S TATISTICAL INFERENCE
FOR
VIRAL DISEASES
USING
EPIDEMIOLOGICAL
AND
GENETIC
SUMMARY STATISTICS
Oliver Ratmann (Duke Biology)
Christophe Fraser (DIDE Imperial), Ge Donker (NIVEL), Katia Koelle (Duke Biology)
oliver.ratmann@duke.edu
INFER 03-2011
1 / 19
Epidemiological & evolutionary dynamics of influenza A (H3N2):
interact −→
reflect
80
60
40
20
0
ILI/1e5
100 120 140
overlap −→
1970
1975
1980
1985
1990
1995
2000
time [yrs]
HK68
EN72
VI75
TX77
BK79
SI87
BE89
BE92
WU95
SY97
FU02
CA04
oliver.ratmann@duke.edu
INFER 03-2011
2 / 19
Statistical inference using epidemiological and genetic data
Bayesian inference
• x0 observed incidence time series AND viral phylogeny
• phylodynamic model that defines likelihood f (x0 |θ) implicitly
⇒ Bayes’ posterior density
f (θ|x0 ) = f (x0 |θ)π(θ) / f (x0 )
Approximate Bayesian Computation
circumvent evaluation of f (x0 |θ) in two steps:
• simulate from likelihood, x ∼ f ( · |θ)
• weight simulation under θ by degree ε with which x and x0 match
oliver.ratmann@duke.edu
INFER 03-2011
3 / 19
Statistical inference using epidemiological and genetic data
Bayesian inference
• x0 observed incidence time series AND viral phylogeny
• phylodynamic model that defines likelihood f (x0 |θ) implicitly
⇒ Bayes’ posterior density
f (θ|x0 ) = f (x0 |θ)π(θ) / f (x0 )
Approximate Bayesian Computation
circumvent evaluation of f (x0 |θ) in two steps:
• simulate from likelihood, x ∼ f ( · |θ)
• weight simulation under θ by degree ε with which x and x0 match
oliver.ratmann@duke.edu
INFER 03-2011
3 / 19
• A PPROXIMATE B AYESIAN C OMPUTATION •
I NFLUENZA A (H3N2): SUMMARIES
I NFLUENZA A (H3N2): RESULTS
oliver.ratmann@duke.edu
INFER 03-2011
4 / 19
Approximate Bayesian Computation
• eg S1 : # antigenic clusters
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16
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• set ABC kernel κτ (ε) eg to 1/τ 1 |ε| ≤ τ /2
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14
12
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10
DIAM
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Rejection-sampler
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1
Sample θ ∼ π(θ|M)
2
Simulate x ∼ f (x|θ), compute summaries
S(x) = {S1 (x), . . . , SK (x)}
3
Compute auxiliary errors εk = ρk Sk (x), Sk (x0 )
4
Accept (θ, ε) with prob proportional to
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0.2
0.4
0.6
0.8
1.0
1
approx. posterior
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K
Y
8
10
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κτk (εk )
6
k =1
4
DIAM
12
0.5
16
α
14
1.5
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0
●
0.0
0.2
0.4
0.6
0.8
1.0
θα
oliver.ratmann@duke.edu
INFER 03-2011
5 / 19
Approximate Bayesian Computation
ABC: a particular auxiliary variable Monte Carlo method
• ABC projection ξx0 : x → (ε1 , . . . , εK ),
εk = ρk Sk (x), Sk (x0 )
• for given θ, errors are distributed according to
ξx0 ,θ (E1 × . . . × EK )
Z θ,
M
=
1 x ∈ ξx−1
(E1 × . . . × EK ) f (dx|θ, M)
= f ξx−1
(E
×
.
.
.
×
E
)
1
K
0
0
• augmented sampling density of ABC is
fABC (θ, ε|x0 )
∝
K
Y
κτk (εk ) × ξx0 ,θ (ε1 , . . . , εK ) π(θ)
k =1
ABC kernel × prior predictive error density given θ
• .. augmented likelihood still cannot be computed pointwise for z = (θ, ε)
• .. and interested in auxiliary variable for model criticism (Ratmann PNAS 2009)
oliver.ratmann@duke.edu
INFER 03-2011
6 / 19
Approximate Bayesian Computation
ABC: a particular auxiliary variable Monte Carlo method
• ABC projection ξx0 : x → (ε1 , . . . , εK ),
εk = ρk Sk (x), Sk (x0 )
• for given θ, errors are distributed according to
ξx0 ,θ (E1 × . . . × EK )
Z = f ξx−1
(E
×
.
.
.
×
E
)
θ,
M
=
1 x ∈ ξx−1
(E1 × . . . × EK ) f (dx|θ, M)
1
K
0
0
• augmented sampling density of ABC is
fABC (θ, ε|x0 )
∝
K
Y
κτk (εk ) × ξx0 ,θ (ε1 , . . . , εK ) π(θ)
k =1
ABC kernel × prior predictive error density given θ
• .. augmented likelihood still cannot be computed pointwise for z = (θ, ε)
• .. and interested in auxiliary variable for model criticism (Ratmann PNAS 2009)
oliver.ratmann@duke.edu
INFER 03-2011
6 / 19
Approximate Bayesian Computation
ABC: a particular auxiliary variable Monte Carlo method
• ABC projection ξx0 : x → (ε1 , . . . , εK ),
εk = ρk Sk (x), Sk (x0 )
• for given θ, errors are distributed according to
ξx0 ,θ (E1 × . . . × EK )
Z = f ξx−1
(E
×
.
.
.
×
E
)
θ,
M
=
1 x ∈ ξx−1
(E1 × . . . × EK ) f (dx|θ, M)
1
K
0
0
• augmented sampling density of ABC is
fABC (θ, ε|x0 )
∝
K
Y
κτk (εk ) × ξx0 ,θ (ε1 , . . . , εK ) π(θ)
k =1
ABC kernel × prior predictive error density given θ
• .. augmented likelihood still cannot be computed pointwise for z = (θ, ε)
• .. and interested in auxiliary variable for model criticism (Ratmann PNAS 2009)
oliver.ratmann@duke.edu
INFER 03-2011
6 / 19
Approximate Bayesian Computation
ABC: no need to calculate the augmented likelihood
• if we propose from the intractable component
MCMC-sampler
1
Propose θ0 ∼ q(θ → · ) and propose ε0 ∼ ξx0 ,θ0
2
Accept z 0 = (θ0 , ε0 ) with probability
min
1 , mh(z → z 0 ) ,
mh(z → z 0 ) =
Q
0
π(θ0 ) K
q(θ0 → θ)
k =1 κτk (εk )
×
Q
K
0
q(θ → θ )
π(θ) k =1 κτk (εk )
and otherwise stay at z.
• Because, for q(z → z 0 ) = q(θ → θ0 )ξx0 ,θ0 (ε0 ),
{q(z → z 0 )mh(z → z 0 )} / {q(z 0 → z)mh(z 0 → z)} = fABC (z 0 |x0 )/fABC (z|x0 )
oliver.ratmann@duke.edu
INFER 03-2011
7 / 19
Approximate Bayesian Computation
ABC: no need to calculate the augmented likelihood
• if we propose from the intractable component
MCMC-sampler
1
Propose θ0 ∼ q(θ → · ) and propose ε0 ∼ ξx0 ,θ0
2
Accept z 0 = (θ0 , ε0 ) with probability
min
1 , mh(z → z 0 ) ,
mh(z → z 0 ) =
Q
0
π(θ0 ) K
q(θ0 → θ)
k =1 κτk (εk )
×
Q
K
0
q(θ → θ )
π(θ) k =1 κτk (εk )
and otherwise stay at z.
• Because, for q(z → z 0 ) = q(θ → θ0 )ξx0 ,θ0 (ε0 ),
{q(z → z 0 )mh(z → z 0 )} / {q(z 0 → z)mh(z 0 → z)} = fABC (z 0 |x0 )/fABC (z|x0 )
oliver.ratmann@duke.edu
INFER 03-2011
7 / 19
Approximate Bayesian Computation
ABC: no need to calculate the augmented likelihood
• if we propose from the intractable component
MCMC-sampler
1
Propose θ0 ∼ q(θ → · ) and propose ε0 ∼ ξx0 ,θ0
2
Accept z 0 = (θ0 , ε0 ) with probability
min
1 , mh(z → z 0 ) ,
mh(z → z 0 ) =
Q
0
π(θ0 ) K
q(θ0 → θ)
k =1 κτk (εk )
×
Q
K
0
q(θ → θ )
π(θ) k =1 κτk (εk )
and otherwise stay at z.
