### SIS Immortality Transition for KPS spring meeting 2015

1. The SIS immortality transition in small networks Petter Holme Sungkyunkwan University
2. The SIS model Models diseases where re-infection is possible Gonorrhea, Chlamydia, are exampled from sexually transmitted infections (and thus appro- priate for network epidemiology) A population of susceptible (S) and infectious (I) When S meets I, there is a probability λ that S will become I I becomes S again after some time, or with some chance per unit of time
3. Two areas of current research 1.The epidemic threshold (phase transition in λ). 2.The extinction probability as a function of λ. Both points when N → ∞
4. The immortality transition There is another phase transition (threshold)— when λ = 1. The mean time to extinction diverges at this point. It may seem trivial (since it is not an emergent property in the N → ∞), but we will pretend it is not.
5. Our example networks We could take any small networks with a variety of network structures, but to honor the network epidemiology pioneers we use: D. M. Auerbach, W. W. Darrow, H. W. Jaffe, and J. W. Curran, Am. J. Med. 76, 487 (1984). S. Haraldsdottir, S. Gupta, and R. M. Anderson, J. Acquir. Immune Defic. Syndr. 5, 374 (1992).
6. America
7. Iceland
8. Survival probability vs. λ America 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.25 0.5 0.75 0.1 0.15 0.2 0.25 λ ξ λ ξ
9. Survival probability vs. λ Iceland 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.25 0.5 0 0.05 0.1 λ ξ λ ξ
10. Survival probability vs. time 0 5 100 5 10 0.1 1 10 –6 10 –5 10 –4 10 –3 10 –2 0.1 1 10 –6 10 –5 10 –4 10 –3 10 –2 ×10 3 ×10 3 t t ξ ξ λ = 0.07 λ = 0.065 λ = 0.06 λ = 0.18 λ = 0.17 λ = 0.16 America Iceland
11. Time constant vs. λ 0.05 0.1 0.15 0.2 0.25 0.02 0.04 0.06 0.08 0.1 10 6 10 5 10 4 10 3 100 10 10 6 10 5 10 4 10 3 100 10 λλ τ τ America Iceland τ = A exp(λ / l) +B (1 – λ) –ζ
12. Contribution of individual nodes Measure America Iceland 0-param. ki 0.73(4) 0.974(2) ni 0.82(4) 0.75(5) mi 0.83(3) 0.965(2) i 0.64(4) 0.917(6) 1-param. max Ki 0.76(5) 0.98(2) for α 0.17(8) 0.038(5) max Ri 0.72(6) 0.97(4) for d 0.99(1) 0.99(1) ε a = ζ(G ) / ζ(G)i i
13. Contribution of individual nodes a = ζ(G ) / ζ(G)i i 1 2 1 3 3 2 America Iceland
14. Thanks to 1) You, for listening. 2) National Research Foundation of Korea for funding. Preprint at: arXiv:1503.01909