Combining social contact data with spatio-temporal
models for infectious diseases
Leonhard Held
ISCB 2015, Utrecht, 26 August 2015
Joint work with Sebastian Meyer
Financial support by the Swiss National Science Foundation
Social Contact Data
POLYMOD study estimated contact matrices in eight EU countries.
Mossong et al. (2008)
Leonhard Held (University of Zurich)
Modelling infectious diseases
26 August 2015
2 / 24
Statistical Modelling of Infectious Disease Spread
I
Spatio-temporal models have been proposed for counts of infectious
diseases
Schrödle et al. (2011)
Meyer and Held (2014)
I
Contact data has been used to analyse infectious disease spread
between age groups
Goeyvaerts et al. (2015)
→ Combine social contact data with spatio-temporal time series models
for infectious disease counts Ygrt :
I
I
I
Age Group g
Region r
Time t
Leonhard Held (University of Zurich)
Modelling infectious diseases
26 August 2015
3 / 24
Case Study: Noroviral Gastroenteritis
I
Generation time similar to seasonal influenza: 3 − 4 days
I
Highly infectious via droplets
I
No vaccination available
Weekly counts downloaded from https://survstat.rki.de/
I
Age Group:
Region:
Time:
Stratification in 5-year intervals
Berlin
Week 2011/27 to 2014/26
→ 15 age groups
→ 12 city districts
→ 156 weeks
I Latest revision of reporting scheme in 2011
→ Only laboratory-confirmed cases are notifiable
I Lower reporting rates during Christmas break
Leonhard Held (University of Zurich)
Modelling infectious diseases
26 August 2015
4 / 24
Weekly Counts (All Districts and Age Groups)
| Christmas break
200
No. infected
150
100
50
0
2011 2011
III
IV
Leonhard Held (University of Zurich)
2012 2012 2012
II
III
2013 2013 2013
IV
II
Time [week]
Modelling infectious diseases
III
IV
2014
II
26 August 2015
5 / 24
Mean Incidence by District
52.65°N
52.6°N
Pankow
Reinickendorf
52.55°N
Spandau
Charlottenburg−
Wilmersdorf
52.5°N
52.45°N
Steglitz−
Zehlendorf
Mitte
Lichtenberg
Marzahn−
Hellersdorf
Friedrichshain−
Kreuzberg
Tempelhof−
Neukoelln
Schoeneberg
Treptow−
Koepenick
52.4°N
52.35°N
13.1°E 13.2°E 13.3°E 13.4°E 13.5°E 13.6°E 13.7°E
49.00 64.00 81.00 100.00
121.00
144.00
169.00
Mean yearly incidence [per 100 000 inhabitants]
Leonhard Held (University of Zurich)
Modelling infectious diseases
26 August 2015
6 / 24
300
200
100
+
70
−6
9
−6
4
65
−5
9
60
55
−5
4
−4
9
Modelling infectious diseases
50
−4
4
45
40
−3
9
−3
4
35
−2
9
30
25
−2
4
−1
9
Leonhard Held (University of Zurich)
20
−1
4
15
10
−0
9
00
−0
4
0
05
Mean yearly incidence [per 100 000 inhabitants]
Mean Incidence by Age Group
26 August 2015
7 / 24
Aggregate into Larger Age Groups
Aggregate into 5 age groups: 00-04, 05-14, 15-49, 50-64, 65+
70+
65−69
60−64
55−59
50−54
45−49
40−44
35−39
30−34
25−29
20−24
15−19
10−14
05−09
00−04
4.0
6
65+
3.0
2.5
2.0
1.5
1.0
age group of participant
3.5
5
50−64
4
15−49
3
2
05−14
0.5
1
00−04
0.0
00
−
05 04
−
10 09
−
15 14
−
20 19
−
25 24
−
30 29
−
35 34
−
40 39
−
45 44
−
50 49
−
55 54
−
60 59
−
65 64
−6
9
70
+
age group of participant
I
0
00−04
15−49
50−64
65+
age group of contact
age group of contact
Leonhard Held (University of Zurich)
05−14
Modelling infectious diseases
26 August 2015
8 / 24
Weekly Incidence by Age Group
Aggregated across districts, per 100 000 inhabitants
30
00−04
30
05−14
30
25
25
25
20
20
20
15
15
15
10
10
10
5
5
5
0
0
0
30
50−64
30
25
25
20
20
15
15
10
10
5
5
0
0
Leonhard Held (University of Zurich)
15−49
65+
|
Christmas break
Modelling infectious diseases
26 August 2015
9 / 24
Regression Model for Infectious Disease Counts
I
Additive endemic-epidemic decomposition of disease incidence
Held et al. (2005)
Paul et al. (2008)
Endemic
⊕
Epidemic
I
seasonality, population, socio-demography,
climate, . . .
