how the information content of your contact pattern representation affects predictability of...
TRANSCRIPT
How the informationcontent of your contactpattern representationa!ects predictability ofepidemics
Petter Holme
Sungkyunkwan UniversityUmeå University
HONS workshop, NetSci 2015
Zaragoza, Spain
June 2, 2015
Title
Presenter
Affiliation
Occasion
Place
Date
P Holme
Information content of contact-patternrepresentations and the predictabilityof epidemic outbreaks
arxiv:1503.06583
Compartmental models Contact structure
To start with: use canonical compartmental models.
SIR with !xed disease duration (and discrete time).
probability !
time "
Compartmental models Contact structure
Fully mixed Network Temporal network
information
Background / Motivation
Background / Motivation
… 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.
“Weather is hard to predict because it is chaotic”
“Weather is hard to predict because it is modeled by equations that show chaotic behavior”
“Disease outbreaks are hard to predict because human contact structure has this-or-that structure”
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.
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
0.2
0.3
0.1
0
0.2
0.6
0.4
0
0.8
0.5
0
1
Temporal network
Static network
Fully mixed
P(Ω
)P(
Ω)
P(Ω
)
! / T
"0.001 0.01 0.1 1
0.001
0.01
0.1
1
! / T
"0.001 0.01 0.1 1
0.001
0.01
0.1
1
! / T
"0.001 0.01 0.1 1
0.001
0.01
0.1
1
the k
nown
stuf
f:di
fferen
ce in
outb
reak
size
0.2
0.3
0.1
0
0.2
0.6
0.4
0
0.8
0.5
0
1
Temporal network
Static network
Fully mixed
P(Ω
)P(
Ω)
P(Ω
)
! / T
"0.001 0.01 0.1 1
0.001
0.01
0.1
1
! / T
"0.001 0.01 0.1 1
0.001
0.01
0.1
1
! / T
"0.001 0.01 0.1 1
0.001
0.01
0.1
1
the k
nown
stuf
f:di
fferen
ce in
outb
reak
size
Time
Num
ber o
f inf
ecte
d+ time
of infection+ time
of infection+ time
of infection
Time
Num
ber o
f inf
ecte
d
Time
Num
ber o
f inf
ecte
d
Time
s.d.
Results
Example Temporal networksSociopatterns’ hospital data
! = 0.6, " = 0.1
0
10
20
30
40
50
60
70
0 1 2 3 4Time (days)
Num
ber o
f inf
ecte
d
breaking time: 1h
Example Temporal networksSociopatterns’ hospital data
! = 0.6, " = 0.1
0
10
20
30
40
50
60
70
0 1 2 3 4Time (days)
Num
ber o
f inf
ecte
d
breaking time: 2h
Example Temporal networksSociopatterns’ hospital data
! = 0.6, " = 0.1
0
10
20
30
40
50
60
70
0 1 2 3 4Time (days)
Num
ber o
f inf
ecte
d
breaking time: 3h
Example Temporal networksSociopatterns’ hospital data
! = 0.6, " = 0.1
0
10
20
30
40
50
60
70
0 1 2 3 4Time (days)
Num
ber o
f inf
ecte
d
breaking time: 4h
Example Temporal networksSociopatterns’ hospital data
! = 0.6, " = 0.1
0
10
20
30
40
50
60
70
0 1 2 3 4Time (days)
Num
ber o
f inf
ecte
d
breaking time: 6h
Example Temporal networksSociopatterns’ hospital data
! = 0.6, " = 0.1
0
10
20
30
40
50
60
70
0 1 2 3 4Time (days)
Num
ber o
f inf
ecte
d
breaking time: 12h
Example Temporal networksSociopatterns’ hospital data
! = 0.6, " = 0.1
0
10
20
30
40
50
60
70
0 1 2 3 4Time (days)
Num
ber o
f inf
ecte
d
breaking time: 24h
Example Temporal networksSociopatterns’ hospital data
! = 0.6, " = 0.1
0
10
20
30
40
50
60
70
0 1 2 3 4Time (days)
Num
ber o
f inf
ecte
d
breaking time: 36h
Example Temporal networksSociopatterns’ hospital data
! = 0.6, " = 0.1
breaking time: 48h
0
10
20
30
40
50
60
70
0 1 2 3 4Time (days)
Num
ber o
f inf
ecte
d
A ! = 0.00428, " = 0.695Temporal network
0 0.25 0.5 0.75 1t
0
0.25
0.5
0.75
1!
