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Epidemic spreading

Epidemic spreading. Speaker: Ao Weng Chon Advisor: Kwang -Cheng Chen. Outline. Framework SIS SIR Bond-percolation model Conclusion Reference. Framework. Fully mixed model:

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Epidemic spreading

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  1. Epidemic spreading Speaker: AoWeng Chon Advisor: Kwang-Cheng Chen

  2. Outline • Framework • SIS • SIR • Bond-percolation model • Conclusion • Reference

  3. Framework • Fully mixed model: • The individual with whom a susceptible individual has contact are chosen at random form the whole population. • It allows one to write differential equations for the time evolution of the disease. • SIS, SIR • Bond-percolation model: • Incorporate a full network structure of the contact network.

  4. SIS • Susceptible(S) : they don’t have the disease but can catch it if exposed to someone who does. • Infected(I): they have the disease and can pass it on, and recovered, being susceptible again. • Infection spreading rate λ • The average number of contacts • Recover rate γ, set it as 1 w.l.o.g.

  5. SIS • Consider a vertex of degree k • Θis the probability that the vertex at the end of an edge is infective

  6. SIS • In the stationary state dik/dt=0 • A nontrivial solution is allowed when

  7. SIS • The value λ yielding the equality defines the critical epidemic threshold λc • The result implies that in scale-free networks with degree exponent 2<γ≦3 , pk~k-γ, for which <k2>→∞,we have λc=0. • For any positive value of λ, the infection can pervade the system with a finite prevalence , in a sufficiently large network

  8. SIR • Susceptible(S): they don’t have the disease but can catch it if exposed to someone who does. • Infective(I): they have the disease and can pass it on, and recovered. • Recovered(R): they have recovered from the disease and have permanent immunity, so that they can never get it again or pass it on.

  9. SIR • critical epidemic threshold λc =<k>/<k2>, vanishing as <k2>→∞

  10. Bond percolation model • Bond occupation probability T • r is the rate of disease-causing contacts between a pair of connected infective and susceptible individuals • τis the time for an infective individual remains infective

  11. Bond percolation model • Extraction of predictions about epidemics from percolation model • Distribution of percolation clusters: distribution of the sizes of disease outbreaks that start with a randomly chosen initial carrier • Percolation transition: epidemic threshold of epidemiology, above which an epidemic outbreak is possible • Size of the giant component : size of the epidemic

  12. conclusion • The absence of an epidemic threshold and its associated critical behavior implies that scale-free networks are prone to the spreading and the persistence of infections.

  13. Reference • [1] Pastor-Satorras, R. and Vespignani, A., Immunization of complex networks, Phys. Rev. E 65, 036104 (2002). • [2] Pastor-Satorras, R. and Vespignani, A., Epidemic Spreading in Scale-Free Networks, Phys. Rev. L 86, 14 (2001). • [3] Newman, M. E. J., Spread of epidemic disease on networks, Phys. Rev. E 66, 016128 (2002) • [4] Moreno, Y., Pastor-Satorras, R., and Vespignani, A., Epidemic outbreaks in complex heterogeneous networks, Eur. Phys. J. B 26, 512-529 (2002)

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