• Because, for q(z → z 0 ) = q(θ → θ0 )ξx0 ,θ0 (ε0 ),
{q(z → z 0 )mh(z → z 0 )} / {q(z 0 → z)mh(z 0 → z)} = fABC (z 0 |x0 )/fABC (z|x0 )
oliver.ratmann@duke.edu
INFER 03-2011
7 / 19
A PPROXIMATE B AYESIAN C OMPUTATION
• I NFLUENZA A (H3N2): SUMMARIES •
I NFLUENZA A (H3N2): RESULTS
oliver.ratmann@duke.edu
INFER 03-2011
8 / 19
140
Summaries characterizing seasonal influenza A (H3N2) incidence
• interannual variability
60
80
• periodicity
20
40
• explosiveness
• overall magnitude
0
ILI/1e5
100
120
H3N2
H1N1
B
1970
1975
1980
1985
1990
1995
2000
time [yrs]
oliver.ratmann@duke.edu
INFER 03-2011
9 / 19
140
Summaries characterizing seasonal influenza A (H3N2) incidence
• interannual variability
100
120
H3N2
H1N1
B
• overall magnitude
1980
1985
0
1975
1990
1995
INC / 1e5
ILI/1e5
0
1970
• explosiveness
200 400 600 800
20
40
60
ILI/1e5
80
• periodicity
2000
time [yrs]
1970
1975
1980
1985
1990
1995
2000
cdf
0.04
1975/76
1980/81
1500
differences in annual attack rate (∆y )
1985/86
500
1970/71
INC / 1e5
0.00
200 400 600 800
• interannual variability, eg:
1990
1995
2000
1970
1985/86
1990/91
20
probability
⇒
1995/96
1985
0.00
0.02
0.04
0.08
0.08
0.00
0.5
0.0
acf fod att.r
1981/82
1986/87
1991/92
1996/97
1
2
lags
0.0020
time [yrs]
INFER 03-2011
ility
.6 0.8 1.0
last year
1976/77
fod attack rate
−1.0
0.8
1971/72
0.4
0.05
acf per week
0.00
fod attack rate
oliver.ratmann@duke.edu
0.05
−0.5
0.00
fod attack rate
5
−0.05
15
attack rate/ year
−0.05
0.8
0.06
−0.05
0.05
0.06
20
25
0.04
0
0.08
eek
0.06
cdf
0.04
attack rate/ year
.2 0.4 0.6 0.8 1.0
0.02
2000
attack rate
0.02
10
probability
50
probability
30
10
0
0.00
1995
3
incidence last year
time [yrs]
0.00
1990
time [yrs]
0
1980/81
1980
5
cdf
10
0
1975/76
0.0 0.2 0.4 0.6 0.8 1.0
50
30
probability
0.08
0.04
0.00
1970/71
1975
fod attack rate
1985
25
1980
15
1975
time [yrs]
.6 0.8 1.0
1995/96
10
1970
ILI attack rate
1990/91
time [yrs]
0
0
ILI/1e5
ILI attack rate
0.08
time [yrs]
9 / 19
0.0
ILI att
0.00
Summaries characterizing influenza A (H3N2) antigenic evolution
1970/71
1975/76
1980/81
1985/86
• No
large changes
in annual
attack rate
at transition
yrs
0.00
0.02
0.04
0.06
0.08
15
0
0
5
10
probability
20
• Number of antigenic clusters (Smith et al 2004)
10
probability
30
20
40
25
time [yrs]
−0.0
attack rate/ year
1.5
●
●
●
●
●
●
●
●
●
●
EN72 ●
1971/72
VI75
TX77
1976/77
BK79
SI87
1981/82
1986/87
BE89BE92
WU95
1991/92
SY97
1.0
0.00
●
probability
●
●
●
0.5
●
●
●
0.0
●
−0.10
fd.attr[t−1] − fd.attr[t]
●
●
1996/97
0.0
0.2
0.0
acf fod peaks
−0.4
−0.2
0.0010
0.0000
probability
time [yrs]
−500
0
500
1
0.6 0.8 1.0
INFER 03-2011
df
oliver.ratmann@duke.edu
02
12
bility
0.004
fod peaks
10 / 19
0.0
ILI att
0.00
Summaries characterizing influenza A (H3N2) antigenic evolution
1970/71
1975/76
1980/81
25
20
40
10
15
probability
1970
1975
1980
1985
●
TX77
1.5
●
BK79
0.0
0.2
0.04
attack rate
cdf
−0.2
25
20
15
VI75
0.02
●
SI87
BE89BE92
WU95
SY97
SY97
1996/97
0.5
1.0
1.5
0
0.2
0.0000
0.0
acf fod peaks
0
fod peaks
−0.4
0.004
500
bility
12
1
2
3
4
5
−500
4
1
0
fod peaks
oliver.ratmann@duke.edu
3
500
lags
0.4 0.5
0
fod peaks
2
years
−500
−0.2
0.0010
−500
1
fd attr around antigenic transition seasons
0.2
0.8
acf per week
0.0
0.004
WU95
1991/92
0.0
BE89BE92
time [yrs]
INFER 03-2011
0.000
1986/87
acf fod peaks
SI87
1981/82
−0.2
BK79
−0.4
TX77
1976/77
0.8 1.0
VI75
02
EN72 ●
1971/72
0.6 0.8 1.0
−0.10
●
3
0.0
●
●
●
1.0
●
●
0.5
●
●
●
0.0010
0.00
●
●
probability
probability
●
●
0.0
2
lags
0.0
●
●
1996/97
1
1.5
●
●
1991/92
time [yrs]
fod attack rate
●
●
1986/87
0.05
% incidence last year
0.00
attack rate/ year
●
1981/82
−0.05
probability
1976/77
−0.6
1971/72
0.08
df
0.06
cdf
0.04
0.0 0.2 0.4 0.6 0.8 1.0
0.02
ty
0.00
0.4
0
0
5
EN72 ●
●
●
●
●
−0.10
probability
●
0.00 ●
●
●
10
40
30
20
10
●
●
●
0.0 0.2 0.4 0.6 0.8 1.0
●
time [yrs]
1995/96
●
1.0
●
1990/91
probability
●
0.5
●
1985/86
acf fod att.r
1980/81
0.00
fd.attr[t−1] − fd.attr[t]
0.00
1975/76
−0.0
0.0
cdf
0.04
●
1970/71
0.0 0.2 0.4 0.6 0.8 1.0
0.08
attack rate/ year
time [yrs]
0.0 0.2 0.4 0.6 0.8 1.0
2000
0.002
1995
• No
large changes
in annual
attack rate
at transition
yrs
0.00
0.02
0.04
0.06
0.08
time [yrs]
500
10 / 19
0.8 1.0
1990
5
0 500
1985
0
1980
0
1975
1500
INC / 1e5
2500
30
20
10
probability
200 400 600 800
• Number of antigenic clusters (Smith et al 2004)
0
1970
0.0000
0.004
1985/86
time [yrs]
50
acf f
−0.4
0
−0.8
1
2
3
1970
1975
1980
1985
lags
1990
1995
2000
time [yrs]
cdf
−0.05
0.00
0.0 0.2 0.4 0.6 0.8 1.0
• pw diversity between strains collected in season
0.05
0
200
400
600
●
●
●
BK79
1976/77
SI87
1981/82
BE89
BE92
1986/87
WU95
1991/92
SY97
1996/97
1970
1980
1990
●
1
200
400
600
101980
1990
20
1000
0
200
400
2000
30
2010
50
40
0
200
400
600
800
1000
diff of time of peaks to winter solstice
probability
cdf
2
0.00
1970
800
0.0 0.2 0.4 0.6 0.8 1.0
0.08
8
0.12
10
●
peaks
0.04
6
●●
0.000
0
n most prev clusters
4
●● ●
probability
0.004
● ● ●●
●
0.002
probability
●● ● ●
0.000
% incidence last year
0.0 0.2 0.4 0.6 0.8 1.0
1000
2000
time [yrs]
●
●
●
●●●●
0.004
VI75 TX77
0.002
●
EN72
40
50
●
●
30
●
●
●
●
●
20
●
●
●
branch lengths
●
●
10
0.00
●
●
time [yrs]
probability
1000
0
0.10
●
1971/72
0
800
fod peaks
●
●
−0.10
fd.attr[t−1] − fd.attr[t]
fod attack rate
600
800
0.0 0.1 0.2 0.3 0.4 0.5
cdf
0.0 0.2 0.4 0.6 0.8 1.0
Summaries characterizing the influenza A (H3N2) HA phylogeny
1000
2
4
6
8
10
12
years
5
0.10
0.12
0
0.08
max br lengths
●●
●●●● ●
●●●●●●
●
● ●●
●●
●
●
●
●
peaks
●
10
2
●●●● ●●●●●●● ●●●● ●●●●●●●●●●● ● ● ●●●● ●
●
●
4
●●
6
●
●
15
mean br lengths
8
sd br lengths
20
10
25
12
●
14
2
●
epidemic weeks of peaksize/2
0.4
● ●●
0.3
●
0.2
●
probability
● ● ●●
0.1
●
0.0
●
cdf
0.06
0.00
● ●
0.0 0.2 0.4 0.6 0.8 1.0
● ● ● ●●
0.08
cdf
●
0.02
0.04
● ●
0.0 0.2 0.4 0.6 0.8 1.0
0.10
●
4
6
8
epidemic weeks of peaksize/2
oliver.ratmann@duke.edu
10
12
1
2
3
4
5
6
cluster waiting time
INFER 03-2011
11 / 19
0.0
−0.8
50
0.02
80
100
60
1980
0.08
20
−0.05
1
0
0.00
2
1990
time [yrs]
1995
2000
●
●
●
●
0.08
probability
probability
0e+00
4e−04
8e−04
0
5
10 15 20
25
20
1.5
1
2000
30
2010
50
40
●
●
1515
2020
0
21
4
2
mean
br br
lengths
mean
lengths
600
800
probability
peaks
10
15
20
200
400
600
0
800
●
●
●
●
●
1000
●
●
●
●
●
●
●
VI75 TX77
1971/72
BK79
SI87
1981/82 1
1976/77
1986/87
BE89
BE92
WU95
1991/92
1996/97
3 0
4
5
500
2000
1000
10
20
30
1
40
50
brprev
lengths
nmax
most
clusters
6
38
10 4
12
5 14
0
2
10
4
620
sd br lengths
lags
0
8
30
10
12
40
14
sd
br lengths
max
br lengths
200
400
600
800
1000
diff of time of peaks to winter solstice
2
4
mean br lengths
0
●
●
●
lags
fod
peaks
1000
0.00
5
1996/97
n most prev
time clusters
[yrs]
0.30
400
●
●
EN72
1000
0.05
1000
cdf
1010
200
1990
−5002
●
SY
1991/92 0.05
cdf
probability
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.04
0.08
0.12
●●
probability
0.004
1990
20
−1000
time [yrs]
●● ●
0.002
●
1986/87
0.00
WU95
probability
% incidence last year
0.00
0.04
0.08
0.12
0.0 0.2 0.4 0.6 0.8 1.0
1980
15
1.0
● ● ●●
●
probability
acf fod peaks
0.00 0.10 0.20 0.30
−0.6
−0.2
0.2
cdf
probability
probability
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.10
0.20
cdf
cdf
101980
0.0 0.2 0.4 0.6 0.8 1.0
0.000
55
