Post on 10-Jul-2020
Gradual & Punctuated antigenic drift for influenza
evolutionA quantitative approach based on time series analysis
Anton Camacho, Sébastien Ballesteros, Katia Koelle & Bernard Cazelles
Department of BiologyUMR 7625, UPMC-CNRS-ENS
camacho@biologie.ens.fr
Influenza incidence time series in the Netherlands from 1970 to 2000
Year
Incid
ence
per
10
000
hab
1970 1975 1980 1985 1990 1995
0
20
40
60
80
100
120
140 A/H3N2A/H1N1B
•One epidemic each year with variable amplitude•Type/subtype alternate
Thanks to Yingcun Xia
Infection/Replication/Selection
High selective pressure on hemagglutinin (HA) to escape
immune recognition
Mc Hardy and Adams (2009)
Epochal evolution of A/H3N2’s HA
Gradual genetic evolution but punctuated antigenic evolution (Smith et al. 2004)
Genetic evolution Antigenic evolution
Antigenic clusters replace one another
Positive selectionhas been ongoing most of the time
(Shih et al. 2007)
Antigenic clusters replace one another
‘‘Antigenic change accumulates most of the
time with occasional large changes’’
Implication for vaccination strategies
A/H3N2 time seriesIn
ciden
ce p
er 1
0 00
0 ha
b
1970 1975 1980 1985 1990 1995
0
20
40
60
80
100
120
140HK/68 EN/72 VI/75TX/77 BK/79 SI/87 BE/89 BE/92 WU/95SY/97
• High epidemics correlate with cluster transitions (at least at the beginning) and are followed by refractory periods • High epidemics occur also between two transitions • Epidemic sizes (seem to) disminish with time
A/H3N2 time seriesIn
ciden
ce p
er 1
0 00
0 ha
b
1970 1975 1980 1985 1990 1995
0
20
40
60
80
100
120
140HK/68 EN/72 VI/75TX/77 BK/79 SI/87 BE/89 BE/92 WU/95SY/97
• Can we reproduce these patterns with a simple model including both gradual and punctual antigenic drift?• Can we assess the relative importance of gradual/punctual antigenic drift by fitting this time series?
The SIRX model: 1 cluster
S XRIλ ν γ
σin λ
The SIRX model: 1 cluster
S XRIλ ν γ
σin λ
1/γ is the time scale of gradual antigenic drift
σin is the % of gradual antigenic drift
σin ∈ [0;1]
The SIRX model: 1 cluster
S XRIλ ν γ
σin λ
1/γ is the time scale of gradual antigenic drift
σin is the % of gradual antigenic drift
σin ∈ [0;1]
SIRX 1/γ ➔ 0 SIRISIRS σin ➔ 0
The SIRX model: 2 clusters
S XRI1
σin
The SIRX model: 2 clusters
I2
S XRI1
σin
σex
The SIRX model: 2 clusters
XRI2
σin
S XRI1
σin
σex
σex is the % of punctual
antigenic drift σex ≥ σin
The SIRX model: N clusters
• History based model with two levels of immunity (intra & extra cluster) ⇒ O(N3N) for complexity (in practice N < 10)
• Stochastic framework via Euler multinomial scheme (δt = 6 hrs)
• With or without coinfection
• Circulating clusters give rise to new ones according to a Weibull hazard function (Koelle et al. 2010)
• Extra-cluster cross-immunity between i & j: Cij = f(Kij), Kij is the kinship level between i & j
• Cross-protection conferred by an immune repertoire R against infection by a cluster i: CRi = g({Cki}k∈R) ⇒ σRi = 1- CRi
• Cross-protection can act by: reduced susceptibility, reduced infectiosity, polarised infectiosity
Biological interpretationσin>0 & σex>>σin ⇒ gradual & punctuated antigenic drift
σin
σin
σin
σex1
σex1
σex2
Biological interpretationσin = 0 ⇒ punctuated antigenic drift
σex1
σex1
σex2
Previous works• Koelle et al. 2006: model the genetic-antigenic map
and find cluster replacement
• Gökaydin et al. 2007: SIRI limit of the SIRX model (γ = 0) and N=2. Cluster replacement occurs below the reinfection threshold (R0=1/σin) for σex>>σin (numerical study)
• Koelle et al. 2010: SIRS limit of the SIRX model (σin = 1) within a status-based framework. Both gradual and punctuated antigenic drift are necessary to reproduce influenza time-series (not fitted)
Preliminary study
• Replacement of HK/68 by EN/72
• Can we reproduce this non-trivial dynamics?
