SIS epidemics on Networks

Transcription

1 SIS epidemics on etworks Piet Van Mieghem in collaboration with Eric Cator, Ruud van de Bovenkamp, Cong Li, Stojan Trajanovski, Dongchao Guo, Annalisa Socievole and Huijuan Wang 1 EURADOM 30 June-2 July, 2013

2 Outline Exact SIS model IMFA: -intertwined MF approximation Expected survival time on-markovian epidemics

3 Simple SIS model on networks Homogeneous birth (infection) rate β on all links between infected and susceptible nodes Homogeneous death (curing) rate δ for infected nodes τ = β /δ : effective spreading rate δ 1 Healthy 3 β 0 Infected 2 Infected Infection and curing are independent Poisson processes 3

4 SIS model on networks (1) Each node j can be in either of the two states: 0 : healthy 1 : infected Markov continuous time: infection rate β curing rate δ At time t: X j (t) is the state of node j " infinitesimal generator Q j ( t) = q 0 j q $ 0 j $ q # 1 j 0 β δ q 1 j 1 % " ' ' = q 0 j q $ 0 j & # $ δ δ 4 % ' &'

5 SIS model on networks (2) odes are interconnected in graph: " Q j ( t) = q 0 j q % $ 0 j ' # $ δ δ &' where the infection rate is due all infected neighbors of node j: q 0 j = k=1 ( t) = β a jk X k t a a a 1 a a 2 a a ( ) and where the adjacency matrix of the graph is a a A q 0j δ 1 5

6 SIS model on networks (3) Markov theory requires that the infinitesimal generator is a matrix whose elements are not random variables However, this is not the case in our simple model: q 0 j k=1 ( t) = β a jk X k t By conditioning to each possible combination of infected states, we finally arrive to the exact Markov continuous SIS model Drawback: this exact model has 2 states, where is the number of nodes in the network. ( ) 6

7 SIS Markovian process on a graph % ' & ' ( k I for j I : I I { j} at rate β a +ε ki for i I : I I \ { i} at rate δ I: infected subgraph (containing infected nodes) ε: nodal self-infection 7

8 Exact SIS model = 4 nodes Absorbing state states! P. Van Mieghem, J. Omic, R. E. Kooij, Virus Spread in etworks, IEEE/ACM Transaction on etworking, Vol. 17, o. 1, pp. 1-14, (2009).

9 Governing SIS equation for node j de[x j ] dt = E δx j + (1 X j )β a kj X k k=1 time-change of E[X j ] = Pr[X j = 1], probability that node j is infected de[x j ] dt if infected: probability of curing per unit time if not infected (healthy): probability of infection per unit time = δe X j + β a kj E [ X ] k β a kj E X j X k k=1 k=1 9

10 Joint probabilities de!" X i X j # $ dt!' * = E ( δx i + β(1 X i ) a ik X k + ) k=1, X + X ' *# - j i ( δx j + β(1 X j ) a jk X k +. " ) k=1, $ = 2δE"# X i X j $ % + β a ik E"# X j X k $ % + β a jk E [ X i X k ] β ( a jk + a ik )E"# X i X j X k $ % k=1 k=1 k=1! 3 ext, we need the # & differential equations for E[X i X j X k ]... " In total, the SIS process is defined by linear equations $ % 2 = k=1! # " k $ &+1 % E. Cator and P. Van Mieghem, 2012, "Second-order mean-field susceptible -infected-susceptible epidemic threshold", Physical Review E, vol. 85, o. 5, May, p

11 Markov Theory SIS model is exactly described as a continuous-time Markov process on 2 states, with infinitesimal generator Q. Drawbacks: no easy structure in Q computationally intractable for >20 steady-state is the absorbing state (reached after unrealistically long time) very few exact results... The mathematical community (e.g. Liggett, Durrett,...) uses: duality principle & coupling & asymptotics graphical representation of the Poisson infection and recovery events 12

12 Outline Exact SIS model IMFA: -intertwined MF approximation Expected survival time on-markovian epidemics

13 IMFA: -intertwined mean-field approxim. de[x j ] dt = δe X j + β a kj E [ X ] k β a kj E X j X k k=1 E X j X k = Pr X j =1, X k =1 = Pr X j =1 X k =1 Pr[ X k =1] and k=1 Pr X j =1 X k =1 Pr X j =1 E [ X i X ] k Pr[ X i =1]Pr [ X k =1] = E [ X i ]E [ X ] k de[x j ] dt k=1 δe X j + β a kj E X k [ ] k=1 βe X j a kj E X k [ ] IMFA (= equality above) upper bounds the prob. of infection E. Cator and P. Van Mieghem, 2014, odal infection in Markovian SIS and SIR epidemics on networks are non-negatively correlated, Physical Review E, Vol. 89, o. 5, p