0
0.0 0.2 0.4 0.6 0.8 1.0
% incidence last year
0.0 0.2 0.4 0.6 0.8 1.0
0.00
2
0.04
6
0.08
8
0.12
10
n most prev clusters
4
10
0.5
mean
br lengths
fd attr around
antigenic
transition seasons
●● ● ●
●
●
●
BE89
BE92
●
●
800
0.0 0.2 0.4 0.6 0.8 1.0
1970
5
0.0
●
●
●
●●●●
500
0.004
SY97
1996/97
0 0.00
0.002
1991/92
−500
−0.05
●
●
SI87
1981/82
peaks rate
fod attack
0
WU95
1
1970
−1000
0.08
probability
1986/87
1000
probability
1.5
10
1981/82
BE89
BE92
time [yrs]
0
0.06
0.20
1976/77
1.0
0.10
1971/72
SI87
1976/77
−0.05
●
●
●
●
●
BK79
timeattack
[yrs] rate
fod
600
0.000
●
●
BK79
1971/72
fod peaks
0.0 0.1 0.2 0.3 0.4 0.5
●
●
●
●
0.04
400
fd attr around antigenic
transition
attack rate/
year seasons
20
●
●
●
VI75 TX77
0.05
0.08
VI75 TX77
fod attack rate
acf fod peaks
probability
−0.6
−0.2
0.2
0e+00
4e−04
8e−04
●
●
0.5
0.02
30
●
●
●
●
EN72
0.0
0.00
40
50
0.0
0
200
probability
probability
branch lengths
0.10
0.20
0.4 0.6 0.8 1.0
●
0.00
0.06
attack rate
0
0.00
0.0 0.2
0.10
0.00
●
●
−0.10
fd.attr[t−1] − fd.attr[t]
●
●
−0.05
0.04
●
EN72
% incidence last year
fd.attr[t−1] − fd.attr[t]
0.0 0.2 0.4 0.6 0.8 1.0
−0.10
0.00
0.10
0.06
0.02
●
●
●
• pw diversity between strains collected in season
0.05
fod attack rate
●
10
0
5
0.0 0.2 0.4 0.6 0.8 1.0
cdf
0.00
0.04
attack rate/ year
probability
probability
0.2 0.4 0.6 0.8 1.0
10
20
30
40
cdf
0.0 0.2 0.4 0.6 0.8 1.0
−0.05
0.02
●
●
●
Summaries characterizing the influenza A (H3N2) HA phylogeny
0.00 0.00
0.05
3
fod attack
lagsrate
1985
fd.attr[t−1] − fd.attr[t]
cdf
−0.10
0.00
0.10
0.0 0.2 0.4 0.6 0.8 1.0
1975
20
1970
probability
lags
0.06
40
25
0
3
0.04
attack rate
probability
cdf
0
10
20
30
40
0.0 0.2 0.4 0.6 0.8 1.0
2
15
0.0
acf f
−0.4
−0.8
0.00
1
6
8
10
12
14
0
2
4
6
sd br lengths
8
10
12
sd br lengths
2
4
6
8
10
12
years
5
0.10
0.12
0
0.08
max br lengths
●●
●●●● ●
●●●●●●
●
● ●●
●●
●
●
●
●
peaks
●
10
2
●●●● ●●●●●●● ●●●● ●●●●●●●●●●● ● ● ●●●● ●
●
●
4
●●
6
●
●
15
mean br lengths
8
sd br lengths
20
10
25
12
●
14
2
●
epidemic weeks of peaksize/2
0.4
● ●●
0.3
●
0.2
●
probability
● ● ●●
0.1
●
0.0
●
cdf
0.06
0.00
● ●
0.0 0.2 0.4 0.6 0.8 1.0
● ● ● ●●
0.08
cdf
●
0.02
0.04
● ●
0.0 0.2 0.4 0.6 0.8 1.0
0.10
●
4
6
8
epidemic weeks of peaksize/2
oliver.ratmann@duke.edu
10
12
1
2
3
4
5
6
cluster waiting time
INFER 03-2011
11 / 19
14
50
0.0
−0.8
50
0.02
80
100
60
1980
0.08
20
−0.05
1
0
0.00
2
1990
time [yrs]
1995
2000
●
●
●
●
0.08
probability
probability
0e+00
4e−04
8e−04
0
5
10 15 20
25
20
1.5
●●
1
2000
30
●
1515
2020
0
21
4
2
mean
br br
lengths
mean
lengths
600
800
probability
peaks
6
20
200
400
600
0
800
●
●
BK79
●
●
●
●
●
●
SI87
1981/82 1
1976/77
1986/87
BE89
BE92
WU95
1991/92
1996/97
4
5
500
2000
1000
10
20
30
1
40
50
brprev
lengths
nmax
most
clusters
38
10 4
12
5 14
0
2
10
4
620
8
30
10
12
40
14
sd
br lengths
max
br lengths
200
400
600
800
1000
diff of time of peaks to winter solstice
2
4
mean br lengths
0
●
●
1000
●
VI75 TX77
probability
% incidence last year
0.00
0.04
0.08
0.12
0.0 0.2 0.4 0.6 0.8 1.0
3 0
0
probability
15
1996/97
●
●
EN72
sd br lengths
lags
1000
0.00
10
●
●
●
lags
fod
peaks
• TMRCA of strains collected in season
5
SY
1991/92 0.05
n most prev
time clusters
[yrs]
0.30
400
●
1971/72
• divergence of serially sampled strains to root
2010
50
40
1010
200
1990
−5002
●
WU95
cdf
probability
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.04
0.08
0.12
0.004
1990
20
−1000
time [yrs]
●● ●
0.002
●
1986/87
0.00
●
1000
0.05
1000
6
8
10
12
cdf
1980
15
1.0
● ● ●●
●
probability
acf fod peaks
0.00 0.10 0.20 0.30
−0.6
−0.2
0.2
cdf
probability
probability
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.10
0.20
cdf
cdf
101980
0.0 0.2 0.4 0.6 0.8 1.0
0.000
55
0
0.0 0.2 0.4 0.6 0.8 1.0
% incidence last year
0.0 0.2 0.4 0.6 0.8 1.0
0.00
2
0.04
6
0.08
8
0.12
10
n most prev clusters
4
10
0.5
mean
br lengths
fd attr around
antigenic
transition seasons
●● ● ●
●
●
●
BE89
BE92
●
●
800
0.0 0.2 0.4 0.6 0.8 1.0
1970
5
0.0
●
●
●
●●●●
500
0.004
SY97
1996/97
0 0.00
0.002
1991/92
−500
−0.05
●
●
SI87
1981/82
peaks rate
fod attack
0
WU95
1
1970
−1000
0.08
probability
1986/87
1000
probability
1.5
10
1981/82
BE89
BE92
time [yrs]
0
0.06
0.20
1976/77
1.0
0.10
1971/72
SI87
1976/77
−0.05
●
●
●
●
●
BK79
timeattack
[yrs] rate
fod
600
0.000
●
●
BK79
1971/72
fod peaks
0.0 0.1 0.2 0.3 0.4 0.5
●
●
●
●
0.04
400
fd attr around antigenic
transition
attack rate/
year seasons
20
●
●
●
VI75 TX77
0.05
0.08
VI75 TX77
fod attack rate
acf fod peaks
probability
−0.6
−0.2
0.2
0e+00
4e−04
8e−04
●
●
0.5
0.02
30
●
●
●
●
EN72
0.0
0.00
40
50
0.0
0
200
probability
probability
branch lengths
0.10
0.20
0.4 0.6 0.8 1.0
●
0.00
0.06
attack rate
0
0.00
0.0 0.2
0.10
0.00
●
●
−0.10
fd.attr[t−1] − fd.attr[t]
●
●
−0.05
0.04
●
EN72
% incidence last year
fd.attr[t−1] − fd.attr[t]
0.0 0.2 0.4 0.6 0.8 1.0
−0.10
0.00
0.10
0.06
0.02
●
●
●
• pw diversity between strains collected in season
0.05
fod attack rate
●
10
0
5
0.0 0.2 0.4 0.6 0.8 1.0
cdf
0.00
0.04
attack rate/ year
probability
probability
0.2 0.4 0.6 0.8 1.0
10
20
30
40
cdf
0.0 0.2 0.4 0.6 0.8 1.0
−0.05
0.02
●
●
●
Summaries characterizing the influenza A (H3N2) HA phylogeny
0.00 0.00
0.05
3
fod attack
lagsrate
1985
fd.attr[t−1] − fd.attr[t]
cdf
−0.10
0.00
0.10
0.0 0.2 0.4 0.6 0.8 1.0
1975
20
1970
probability
lags
0.06
40
25
0
3
0.04
attack rate
probability
cdf
0
10
20
30
40
0.0 0.2 0.4 0.6 0.8 1.0
2
15
0.0
acf f
−0.4
−0.8
0.00
1
14
0
2
4
6
sd br lengths
8
10
12
sd br lengths
2
4
6
8
10
12
years
5
0.10
0.12
0
0.08
max br lengths
●●
●●●● ●
●●●●●●
●
● ●●
●●
●
●
●
●
peaks
●
10
2
●●●● ●●●●●●● ●●●● ●●●●●●●●●●● ● ● ●●●● ●
●
●
4
●●
6
●
●
15
mean br lengths
8
sd br lengths
20
10
25
12
●
14
2
●
epidemic weeks of peaksize/2
0.4
● ●●
0.3
●
0.2
●
probability
● ● ●●
0.1
●
0.0
●
cdf
0.06
0.00
● ●
0.0 0.2 0.4 0.6 0.8 1.0
● ● ● ●●
0.08
cdf
●
0.02
0.04
● ●
0.0 0.2 0.4 0.6 0.8 1.0
0.10
●
4
6
8
epidemic weeks of peaksize/2
oliver.ratmann@duke.edu
10
12
1
2
3
4
5
6
cluster waiting time
INFER 03-2011
11 / 19
14
50
0.0
−0.8
50
0.02
80
100
60
1980
0.08
20
−0.05
1
0
0.00
2
1990
time [yrs]
1995
2000
●
●
●
●
0.08
probability
probability
0e+00
4e−04
8e−04
0
5
10 15 20
25
20
1.5
●●
1
2000
30
●
1515
2020
0
21
4
2
mean
br br
lengths
mean
lengths
600
800
probability
peaks
6
20
200
400
600
0
800
●
●
BK79
●
●
●
●
●
●
SI87
1981/82 1
1976/77
1986/87
BE89
BE92
WU95
1991/92
1996/97
4
5
500
2000
1000
10
20
30
1
40
50
brprev
lengths
nmax
most
clusters
38
10 4
12
5 14
0
2
10
4
620
8
30
10
12
40
14
sd
br lengths
max
br lengths
200
400
600
800
1000
diff of time of peaks to winter solstice
2
4
mean br lengths
0
●
●
1000
●
VI75 TX77
probability
% incidence last year
0.00
0.04
0.08
0.12
0.0 0.2 0.4 0.6 0.8 1.0
3 0
0
probability
15
1996/97
●
●
EN72
sd br lengths
lags
1000
0.00
10
●
●
●
lags
fod
peaks
• TMRCA of strains collected in season
5
SY
1991/92 0.05
n most prev
time clusters
[yrs]
0.30
400
●
1971/72
• divergence of serially sampled strains to root
2010
50
40
1010
200
1990
−5002
●
WU95
cdf
probability
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.04
0.08
0.12
0.004
1990
20
−1000
time [yrs]
●● ●
0.002
●
1986/87
0.00
●
1000
0.05
1000
6
8
10
12
cdf
1980
15
1.0
● ● ●●
●
probability
acf fod peaks
