Infectious diseases are a major burden to global health. Understanding their mechanisms and being able to predict and intervene epidemic outbreaks is an important challenge for researchers and decision makers alike. It should not be too hard either―if we include human contact patterns, the mechanisms of contagion and the typical features of the disease, we could model most infectious-disease related phenomena. Of these three components, the network epidemiology of the last decade has shown that our limited understanding of human contact patterns is probably the most important focus are for advancing infectious disease epidemiology. We will discuss what is known about human contact patterns and how to include this knowledge in epidemic modeling. First, we discuss recent work on what the epidemiologically most important temporal structures of human contacts are. We use about 80 empirical temporal network datasets, several arguably important for disease spreading, and scan the entire parameter space of disease-spreading models. By comparing to null-models, we identify important, simple temporal patterns that affect disease spreading stronger than the bursty interevent time distributions. Furthermore, we investigate how to eliminate the temporal information to make an as relevant static network as possible. After all, static network epidemiology has more methods and results than temporal network epidemiology and it for some purposes it is necessary. We find that an “exponential threshold” representation almost always the best performance, but time-sliced network (with a carefully chosen window, usually considerably different than the sampling time of the data) works almost as good. In contrast, networks of concurrent contacts do not seem to carry so important information.

1. TNETS edition

2. Susceptible
Infectious
Infectious
Compartmental
models

3. Infectious
Recovered
Susceptible

4. Contact
patterns

5. ID1 ID2 Time
1 2 34
3 4 55
1 5 56
4 2 70
5 4 77
6 1 102
5 6 110
5 7 122
6 7 130
2 5 198
3 4 205
4 2 210
2 7 230

6. Sociopatterns gallery
P H Y S I C A L P R O X I M I T Y
Prostitution
Sociopatterns conference
Hospital system
N = 16,730, L = 50,632, T = 6.0y
N = 113, L = 20,818, T = 59h
N = 159(8), L = 6,027(350), T = 7.3(1)h
N = 293,878, L = 64,625,283, T = 3,570dReality mining
N = 63, L = 26,260, T = 8.6h

7. ELECTRONIC COMMUNICATION
N = 57,189, L = 444,162, T = 112.0d
Bornholdt’s e-mail
Eckmann’s e-mail
N = 3,188, L = 115,684, T = 81.6d
Filmtipset forum
N = 7,084, L = 1,412,401, T = 8.61y
Filmtipset messages
Pussokram dating
N = 28,972, L = 529,890, T = 512.0d
QX dating
N = 80,683, L = 4,337,203, T = 63.7d
N = 35,624, L = 472,496, T = 8.27y
Facebook wall posts
N = 293,878, L = 876,993, T = 1591d

8. TEMPORAL TO
STATIC

9. tstart
tstop
0 5 10 15 20
1
2
3
4
5
6
t
1
2
3
4
5
6
time-slice networks

10. tstart
tstop
0 5 10 15 20
1
2
3
4
5
6
t
1
2
3
4
5
6
ongoing networks

11. 0 5 10 15 20
1
2
3
4
5
6
t
1
2
3
4
5
6
1
2
3
4
5
6
Ω
=
2.5
τ = 10
exponential-threshold networks

12. GOOD REPRESENTATION:
RANKING OF IMPORTANT
VERTICES CONSERVED
FOR ALL PARAMETER VALUES:
MEASURE AVG OUTBREAK SIZE
WHEN SPREADING STARTS AT i
FOR ALL PARAMETER VALUES:
MEASURE DEGREE OF i
FOR ALL PARAMETER VALUES:
MEASURE CORENESS OF i
degree 4
coreness 0
coreness 2
coreness 3
coreness 4
static importance
optimal params.
dynamic
importance
Spearman
rank correlation
coefficent
=
Quality of
representation

13. E-mail 1
E-mail 2
Dating
Gallery
Conference
Prostitution
Results, Degree
Time-slice Ongoing Exponential-threshold Accumulated

14. E-mail 1
E-mail 2
Dating
Gallery
Conference
Prostitution
Time-slice Ongoing Exponential-threshold Accumulated
Results, Coreness

