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
×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

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

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

Contribution of individual nodes
a = ζ(G ) / ζ(G)i i
1
2
1
3
3
2
America Iceland

Thanks to
1) You, for listening.
2) National Research Foundation of Korea for funding.
Preprint at: arXiv:1503.01909