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