have R 1 P()I ( ) [ +1 1 am2 2m t ( 1 am t P()S ( ) 3 e λφ( ) + (1 a) 2 2m 3 u m e λφ( ) ] ) 2 2m e λφ( ) d a e λφ( ) d (15) In the above equation, we assume that u is of a very large or infinite value, whih is a reasonable approximation in ultralarge sale-free networs [28,29] Denote z 1 λ t φ( ) and z 2 λ +1 φ( ) For a spreading rate lose to the epidemi threshold, we have z 1 0, and z 2 0 Therefore, Eq (15) an be re-written as R 1 am m 2 ( z t z 1 x 3 e x dx a z x 3 e x dx z 2 ) (16) The integrations in Eq (16) are in the form of inomplete gamma funtion Ɣ(a, z) t a 1 e t with the property z that for z 0, Ɣ(a + 1, z) aɣ(a, z) + z a e z, Ɣ(0, z) r E ln(z) + z + o(z 2 ), where r E is the Euler onstant [35] By negleting the higher-order terms in Eqs (16) (18), Eq (16) an be expressed as [ ( ) R 1 am2 2m t 2 z 1 a 1 ( ) ] z 2 +1 ( mλφ( ) 2 am ) (19) +1 The only unnown term in Eq (19) is φ( ) From the self-onsistent Eq (11), we have φ( ) 1 1 u P()I (t) 1 P()S ( ) +1 ( 1 ma u ) u m 2 e λφ( ) d a 2 e λφ( ) d (20) +1 t +1 The integrations are again in the form of inomplete gamma funtion Let z 1 λ t φ( ) and z 2 λ +1 φ( ) Eq (20) an be re-written as: φ( ) 1 ma ( z1 m x 2 e x dx a z ) 2 x 2 e x dx (21) +1 t z 1 +1 z 2 (17) (18)

6 2540 Y Wang et al / Physia A 388 (2009) Applying the properties of the inomplete gamma funtion and its expansion as shown in Eqs (17) and (18), we have φ( ) z 1 r E z 1 z 1 ln z 1 ma z 2 + ma r E z 2 + ma z 2 ln z 2 (22) Replaing z 1 and z 2 by their respetive expressions, we have Aφ( ) + Bφ( ) ln φ( ) 0, (23) where A [1 (1 a)λm+(1 a)r E λm+λm ln(λm) aλm ln(λ +1 )] and B (1 a)λm Hene φ( ) an be alulated as ( φ( ) exp A ) (24) B Combining Eqs (19) and (24), we an predit the networ AOS as a funtion of immunization rate, spreading rate and immunization ut-off for an infinite-size sale-free networ For finite-size networs, the analytial result reflets the trend of AOS hanges in PNI as long as the networ is large enough 32 Partially Effetive Immunization (PEI) The same set of symbols has been adopted in the analysis of PEI Still by applying the dynamial mean-field theory, we desribe PEI by the following set of oupled differential equations: dρ (t) ρ (t) + (1 a)λs (t)θ(t) if >, (25) ρ (t) + λs (t)θ(t) if ; ds (t) (1 a)λs (t)θ(t) if >, (26) λs (t)θ(t) if ; dr (t) ρ (t) (27) Eqs (25) (27) are valid in evaluating the onset of infetions lose to the epidemi threshold where ρ (t) 1 and θ(t) 1 Similarly to that in Eqs (3) and (4), (1 a)λs (t)1 [1 θ(t)] } is replaed by (1 a)λs (t)θ(t) The probability that a randomly seleted lin is onneted to an infeted node an still be expressed as P()ρ (t) θ(t) (28) Still assume the initial ondition as R (0) 0, ρ (t) 0, and S (0) 1 ρ (0) 1 We have e S (t) (1 a)λ t 0 θ(u)du if >, e λ t 0 θ(u)du if Define an auxiliary funtion φ(t) t 0 θ(u)du, φ(t) 1 t P()ρ (u)du 1 t P() ρ (u)du P()R (t) Taing differentiation of φ(t), we have the self-onsistent equation of φ(t) where dφ(t) 1 P()ρ (t) 1 P()[1 R (t) S (t)] 1 φ(t) 1 P()S (t) (31) dφ(t) At the steady state where t, all the infeted nodes are removed Therefore we have ρ ( ) 0 and lim t 0 From Eq (31), we obtain φ(t) as φ( ) 1 1 P()S ( ) From Eqs (28) and (32), we have [ φ( ) 1 1 u P()e (1 a)λφ( ) + +1 P()e λφ( ) t ] (29) (30) (32) (33)

