1. How the information
content of your contact
pattern representation
affects predictability of
epidemics
Petter Holme
Sungkyunkwan University
Umeå University
HONS workshop, NetSci 2015
Zaragoza, Spain
June 2, 2015
Title
Presenter
Affiliation
Occasion
Place
Date

2. P Holme
Information content of contact-pattern
representations and the predictability
of epidemic outbreaks
arxiv:1503.06583

6. Compartmental models Contact structure
To start with: use canonical compartmental models.
SIR with fixed disease duration (and discrete time).
probability λ
time δ

7. Compartmental models Contact structure
Fully mixed Network Temporal network
information

8. Background / Motivation

9. Background / Motivation

12. … conditional on a large outbreak,
the evolutions of certain quantities of
interest, such as the fraction of
infective vertices, converge to
deterministic functions of time.

13. “Weather is
hard to predict
because it is
chaotic”

14. “Weather is
hard to predict
because it is
modeled by
equations that
show chaotic
behavior”

15. “Disease
outbreaks are
hard to predict
because human
contact
structure has
this-or-that
structure”

16. predictability … in what sense?
Assume we know the present, and can
predict future contacts, then how well
can we predict the final outbreak size?
… so it’s about the uncertainty of the
SIR model rather than the contacts.

17. Datasets
Human proximity data: who is close to whom at what time
From the Sociopatterns project (RFID sensors, ~1.5m range,
N = 75~250), T = 10h~5days
From the Reality mining project N = 64, T = 9 hrs
From Brazilian online prostitution N = 16,730, T = 6 hrs

18. 0.2
0.3
0.1
0
0.2
0.6
0.4
0
0.8
Temporal network
Static network
P(Ω)P(Ω)
δ/T
λ
0.001 0.01 0.1 1
0.001
0.01
0.1
1
δ/T
λ
0.001 0.01 0.1 1
0.001
0.01
0.1
1
theknownstuff:
differenceinoutbreaksize

19. 0
0.2
0.6
0.4
0
0.8
0.5
0
1
Static network
Fully mixed
P(Ω)P(Ω)
λ
0.001 0.01 0.1 1
0.001
δ/T
λ
0.001 0.01 0.1 1
0.001
0.01
0.1
1
δ/T
λ
0.001 0.01 0.1 1
0.001
0.01
0.1
1
theknownstuff:
differenceinoutbreaksize

20. Time
Numberofinfected
+ time
of infection
+ time
of infection
+ time
of infection

21. Time
Numberofinfected

22. Time
Numberofinfected
Time
s.d.

23. Results

24. Example Temporal networks
Sociopatterns’ hospital data
δ = 0.6, λ = 0.1
0
10
20
30
40
50
60
70
0 1 2 3 4
Time (days)
Numberofinfected
breaking time: 1h

25. Example Temporal networks
Sociopatterns’ hospital data
δ = 0.6, λ = 0.1
0
10
20
30
40
50
60
70
0 1 2 3 4
Time (days)
Numberofinfected
breaking time: 2h

26. Example Temporal networks
Sociopatterns’ hospital data
δ = 0.6, λ = 0.1
0
10
20
30
40
50
60
70
0 1 2 3 4
Time (days)
Numberofinfected
breaking time: 3h

27. Example Temporal networks
Sociopatterns’ hospital data
δ = 0.6, λ = 0.1
0
10
20
30
40
50
60
70
0 1 2 3 4
Time (days)
Numberofinfected
breaking time: 4h

28. Example Temporal networks
Sociopatterns’ hospital data
δ = 0.6, λ = 0.1
0
10
20
30
40
50
60
70
0 1 2 3 4
Time (days)
Numberofinfected
breaking time: 6h

29. Example Temporal networks
Sociopatterns’ hospital data
δ = 0.6, λ = 0.1
0
10
20
30
40
50
60
70
0 1 2 3 4
Time (days)
Numberofinfected
breaking time: 12h

30. Example Temporal networks
Sociopatterns’ hospital data
δ = 0.6, λ = 0.1
0
10
20
30
40
50
60
70
0 1 2 3 4
Time (days)
Numberofinfected
breaking time: 24h

31. Example Temporal networks
Sociopatterns’ hospital data
δ = 0.6, λ = 0.1
0
10
20
30
40
50
60
70
0 1 2 3 4
Time (days)
Numberofinfected
breaking time: 36h

32. Example Temporal networks
Sociopatterns’ hospital data
δ = 0.6, λ = 0.1
breaking time: 48h
0
10
20
30
40
50
60
70
0 1 2 3 4
Time (days)
Numberofinfected