B ! = 0.0127, " = 0.233Temporal network
0 0.25 0.5 0.75 1t
0
0.25
0.5
0.75
1
!
C ! = 0.233, " = 0.112Temporal network
0 0.25 0.5 0.75 1t
0
0.25
0.5
0.75
1
!
D ! = 0.233, " = 0.162Temporal network
0 0.25 0.5 0.75 1t
0
0.25
0.5
0.75
1
!
E ! = 0.00428, " = 0.695Static network
0 0.25 0.5 0.75 1t
0
0.25
0.5
0.75
1
!
F ! = 0.0263, " = 0.112Static network
0 0.25 0.5 0.75 1t
0
0.25
0.5
0.75
1
!
00.
050.
10.
150.
2P(
!)
0
2
4
6
8
10
D Prostitution
0 0.2 0.4 0.6 0.8 1t / 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 1t / T
∆Ω
0
0.01
0.02
0.03
B Gallery
0 0.2 0.4 0.6 0.8 1t / T
∆Ω0
0.02
0.04
0.06
0.08
C Hospital
0 0.2 0.4 0.6 0.8 1t / T
∆Ω
0
0.01
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0.03
0.04
0.05
0.06
F School
0 0.2 0.4 0.6 0.8 1t / T
∆Ω
0
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0.06
0.08
E Reality
0 0.2 0.4 0.6 0.8 1t / T
∆Ω
0($1
'(9,$7,21
0
0.1
0.2
0.3
0.4
max
∆Ω
0 0.2 0.4 0.6 0.8 1t / T
B Gallery
E Reality
0
0.1
0.2
0.3
0.4
max
∆Ω
0 0.2 0.4 0.6 0.8 1t / T
F School
0 0.2 0.4 0.6 0.8 1t / T
0
0.1
0.2
0.3
0.4
max
∆Ω
Static network
Temporal network
Fully mixed
A Conference
0 0.2 0.4 0.6 0.8 1t / T
0
0.1
0.2
0.3m
ax ∆
Ω
0
1
2
3 D Prostitution
0 0.2 0.4 0.6 0.8 1t / T
×10
max
∆Ω
–4
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0.2
0.3
0 0.2 0.4 0.6 0.8 1t / T
0
max
∆Ω
C Hospital
0$;
'(9,$7,21
Tem
pora
l net
wor
k, S
ocio
patte
rns’
hosp
ital d
ata
! / T
"
"
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
A Temporal network
B Static network
0.5
0
t p/ T
0.5
0
1
t p/ T
0
t p/ T
! / T
0.001 0.01 0.1 10.001
0.01
0.1
1
"
! / T
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Tem
pora
l net
wor
k, S
ocio
patte
rns’
hosp
ital d
ata
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"
"
C Fully mixed
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0.001 0.01 0.1 1
0.001
0.01
0.1
1
0.001
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1
A Temporal network
B Static network
0.5
0
t p/ T
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0
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0
t p/ T
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"
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1
P Holme, N Masuda
The basic reproduction numberas a predictor for epidemicoutbreaks in temporal networks
PLOS ONE 10: e0120567 (2015)
R₀ — basic reproductive number, reproduction ratio, reproductive ratio, ...
The expected number of secondary infections of an infectious individual in a population of susceptible individuals.
One of few concepts that went from mathematical to medical epidemiology
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
SIR model
dsdt = –βsi—
didt = βsi – νi—
= νidrdt—
S I I I
I R
Ω = r(∞) = 1 – exp[–R₀ Ω]
where R₀ = β/ν
Ω > 0 if and only if R₀ > 1The epidemic threshold
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
PlanUse 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
1
0.8
0.6
0.4
0.2
00 0.5 1 1.5 2 2.5 3 3.5 4
Aver
age
outb
reak
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, Ω
Basic reproductive number, R!
1
0.8
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00 0.5 1 1.5 2 2.5 3 3.5 4
Aver
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Basic reproductive number, R!
