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The SIS immortality transition in small networks
Petter Holme
Sungkyunkwan University
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
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 → ∞
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.
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).
America
Iceland
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
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
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
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
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
Contribution of individual nodesa = ζ(G ) / ζ(G) i i
1
2
1
3
32
America Iceland
Thanks to
1) You, for listening.
2) National Research Foundation of Korea for funding.
Preprint at: arXiv:1503.01909
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