• Fit the SIRX model with N=2 clusters, protection acts by reducing susceptibility
• Seasonal forcing: simple sinusoidal
• Simplification: the second cluster is introduced in october 1972 by 1 individual
Incid
ence
per
10
000
hab
1969 1970 1971 1972 1973 1974
0
20
40
60
80
100
120
140HK/68 EN/72 VI/75
Maximum likelihood via Iterated Filtering (Ionides 2006)
Incid
ence
per
10
000
hab
1970.0 1970.2 1970.4 1970.6 1970.8 1971.0
10
20
30
40
50
60
Parameters (θt)
State space (Xt)
Maximum likelihood via Iterated Filtering (Ionides 2006)
Incid
ence
per
10
000
hab
1970.0 1970.2 1970.4 1970.6 1970.8 1971.0
10
20
30
40
50
60
Parameters (θt)
State space (Xt)
GenerateN particles
with random θ
Maximum likelihood via Iterated Filtering (Ionides 2006)
Incid
ence
per
10
000
hab
1970.0 1970.2 1970.4 1970.6 1970.8 1971.0
10
20
30
40
50
60
Parameters (θt)
State space (Xt)
Prediction
Maximum likelihood via Iterated Filtering (Ionides 2006)
Incid
ence
per
10
000
hab
1970.0 1970.2 1970.4 1970.6 1970.8 1971.0
10
20
30
40
50
60
SelectionMutation (θ)Prediction
Parameters (θt)
State space (Xt)
Prediction
• Selection: particles reproduce proportionally to their likelihood Prob( data | prediction ) i.e. negative binomial observation process• Mutation: θt+1 ➥ N(θt , m)
✗
Maximum likelihood via Iterated Filtering (Ionides 2006)
Incid
ence
per
10
000
hab
1970.0 1970.2 1970.4 1970.6 1970.8 1971.0
10
20
30
40
50
60
SelectionMutation (θ)Prediction
Parameters (θt)
State space (Xt)
Prediction
SelectionMutation (θ)Prediction
• Selection: particles reproduce proportionally to their likelihood Prob( data | prediction ) i.e. negative binomial observation process• Mutation: θt+1 ➥ N(θt , m)
Maximum likelihood via Iterated Filtering (Ionides 2006)
Incid
ence
per
10
000
hab
1970.0 1970.2 1970.4 1970.6 1970.8 1971.0
10
20
30
40
50
60
SelectionMutation (θ)Prediction
Parameters (θt)
State space (Xt)
Prediction
SelectionMutation (θ)Prediction
SM(θ)
P
• Selection: particles reproduce proportionally to their likelihood Prob( data | prediction ) i.e. negative binomial observation process• Mutation: θt+1 ➥ N(θt , m)
Maximum likelihood via Iterated Filtering (Ionides 2006)
Incid
ence
per
10
000
hab
1970.0 1970.2 1970.4 1970.6 1970.8 1971.0
10
20
30
40
50
60
SelectionMutation (θ)Prediction
Parameters (θt)
State space (Xt)
Prediction
SelectionMutation (θ)Prediction
SM(θ)
P
SM(θ)
P
• Selection: particles reproduce proportionally to their likelihood Prob( data | prediction ) i.e. negative binomial observation process• Mutation: θt+1 ➥ N(θt , m)
Maximum likelihood via Iterated Filtering (Ionides 2006)
Incid
ence
per
10
000
hab
1970.0 1970.2 1970.4 1970.6 1970.8 1971.0
10
20
30
40
50
60
SelectionMutation (θ)Prediction
Parameters (θt)
State space (Xt)
Prediction
SelectionMutation (θ)Prediction