Conference Hospital
Forum
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00 0.5 1 1.5 2 2.5 3 3.5 4
Aver
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School, day 2
1
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Aver
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Basic reproductive number, R!
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00 0.5 1 1.5 2 2.5 3 3.5 4
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Basic reproductive number, R!
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00 0.5 1 1.5 2 2.5 3 3.5 4
Aver
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reak
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Basic reproductive number, R!
Conference Hospital
Forum
1
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00 0.5 1 1.5 2 2.5 3 3.5 4
Aver
age
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reak
size
, Ω
Basic reproductive number, R!
School, day 2
0.001 0.01 0.1 1
1
0.1
0.01
0.001
transmission probability
dise
ase
dura
tion
1
0.8
0.6
0.4
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00 0.5 1 1.5 2 2.5 3 3.5 4
Aver
age
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1
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00 0.5 1 1.5 2 2.5 3 3.5 4
Aver
age
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reak
size
, Ω
Basic reproductive number, R!
1
0.8
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0.4
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00 0.5 1 1.5 2 2.5 3 3.5 4
Aver
age
outb
reak
size
, Ω
Basic reproductive number, R!
Conference Hospital
Forum
1
0.8
0.6
0.4
0.2
00 0.5 1 1.5 2 2.5 3 3.5 4
Aver
age
outb
reak
size
, Ω
Basic reproductive number, R!
School, day 2
Shape index (example)—discordant pair separation in Ω
1.0
0.8
0.6
0.4
0.2
0.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Basic reproductive number, R0
Aver
age
outb
reak
size
, Ω
μ Ω=
0.30
4
ρΩ= 2.663
avg. fraction of nodes present when 50% of contact happenedavg. fraction of links present when 50% of contact happenedavg. fraction of nodes present at 50% of the sampling timeavg. fraction of links present at 50% of the sampling timefrac. of nodes present 1st and last 10% of the contactsfrac. of links present 1st and last 10% of the contactsfrac. of nodes present 1st and last 10% of the sampling timefrac. of links present 1st and last 10% of the sampling time
Time evolution
degree distribution, meandegree distribution, s.d.degree distribution, coe!cient of variationdegree distribution, skew
Degree distribution
link duration, meanlink duration, s.d.link duration, coe!cient of variationlink duration, skewlink interevent time, meanlink interevent time, s.d.link interevent time, coe!cient of variationlink interevent time, skew
Link activity
Node activitynode duration, meannode duration, s.d.node duration, coe!cient of variationnode duration, skewnode interevent time, meannode interevent time, s.d.node interevent time, coe!cient of variationnode interevent time, skew
Other network structurenumber of nodesclustering coe!cientassortativity
Temporal network structure
Correlation between point-cloud shape &temporal network structure
*
*
** ** ** ****
*
**** **
*
!R"
0
0.2
0.4
0.6
0.8
1
R#
Time evolutionNode activity Link activity
Degreedistribution
Networkstructure
fLTfNTfLCfNC FLTFNTFLCFNC !Nt"Nt cNtµNt !N!"N! cN!µN! !Lt"Lt cLtµLt !L!"L! cL!µL! !k"k ckµk N C r
***
**
!Ω
0
0.2
0.4
0.6
0.8
1
R"
Time evolution
Node activity
Link activity
Networkstructure
fLTfNTfLCfNC FLTFNTFLCFNC γNtσNt cNtµNt γN!σN! cN!µN! γLtσLt cLtµLt γL!σL! cL!µL! γkσk ckµk N C r
Deg
ree
dist
ribu
tion
Correlation between point-cloud shape &temporal network structure
Holme & Masuda, 2015,PLoS ONE 10:e0120567.
P Holme, T Takaguchi
Time evolution of predictabilityof epidemics on networks
Phys. Rev. E 91: 042811 (2015)
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
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(a) L
arge
wor
ld
(b) S
mal
l wor
ld
(c) R
ando
m re
gula
r
(d) 6
FDOH�IUHH��Ȗ� ��
(f) 6
FDOH�IUHH��Ȗ� ��
(e) 6
FDOH�IUHH��Ȗ� ����
4
�
6
2
4
6
8
Thank you!