force of previously infected individuals ⇒
spatio-temporal interaction
Multivariate branching process formulation → epidemic proportion λ
Leonhard Held (University of Zurich)
Modelling infectious diseases
26 August 2015
10 / 24
Spatio-temporal Model for Infectious Disease Counts
I
Time series model for weekly counts Yrt in region r and week t
Meyer and Held (2014)
I
Negative binomial likelihood with mean
X
bwr 0 r c Yr 0 ,t−1
µrt = ert νrt + φrt
r0
ert
νrt , φrt
wr 0 r
bwr 0 r c
known population fractions
log-linear predictors
weights for r 0 to r transmission, e. g . power law:
wr 0 r = (or 0 r + 1)−d with adjacency order or 0 r
and unknown decay parameterPd
normalized weights such that r bwr 0 r c = 1
Leonhard Held (University of Zurich)
Modelling infectious diseases
26 August 2015
11 / 24
Specific Model Formulation
µrt = er exp{α + βxt + γ sin(ωt) + δ cos(ωt)}
X
b(or 0 r + 1)−d cYr 0 ,t−1
+ φr
r0
I
I
I
I
xt : Christmas break indicator
Sinusoidal log-rate with frequency ω = 2π/52
Power-law distance decay
Model selection:
Model
endemic-only
+ power-law
Leonhard Held (University of Zurich)
dim
5
18
∆ AIC
0.0
−537.3
Modelling infectious diseases
d̂
–
2.5
λ̂
0
0.64
26 August 2015
12 / 24
Fitted Mean by District
Charlottenburg−Wilmersdorf
30
epidemic
endemic
25
20
●
15
●
●
●
●
●
25
10
0
●
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30
●
20
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●
25
20
●
5
Lichtenberg
30
15
●
●
10
Friedrichshain−Kreuzberg
30
●
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5
0
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10
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Mitte
30
Neukoelln
25
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Pankow
30
● ●
25
●
●
20
●
Reinickendorf
●
30
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25
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30
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20
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Spandau
Steglitz−Zehlendorf
●
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30
30
●
30
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30
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25
25
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Leonhard Held (University of Zurich)
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Modelling infectious diseases
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26 August 2015
13 / 24
Power Law Weights
1.0
●
●
weight
0.8
power law
unconstrained
0.6
0.4
0.2
●
●
0.0
0
1
2
●
●
3
4
adjacency order o
Leonhard Held (University of Zurich)
Modelling infectious diseases
26 August 2015
14 / 24
Normalized Power Law Weights
Pankow
46.8%
8.4%
1.5%
3.0%
1.4%
Leonhard Held (University of Zurich)
7.9%
7.9%
3.0%
3.0%
2.9%
44.4%
2.9%
3.0%
8.4%
3.0%
7.9%
7.9%
8.4%
8.4%
3.0%
Mitte
2.9%
Modelling infectious diseases
7.9%
2.9%
2.9%
26 August 2015
15 / 24
Age-Stratified Spatio-temporal Model
µgrt = egr exp{αgG + αrR + βxt + γg sin(ωt) + δg cos(ωt)}
X
R
+ φG
φ