7 Y Wang et al / Physia A 388 (2009) Similarly to the ase of PNI, we define an auxiliary funtion [ ] F (φ( )) 1 1 u P()e (1 a)λφ( ) + P()e λφ( ) (34) +1 t To ensure that φ( ) F (φ( )) has a non-zero root between 0 and 1, subjet to the onstraint that φ( ) 0 is a solution of this equation, the following ondition has to be satisfied [13,16]: df (φ( )) dφ( ) > 1 (35) φ( )0 Therefore, [ 1 u P()(1 a)λ + +1 t P()λ ] > 1 (36) Similar to Eq (10), Eq (34) defines the epidemi threshold, whih an be alulated as [ ] λ 1 1 u u 2 P() a 2 P() (37) t +1 Considering the partiular ase of an N-node BA model, P() 2m 2 / 3, t m and u mn 1/2, we have from Eq (37) that λ 1 m ln 1 a u m a /2 mn m ln N am ln (38) 2 +1 Finally, to alulate the AOS in an infinitely large sale-free networ, we have R 1 P()S ( ) ( 2m e λφ( ) + 1 t ( t 2 2m 3 u +1 2m 2 3 e (1 a)λφ( ) e λφ( ) d 2 2m e λφ( ) d ) 2 2m 3 e (1 a)λφ( ) d ) (39) Sine the integrations in Eq (39) are in the form of inomplete gamma funtion, applying Eqs (17) and (18), we have ( 1 R 2m 2 λφ( ) m a ) (40) +1 Also we an derive the expression of φ( ) from Eq (32) that φ( ) 1 1 P()S ( ) ( ) 1 m 2 e λφ( ) d 2 e λφ( ) d + 2 e (1 a)λφ( ) d (41) t Again, the integrations in Eq (41) an be onsidered as inomplete gamma funtions Therefore we have the simplified equation of φ( ) that 1 ln(φ( )) (1 a)λm ( 1 + λm(1 a)(1 r E) ln m (1 a) ln[(1 a)λ] a ln +1 }), (42) and onsequently, [ ] 1 φ( ) exp (1 a)λm ( 1 + λm(1 a)(1 r E) ln m (1 a) ln[(1 a)λ] a ln +1 }) (43) Together with Eq (39), the networ AOS an be alulated Sine a simple losed-form solution of φ( ) is diffiult to ahieve, we plot figures in Setion 4 to demonstrate the respetive effets of immunization rate and spreading rate on AOS

8 2542 Y Wang et al / Physia A 388 (2009) Partial Lin Immunization (PLI) The mean-field level differential equations for PLI an be presented as: dρ (t) ρ (t) + (1 a)λs (t)θ(t) if >, ρ (t) + λs (t)θ(t) if ; ds (t) (1 a)λs (t)θ(t) if >, λs (t)θ(t) if ; dr (t) ρ (t) (46) Again, the term (1 a)λs (t)θ(t) in Eqs (44) and (45) denotes the perentage of -degree hub nodes that are newly infeted at a lose-to-threshold spreading rate Similar to Eqs (6) and (28) for PNI and PEI respetively, the probability that a randomly seleted lin is onneted to an infeted node an be expressed as P()ρ (t) θ(t) (47) We see that Eqs (44) (47) are exatly the same as Eqs (25) (28) Therefore, the epidemi threshold and AOS at a spreading rate lose to epidemi threshold are the same for PEI and PLI The AOS of PLI under high spreading rates will be studied by numerial simulations 34 Imperfet targeted immunization in degree-orrelated SIR model To analyze imperfet targeted immunization in the degree-orrelated SIR model, the mean-field level equations for PNI an be re-written as follows: dρ (t) ρ (t) + λa()s (t)θ(t); (48) ds (t) λa()s (t)θ(t); (49) dr (t) ρ (t) (50) For PEI and PLI, we have dρ (t) ρ (t) + (1 a)λa()s (t)θ(t) if >, (51) ds (t) ρ (t) + λa()s (t)θ(t) if ; (1 a)λa()s (t)θ(t) if >, λa()s (t)θ(t) if ; dr (t) ρ (t) (53) In all the above equations, T()P()ρ (t) θ(t) (54) Meanwhile, all the initial onditions and assumptions in Setions remain valid Following the same proedure as that in Setions 31 33, we an easily derive a general expression of the epidemi threshold that [ ] λ 1 1 u u 2 P()A()T() a 2 P()A()T() (55) t +1 Clearly, there still exists a linear relationship between the inverse of the epidemi threshold and the immunization rate An example ase where A() ε and T() 1 is illustrated in Fig 2 Sine it is diffiult to find a losed-form expression of AOS, numerial results will be presented in the next setion 4 Simulation results and disussions Numerial simulations have been implemented on a 10,000-node BA model and the AS-level Internet model For eah simulation, a single infeted node is randomly seleted at the starting point Following the SIR sheme, the system finally reahes the steady state when all the infeted nodes are removed All the results displayed are averaged from at least 10,000 (44) (45) (52)