0.00 0.10 0.20 0.30
−0.6
−0.2
0.2
cdf
probability
probability
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.10
0.20
cdf
cdf
101980
0.0 0.2 0.4 0.6 0.8 1.0
0.000
55
0
0.0 0.2 0.4 0.6 0.8 1.0
% incidence last year
0.0 0.2 0.4 0.6 0.8 1.0
0.00
2
0.04
6
0.08
8
0.12
10
n most prev clusters
4
10
0.5
mean
br lengths
fd attr around
antigenic
transition seasons
●● ● ●
●
●
●
BE89
BE92
●
●
800
0.0 0.2 0.4 0.6 0.8 1.0
1970
5
0.0
●
●
●
●●●●
500
0.004
SY97
1996/97
0 0.00
0.002
1991/92
−500
−0.05
●
●
SI87
1981/82
peaks rate
fod attack
0
WU95
1
1970
−1000
0.08
probability
1986/87
1000
probability
1.5
10
1981/82
BE89
BE92
time [yrs]
0
0.06
0.20
1976/77
1.0
0.10
1971/72
SI87
1976/77
−0.05
●
●
●
●
●
BK79
timeattack
[yrs] rate
fod
600
0.000
●
●
BK79
1971/72
fod peaks
0.0 0.1 0.2 0.3 0.4 0.5
●
●
●
●
0.04
400
fd attr around antigenic
transition
attack rate/
year seasons
20
●
●
●
VI75 TX77
0.05
0.08
VI75 TX77
fod attack rate
acf fod peaks
probability
−0.6
−0.2
0.2
0e+00
4e−04
8e−04
●
●
0.5
0.02
30
●
●
●
●
EN72
0.0
0.00
40
50
0.0
0
200
probability
probability
branch lengths
0.10
0.20
0.4 0.6 0.8 1.0
●
0.00
0.06
attack rate
0
0.00
0.0 0.2
0.10
0.00
●
●
−0.10
fd.attr[t−1] − fd.attr[t]
●
●
−0.05
0.04
●
EN72
% incidence last year
fd.attr[t−1] − fd.attr[t]
0.0 0.2 0.4 0.6 0.8 1.0
−0.10
0.00
0.10
0.06
0.02
●
●
●
• pw diversity between strains collected in season
0.05
fod attack rate
●
10
0
5
0.0 0.2 0.4 0.6 0.8 1.0
cdf
0.00
0.04
attack rate/ year
probability
probability
0.2 0.4 0.6 0.8 1.0
10
20
30
40
cdf
0.0 0.2 0.4 0.6 0.8 1.0
−0.05
0.02
●
●
●
Summaries characterizing the influenza A (H3N2) HA phylogeny
0.00 0.00
0.05
3
fod attack
lagsrate
1985
fd.attr[t−1] − fd.attr[t]
cdf
−0.10
0.00
0.10
0.0 0.2 0.4 0.6 0.8 1.0
1975
20
1970
probability
lags
0.06
40
25
0
3
0.04
attack rate
probability
cdf
0
10
20
30
40
0.0 0.2 0.4 0.6 0.8 1.0
2
15
0.0
acf f
−0.4
−0.8
0.00
1
14
0
2
4
6
sd br lengths
8
10
12
sd br lengths
2
4
6
8
10
12
years
5
0.12
0.10
●
●
● ●●
max br lengths
●●
●●●● ●
●●●●●●
●
● ●●
●●
●
●
●
●
peaks
●
epidemic weeks of peaksize/2
0.4
● ● ●●
0.3
●
10
2
●●●● ●●●●●●● ●●●● ●●●●●●●●●●● ● ● ●●●● ●
●
●
4
●●
6
●
●
15
mean br lengths
8
sd br lengths
20
10
25
12
●
14
2
●
0.2
0.0
0.1
probability
• substantial # pilot runs to determine which summary to include
based on ability to further constrain posterior Θ (Nunes Balding 2010)
0
0.08
●
cdf
0.06
0.00
● ●
0.0 0.2 0.4 0.6 0.8 1.0
● ● ● ●●
0.08
cdf
●
0.02
0.04
● ●
0.0 0.2 0.4 0.6 0.8 1.0
0.10
●
4
6
8
epidemic weeks of peaksize/2
oliver.ratmann@duke.edu
10
12
1
2
3
4
5
6
cluster waiting time
INFER 03-2011
11 / 19
14
50
A PPROXIMATE B AYESIAN C OMPUTATION
I NFLUENZA A (H3N2): SUMMARIES
• I NFLUENZA A (H3N2): RESULTS •
oliver.ratmann@duke.edu
INFER 03-2011
12 / 19
SIRS with sinusoidal seasonal forcing
• MCMC using epidemiological summaries
fix: demography, birth/death rate, low migration
oliver.ratmann@duke.edu
INFER 03-2011
13 / 19
3.5
R0
1.5
1.0e+07
fix: demography, birth/death rate, low migration
0
5000
migrPPYr
durImm
5
0.25
10000
iterations
2.5
1.5
2.0
wSh
0.15
0
5000
1.0
15000
10000
15000
5000
0
10000
iterations
0
1
chain 1
chain 2
chain 3
chain 4
5000
10000
15000
iterations
−2
−1
15000
iterations
MED.ANN.ATT.R
0.4
10000
0
chain 1
chain 2
chain 3
chain 4
0.8
CFD.PKS
iterations
5000
5000 15000
iterations
chain 1
chain 2
chain 3
chain 4
0
10000
chain 1
chain 2
chain 3
chain 4
0.6
ACF.FD.PKS
−0.2 0.0
INFER 03-2011
−0.6
0.2
chain 1
chain 2
chain 3
0
chain
4
5000
10000
iterations
chain 1
chain 2
chain 1
chain 2
chain 3
15000chain 4
0
5000
10000
iterations
1.4
0.5
−0.5
PKS
15000
10000
3
−1.5
repProb
10000
iterations
5000
0
chain 1
chain 2
chain 3
chain 4
0.05
5000
10000
1.0e+07
0.6
10 15 20 25 30
1.0
0.9
0.20
0.8
seasonality
xImm
0.7
1.0
0.04
0.5
0.02
0.0
selA
1
2
chain 3
chain 4
5
oliver.ratmann@duke.edu
15000
15000
CPKS
5000
10000
5000
10000
iterations
iterations
chain 1
chain21
chain
chain
chain32
chain
chain43
chain 4
5000
iterations
chain 1
chain chain
2
chain chain
3
chain 4
0
5000
chain 1
chain 2
chain 3
chain 4
chain 1
chain 2
chain 3
chain 4
15000
0
2
0
0
10000
15000
1
15000
15000
0.8
10000
10000
iterations
iterations
5000 10000
5000
chain 1
chain 2
chain 3
chain 4
0
iPer
CFD.PKS
5000
5000
chain 1
chain
chain 12
chain 23
chain
chain 34
chain
chain 4
15000
15000
0.6
0
0
0
0
chain 1
chain21
chain
chain32
chain
chain43
chain
chain 4
0.4
0.5 30
25
PKS
extWrld.N
15000
15000
10000
10000
0.00
repODisp
5000
5000
iterations
iterations
−1.5
5 10
10000
10000
iterations
iterations
ACF.FD.ATT.R
SD.FD.ATT.R
5000
5000
chain 1
chain
chain1 2
chain2 3
chain
chain3 4
chain
chain 4
0
0
0
iterationsiterations
4
15000
15000
iterations
iterations
0.0−2 0.2 00.41 0.6
2 3 0.84
ANN.ATT.R
MED.ANN.ATT.R
1.0
2.0
−1
0
1
−2
10000
10000
0
−1.0 −0.5
7 0.25
8
mutR
repProb
5000
5000
chain 1
chain
chain 12
chain
chain 23
chain
chain 34
chain 4
−0.5
15 20
iPer
wSh
1.0 1.5 2.0 2.5 3.0 3.5
1.5
2.0
2.5
0
0
15000
15000
0.05 4
0
0
chain 1
chain 2
3
4
0.2
15000
15000
10000
10000
ACF.FD.ATT.R
10000
10000
5 0.156
0.04 550
450
selA
rScPhRob
5000
5000
chain 1
chain
chain1 2
chain
chain2 3
chain
chain3 4
chain 4
5000
5000
chain
chain11
chain
chain22
chain
chain33
chain
chain44
1.0
2.0 2.5
0.4 1.5 0.6
0.8 3.01.03.5
0
15000
iterations
chain chain
1
chain chain
2
chain 3
chain 4
iterations
iterations
chain 11
chain
chain 22
chain
chain 33
chain
chain 44
chain
iterations
iterations
0.0
00
0.00
250
0
15000
15000
15000
15000
ANN.ATT.R
ACF.FD.PKS
15000
15000
10000
10000
iterations
iterations
0.0
5000
5000
10000
iterations
0.10
lifespan
durImm
505
00
0.02
350
8
mutR
migrPPYr
4
5
6
7
1.0e+07
1.6e+07
chain1 1
chain
chain2 2
chain
chain3 3
chain
chain4 4
chain
chain11
chain
chain
chain22
chain
chain33
chain
chain44
.2
15000
15000
2.0
0.2
10000
10000
iterations
iterations
0.0 −0.21.00.0
5000
5000
0.6
00
90 25 110
107015 20
30
1.0
7
xImm
D
0.6 2 0.7
3
50
1.5
15000
15000
chain
chain 11
chain 22
chain
chain 33
chain
chain 44
chain
40.8 5 0.96
lifespan
R0
70
90
110
2.5
3.5
0.30
⇒ strong seasonal forcing to explain interannual seasonal variation
chain
chain1 1
chain2 2
chain
chain3 3
chain
chain4 4
chain
chain 1
chain 2
chain 3
chain 4
1.6e+07
N
1.6e+07
• MCMC using epidemiological summaries
chain 1
chain 2
chain 3
chain 4
2.5
2.2e+07
SIRS with sinusoidal seasonal forcing
chain 1
chain 2
13 / 19
●
3.5
R0
repProb
0.15
migrPPYr
durImm
iterations
15000
5
1.0e+07
durImm
5
0.25
0.7
0.10
0.6
10 15 20 25 30
1.0
0.9
0.20
0.8
10 15 20 25 30
1.5
1.0e+07
0.30
●●● ● ●
15000
0
5000
0.25
2.5
0.15
0
10000
15000
5000 15000
10000
iterations
chain 1
chain 2
chain 3
chain 4
2.0
iterations
wSh
repProb
1.0
chain 1
chain 2
chain 3
chain 4
15000
1.5
0.05
0.8
0.6
10000
0.8
0.6
0
1
0
5000
10000
0.04
chain 1
chain 2
lags
INFER 03-2011
15000
0
3
0.06
5000
10000
iterations
−0.05
1.4
2
−0.6
1
iterations
cdf
0.2
10000 5000
iterations
−0.2 0.0
0
ACF.FD.PKS
−0.5
15000
5000
iterations
chain 1
chain 2
chain 3
10000
15000
chain 4
15000
attack rate
iterations
−2
0.02
chain 1
chain 2
chain 3
chain 4
0.0 0.2 0.4 0.6 0.8 1.0
1.0
chain 1
chain 2
chain 3
chain 4
15000
10000
1990
iterations
time [yrs]
15000
iterations
0.4
CFD.PKS
5000
−0.6
chain 1
