Spreading dynamics on small-world networks with a power law degree distribution Alexei Vazquez The Simons Center for Systems Biology Institute for Advanced Study Alexei Vazquez The Simons Center for Systems Biology Institute for Advanced Study

Epidemic outbreak Population External source

Population structure N individuals p k connectivity distribution D average distance Contact graph

Sexual contacts p k ~k - , 2<  <5 Liljeros et al. Nature (2001) Jones & Handcook, Nature (2003) Schneeberger et al, Sex Transm Dis (2004) Sexually transmitted diseases 1 yearlifetime Sweden  -1

Sexual contacts Colorado Springs HIV network Potterat et al, Sex. Transm Infect 2002 STD N=250 D  8 k -2

Physical contact or proximity Barrat et al, PNAS 2004 N=3,880 Eubank et al, Nature day Portland k USA D  4.37 city nation/world

Physical contact or proximity

Branching process model Spanning tree Generation 0root pkpk kp k / k-1 k

d d+1 generation t t+T 1 t+T 2 t+T 3 time Generation time T Distribution G(  )=Pr(T  ) Branching process model Timming

1.The process start with a node (d=0) that generates k sons with probability distribution p k. 2.Each son at generation 0. 3.Nodes at generation D does not generate any son. 4.The generation times are independent random variables with distribution function G(  ). Note: Galton-Watson, Newman Bellman-Harris, Crum-Mode-Jagers Branching process model

Recursive calculation t=0 d d T2T2 T1T1 d+1

ResultsResults Constant transmission rate : G(  )=1-e -  Vazquez, Phys. Rev. Lett Reproductive number Time scale Incidence I(t): expected rate of new infections at time t

p k ~k - , k max ~N   >3,t<<t 0 (t 0  when N  )  >t 0 (t 0  0 when N  ) Vazquez, Phys. Rev. Lett. 2006

Numerical simulations Network: random graph with a given degree distribution. p k ~k -  Constant transmission rate N=1000, 10000, graph realizations, outbreaks

I(t)/N t t e (K-1) t t D-1 e - t  1,000  10,000  100,000 log-loglinear-log Numerical simulations

Case study: AIDS epidemics  New York - HOM  New York - HET  San Francisco - HOM  South Africa  Kenya  Georgia Latvia  Lithuania Szendroi & Czanyi, Proc. R Soc. Lond. B 2004 t2t2 t3t3 t3t3 t3t3 t2t2 t (years) Cumulative number exponential

GeneralizationsGeneralizations Degree correlations Multitype

Degree correlations k’ k

Degree correlations kk KkKk KkKk

N( t) D-1 e - t e (R*-1) t  Vazquez, Phys. Rev. E 74, (2006) k’ k

Multi-typeMulti-type i=1,…,M types N i number of type i agents p (i) k type i degree distribution e ij mixing matrix D average distance Reproductive number matrix  : largest eigenvalue

Multi-typeMulti-type Type 1 Type 2 Type 3 Type 4 e ij Strongly connected type-networks Vazquez, Phys. Rev. E (In press); Type-network e ii

GeneralizationsGeneralizations Non-exponential generating time distributions

Intermediate states Vazquez, DIMACS Series in Discrete Mathematics… 70, 163 (2006)

Long time behavior: worms Receive infected time Sent infected s generating time (residual waiting time) Generating time probability density In collaboration with R. Balazs, L. Andras and A.-L. Barabasi

activity patterns Left: University server 3,188 users 129,135 s sent ~1 day  E ~25 days Right: Comercial server ~1,7 millions users ~39 millions s sent ~4 days  E ~9 months TT

Incidence: model

Prevalence: Prevalence: Prevalence data Decay time ~ 1 year data  E ~25 days - University  E ~9 months - Comercial Poisson model ~1 day - University ~4 days - Comercial I(t)I(t) I(t)I(t) I(t)I(t)

ConclusionsConclusions Truncated branching processes are a suitable framework to model spreading processess on real networks. There are two spreading regimes. –Exponential growth. –Polynomial growth followed by an exponential decay. The time scale separating them is determined by D/R. The small-world property and the connectivity fluctuations favor the polynomial regime. Intermediate states favor the exponential regime.