9 Y Wang et al / Physia A 388 (2009) Fig 2 Dependene of inverse epidemi threshold on immunization rate for orrelation funtions A() ε and T() 1 where ε 0 and ε 025, respetively, on top of 10,000-node BA model Fig 3 Dependene of the epidemi threshold on the immunization rate in imperfet targeted immunizations in the BA model The embedded figure shows the inverse epidemi thresholds The solid lines in the major and embedded figures show the analytial results realizations, eah of whih with a randomly seleted infeted node at the starting point Different realizations may still have the same starting infeted node; however, the random proess of disease propagation, even under suh ases, will not be exatly repeated 41 Epidemi threshold Simulation results for the BA model are presented in Fig 3, where we set 4 As we an see, the epidemi threshold inreases with the immunization rate a At the beginning, the inreasing speed is slow, and then beomes muh faster With the three different types of imperfet targeted immunizations, the aptured epidemi thresholds almost equal to eah other ) versus the orresponding value of a are plotted in the embedded figure We see that the relationship losely resembles a linear funtion, whih verifies the analytial results Simulation results for the AS-level Internet model are presented in Fig 4 We see that in this orrelated networ, there is still a roughly linear relationship between the inverse epidemi threshold and the immunization rate The results suggest that in orrelated and unorrelated large-size sale-free networs, to signifiantly inrease the epidemi threshold, the immunization rate has to be rather high The inverse of the aptured epidemi thresholds (ie, the simulated λ 1 42 Average Outbrea Size (AOS) Eqs (24) and (43) show the relationship between AOS and immunization rate As in almost all the existing studies, the equations are derived for the ase with an infinitely large networ size and a lose-to-threshold low spreading rate The analytial results for the ase are plotted in Fig 5 where we set λ 004 and ut-off 4 for the BA model Note that the PEI and PLI have the same results It is learly shown that even a low immunization rate signifiantly dereases AOS, although an epidemi outbrea may still happen Moreover, under a low spreading rate, the AOSs for the three different immunization shemes are almost the same Under a higher spreading rate, however, different immunization shemes have different impats on AOS, as illustrated in Fig 5(b) To address the hanges of AOS in finite-size networs (with or without orrelations), numerial simulations have been onduted for 10,000-node BA and AS-level Internet models The results are presented in Fig 6, where we set the spreading

10 2544 Y Wang et al / Physia A 388 (2009) Fig 4 Dependene of the epidemi threshold on the immunization rate in imperfet targeted immunizations in the AS-level Internet model The solid line in the embedded figure omes from the least-square linear regression alulations of the aptured inverse epidemi thresholds Fig 5 Analytial results on the dependene of AOS on immunization rate in the infinitely large BA model The spreading rates are set to be 004 and 008 for Fig 5(a) and (b), respetively Cut-off is set to be 4 for both of them (a) BA model (b) AS-level Internet model Fig 6 Dependene of AOS on different immunization rates in finite-size networs rate and immunization ut-off as 025 and 4, respetively Apparently, a low immunization rate still helps to signifiantly lower AOS even when the spreading rate is relatively high Furthermore, we notie that the three imperfet targeted immunizations have different impats on AOS: PNI and PLI an be viewed as epidemi spreading in sale-free networs with some of their hub nodes and/or lins onneted to them being removed [36 39] Compared to PLI, PNI leads to a removal of more hub nodes (aa super-spreader [13,40]) and their lins, and onsequently a smaller AOS In PEI, similarly to PNI, some of the hub nodes are proteted in eah time slot However, suh a protetion is not permanent: a node being proteted in this time slot an still be affeted later on As a result, PEI leads to a larger AOS than PNI 43 Imperfet targeted immunization in degree-orrelated SIR model To evaluate the performane of imperfet targeted immunization against degree-orrelated transmission shemes, as that in Ref [24] we investigate the speial ase where A() ε and T() 1 A larger value of ε denotes a stronger degree orrelation, while ε 0 redues to the lassi SIR model with no orrelation Fig 7 illustrates two different ases