chain 2
chain 3
10000
0
chain
4
15000
5000
1985
−1
0.4
0
0.2
−0.5
5
10000
0
chain 1
chain 2
chain 3
10000
chain 4
MED.ANN.ATT.R
5000
0.00
15000
1980
iterations
1
0.8
0
0.6
0.0
5000
iterations
15000
chain 1
chain 2
chain 3
chain 4
0
0.4
0
10000
1975
−0.2 0.0
0.5
10000
5000
0.2
ACF.FD.ATT.R
acf fod att.r
chain 71
chain 2
chain 3
● ●chain
●● ● 4
chain 1
chain 2
chain 3
10000
chain 4
5000
iterations
0
chain 1
chain 2
chain 3
10000
chain 4
4
−1.5
CFD.PKS
5000
3
cdf
0.0 0.2 0.4 0.6 0.8 1.0
CPKS
0.05
1.0
0.04
0.02
selA
0.5
0
iterations
0.5
60.05
5000
10000
chain 1
chain 2
chain 3
chain 4
iterations
0
1970
2
−1.5
PKS
●
chain 1
chain 2
chain 3
chain 1
chain 2
chain 3
10000
chain 4
chain 1
chain 2
1.0
●
5000
chain 3
chain 4
0.0
repODisp
INC / 1e5
500
−0.5
chain 1
chain chain
2
1
chain chain
3
10000
15000
2
chain 4
0
●
0.6
0
5000
iterations
chain 4
10000
15000
5000 10000
5000
0.2
15000
attack
epidemicfod
weeks
ofrate
peaksize/2
● ●● ●
0
iterations
●●●
● ●●● ●
●
●●
●
5
0
chain 4
simu_N_16600000_R0_2,1597_D_1,8824_lif_80_xIm_0,8397_dur_14,3518_sea_0,2619_mig_1,5e+07_rSc_400_mut_5,7_sel_0,02_rep_0,0887_rep_0_wSh_2_ext_10_eco_15_eco_42_dt_0,5_log_1_nLo_1_evo_1_max_20
0.0
0.5
●●
0.4
ACF.FD.ATT.R
●
● ●
oliver.ratmann@duke.edu
●
5000
chain 1
chain 2
chain 3
chain 4
15000
800
15000
5000
10000
600
5000 1995/96
10000
iterations
iterations
40.00
3
2000
0
0.8
2.0
0.2
ANN.ATT.R
ACF.FD.PKS
15000
● ●
chain 1
chain21
chain
chain
chain32
5000
10000
chain
chain43
iterations chain 4
−0.05
2
chain
1
chain 2
chain 3
4
●● chain●●
0.05
0.0
400
0.0 −0.21.00.0
1
1995
1.0
1.0
0
0
●
0.6
cdf
iPer
CFD.PKS
2.0
ANN.ATT.R
25
150.420
probability
5 0.2
10
0.0
0
● ● ●●●●●
0
1.0
800 0.08
1990/91
fod peaks
●●
●●●
15000
012
6000.06
chain 1
chain 2
chain 3
chain 4
15000
200
15000
10000
1985/86
0
10000
time [yrs]
iterations
iterations
L
chain 1
chain 2
attackpeaks
rate/ year
chain 3
chain 4
chain 1
chain
chain 12
chain 23
chain
5000
10000
chain 34
chain
iterations chain 4
ACF.FD.ATT.R
SD.FD.ATT.R
0
0.04
400
5000
5000
● ●●
● ●
chain 1
chain 2
chain 3
chain 4
15000
15000
10000
10000
iterations
iterations
0.0
−1.5
5 10
5
0.0 0.2 0.4 0.6 0.8 1.0
−0.5
15 20
PKS
extWrld.N
5
4
3
2
15000
0.02
200
0
0
1980/81
4
0
●● ●● ● ●
−2
● ●●
1
10000
1975/76 3
2
10000
lags
iterations
iterations
●
0.0−2 0.2 00.41 0.6
2 3 0.84
0.0
−2
0.00
0
SD.FD.ATT.R
5000
5000
chain 1
chain
chain1 2
chain2 3
chain
5000
10000
chain3 4
chain
iterations chain 4
3.0
ANN.ATT.R
probability
cdf
MED.ANN.ATT.R
0 0.2
10 20
0.0
0.4
0.6 40
0.8 50
1.0
1.0
2.030
−1
0
1
1970/71
1
5
wSh
2.0
1.5
iPer
ILI attack
rate
acf
fod peaks
0.001.0
0.04
0.12
−0.5
1.50.00.08
2.0 0.5
2.5
0
●
●
●
●●● ●●●
chain 1
chain 2
chain 3
chain 4
15000
15000
PKS
5000
5000
0
iterations
0.00
0.5
0.05 4
0
0
ME
iterations
iterations
5000
1500
5 0.156
mutR
repProb
chain 1
chain 2
chain 3
chain 4
15000
15000
10000
10000
1.0
2.0 2.5
0.4 1.5 0.6
0.8 3.01.03.5
cdf
5000
5000
0.5 30
25
iterations
iterations
0
0
iPer
10000
10000
0.00
250
25
20
5000
5000
15
extWrld.N
0
1970
0
1.0
0
0.08
1975
1980
1985
1990
2
3 chain 1
−0.05
0.00 1
chain 11
chain
chain
chain
chain21
chain1 2
chain 12
time [yrs]
lags
fod chain
attack rate
chain
chain
chain32
chain2 3
chain 23
chain
5000
10000
15000
0
5000
10000
15000
0
5000
10000
chain
chain
chain43
chain3 4
chain 34
chain
simu_N_16600000_R0_2,1597_D_1,8824_lif_80_xIm_0,8397_dur_14,3518_sea_0,2619_mig_1,5e+07_rSc_400_mut_5,7_sel_0,02_rep_0,0887_rep_0_wSh_2_ext_10_eco_15_eco_42_dt_0,5_log_1_nLo_1_evo_1_max_200
chain 4
chain 4
chain 4
iterations
iterations
iterations
chain 1
chain 2
chain 3
chain 4
15000
15000
15000
0.06
iterations
−1.0 −0.5
7 0.25
8
chain
chain11
chain
chain22
chain
10000
chain33
chain
chain44
5000
15000
chain 2
iterationsiterations
0.5
0.0 0.2 0.4 0.6 0.8 1.0
attack rate
1.0 1.5 2.0 2.5 3.0 3.5
30
chain 1
chain 2
chain 3
chain 4
15000
15000
0
simu_N_16600000_R0_2,1597_D_1,8824_lif_80_xIm_0,8397_dur_14,3518_sea_0,2619_mig_1,5e+07_rSc_400_mut_5,7_sel_0,02_rep_0,0887_rep_0_wSh_2_ext_10_eco_15_eco_42_dt_0,5_log_1_nLo_1_evo_1_max_200
0
3.0 3.5
2.5
15000
0.04
iterations
10
1.0e+07
0.02
0
acfILI/1e5
fod att.r
200 400
−0.5
0.0 600 0.5
4800
chain 1
chain 2
chain 3
chain 4
15000
15000
5000
selA
rScPhRob
0
0.02
350
15000
0.00
iterations
chain 1
chain 2
0
0.00
iterations
iterations
10000
chain 1
iterations
.2
lifespan
durImm
xImm
10000
10000
selA
5000
5000
⇒ too regular and too strong sustained oscillations
5000
5000
chain 1
chain2000
2
chain 3
chain 4
15000
15000
0.04
1995
00
4
chain 11
chain
chain 22
chain
chain 33
10000
chain
chain 44
chain
0
chain 1
chain 2
chain 3
chain 4
iterations chain 3
3
chain chain
1
4
chain chain
2
chain 3
10000
15000
chain 4
0.02
6
mutR
iterations
iterations
505
chain 1
chain 2 1990
chain 3
chain 4
15000
15000
10000
10000
15000
iterations
5
5000
5000
seasonality
0.7
chain11
chain
chain
chain22
chain
chain33
10000
chain
chain44
0.6
90 25 110
107015 20
30
5000
8
1985
time [yrs]
00
0.04 550
450
chain1 1
chain
chain2 2
chain
chain3 3
10000
chain
chain4 4
chain
0
7
1980
250
iterations
iterations
0.6 2 0.7
3
450
rScPhRob
10000
10000
350
5000
5000
15000
iterations
chain 1
chain 2
chain 3
chain 4
15000
15000
1975
00
0.4
8 0.6 0.8 1.0
50
1.0
7
5000
550
0
1970
mutR
migrPPYr
5
6 0.0 70.2
1.6e+07
cdf
50
1.5
chain 1
20002
chain
chain 3
chain 4
15000
15000
chain
chain 11
chain 22
chain
chain 33
chain
10000
chain 44
chain
40.8 5 0.96
0
xImm
D
15000
iterations
chain 1
chain 2
chain 3
chain 4
5000
0.8
xImm
90
lifespan
70
5
D
4
3
2
lifespan
INC / 1e5
R0
500
1500
70
90
110
2.5
3.5
chain
chain1 1
chain2 2
chain
chain3 3
chain
10000
chain4 4
chain
simu_N_16600000_R0_2,1597_D_1,8824_lif_80_xIm_0,8397_dur_14,3518_sea_0,2619_mig_1,5e+07_rSc_400_mut_5,7_sel_0,02_rep_0,0887_rep_0_wSh_2_ext_10_eco_15_eco_42_dt_0,5_log_1_nLo_1_evo_1_max_200
5000
●● ●
0
chain 1
chain 2
chain 3
chain 4
⇒ strong seasonal forcing to explain interannual seasonal variation
ACF.FD.PKS
chain 1
chain 2
chain 3
chain 4
0.9
6
chain 1
chain 2
chain 3
chain 4
1.0
110
7
fix: demography, birth/death rate, low migration
chain 1
chain 2
chain 3
chain 4
1.6e+07
N
1.6e+07
• MCMC using epidemiological summaries
chain 1
chain 2
chain 3
chain 4
2.5
2.2e+07
SIRS with sinusoidal seasonal forcing
13 / 19
Antigenic tempo model (Koelle et al JRoySoc 2010)
• track status of infection with multiple phenot distinct variants
i = 1, . . . , n :
n
dSi
Si X
= µ(N − Si ) − βt
σij Ij + γ(N − Si − Ii )
dt
N
j=1
Si
dIi
= βt Ii − (µ + ν)Ii
dt
N
• specify only tempo with which variants emerge
HK68
EN72
VI75
TX77
BK79
SI87
BE89
BE92
WU95
SY97
FU02
CA04
dIi
Si
= βt Ii − (µ + ν)Ii + h(agei )Ii
dt
N
h(a) = κ/λ (a/λ)κ−1
• simulate strains of each variant
oliver.ratmann@duke.edu
INFER 03-2011
14 / 19
Antigenic tempo model
• MCMC using epidemiological and immunological summaries
fix: demography, birth/death rate, linear aging, low migration; not shown: λ, report prob
oliver.ratmann@duke.edu
INFER 03-2011
15 / 19
2.2e+07
5
4
R0
3
2
N
1.6e+07
1.0e+07
2000
4000
6000
8000
10000
8000
10000
8000
10000
8000
10000
10000
10000
0
0
2000
2000
iterations
iterations
ACF.FD.ATT.R
ACF.FD.PKS
4000
4000
6000
6000
iterations
iterations
8000
8000
10000
10000
0