14 IMFA non-linear equations dv 1 dt = (1 v 1)β a 1k v k δv 1 dv 2 dt dv dt k=1 = (1 v 2 )β a 2k v k δv 2 k=1 k=1 = (1 v )β a k v k δv In matrix form: ( ) dt dv t = βa.v t where the viral probability of infection is v k ( ) diag v i ( t) ( t) = E[X k (t)] = Pr!" X k ( t) =1# $ ( )( βa.v ( t) + δu) where the vector u T =[ ] and V T = [v 1 v 2... v ] P. Van Mieghem, J. Omic, R. E. Kooij, Virus Spread in etworks, IEEE/ACM Transaction on etworking, Vol. 17, o. 1, pp. 1-14, (2009). 15

15 Lower bound for the epidemic threshold dv j (t) dt = δv j + β a kj v k β a kj E X i X k v k t k=1 k=1 Ignoring the correlation terms dv(t) dt ( δi + βa)v(t) [ ] ( ) = E[X k (t)] If all eigenvalues of βa δi are negative, v j tends exponentially fast to zero with t. Hence, if βλ 1 (A) δ < 0 τ = β δ < 1 λ 1 (A) < τ c The IMFA epidemic threshold is precisely τ (1) 1 c = λ 1 (A) < τ c τ (1) 1 c = λ 1 (A) < τ (2) 1 c = λ 1 (H ) < τ c ( V(t) e δi+βa)t V(0) 16

16 What is so interesting about epidemics? network protection self-replicating objects (worms) propagation errors rumors (social nets) epidemic algorithms (gossiping) cybercrime : network robustnes & security max E D [ ] 1+ Var[D] ( E[D] ), d 2 max λ 1 ( A ) d max τ c = λ 1 1 A ( ) Epidemic threshold 18

17 Transformation s = 1 τ & principal eigenvector dy (s) ds s=0 = 1 j=1 1 1 d j 2L convex dy (s) ds s=λ 1 = 1 j=1 j=1 (x 1 ) j (x 1 ) j 3 19 Van Mieghem, P., 2012, "Epidemic Phase Transition of the SIS-type in etworks", Europhysics Letters (EPL), Vol. 97, Februari, p s c = λ 1

18 Extensions of the IMFA In-homogeneous: each node i has own β i and δ i : P. Van Mieghem and J. Omic, 2008, In-homogeneous Virus Spread in etworks, (arxiv.org/ ) SAIS (Infected, Susceptible, Alert) and SIR instead of SIS: F. Darabi Sahneh and C. Scoglio, 2011,"Epidemic Spread in Human etworks, 50 th IEEE Conf. Decision and Contol, Orlando, Florida. "M. Youssef and C. Scoglio, 2011, An individual-based approach to SIR epidemics in contact networks Journal of Theoretical Biology 283, pp GEMF: very general extension: m compartments (includes both SIS, SAIS, SIR,...): F. Darabi Sahneh, C. Scoglio, P. Van Mieghem, 2013, "Generalized Epidemic Mean-Field Model for Spreading Processes over Multi-Layer Complex etworks",, IEEE/ACM Transactions on etworking, Vol. 21, o. 5, pp Interdependent networks Wang, H., Q. Li, G. D'Agostino, S. Havlin, H. E. Stanley and P. Van Mieghem, 2013, "Effect of the Interconnected etwork Structure on the Epidemic Threshold, Physical Review E, Vol. 88, o. 2, August, p

19 Time-dependent rates in IMFA for regular graphs dv( t) dt v( t) = = rβ ( t)v t 1 v 0 t ( ) + rβ s 0 ( ) 1 v( t) ( ) δ t ( )v t ( ) # t & exp% { rβ ( u) δ ( u) }du( $ 0 ' # s & ( )exp% { rβ ( u) δ ( u) }du(ds $ ' 0 Classical case (constant rates): Kephart & White (1992) exp( { rβ δ}t) v( t) = 1 v( 0) + 1 { 1 1 ( exp( { rβ δ}t) 1) } rτ P. Van Mieghem, SIS epidemics with time-dependent rates describing ageing of information spread and mutation of pathogens, unpublished. 21

20 Mutations 22

21 Outline Exact SIS model IMFA: -intertwined MF approximation Expected survival time on-markovian epidemics

22 SIS epidemics on the complete graph λ 0 λ j-1 λ j 0... j-1 j j+1... µ 1 µ j µ j+1 " $ # %$ λ j = ( β j +ε) ( j) µ j = δ j Birth-death processes quadratic in state j E. Cator and P. Van Mieghem, 2012, SIS epidemics on the complete graph and the star graph: exact analysis, PRE, vol. 87, p