15. Time sliceTime sliceTime slice OngoingOngoingOngoing Expo. thresholdExpo. thresholdExpo. threshold Acc.
ρmax tstart tstop ρmax tstart tstop ρmax τ ΩΩ ρ
E-mail 1 0.73 0 0.42 0.50 0.25 0.25 0.77 0.40 0.30 0.46
E-mail 2 0.91 0 0.25 0.91 0.20 0.20 0.93 1.0 0.26 0.88
Dating 0.82 0 0.65 0.42 0.25 0.25 0.86 0.10 0.16 0.71
Gallery 0.77 0 0.72 0.53 0.39 0.39 0.87 0.70 0.71 0.76
Conference 0.79 0 0.10 0.74 0.10 0.11 0.77 0.04 0.02 0.53
Prostitution 0.71 0 0.77 0.30 0.60 0.60 0.72 0.04 0.20 0.49
Performance & parameter values
Degree

16. Parameter dependence of performance
0 0.2 0.4 0.6 0.8 1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
00
0.2
0.4
0.6
0.8
1
ρ
tstart/ T
tstop/T
Time slice

17. Parameter dependence of performance
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
0.3
ρ
tstart/ T
tstop/T
Concurrency

18. Parameter dependence of performance
0 0.5 1 1.5 2 2.5 3 3.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
0.5
1
1.5
2
ρ
Ω
τ/T
Exponential threshold

20. STEP 1 Assign stubs to vertices from a
random number distribution.
1
2
3
4
5
6

21. STEP 2 Connect random pairs of stubs
to form a simple graph.
1
2
3
4
5
6

22. STEP 3 Create active intervals for each
edge.
(1,2)
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
(5,6)
time

23. STEP 4 Create a time series of contacts
from some interevent-time
distribution.
time

24. STEP 5 Split the time series into
segments proportional to the
intervals and impose the
contacts of the segments to the
intervals.
(1,2)
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
(5,6)
time

25. STEP 6 Forget the active intervals.
(1,2)
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
(5,6)
time

26. 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.1 1µ
ρ
max
0.50.05
Exponential threshold
Time-slice
Accumulated
Ongoing

27. break

28. STATIC TO
TEMPORAL

31. (1,2)
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
(5,6)

32. (1,2)
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
(5,6)
time
time
(1,2)
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
(5,6)
ONGOINGLINKPICTURE

33. time
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
(5,6)
(1,2)
(1,2)
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
(5,6)
time
LINKTURNOVERPICTURE

34. time
Beginning time Interevent times End time
0 T
t1
t2
t3
t4
t5
t6
t7
t8
tB
tE
DEFI
NITI
ONS

36. Compensate for the size bias on intervals because of finite
T0
t’
t
sampling time (t’ would only be recorded if it starts within [0,T–t’])

37. Compensate for the size bias on intervals because of finite
T0
t’
t
sampling time (t’ would only be recorded if it starts within [0,T–t’])
Compensate for the chance an interevent time t is active
0
t
at the start of the sampling is proportional to t

38. Compensate for the size bias on intervals because of finite
T0
t’
t
sampling time (t’ would only be recorded if it starts within [0,T–t’])
Compensate for the chance an interevent time t is active
0
t
at the start of the sampling is proportional to t
ti
T–tii: ti≥t
∑ /
ti
T–tii
∑
Sum up and normalize

39. 0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000
time t (days)
predicted from
interevent times
end times
beginning times
PROSTITUTION
P(t)B

40. Dating 2
1
0
0.5
0.75
0.25
Dating 1
Forum
1
0
0.5
0.75
0.25
1
0
0.5
0.75
0.25
Prostitution
Hospital
1
0
0.5
0.75
0.25
1
0
0.5
0.75
0.25
E-mail 2
Facebook
1
0
0.5
0.75
0.25
1
0
0.5
0.75
0.25
E-mail 1
Film
1
0
0.5
0.75
0.25
1
0
0.5
0.75
0.25
Conference
Gallery
1
0
0.5
0.75
0.25
1
0
0.5
0.75
0.25
End Times
Beginning Times
Predictable
edges w.r.t.
beginning /
end times

41. (1,2)
(1,3)
(1,4)
(2,3)
reference networks

42. (1,2)
(1,3)
(1,4)
(2,3)
(1,2)
(1,3)
(1,4)
(2,3)
reference network:
identical interevent times

43. (1,2)
(1,3)
(1,4)
(2,3)
(1,2)
(1,3)
(1,4)
(2,3)
reference network:
identical beginning times

44. (1,2)
(1,3)
(1,4)
(2,3)
(1,2)
(1,3)
(1,4)
(2,3)
reference network:
identical end times

45. 0
0.1
0.2
0.3
0.1 0.2 0.90.8 10.70.60.50.40.3
0.1
1
0.01
0.001
per-contact transmission probability
durationofinfectivestage
fractionofinfectives
O r i g i n a l d a t a S I R