0
2000
2000
4000
4000
6000
6000
8000 10000
8000 10000
migrPPYr
1.0e+07
20
0.25
2.5
1.5
2.0
wSh
1.0
0.20
0.15
iterations
chain 1
chain 2
chain 3
0
2000chain
40004
1.5
repProb
0.5
0.0
5
4
3
CPKS
4000
6000
8000 10000
4000
6000
8000 10000
chain 1
chain 1
chain 2
chain 2
chain 3
chain 3
chain 4
chain 4
6000
8000
10000
iterations
chain 1
chain 2
chain 3
chain 4
0
2000
4000
6000
iterations
0
0
2000
2000
iterations
iterations
oliver.ratmann@duke.edu
chain 1
15
durImm
10
5
2000
2000
0
1
0
iterations
iterations
chain 1
chain 1
chain 2
chain 2
chain 3
chain 3
chain 4
chain 4
3
chain 1
0
iterations
iterations
chain 1
chain 1
chain 2
chain 2
chain 3
chain 3
chain 4
chain 4
0.20
seasonality
0.30
8000 10000
8000 10000
10000
6000
0.5
6000
6000
8000
4000
2
2
1
4 5
3
0
2
−1
1
AMED.FD.PKS
CPKS
4000
4000
6000
iterations
SD.FD.ATT.R
2000
2000
4000
2000
−0.5
0
0
2000
chain
0 1
chain 2
chain 3
chain 4
−1.5
10000
10000
1
2
chain 3
chain 4
−1.0
8000
8000
MED.ANN.ATT.R
6000
6000
0
iterations
iterations
0.0
4000
4000
0.10
4000
4000 6000
6000 8000
8000 10000
10000
AMS.SFD.ATT.R
2000
2000
0.05
repODisp
2000
2000
chain
chain1 1
chain
chain2 2
chain
chain3 3
chain
chain4 4
−1.0 −0.5
00
6000
10000
iterations
chain 1
chain 2
chain 3 chain
chain 4 chain
0.0
8000
8000 10000
10000
−2.0
6000
6000
iterations
iterations
−0.5
4000
4000
chain
chain11
chain
chain22
chain
chain33
chain
chain44
−1.0
2000
2000
0.15
xImm
durImm
5
selA
repProb
0.00
0.05
00
40008000
6000
4000
6000
8000 10000
4000
6000
8000 10000
iterations
iterations
chain 1
0
2000
4000
6000
chainiterations
1
chain 2
INFER 03-2011 chain 3
chain 1
8000
10000
1.0
8000
8000
chain
chain1 1
chain
chain2 2
chain
chain3 3
chain
chain4 4
2000
4000
15 / 19
S
6000
6000
10000
10000
−0.2
−0.4
ANN.ATT.R
ACF.FD.ATT.R
4000
4000
0.70
10
8000
8000
02000
0.0
2000
2000
chain 1
0.80
0.90
15
20
110
7
0.04
6000
6000
iterations
iterations
0.5 0.4
1.0 0.6
1.5 0.8
2.0
−0.2 0.0 0.2
1.5
2.0
0
0
5
6
0.02
mutR
selA
4000
4000
0
iterations
0
0
0
iterations
iterations
chain 1
chain 1
chain 2
chain 2
chain 3
chain 3
chain 4
chain 4
4000
4000 6000
6000 8000
8000 10000
10000
1.0
10000
10000
2000
2000
HS
8000
8000
00
4
2000
2000
chain
chain11
chain
chain22
chain
chain33
chain
chain44
chain 1
chain 2
chain 3
4
chain 1 chain
chain 2
chain 3
chain 4
iterations
iterations
0.00
00
30
6000
6000
8000
8000 10000
10000
chain
chain11
chain
chain22
chain
chain33
chain
chain44
iterations
iterations
extWrld.N
MAX.PKS
4000
4000
6000
6000
0.0 0.2 0.4
−0.5
0.0
10000
10000
4000
4000
iterations
iterations
5−1.0
10
2000
2000
2000
2000
ACF.FD.PKS
AMS.SFD.ATT.R
8000
8000
15 0.0
20 0.5
25
2.5 30
25
1.5 15 2.0
10
20
0
0
00
0.0 0.5
1
2
6000
6000
90
0.90
10000
10000
8
1.4
8
rScPhRob
mutR
4000
4000
chain
chain 11
chain
chain 22
chain
chain 33
chain
chain 44
5
wSh
extWrld.N
8000
8000
MAX.PKS
AMED.FD.PKS
2000
2000
iterations
iterations
10000
10000
6000
6000
0.2
4
00
10000
10000
4000
4000
0.02
0.04
0.10
0.15
0.20
2000
2000
chain
chain11
chain
chain22
chain
chain33
chain
chain44
50.6 6 1.07
1.4
migrPPYr
rScPhRob
1.0e+07
0.2
0.6 1.6e+07
1.0
10000
10000
lifespan
xImm
50
0.70
00
iterations
iterations
chain
chain 11
chain
chain 22
chain
chain 33
chain
chain 44
chain
chain1 1
chain
chain2 2
chain
chain3 3
chain
chain4 4
−0.4
−1.0
10000
10000
0.6
1.0
8000
8000
0
6000
6000
iterations
iterations
−1.0
−1
4000
4000
0.2 0.4 0.6 0.8
0.0 0.2 0.4
2000
2000
5
.6
00
chain
chain11
chain
chain22
chain
chain33
chain
chain44
70
0.80
6
110
4 90 5
D
lifespan
50 2
22
10000
10000
chain
chain11
chain
chain22
chain
chain33
chain
chain44
3
70
4 4 5 56
chain
chain 11
chain
chain 22
chain
chain 33
chain
chain 44
33
R0
D
⇒ intrinsic dynamics contribute to overall seasonality
1.6e+07
iterations
0.6
0
00
0
SD.FD.ATT.R
ANN.ATT.R
0
00
chain 1
chain 2
chain 3
chain 4
fix: demography, birth/death rate, linear aging, low migration; not shown: λ, report prob
−1.5
0.0 −0.5
0.5 1.0 0.51.5
0
00
• MCMC using epidemiological and immunological summaries
0.6
0
00
Antigenic tempo model
2.2e+07
5
4
R0
3
0.12
ILI attack rate
simu_N_16600000_R0_2,8433_D_2,802_lif_80_xIm_
2
1.0e+07
2000
4000
chain 1
2
10000
10000
4
lags
6
7
8000 10000
8000 10000
iterations
iterations
oliver.ratmann@duke.edu
chain 1
chain 1
1.0
migrPPYr
30
probability
20
1.0e+07
2.5
0.15
repProb
10
5
0
0.25
20
15
0.20
durImm
1
2000
2000
0
−0.05
2
cdf
attack rate/ year
0.00
4
5
fod attack rate
2000
6
4000
0.05
7
6000
iterations
chain 2
chain 1
0.10
8
epidemic weeks of0peaksize/2
4000
6000
8000 10000
2000
4000
6000
8000
4000● ● 6000
8000
●● ● ●10000
●●
●
● ● ●● ●
●● ● ●
● ●●
●
●
chainiterations
1
iterations
iterations
INFER 03-2011 chain 3
0.00 0.8
0.6
fd.attr[t−1]
− fd.attr[t]
probability
0
2.0
wSh
1.5
10000
0.04
0
●
●
●
0.06
200
peaks
chain 1
chain 2
chain 3
chain 4
0.5
8000
800
SD.FD.ATT.R
probability
6000
iterations
● ●
●●
●● ●●
●
●●
●●
−0.5
600
● ● ●●● ●
●●
●●●●
diff o
−1.5
●
●
0.000
0
chain 1
chain 2
chain 3
2000chain
40004
1990/91
10000−500
6000
time [yrs] 8000
●●●● ●
iterations
0.004
●
●
● ●●
1.5
0.0000
1985/86
4000
●
0.0010
probability
0.0
1980/81
●
●●
10000
iterations
2
2000
0
0.02
400
3
8000
0.0 −0.10
0.2 0.4
200
6000
0.002
0.05
5
4
6000
8000 10000
6000
8000 10000
AMS.SFD.ATT.R
cdf cdf
8
0
0.5
0.20
0.0 0.00.2 0.20.4 0.40.6 0.60.8 0.81.0 1.0
3
5
epidemic weeks
2000
4000of peaksize/2
6000
2000
4000
6000
0.0 0.2 0.4
−0.5
0.0
ACF.FD.PKS
AMS.SFD.ATT.R
2
3
0
0
−0.05
●
●
●●
●
●
●
●
●
●
●
●
●
●
8000
●
10000
EN72 ●
1971/72
1
VI75
TX77
BK79
1976/77
2
1
10000
1.0
8000
8000
0.10
repODisp
4000
4000
0.00
0
4000
chain
0 1
chain 2
3
4
3
CPKS
● ●
●●
2000
n most prev clusters
chain
●
● ●
chain
●● ●● ●
●●
● ●
●●
●●
chain 1
chain 1
chain 2
chain 2
chain 3
chain 3
chain 4
chain 4
●
3
2
● ●● ● ●
●●
●●●● ●
●●
−1.0 −0.5
1975/76
1
iterations
iterations
●
0.0010
6000
6000
iterations
iterations
●
10000
0.2
500
●
bability
4000
4000●
chain 1 ●
chain 1
chain 2
chain 2
chain 3
chain 3
chain 4
chain 4
3
NA NA
NA NA
NA NA
2000
2000
●
1
1
800
0
0
●
−0.4
−1.0
0.05
600
0.2 0.4 0.6 0.8
0.0 0.2 0.4
●
chain 1
chain 1
chain 2
chain 2
chain 3
chain 3
chain 4
chain 4
0.6
1.0
0.00
400
fod attack rate
4000
6000
8000
peaks10000
4000 ● ●
6000
10000
●
●8000
●
iterations
TOTAL.ILI NA iterations MED.WIDTHS −0.55
MED.ANN.ATT.R −0.71
CWIDTHS 0.85
ANN.ATT.R 0.21
WAITINGTIMES 0.31
chain 1
● ●●
●
●
●●●
ACF.FD.ATT.R
ACF.FD.PKS
●
8000
● ●
1
2000
2000
0
1970/71
fod peaks
lags
●● ●●
10
5
0
iterations
0.2
0
iterations
iterations
seasonality
0.30
8000 10000
8000 10000
0.08
1
2
chain 3
chain 4
0.0
6000
6000
●
0.00
40008000 6000
6000
10000
od peaks
−500
iterations
attack rate
iterations
4000
4000
2000
4000
chain 1
chain 2
chain 3 chain
chain 4 chain
−1.0
2000
2000
iterations
iterations
MED.ANN.ATT.R
0.06
0
0.0
0
0
4000
4000 6000
6000 8000
8000 10000
10000
−2.0
10000
10000
0.2
probability
probability
0
8000
8000
● ●
simu_N_16600000_R0_2,8433_D_2,802_lif_80_xIm_0,7871_dur_12,1027_sea_0,1629_mig_1,5e+07_rSc_0,5217_mut_5,7_sel_0,02_rep_0,0574_rep_0_wSh_2_ext_10_eco_15_eco_42_dt_0,5_log_1_nLo_1_evo_1_max_2
−0.5
2
1
4 5
3
0
AMED.FD.PKS
CPKS
0
chain
chain1 1
chain
chain2 2
chain
chain3 3
chain
chain4 4
2
0.0 0.5
1
2
2000
1.5
−1.0
−1
MAX.PKS
AMED.FD.PKS
1995
1.0