bCg 0 g (or 0 r + 1)−d c Yg 0 ,r 0 ,t−1
g r
(g 0 ,r 0 )
I
Age group and region-specific effects αgG and αrR
I
Age group-specific seasonality γg , δg
I
R
Parsimonious “main effects” decomposition: φgr = φG
g φr
I
Cg 0 g : Number of social contacts of participant in age group g 0 with
age group g
I
Power law weights (or 0 r + 1)−d
Leonhard Held (University of Zurich)
Modelling infectious diseases
26 August 2015
16 / 24
Model Selection
Model
endemic-only with age group and region-specific effects
+ age group-specific seasonality
+ power-law, homogeneous mixing between age groups
+ power-law with social contact data
+ power-law, no contact between age groups
Leonhard Held (University of Zurich)
Modelling infectious diseases
dim
20
28
45
45
45
∆ AIC
0.0
−604.8
−944.1
−1095.4
−1052.3
d̂
–
–
2.8
2.5
2.2
26 August 2015
λ̂
–
–
0.47
0.68
0.7
17 / 24
Fitted Mean By Age Group
Aggregated across districts
00−04
140
05−14
15−49
140
140
120
120
100
100
80
80
80
60
60
60
from other groups
within group
endemic
120
100
●
40
20
0
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●
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40
40
20
20
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0
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50−64
0
●
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●
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●
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●
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●
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●
●●
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●
●
65+
140
140
120
120
100
100
●
●●
●
●
●
●
●
●
80
●
●
●
● ●●●
80
●
●
●
●
●
●
●
●
●
●●
60
60
40
40
●
20
0
Leonhard Held (University of Zurich)
20
0
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Modelling infectious diseases
●
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26 August 2015
18 / 24
Regional Effects
Endemic
Epidemic
0.6
0.4
Pankow
Reinickendorf
0.6
0.4
Pankow
Reinickendorf
0.2
Spandau
Charlottenburg−
Wilmersdorf
Mitte
Lichtenberg
Marzahn−
Friedrichshain−
Hellersdorf
Kreuzberg
0.0
0.2
Spandau
Charlottenburg−
Wilmersdorf
Mitte
Lichtenberg
Marzahn−
Friedrichshain−
Hellersdorf
Kreuzberg
−0.2
Steglitz−
Zehlendorf
Tempelhof−
Neukoelln
Schoeneberg
−0.2
Treptow−
Steglitz−
Koepenick
Zehlendorf
−0.4
Tempelhof−
Neukoelln
Schoeneberg
Treptow−
Koepenick
−0.6
Leonhard Held (University of Zurich)
0.0
Modelling infectious diseases
−0.4
−0.6
26 August 2015
19 / 24
Normalized Power-Law and Contact Weights
Transmission pattern from a case in Mitte aged 15-49 years
00−04
0.1%
0.1%
0.1%
05−14
0.1%
0.5%
0.1%
0.8%
0.0%
0.1%
0.1%
0.1%0.1%
0.1%
0.2%
0.5%
1.0%
1.0%0.4%
5.8%
0.2%
0.5%0.2%
0.2%
2.1%
5.8%
2.1%
5.8%
2.1%
32.9%
1.0%
5.8%
5.8%2.1%
2.1%
65+
0.4%
0.4%
5.8%
0.4%
0.2%
1.0%
1.0%
0.4%
0.5%
0.2%
3.1%
0.1%
0.5%
50−64
1.0%
15−49
0.4%
Leonhard Held (University of Zurich)
0.4%
0.1%
2.1%
0.1%
0.1%
0.4%
0.4%
0.1%
0.4%0.1%
0.1%
Modelling infectious diseases
26 August 2015
20 / 24
Normalized Power-Law and Contact Weights