33. A λ = 0.00428, δ = 0.695
Temporal network
0 0.25 0.5 0.75 1
t
0
0.25
0.5
0.75
1∆
B λ = 0.0127, δ = 0.233
Temporal network
0 0.25 0.5 0.75 1
t
0
0.25
0.5
0.75
1
∆
C λ = 0.233, δ = 0.112
Temporal network
0 0.25 0.5 0.75 1
t
0
0.25
0.5
0.75
1
∆
D λ = 0.233, δ = 0.162
Temporal network
0 0.25 0.5 0.75 1
t
0
0.25
0.5
0.75
1
∆
E λ = 0.00428, δ = 0.695
Static network
0 0.25 0.5 0.75 1
t
0
0.25
0.5
0.75
1
∆
F λ = 0.0263, δ = 0.112
Static network
0 0.25 0.5 0.75 1
t
0
0.25
0.5
0.75
1
∆
00.050.10.150.2
P(∆)

34. 0
2
4
6
8
10
D Prostitution
0 0.2 0.4 0.6 0.8 1
t / T
–5
×10
∆Ω
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Static network
Temporal network
Fully mixed
A Conference
0 0.2 0.4 0.6 0.8 1
t / T
∆Ω
0
0.01
0.02
0.03
B Gallery
0 0.2 0.4 0.6 0.8 1
t / T
∆Ω
0
0.02
0.04
0.06
0.08
C Hospital
0 0.2 0.4 0.6 0.8 1
t / T
∆Ω
0
0.01
0.02
0.03
0.04
0.05
0.06
F School
0 0.2 0.4 0.6 0.8 1
t / T
∆Ω
0
0.02
0.04
0.06
0.08
E Reality
0 0.2 0.4 0.6 0.8 1
t / T
∆Ω

35. 0
0.1
0.2
0.3
0.4
max∆Ω
0 0.2 0.4 0.6 0.8 1
t / T
B Gallery
E Reality
0
0.1
0.2
0.3
0.4
max∆Ω
0 0.2 0.4 0.6 0.8 1
t / T
F School
0 0.2 0.4 0.6 0.8 1
t / T
0
0.1
0.2
0.3
0.4
max∆Ω
Static network
Temporal network
Fully mixed
A Conference
0 0.2 0.4 0.6 0.8 1
t / T
0
0.1
0.2
0.3max∆Ω
0
1
2
3 D Prostitution
0 0.2 0.4 0.6 0.8 1
t / T
×10
max∆Ω
–4
0.1
0.2
0.3
0 0.2 0.4 0.6 0.8 1
t / T
0
max∆Ω
C Hospital

36. Temporalnetwork,Sociopatterns’hospitaldata
λ
0.001 0.01 0.1 1
0.001
0.01
0.1
1
A Temporal network
B Static network
0.5
0
1
tp/T
0
tp/T
δ/T
0.001 0.01 0.1 1
0.001
0.01
0.1
1
λ
δ/T
0.8
0.6
0.4
0.2

37. Temporalnetwork,Sociopatterns’hospitaldata
δ/T
λ
λ
C Fully mixed
0.001 0.01 0.1 1
0.001 0.01 0.1 1
0.001
0.01
0.1
1
0.001
0.01
0.1
1
B Static network
0.5
0
tp/T
0.5
0
1
tp/T
δ/T
1

38. P Holme, N Masuda
The basic reproduction number
as a predictor for epidemic
outbreaks in temporal networks
PLOS ONE 10: e0120567 (2015)

39. R₀ — basic reproductive number,
reproduction ratio, reproductive ratio, ...
The expected number of secondary
infections of an infectious individual in a
population of susceptible individuals.

40. One of few concepts that
went from mathematical
to medical epidemiology

41. Disease R₀
Measles 12–18
Pertussis 12–17
Diphtheria 6–7
Smallpox 5–7
Polio 5–7
Rubella 5–7
Mumps 4–7
SARS 2–5
Influenza 2–4
Ebola 1–2

42. SIR model
ds
dt
= –βsi—
di
dt
= βsi – νi—
= νi
dr
dt
—
S I I I
I R
Ω = r(∞) = 1 – exp[–R₀ Ω]
where R₀ = β/ν
Ω > 0 if and only if R₀ > 1
The epidemic threshold

43. Problems with R₀
Hard to
estimate
Can be hard for
models
& even harder for outbreak data
and many datasets lack the
important early period
The threshold isn’t R₀ = 1 in practice
The meaning of a threshold in a finite population.
In temporal networks, the outbreak size
needn’t be a monotonous function of R₀

44. Plan
Use empirical
contact data
Simulate the entire
parameter space of
the SIR model
Plot Ω vs R₀
Figure out what temporal network
structure that creates the deviations

45. 1
0.8
0.6
0.4
0.2
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Averageoutbreaksize,Ω
Basic reproductive number, R0
Conference

46. 1
0.8
0.6
0.4
0.2
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Averageoutbreaksize,Ω
Basic reproductive number, R0
Conference
0.001 0.01 0.1 1
1
0.1
0.01
0.001
transmission probability
diseaseduration