11 Y Wang et al / Physia A 388 (2009) (a) ε 0 (b) ε 025 Fig 7 Dependene of AOS on different immunization rates in BA model with a degree-orrelated transmission sheme where A() ε and T() 1: (a) ε 0; and (b) ε 025 Spreading rate and ut-off are set to 05 and 4, respetively with ε 0 and ε 025, respetively We observe that (i) a larger value of ε leads to a smaller AOS; and (ii) for both ases, imperfet targeted immunization remains as effetive in reduing AOS even at a moderate immunization rate Therefore, networ robustness is still signifiantly enhaned 5 Conlusion In this paper, we evaluated the effetiveness of imperfet targeted immunization in sale-free networs with lassi as well as degree-orrelated SIR models Three different ases have been proposed and analyzed We found that under the same immunization rate, the three different ases result in the same epidemi threshold but different AOS A linear relationship between the inverse epidemi threshold and the immunization rate has been identified, whih shows that the possibility of having an epidemi outbrea annot be signifiantly lowered unless the protetion is reasonably strong On the other hand, even a low immunization rate lowers networ AOS signifiantly Our studies have revealed that similar onlusions hold for imperfet targeted immunization on SIS model Speifially, there exists a linear relationship between the inverse of the epidemi threshold and the immunization rate; meanwhile the number of infetions when the system reahes its steady state (termed as prevalene [10]) an be signifiantly redued even with a moderate immunization rate Due to page limit, further details have to be presented in a separate report A reent trend in related researh is to study omplex adaptive systems and dynami networs (eg, Refs [41,42]) It is important to realize that many real-life systems will not eep stati during epidemi spreading Understanding the effets of perfet and imperfet targeted immunizations in suh systems will be our future researh interest Anowledgment This wor is supported in part by the Singapore A*STAR grant BMRC 06/1/21/19/457 Referenes [1] A-L Barabási, Lined: How Everything is Conneted to Everything Else, Plume, 2004 [2] S Bornhol, HG Shuster (Eds), Handboo of Graphs and Networs: From the Genome to the Internet, Wiley-VCH, 2003 [3] M Faloutsos, P Faloutsos, C Faloutsos, SIGCOMM (1999) 251 [4] R Albert, H Jeong, A-L Barabási, Nature 401 (1999) 130 [5] H Jeon, B Tombor, R Albert, ZN Oltval, A-L Barabási, Nature 407 (2000) 651 [6] RJ Williams, EL Berlow, JA Dunne, A-L Barabási, ND Martinez, Pro Natl Aad Si USA 99 (2002) [7] D Garlashelli, G Caldarelli, L Pietronero, Nature 423 (2003) 165 [8] F Liljeros, CR Edling, LAN Amaral, HE Stanley, Y Aberg, Nature 411 (2001) 907 [9] SH Strogatz, Nature 410 (2001) 268 [10] R Pastor-Satorras, A Vespignani, Phy Rev Lett 86 (2001) 3200 [11] AL Lloyd, RM May, Siene 292 (2001) 1316 [12] MEJ Newman, Phy Rev E 66 (2002) [13] Y Moreno, R Pastor-Satorras, A Vespignani, Eur Phys J B 26 (2002) 521 [14] M Boguñá, R Pastor-Satorras, Phy Rev E 66 (2002) [15] R Pastor-Satorras, A Vespignani, Phy Rev E 65 (2002) (R) [16] R Pastor-Satorras, A Vespignani, Phy Rev E 65 (2002) [17] Z Dezso, A-L Barabási, Phy Rev E 65 (2002) [18] S Xiao, G Xiao, TH Cheng, IEEE Commun Mag 46 (2008) 146 [19] S Gandon, M Mainnon, S Nee, A Read, Pro R So Lond B 270 (2003) 1129 [20] VV Ganusov, R Antia, Evolution 60 (5) (2006) 957 [21] P Yip, R Watson, Q Chen, Statist Med 26 (2007) 4475 [22] RM Anderson, RM May, Infetious Diseases in Humans, Oxford University Press, Oxford, 1992 [23] JD Murray, Mathematial Biology, Springer Verlag, Berlin, 1993

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