Transmission pattern from a case in Mitte aged 65+ years
00−04
0.1%
0.0%
0.1%
05−14
0.1%
0.6%
0.0%
0.4%
0.0%
0.1%
0.0%
0.1%0.0%
0.0%
0.2%
0.6%
1.8%
1.8%0.7%
3.2%
0.2%
0.6%0.2%
0.2%
1.2%
3.2%
1.2%
3.2%
1.2%
18.2%
0.6%
3.2%
3.2%1.2%
1.2%
65+
2.3%
0.7%
10.1%
0.7%
0.3%
1.8%
1.8%
0.7%
0.6%
0.2%
3.1%
0.1%
0.6%
50−64
1.8%
15−49
0.7%
Leonhard Held (University of Zurich)
2.3%
0.8%
12.8%
0.8%
0.4%
2.3%
2.3%
0.8%
2.3%0.8%
0.8%
Modelling infectious diseases
26 August 2015
21 / 24
Normalized Power-Law and Contact Weights
Transmission pattern from a case in Mitte aged 00-04 years
00−04
1.7%
0.6%
1.7%
05−14
1.7%
1.0%
0.6%
9.4%
0.3%
1.7%
0.6%
1.7%0.6%
0.6%
0.4%
1.0%
0.8%
0.8%0.3%
4.0%
0.4%
1.0%0.4%
0.4%
1.4%
4.0%
1.4%
4.0%
1.4%
22.4%
0.7%
4.0%
4.0%1.4%
1.4%
65+
0.5%
0.3%
4.5%
0.3%
0.1%
0.8%
0.8%
0.3%
1.0%
0.4%
5.7%
0.2%
1.0%
50−64
0.8%
15−49
0.3%
Leonhard Held (University of Zurich)
0.5%
0.2%
2.8%
0.2%
0.1%
0.5%
0.5%
0.2%
0.5%0.2%
0.2%
Modelling infectious diseases
26 August 2015
22 / 24
Summary and Outlook
I
Endemic-epidemic time series modelling is implemented in the open
package surveillance
source
Meyer et al. (2014)
I
3-dim. model describes disease spread by borrowing strength from
I
I
different regions and
different age groups
Outlook:
√
Age group-specific overdispersion
→ Increases model fit by ≈ 130 AIC units, model order remains the same
I
Comparison with age group-specific modelling
→ Improved predictive performance?
I
Estimation of parameters within the contact matrix?
Leonhard Held (University of Zurich)
Modelling infectious diseases
26 August 2015
23 / 24
References I
Goeyvaerts, N., Willem, L., Kerckhove, K. V., Vandendijck, Y., Hanquet, G., Beutels, P., and
Hens, N. (2015). Estimating dynamic transmission model parameters for seasonal influenza
by fitting to age and season-specific influenza-like illness incidence. Epidemics, 13:1–9.
Held, L., Höhle, M., and Hofmann, M. (2005). A statistical framework for the analysis of
multivariate infectious disease surveillance counts. Statistical Modelling, 5(3):187–199.
Meyer, S. and Held, L. (2014). Power-law models for infectious disease spread. The Annals of
Applied Statistics, 8(3):1612–1639.
Meyer, S., Held, L., and Höhle, M. (2014). Spatio-temporal analysis of epidemic phenomena
using the R package surveillance. arXiv, 1411.0416.
Mossong, J., Hens, N., Jit, M., Beutels, P., Auranen, K., Mikolajczyk, R., Massari, M., Salmaso,
S., Tomba, G. S., Wallinga, J., Heijne, J., Sadkowska-Todys, M., Rosinska, M., and
Edmunds, W. J. (2008). Social contacts and mixing patterns relevant to the spread of
infectious diseases. PLoS Medicine, 5(3):e74.
Paul, M., Held, L., and Toschke, A. (2008). Multivariate modelling of infectious disease
surveillance data. Statistics in Medicine, 27(29):6250–6267.
Schrödle, B., Held, L., and Rue, H. (2011). Assessing the impact of a movement network on the
spatiotemporal spread of infectious diseases. Biometrics, 68:736–744.
Leonhard Held (University of Zurich)
Modelling infectious diseases
26 August 2015
24 / 24