0.5
chain
chain11
chain 2
chain
chain33
chain
chain44
chain
2 antigenic transition seasons
fd attr
around
2000
2000
2
lags
−1.0
00
1
● ●● ● ●
0.08
02000
0.0
8000
8000 10000
10000
−0.2
acf fod att.r
−0.6
0.00
0.05
0.00
6000
6000
chain 1
chain 2
chain 3
4
iterations
1.0
4000
4000
●
ILI attack
% incidence
last rate
year
2000
2000
●
0.04 0.60.08
0.12
0.00.00
0.2 0.4
0.8 1.0
●
20
0.004 30
●
0
10000
1970/71
chain 1 chain
chain 2
chain 3
chain 4
iterations
iterations
chain
chain1 1
chain
chain2 2
chain
chain3 3
chain
chain4 4
4
●● ●
4000
4000 6000
6000 8000
8000 10000
10000
5 0.002
10
0.05
2000
2000
8000
2000
0.15
0.80
0.90
15
20
10
5
00
0.0000
2.0
1.5
4000
6000
4000 0.04 6000
−0.2
−0.4
200
0.00
0.06
attack rate
iterations
iterations
5
.6
−0.05
5
8000
8000 10000
10000
fod attack rate
1990
0.0
cdf
4
xImm
durImm
chain
chain11
chain
chain22
chain
chain33
chain
chain44
00
1985
2000
2000
●
HS
3
●●
●
chain 1
chain 1
chain 2
chain 2
chain 3
chain 3
chain 4
chain 4
0
0
0
10000
10000
time [yrs]
acf fod att.r
probability
● ● ●●
●●
10000
10000
0.02
1.0
SY97
1996/97
chain11
chain
chain22
chain
chain33
chain
chain44
6000
6000
● ●●
●
●
●●●
selA
repProb
0
mutR
selA
5
6
0.02
rScPhRob
mutR
8000
8000
2
iterations
iterations
●●
1.5
2.0
6000
6000
probability
iterations
iterations
WU95
8000
8000
−0.60.2 0.3
−0.20.4 0.5
0.2
0.0 0.1
4000
4000
6000
6000
0.0 0.2 0.4 0.6 0.8 1.0
BE89BE92
1980
1991/92
chain
●
4000
4000
0.5
4000
4000
●
●
SI87
●
0.0
●
●
1986/87
time [yrs]
ANN.ATT.R
ACF.FD.ATT.R
cdf
probability
SD.FD.ATT.R
ANN.ATT.R
2.0
●
●
nce last year
2000
2000
0
2000
0 ● 2000
●●●
●
2000
2000
●
4 0.6 0.8 1.0
0
0
0.00
−1.5
0.0 −0.5
0.5 1.0 0.51.5
0.0 0.2 0.4 0.6 0.8 1.0
0
5
10
15
20
●
●
●
BK79
1975
1981/82
●
●
●
00
●
extWrld.N
MAX.PKS
2.5 30
25
TX77
1976/77
chain
chain 11
chain
chain 22
chain
chain 33
chain
chain 44
●
●
●
●
●
●
5−1.0
10
wSh
extWrld.N
1.5 15 2.0
10
20
acf fod peaks
cdf
VI75
●
●
10000
10000
●
iterations
iterations
EN72 ●
●
●
●
6000
6000 8000
8000
●
5
−0.6
−0.2
0.2
0.0 0.2 0.4 0.6 0.8 1.0
4000
4000
1
10000
10000
●
●
●
●
30
●
●
2000
2000
●
0.2
4
●
●
●
00
●
●
15 0.0
20 0.5
25
●
●
●
●● ●●
iterations
iterations
−0.05
●
●
1970
1971/72
10000
10000
50.6 6 1.07
●
0.5 0.4
1.0 0.6
1.5 0.8
2.0
−0.2 0.0 0.2
fd.attr[t−1] − fd.attr[t]
ILI/1e5
−0.10
0.00
0.10
0
200 400 600 800
10000
10000
2000
2000
simu_N_16600000_R0_2,8433_D_2,802_lif_80_xIm_0,7871_dur_12,1027_sea_0,1629_mig_1,5e+07_rSc_0,5217_mut_5,7_sel_0,02_rep_0,0574_rep_0_wSh_2_ext_10_eco_15_eco_42_dt_0,5_log_1_nLo_1_evo_1_max_20
0.6
migrPPYr
rScPhRob
0.06
attack rate/ year
0.04
0.70
00
●●
●
0.02
0.04
0.10
0.15
0.20
chain
chain11
chain
chain22
chain
chain33
0.08 4
chain
chain
4
90
0.90
lifespan
xImm
50
0.70
10000
10000
5
1.4
8
1.4
chain
chain 11
chain
chain 22
chain
chain 33
chain
chain 44
8000
8000
⇒ consistent with observed summaries
1.0e+07
0.2
0.6 1.6e+07
1.0
0.04
6000
6000
iterations
iterations
chain
chain1 1
chain
chain2 2
chain
chain3 3
chain
chain4 4
−1
1
4000
4000
70
0.80
cdf
2000
2000
0.02
● ● ●●
●●
1995
time [yrs]
8
00
●●
7
0.04
10000
10000
15
8000
8000
20
50 2
6000
6000
chain
chain11
chain
chain22
chain
chain33
chain
chain44
10
4000
4000
0.00
probability
2000
2000
110
0.0 0.2 0.4 0.6 0.8 1.0
6
110
4 90 5
D
lifespan
1995/96
● ●
iterations
iterations
1.5
0
00
●
00
bability
0
00
10000
10000
1.0
0
00
1990/91
● ● ●●●
0.6
0
00
●
●
22
1985/86
time [yrs]
●●●● ●
chain
chain11
chain
chain22
chain
chain33
chain
chain44
3
70
33
R0
D
4 4 5 56
chain
chain 11
chain
chain 22
chain
chain 33
chain
chain 44
1990
0.0 0.2 0.4 0.6 0.8 1.0
1985
0.00
0
1980
⇒ intrinsic dynamics contribute to overall seasonality
ep_0,0574_rep_0_wSh_2_ext_10_eco_15_eco_42_dt_0,5_log_1_nLo_1_evo_1_max_20
6000
iterations
1975
1.6e+07
0
1970
0.04
fix: demography, birth/death rate, linear aging, low migration; not shown: λ, report prob
15 / 19
S
ILI/1e5
simu_N_16600000_R0_2,8433_D_2,802_lif_80_xIm_0,7871_dur_12,1027_sea_0,1629_mig_1,5e+07_rSc_0,5217_mut_5,7_sel_0,02_rep_0,0574_rep_0_wSh_2_ext_10_eco_15_eco_42_dt_0,5_log_1_nLo_1_evo_1_max_20
0.08
N
• MCMC using epidemiological and immunological summaries
200 400 600 800
chain 1
chain 2
chain 3
chain 4
1.6e+07
Antigenic tempo model
Antigenic tempo model with genetic simulations
• MCMC using also genetic summaries
fix: HA nucl mut rate 5.7 × 10−3 /site/yr, low seasonality (mild bottleneck)
oliver.ratmann@duke.edu
INFER 03-2011
16 / 19
p
fod
0.000
−0.05
_R0_2,3_D_1,9_lif_80_xIm_0,8_dur_9,5_sea_0,15_mig_1,5e+07_rSc_1,1872_mut_5,7_sel_0,0675_rep_0,05_rep_0_ext_132,2678_eco_15_eco_42_dt_0,5_log_1_nLo_1_evo_1_max_20
0.06
0.08
1.0
1.5
2.0
2.5
3.0
0
200
400
●●●
●
●
●●
● ● ●● ●
●● ● ● ● ●
●
●
●●
●
●
●
● ●●
●●●
800
●
0.20
●
0.10
probability
simu_N_16600000_R0_2,3_D_1,9_lif_80_xIm_0,8_dur_9,5_sea_0,15_mig_1,5e+07_rSc_1,1872_mut_5,7_sel_0,0675_rep_0,05_rep_0_ext_132,2678_eco_15_eco_42_dt_0,5_log_1_nLo_1_evo_1_max_20
800
600
diff of time of peaks to winter solstice
● ●
1.0
% incidence last year
0.0 0.2 0.4 0.6 0.8 1.0
time [yrs]
0.8
1
0.00
0.6
cdf
3
0.4
600
400
2
4
6
8
0.2
n most prev clusters
0.0 0.2 0.4 0.6 0.8 1.0
0.0
1980
acf fod peaks
1975
0.2
0
1970
1985
1990
1995
2000
0.00
0.02
0.04
10
12
14
sd br lengths
• MCMC using also genetic summaries
0.06
0.08
●
●
●
4
5
0.6
2
lags
200
1
ILI/1e5
Antigenic tempo model with genetic simulations
1
2
200
400
600
800
1
2
3
peaks
0.2
●●●
●●
●●●●
●●
●●●●●●●●●●
●●
●●●●●●●●●●
●●
●●
●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●● ●● ●●●●●●● ●● ●●●● ●●
6
7
epidemic weeks of peaksize/2
50
0.0 0.2 0.4 0.6 0.8 1.0
cdf
0.4
probability
0
0.4
0.00 0.02 0.04 0.06 0.08
0.04
0.2
0.0
cdf
3
lags
● ●● ● ●●
●●
●●●
●
●●
●
●
40
800
0.6
600
fod peaks
●
●●●
●
●
●●
0.8
1.0
attack rate
−0.6
0.12
400
time [yrs]
simu_N_16600000_R0_2,3_D_1,9_lif_80_xIm_0,8_dur_9,5_sea_0,15_mig_1,5e+07_rSc_1,1872_mut_5,7_sel_0,0675_rep_0,05_rep_0_ext_132,2678_eco_15_eco_42_dt_0,5_log_1_nLo_1_evo_1_max_20
0.08
200
ILI attack rate
0
−0.2
fix: HA nucl mut rate 5.7 × 10−3 /site/yr, low seasonality (mild bottleneck)
1970
0.04
1980
0.06
3.0
2
800
30
20
branch lengths
10
1990
●
2000
20
800
5
10
15
20
mean br lengths
● ● ●● ●●●
●
0.00
4
6
8
10
12
14
3
●●●●●●●●●●●●●●●●●●●●
●●●●●●●
●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●
●●●●
●●●●●●●● ●● ●●● ●● ●●●●●●● ●● ●●●● ●●
●
●
3
4
5
probability
0.4
0.6
●
0.0
0.2
0.8
0.6
cdf
0.4
0.2
0
2
0
lags
0.08
● ●● ●● ●● ● ●●●●●●●●●● ●●●● ● ●●● ●●●
50
1
fod peaks
●●●●●
1980
0.20
●● ● ● ● ● ● ●● ●
200
400
600
800
1
peaks
oliver.ratmann@duke.edu
2
6
7
epidemic weeks of peaksize/2
50
600
●
sd br lengths
0.2
−0.6
400
●
15600
40
200
1970
time [yrs]
● ●
● ●
●
1.0
0
6
1.0
●●● ●●● ●
●●● ●
●
● ●●
10 400
200
2
−0.2
acf fod peaks
0.6
0.2
2000
7
0.10
probability
0.8
1
100
0.0
1995
6
mean br lengths
1.0
1990
●
●●●
●●
n most prev clusters
0.8
1985
time [yrs]
0
5
diff of time of peaks to winter solstice
0.6
3
0.4
cdf
1980
0.004
fod peaks
500
150
1975
5
0
1.0
1
● ●
●● ● ●●
●
● ● ● ●●● ●● ● ●● ●
lags
0
−500
0.05
04
0.00
fod attack rate
.8
−0.05
●
time [yrs]
0.0
1998/99
time [yrs]
4
epidemic weeks of peaksize/2
cdf
0.004
2.5
0.002
0.000
probability
probability
2000
2.0
0.4
% incidence last year
0.2
−0.6
−0.2
acf fod att.r
0.05
1971/72