23 Average Time to Absorption (Survival time) Ganesh,Massoulie,Towsley (2005): Mountford et al. (2013): (regular trees w. bounded degree) E [ T ] 1 log +1 δ ( 1 τλ) 1 E [ T ] = O e b a Complete graph K : E [ T ] 1 with ζ = 1 ζ F τ ( ) E [ T ] = O( e c ) " ( ) +O 2 log $ # x 2 1 x = τ τ! F x $ # & ~ 1 x 2π F ( τ ) = 1 j 1 ( j + r)! τ c δ j( j)! " % δ x 1 j=1 r=0 ( ) 2 e τ < τ c τ > τ c % ' &! # log x+ 1 " x 1 $ & % Hitting time (on K ) for all τ: E T [ ] = F τ ( ) 25 P. Van Mieghem, Decay towards the overall-healthy state in SIS epidemics on networks, arxiv (2013).

24 Average survival time in K E [ T ] = F ( τ ) = 1 δ j=1 j 1 r=0 ( j + r)! j( j)! 26

25 Second smallest eigenvalue Q in graphs ζ 1 E T [ ] 28

26 Pdf survival time in K 100 Black dotted τ = τ c 29 f T ( t β = 0) = δi ( 1 e δt ) I 1 e δt (Max I i.i.d. exp.)

27 Outline Exact SIS model IMFA: -intertwined MF approximation Expected survival time on-markovian epidemics

28 Epidemic times are not exponential tweet sent tweet received by follower tweet observed by follower tweet retweeted t 0 t 1 t 2 t 3 time } } } D network D observe D reaction 31 C. Doerr,. Blenn and P. Van Mieghem, Lognormal infection times of Online information spread, PLOS OE, Vol. 8, o. 5, p. e64349, 2013

29 on-markovian infection times f T ( t) = α b! # " x b $ & % Weibull pdf α 1!! exp x $ # # & " " b % α $ & % Same mean E[T]: b = 1 " βγ 1+ 1 % $ ' # α & 32 T is the time to infect a neighboring node

30 on-markovian epidemic threshold ER-graph = 500 p = 2 p c on-exponential infection time has a dramatic influence! 33 P. Van Mieghem and R. van de Bovenkamp, on-markovian infection spread Dramatically alters the SIS epidemic threshold, Physical Review Letters, vol. 110, o. 10, March, p

31 GSIS: SIS with general infection times IMFA is valid provided the effective infection rate τ = β/δ is replaced by the averaged number E[M] of infection events during a healthy period: E[M ] = 1 2πi c+i c i φ T (z)φ R ( z) 1 φ T (z) dz z φ X (z) = E[e zx ] Generalized criterion for the epidemic threshold: E[M c ] = 1 λ 1 Scaling law for large When infection time T is Weibullian: τ c = q(α) λ 1 α 1 q(α) = O(1) 34 E. Cator, R. van de Bovenkamp and P. Van Mieghem, SIS epidemics on networks with general infection and curing times, Physical Review E, Vol. 87, o. 6, p , 2013.

32 GSIS: E[M] gives the right scaling Average fraction of infected nodes simulations GSIS IMFA Complete graph K increasing 3 E(M) 4 5 6x10-3 E[M]: averaged number of infection events received during a healthy period 35

33 Pdf survival time K 100 (Weibull) Increasing α f infection time ( t) = α b! t # $ α 1! " b % &! exp # t $ α $ # " " b % & & % E[M]=

34 Challenges for SIS epidemics on nets Tight upper bound of the epidemic threshold (for any graph) A general mean-field criterion that specifies the graphs for which IMFA is accurate Time-dependent analysis of SIS epidemics Epidemics on evolving and adaptive networks Competing and mutating viruses on networks Measurements of epidemics (e.g. fraction of infected nodes) in real-world networks are scarce 38

35 More to read General overviews P. Van Mieghem, Performance Analysis of Complex etworks and Systems, Cambridge University Press, 2014 (Chapter 17: Epidemics in etworks) R. Pastor-Satorras, C. Castellano, P. Van Mieghem and A. Vespignani, Epidemic processes in complex networks, Review of Modern Physics, 2014 Specialized recent topics (see my website): Adaptive SIS on networks Competing viruses Average survival time of a virus in a network (decay time towards absorption) SIS Epidemics in (two-level) communities 39

36 Books Articles: 40

37 Thank You Piet Van Mieghem AS, TUDelft 41