1. E PIDEMIC SPREADING Speaker: Ao Weng Chon Advisor: Kwang-Cheng Chen 1

2. O UTLINE Framework SIS SIR Bond-percolation model Conclusion Reference 2

3. F RAMEWORK 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. 3

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. 4

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

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

7. SIS The value λ yielding the equality defines the critical epidemic threshold λ c The result implies that in scale-free networks with degree exponent 2 →∞, we have λ c =0. For any positive value of λ, the infection can pervade the system with a finite prevalence, in a sufficiently large network 7

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. 8

9. SIR critical epidemic threshold λ c = /, vanishing as →∞ 9

10. B OND PERCOLATION MODEL Bond occupation probability T r is the rate of disease-causing contacts between a pair of connected infective and susceptible individuals τ is the time for an infective individual remains infective 10

11. B OND 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 11

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. 12

13. R EFERENCE [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) 13