1.5
time [yrs]
50
3
● ●● ● ● ●● ● ● ●●● ●●●
0.04
0.05
1990 1.0
0.08
attack rate/ year
0.08
0.00
attack rate
0.06
fod attack rate
0.04
2
●●● ●●● ●
0.2
50
0.02
−0.05
0.02
0.00
fod shifted att.r
50
5 10
no of lineages
2
40
100
1
1
300.00
max br lengths
150
120
−0.05
50
40
30
20
0.05
100
simu_N_16600000_R0_2,3_D_1,9_lif_80_xIm_0,8_dur_9,5_sea_0,15_mig_1,5e+07_rSc_1,1872_mut_5,7_sel_0,0675_rep_0,05_rep_0_ext_132,2678_eco_15_eco_42_dt_0,5_log_1_nLo_1_evo_1_max_20
0
10
80
0
0.00
fod attack rate
60
no lineages
0.0 0.2 0.4 0.6 0.8 1.0
cdf
−0.05
40
0.12
1995/96
20
0.08
1990/91
0
●●●● ● ●● ●●●●● ●
0.00
1985/86
800
●● ● ● ● ●● ●
10
probability
peaks
0.0
1980/81
600
60
400
20
200
1.0
1975/76
time [yrs]
0
.8
1970/71
0.0
0.00
probability
⇒ in principle, model of punctuated antigenic change can reproduce limited diversity
INFER 03-2011
16 / 19
100
200
300
400
100
100
400
400
0.6
0.4
0.2
MAX.PKS
!0.2
0.0
15
300
300
0
100
200
300
400
0
100
iterations
1.1
ANN.ATT.R
0.2
chain 1
chain 2
chain 3
chain 4
0.9
chain11
chain
chain22
chain
chain33
chain
chain44
chain
xImm
!0.3
!0.35
!0.5
!0.50
AMED.FD.ATT.R
MED.ANN.ATT.R
110
chain 1
chain 2
chain 3
chain 4
90
lifespan
chain 1
chain 2
chain 3
chain 4
0.6
CFD.PKS
2.5
chain 1
chain 2
chain 3
chain 4
2.0
D
0.0
!0.2
AMED.FD.PKS
3.0
2.5
R0
chain 1
chain 2
chain 3
chain 4
100
extWrld.N
200
200
iterations
iterations
Antigenic tempo model with genetic simulations
chain 1
chain 2
chain 3
chain 4
50
0
00
iterations
chain 1
chain 2
chain 3
chain 4
200
iterations
chain 1
chain 2
chain 3
chain 4
0.6
0
chain 1
chain 2
chain 3
chain 4
0.4
400
ACF.FD.ATT.R
300
repODisp
repProb
selA
200
iterations
12
100
10
0
durImm
400
chain 1
chain 2
chain 3
chain 4
0.1
300
chain11
chain
chain22
chain
chain33
chain
chain44
chain
0.0
200
iterations
chain 1
chain 2
chain 3
chain 4
!1.0
0.0 0.06
0.5 0.0
1.0
0.03 !0.5
0.04 0.05
0.00 0.02 0.04 0.06 0.0
8
6
4
5
mutR
7
chain 1
chain 2
chain 3
chain 4
chain 1
chain 2
chain 3
chain 4
400 200
3000
100
400
400200
200
chain 1
chain 1
chain
2
chain 2
chain
3
chain 3
chain
4
chain 4
chain 1
chain 2
chain 3
chain 4
chain 1
chain
chain12
chain
chain23
chain
chain34
chain 4
0
300
300
100
400
400
200 0
chain
chain11
chain
chain22
chain
chain33
chain
chain44
300100
iterations
iterations
iterations
chain 1
chain 2
chain 3
chain 4
400200
3000
100
400
200 0
300100
iterations
iterations
chain 1
chain 2
chain 3
chain 4
0.2
6
0.5
300100
100
0.0
8
0.7
!0.2
!0.7
!0.65
200 00
iterations
iterations
chain 1
chain 2
chain 3
chain 4
1.2
300 100
0.03 0.04 0.05 0.06 0.07
200 0
iterations
0.8
100
400
0.00 0.02 0.04 0.06 0.08
3000
iterations
chain 1
chain 2
chain 3
chain 4
0.8
AMAX.SFD.ATT.R
400200
0.20.6 0.4
300100
3
chain 1
chain21
chain
chain32
chain
chain43
chain
chain 4
50
!0.2
200 0
8
100
400
iterations
1.6e+07
90
110
70
0.2
1.5
fix: HA nucl mut rate 5.7 × 10−3 /site/yr, low seasonality (mild bottleneck)
12
7
3000
iterations
MED.WIDTHS
400200
0.5 0.80.7 1.0 0.9 1.2 1.11.4
1.5
!0.4
2.0
• MCMC using also genetic summaries of strains sampled from northern EU (1968-2009)
400200
iterations
chain 1
chain 2
chain 3
chain 4
00
300
100
100
400
100
400
400
300100
iterations
400200
!0.4
chain 1
chain 2
chain 3
chain 4
300100
400200
iterations
chain 1
chain 2
chain 3
chain 4
!1.2
3000
100
400
200 0
300100
iterations
iterations
chain 1
chain 2
chain 3
chain 4
0.8
200 0
CSD.BR.LEN
!0.35
chain 1
chain 2
chain 3
chain 4
0.4
DIST2ROOT
0.0
100
400
iterations
!0.50
CFD.PKS
1.0
200 0
3000
iterations
0.2
0
300
300
iterations
iterations
chain11
chain
chain22
chain
chain33
chain
chain44
chain
400200
0.6
300100
400200
300
100 400
200
chain 1
chain 2
chain 3
chain 4
SD.BR.LEN
200 00
200
300100
MED.ANN.ATT.R
200 0
!0.8
repProb
0.2
0.0
100
400
400
iterations
0.6
chain
chain11
chain
chain22
chain
chain33
chain
chain44
iterations
iterations
iterations
0
300
300
iterations
iterations
!0.2
chain11
chain
chain22
chain
chain33
chain
chain44
chain
CMED.BR.LEN
MED.BR.LEN
repProb
AMED.FD.PKS
400 200
400
chain 1
chain 2
chain 3
chain 4
300 100 400200
300 100
400 200
0.2
200
200 00
0.2!1.0
0.4 !0.5
0.6 0.80.01.0
300 100
300
iterations
iterations
chain 1
chain 2
chain 3
chain 4
!0.2
0.2
6
200 0
200
100
400
100
iterations
iterations
0.03
0.04 !0.2
0.05 0.06
!0.4
0.0 0.07
0.8
0.0
100
100
400
0
300
0
iterations
chain 1
chain 2
chain 3
chain 4
0.4
MAX.BR.LEN
400
400 200
!0.65
0.6
FNCL
0.4
selA
NCL
CWIDTHS
0.0
0.4
durImm
mutR
4 8 5 106
!0.3
!0.5
0
3000
iterations
chain 1
chain 2
chain 3
chain 4
300
300 100
!0.4
MAX.PKS
selA
!0.2 0.02
0.0 0.04
0.2 0.06
0.4 0.08
0.6
0.00
400
400200
chain
1
chain 1
chain
2
chain 2
chain
3
chain 3
chain
4
chain 4
1.0
0.4
200
200 0
!0.35
0.8
300
300100
400200
iterations
chain 1
chain 2
chain 3
chain 4
200
200
300
300
400
400
0
chain
chain11
chain
chain22
chain
chain33
chain
chain44
oliver.ratmann@duke.edu
0.6
0.4
CWIDTHS
100
iterations
iterations
iterations
200
300
400
0
100
iterations
chain 1
chain 2
chain 3
chain 4
INFER 03-2011
200
300
400
iterations
MED.BR.LEN
100
100
400
0.2
!0.5
00
300
200
!0.3
MED.WIDTHS
!0.500.6
0.4
MED.ANN.ATT.R
AMAX.SFD.ATT.R
400
400
.2 0.4 0.6 0.8 1.0
100
0.8
300
300
0.4
0
!0.65
0.0
0.2
0.5
200
200
iterations
iterations
MAX.BR.LEN
100
400
chain 1
chain 2
2
chain 3
chain
3
chain 4
chain
4
0.0
0
300
iterations
!0.20.6 0.2
400
400
200
CWIDTHS
LINEAGE
300
300
100
0.4
!0.6
0
.0
200
200
1.2
100
100
400
chain11
chain
chain22
chain
chain33
chain
chain44
chain
0.0
END.TIME
e > 100 or eN > 1500m
!1.0 !0.5
ACF.FD.ATT.R
CFD.PKS
0.2
!0.2 0.0 0.2
TMRCA
0.2
To match avg expected diversity and variation in diversity across seasons within 1.5-fold,
!0.2
0.1 0.2
0.0
0.0
!0.2
chain
chain11
chain
chain22
chain
chain33
chain
chain44
100
400
100
iterations
iterations
0.4
0.6 0.6
200
200 0
iterations
iterations
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
rScPhRob
xImm
0.6
0.2
!0.2
LINEAGE
chain 1
chain 2
chain 3
chain 4
!1.0
100
100
400
3000
0
iterations
iterations
iterations
00
300
400200
400
0.8
!0.3
30000
!0.2
!0.4
ANN.ATT.R
AMED.FD.PKS
300100
300
!0.6
150
8
100
7
6
0
4
extWrld.N
mutR
50
5
chain
chain11
chain
chain22
chain
chain33
chain
chain44
iterations
FNCL
AMAX.SFD.ATT.R
200 0
200
iterations
iterations
iterations
400
400
200
0.4
0.0
100
400
100
iterations
400
400
200
0.2
scale system by constant e to see how big the population should be
DIST2ROOT
MED.WIDTHS
30000
0.5 0.4
400200
400
0.0
migrPPYr
lifespan
1.0e+07
50
70
⇒ However, given summer trough, Dutch population (N ≈ 15m) too small to create diversity
chain 1
chain 2
chain 3
chain 4
17 / 19
Take home
Methodological:
• ABC enables the analysis of influenza dynamics with epidemiological, genetic and
immunogenic data
Epidemiological:
• SIRS fails to reproduce influenza A (H3N2)’s irregular seasonality
• modeling abrupt changes in herd immunity within H3N2:
excite dynamics that match H3N2’s irregular seasonality
in principle limit genetic diversity to observed levels
• pop size required suggests spatial model component necessary
oliver.ratmann@duke.edu
INFER 03-2011
18 / 19
Thank you!
and the Wellcome Trust for funding through a Sir Henry Wellcome fellowship
oliver.ratmann@duke.edu
INFER 03-2011
19 / 19