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Spreading dynamics on small-world networks with a power law degree distribution

Spreading dynamics on small-world networks with a power law degree distribution. Alexei Vazquez The Simons Center for Systems Biology Institute for Advanced Study. Epidemic outbreak. External source. Population. Population structure. Contact graph. N individuals

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Spreading dynamics on small-world networks with a power law degree distribution

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

  2. Epidemic outbreak External source Population

  3. Population structure Contact graph N individuals pk connectivity distribution D average distance

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

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

  6. Physical contact or proximity 1 day Portland k-1.8 USA D4.37 city Eubank et al, Nature 2004 nation/world Barrat et al, PNAS 2004 N=3,880

  7. Physical contact or proximity

  8. Branching process model Generation 0 root 1 2 3 4 Spanning tree kpk/<k> k-1 pk k

  9. Branching process model Timming dd+1 generation Generation time T Distribution G()=Pr(T) tt+T1t+T2t+T3 time

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

  11. Recursive calculation T1 t=0 T2 d d+1 t=0 d

  12. Results Reproductive number Time scale Vazquez, Phys. Rev. Lett. 2006 Constant transmission rate : G()=1-e- Incidence I(t): expected rate of new infections at time t

  13. pk~k -, kmax~N >3, t<<t0 (t0 when N ) <3, t>>t0 (t00 when N ) Vazquez, Phys. Rev. Lett. 2006

  14. Numerical simulations • Network: random graph with a given degree distribution. • pk~k - • Constant transmission rate  • N=1000, 10000, 100000 • 100 graph realizations, 10000 outbreaks

  15. Numerical simulations log-log linear-log e(K-1)t I(t)/N I(t)/N  1,000  10,000  100,000 tD-1e-t t t

  16. Case study: AIDS epidemics • New York - HOM • New York - HET • San Francisco - HOM • South Africa • Kenya • Georgia • Latvia • Lithuania t3 t3 t3 Cumulative number t2 t2 exponential t (years) Szendroi & Czanyi, Proc. R Soc. Lond. B 2004

  17. Generalizations Degree correlations Multitype

  18. Degree correlations k’ k

  19. Degree correlations Kk Kk k k

  20. Degree correlations N( t)D-1e-t k’ k   e(R*-1)t  Vazquez, Phys. Rev. E 74, 056101 (2006)

  21. Multi-type i=1,…,M types Ni number of type i agents p(i)k type i degree distribution eijmixing matrix D average distance Reproductive number matrix : largest eigenvalue

  22. Multi-type Strongly connected type-networks Vazquez, Phys. Rev. E (In press); http://arxiv.org/q-bio.PE/0605001  Type 1  Type 2  Type 3  Type 4 Type-network eij eii

  23. Generalizations Non-exponential generating time distributions

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

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

  26. Email activity patterns Left: University server 3,188 users 129,135 emails sent <  >~1 day E~25 days Right: Comercial email server ~1,7 millions users ~39 millions emails sent <  >~4 days E~9 months T T

  27. Incidence: model

  28. Prevalence: http://www.virusbtn.com I(t) I(t) I(t) Prevalence data Decay time ~ 1 year Poisson model <  >~1 day - University <  >~4 days - Comercial Email data E~25 days - University E~9 months - Comercial

  29. Conclusions • 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.

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