sis immortality transition for kps spring meeting 2015
TRANSCRIPT
![Page 1: SIS Immortality Transition for KPS spring meeting 2015](https://reader034.vdocuments.site/reader034/viewer/2022042701/55b88e74bb61ebea2a8b46a1/html5/thumbnails/1.jpg)
The SIS immortality transition in small networks
Petter Holme
Sungkyunkwan University
![Page 2: SIS Immortality Transition for KPS spring meeting 2015](https://reader034.vdocuments.site/reader034/viewer/2022042701/55b88e74bb61ebea2a8b46a1/html5/thumbnails/2.jpg)
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
![Page 3: SIS Immortality Transition for KPS spring meeting 2015](https://reader034.vdocuments.site/reader034/viewer/2022042701/55b88e74bb61ebea2a8b46a1/html5/thumbnails/3.jpg)
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 → ∞
![Page 4: SIS Immortality Transition for KPS spring meeting 2015](https://reader034.vdocuments.site/reader034/viewer/2022042701/55b88e74bb61ebea2a8b46a1/html5/thumbnails/4.jpg)
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.
![Page 5: SIS Immortality Transition for KPS spring meeting 2015](https://reader034.vdocuments.site/reader034/viewer/2022042701/55b88e74bb61ebea2a8b46a1/html5/thumbnails/5.jpg)
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).
![Page 6: SIS Immortality Transition for KPS spring meeting 2015](https://reader034.vdocuments.site/reader034/viewer/2022042701/55b88e74bb61ebea2a8b46a1/html5/thumbnails/6.jpg)
America
![Page 7: SIS Immortality Transition for KPS spring meeting 2015](https://reader034.vdocuments.site/reader034/viewer/2022042701/55b88e74bb61ebea2a8b46a1/html5/thumbnails/7.jpg)
Iceland
![Page 8: SIS Immortality Transition for KPS spring meeting 2015](https://reader034.vdocuments.site/reader034/viewer/2022042701/55b88e74bb61ebea2a8b46a1/html5/thumbnails/8.jpg)
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
![Page 9: SIS Immortality Transition for KPS spring meeting 2015](https://reader034.vdocuments.site/reader034/viewer/2022042701/55b88e74bb61ebea2a8b46a1/html5/thumbnails/9.jpg)
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
![Page 10: SIS Immortality Transition for KPS spring meeting 2015](https://reader034.vdocuments.site/reader034/viewer/2022042701/55b88e74bb61ebea2a8b46a1/html5/thumbnails/10.jpg)
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
×103 ×103t t
ξ ξ
λ = 0.07λ = 0.065λ = 0.06
λ = 0.18λ = 0.17λ = 0.16
America Iceland
![Page 11: SIS Immortality Transition for KPS spring meeting 2015](https://reader034.vdocuments.site/reader034/viewer/2022042701/55b88e74bb61ebea2a8b46a1/html5/thumbnails/11.jpg)
Time constant vs. λ
0.05 0.1 0.15 0.2 0.25 0.02 0.04 0.06 0.08 0.1
106
105
104
103
100
10
106
105
104
103
100
10
λλ
τ τ
America Iceland
τ = A exp(λ / l) +B (1 – λ)–ζ
![Page 12: SIS Immortality Transition for KPS spring meeting 2015](https://reader034.vdocuments.site/reader034/viewer/2022042701/55b88e74bb61ebea2a8b46a1/html5/thumbnails/12.jpg)
Contribution of individual nodes
Measure America Iceland
0-pa
ram
. 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-pa
ram
. 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
![Page 13: SIS Immortality Transition for KPS spring meeting 2015](https://reader034.vdocuments.site/reader034/viewer/2022042701/55b88e74bb61ebea2a8b46a1/html5/thumbnails/13.jpg)
Contribution of individual nodesa = ζ(G ) / ζ(G) i i
1
2
1
3
32
America Iceland
![Page 14: SIS Immortality Transition for KPS spring meeting 2015](https://reader034.vdocuments.site/reader034/viewer/2022042701/55b88e74bb61ebea2a8b46a1/html5/thumbnails/14.jpg)
Thanks to
1) You, for listening.
2) National Research Foundation of Korea for funding.
Preprint at: arXiv:1503.01909