46. 0
0.1
0.2
0.3
0.1 0.2 0.90.8 10.70.60.50.40.3
0.1
1
0.01
0.001
per-contact transmission probability
durationofinfectivestage
fractionofinfectives
I n t e r e v e n t t i m e s S I R

47. 0
0.1
0.2
0.3
0.1 0.2 0.90.8 10.70.60.50.40.3
0.1
1
0.01
0.001
per-contact transmission probability
durationofinfectivestage
fractionofinfectives
B e g i n n i n g t i m e s S I R

48. 0
0.1
0.2
0.3
0.1 0.2 0.90.8 10.70.60.50.40.3
0.1
1
0.01
0.001
per-contact transmission probability
durationofinfectivestage
fractionofinfectives
E n d t i m e s S I R

49. 0
0.02
0.04
0.06
E-mail 1
0.1
0
0.05
Film
0
0.05
0.1
Dating 1
0.05
0.1
0.15
0.2
0
Forum
0
0.02
0.04
0.06
E-mail 2
0
0.02
0.06
0.08
0.04
Facebook
0
0.01
0.02
0.03
0.04
Prostitution
0
0.1
0.2
0.3
Hospital
0
0.04
0.06
0.08
0.02
Gallery
0
0.02
0.04
0.06
Conference
0.05
0.1
0
Dating 2
endtimes
beginningtimes
intereventtimes

50. 0.1 0.2 0.90.8 10.70.60.50.40.3
0.1
1
0.01
0.001
per-contact transmission probability
durationofinfectivestage
O r i g i n a l d a t a
0
0.2
0.3
0.4
averagenumberofinfections
S I S
0.1

51. 0.1 0.2 0.90.8 10.70.60.50.40.3
0.1
1
0.01
0.001
per-contact transmission probability
durationofinfectivestage
I n t e r e v e n t t i m e s
0
0.2
0.3
0.4
averagenumberofinfections
S I S
0.1

52. 0.1 0.2 0.90.8 10.70.60.50.40.3
0.1
1
0.01
0.001
per-contact transmission probability
durationofinfectivestage
B e g i n n i n g t i m e s
0
0.2
0.3
0.4
averagenumberofinfections
S I S
0.1

53. 0
0.2
0.3
0.4
0.1 0.2 0.90.8 10.70.60.50.40.3
0.1
1
0.01
0.001
per-contact transmission probability
durationofinfectivestage
averagenumberofinfections
E n d t i m e s S I S
0.1

54. 0
0.01
0.02
0
0.0005
0.001
0.0015
0
0.05
0.1
0
0.01
0.02
0.03
0.04
0
0.0005
0.001
0.0005
0.001
0 0
0.05
0.1
0
0.001
0.0015
0.0005
0.05
0.1
0.15
0.2
0 0
0.02
0.04
0.06
0.08
0
0.01
0.03
0.04
0.02
E-mail 1
Film
Dating 1
Forum
E-mail 2
Facebook Prostitution
Hospital
Gallery
Conference
Dating 2
endtimes
beginningtimes
intereventtimes

56. Science by: Illustrations by:
Petter Holme Fredrik Liljeros Mi Jin Lee
P Holme, 2013, PLoS Comp. Biol. 9:e1003142.
P Holme, F Liljeros, 2013, arxiv:1307.6436.

57. 0
0.2
0.4
0.6
0.8
E-mail 1
0
0.2
0.6
0.4
Film
0
0.1
0.2
0.3
0.4
E-mail 2
0
0.2
0.4
Facebook
0
0.4
0.6
0.2
Dating 1
0.8
0.2
0.4
0.6
0
ForumDating 2
0.2
0.8
0
0.4
0.6
–0.4
–0.2
0.2
0.8
0
0.4
0.6
Conference
Hospital
0
0.2
0.4
Prostitution
0
0.1
0.2
Gallery
0
0.2
0.8
0.4
0.6endtimes
beginningtimes
intereventtimes

58. 0
0.1
0.2
0.3
0.4
E-mail 1
0
0.2
0.4
0.6
E-mail 2
0
0.4
0.6
0.2
Dating 1
0
0.005
0.01
Facebook
0.8
0.2
0.4
0
ForumDating 2
–0.005
0
0.005
endtimes
–0.5
0
0.5
Conference
Hospital
–0.001
0
0.001
Prostitution
0
0.1
0.2
beginningtimes
intereventtimes
Gallery
0
0.2
0.8
0.4
0.6
0
0.2
0.5
0.4
Film
0.3
0.1