47. 1
0.8
0.6
0.4
0.2
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Averageoutbreaksize,Ω
Basic reproductive number, R0
1
0.8
0.6
0.4
0.2
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Averageoutbreaksize,Ω
Basic reproductive number, R0
1
0.8
0.6
0.4
0.2
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Averageoutbreaksize,Ω
Basic reproductive number, R0
Conference Hospital
Forum
1
0.8
0.6
0.4
0.2
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Averageoutbreaksize,Ω
Basic reproductive number, R0
School, day 2

48. Shape index (example)—
discordant pair separation in Ω
1.0
0.8
0.6
0.4
0.2
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Basic reproductive number, R0
Averageoutbreaksize,Ω
μΩ=0.304
ρΩ = 2.663

49. avg. fraction of nodes present when 50% of contact happened
avg. fraction of links present when 50% of contact happened
avg. fraction of nodes present at 50% of the sampling time
avg. fraction of links present at 50% of the sampling time
frac. of nodes present 1st and last 10% of the contacts
frac. of links present 1st and last 10% of the contacts
frac. of nodes present 1st and last 10% of the sampling time
frac. of links present 1st and last 10% of the sampling time
Time evolution
degree distribution, mean
degree distribution, s.d.
degree distribution, coefficient of variation
degree distribution, skew
Degree distribution
link duration, mean
link duration, s.d.
link duration, coefficient of variation
link duration, skew
link interevent time, mean
link interevent time, s.d.
link interevent time, coefficient of variation
link interevent time, skew
Link activity
Node activity
node duration, mean
node duration, s.d.
node duration, coefficient of variation
node duration, skew
node interevent time, mean
node interevent time, s.d.
node interevent time, coefficient of variation
node interevent time, skew
Other network structure
number of nodes
clustering coefficient
assortativity
Temporal network structure

50. Correlation between point-cloud shape &
temporal network structure
*
*
** ** ** **
**
*
**
**
**
*
∆R0
0
0.2
0.4
0.6
0.8
1
R²
Time evolution
Node activity Link activity
Degree
distribution
Network
structure
fLT
fNT
fLC
fNC
FLT
FNT
FLC
FNC
γNt
σNt
cNt
µNt
γNτ
σNτ
cNτ
µNτ
γLt
σLt
cLt
µLt
γLτ
σLτ
cLτ
µLτ
γk
σk
ck
µk
N C r

51. ***
**
∆Ω
0
0.2
0.4
0.6
0.8
1
R²
Time evolution
Node activity
Link activity
Network
structure
fLT
fNT
fLC
fNC
FLT
FNT
FLC
FNC
γNt
σNt
cNt
µNt
γNτ
σNτ
cNτ
µNτ
γLt
σLt
cLt
µLt
γLτ
σLτ
cLτ
µLτ
γk
σk
ck
µk
N C r
Degreedistribution
Correlation between point-cloud shape &
temporal network structure
Holme & Masuda, 2015,
PLoS ONE 10:e0120567.

52. P Holme, T Takaguchi
Time evolution of predictability
of epidemics on networks
Phys. Rev. E 91: 042811 (2015)

53. Only static networks
Constant recovery rate SIR
Different topologies (RR, SW, LW, SF w expo 2,
2.5, 3)
Two different assumptions of what is known
about the outbreak.
Standard deviation as measure of outbreak
diversity or non-predictability

54. s,t
(a) (b)
(d) (e)
0
t
0
t
0
t
0
t
–7
–9
–7 –7
–7
s,t
s,t
s,t
s,t
s,t

55. (b) (c)
(f)(e)
0
t
0
t
0
t
0
t
–7
–7 –7
–7
s,t
s,t
s,t
s,t

56. 0
1
2
3
4
5
6
0
1
2
3
1 4 161/16 1/4
0
1
2
0
1
2
3
0
1
2
3
0
1
2
3
1 4 161/16 1/4 1 4 161/16 1/4
1 4 161/16 1/41 4 161/16 1/41 4 161/16 1/4
T
T
T
T
T
T
(a) Large world
(b) Small world
(c) Random regular
(d) (f)(e)
4
5

57. 0 50 70 90
t
0 50 70
t
0 5
t
0 5
t
(a) (b)
(d) (e)
|R
|R
|R
|R

58. 0 50 70
t
0 5
t
0 5
t
(b) (c)
(f)(e)
0 50 70 90
t
|R
|R
|R
|R

59. 0
2
4
6
8
10
0
1
2
3
1 4 161/16 1/4
0
0
1
2
3
0
1
2
0
1
2
1 4 161/16 1/4 1 4 161/16 1/4
1 4 161/16 1/41 4 161/16 1/41 4 161/16 1/4
T
T
T
T
T
T
(a)Largeworld
(b)Smallworld
(c)Randomregular
(d)
(f)
(e)
4
6
2
4
6
8

60. Thank you!