SM(θ)
P
SM(θ)
P
SM(θ)
P
• Selection: particles reproduce proportionally to their likelihood Prob( data | prediction ) i.e. negative binomial observation process• Mutation: θt+1 ➥ N(θt , m)
Maximum likelihood via Iterated Filtering (Ionides 2006)
Incid
ence
per
10
000
hab
1970.0 1970.2 1970.4 1970.6 1970.8 1971.0
10
20
30
40
50
60
SelectionMutation (θ)Prediction
Parameters (θt)
State space (Xt)
Prediction
SelectionMutation (θ)Prediction
SM(θ)
P
SM(θ)
P
SM(θ)
P
SM(θ)
P
• Selection: particles reproduce proportionally to their likelihood Prob( data | prediction ) i.e. negative binomial observation process• Mutation: θt+1 ➥ N(θt , m)
Maximum likelihood via Iterated Filtering (Ionides 2006)
Incid
ence
per
10
000
hab
1970.0 1970.2 1970.4 1970.6 1970.8 1971.0
10
20
30
40
50
60
SelectionMutation (θ)Prediction
Parameters (θt)
State space (Xt)
Prediction
SelectionMutation (θ)Prediction
SM(θ)
P
SM(θ)
P
SM(θ)
P
SM(θ)
P
SM(θ)
P
• Selection: particles reproduce proportionally to their likelihood Prob( data | prediction ) i.e. negative binomial observation process• Mutation: θt+1 ➥ N(θt , m)
Maximum likelihood via Iterated Filtering (Ionides 2006)
Incid
ence
per
10
000
hab
1970.0 1970.2 1970.4 1970.6 1970.8 1971.0
10
20
30
40
50
60
SelectionMutation (θ)Prediction
Parameters (θt)
State space (Xt)
Prediction
SelectionMutation (θ) Prediction
SM(θ)
P
SM(θ)
P
SM(θ)
P
SM(θ)
P
SM(θ)
P
{θ1}Mean
Variance
{θ2}Mean
Variance
{θ3}Mean
Variance
Maximum likelihood via Iterated Filtering (Ionides 2006)
Incid
ence
per
10
000
hab
1970.0 1970.2 1970.4 1970.6 1970.8 1971.0
10
20
30
40
50
60
SelectionMutation (θ)Prediction
Parameters (θt)
State space (Xt)
Prediction
SelectionMutation (θ) Prediction
SM(θ)
P
SM(θ)
P
SM(θ)
P
SM(θ)
P
SM(θ)
P
{θ1}Mean
Variance
{θ2}Mean
Variance
{θ3}Mean
Variance Ionides formula Global estimate
for θ
Maximum likelihood via Iterated Filtering (Ionides 2006)
Incid
ence
per
10
000
hab
1970.0 1970.2 1970.4 1970.6 1970.8 1971.0
10
20
30
40
50
60
SelectionMutation (θ)Prediction
Parameters (θt)
State space (Xt)
Prediction
SelectionMutation (θ) Prediction
SM(θ)
P
SM(θ)
P
SM(θ)
P
SM(θ)
P
SM(θ)
P
Global estimate for θGenerate N new particles and iterate filtering
Maximum likelihood via Iterated Filtering (Ionides 2006)
Incid
ence
per
10
000
hab
1970.0 1970.2 1970.4 1970.6 1970.8 1971.0
10
20
30
40
50
60
SelectionMutation (θ)Prediction
Parameters (θt)
State space (Xt)
Prediction
SelectionMutation (θ) Prediction
SM(θ)
P
SM(θ)
P
SM(θ)
P
SM(θ)
P
SM(θ)
P
Mutations N(θ,m→0) decrease at each iteration θ→Maximum Likelihood Estimate
Exploring the likelihood surface
• High dimensional (11) parameter space➽ bound the parameter space within biologically realistic values1. σin∈[0;0.2] (intra-cluster cross-immunity > 80%)2. σex∈[0;0.5] (extra-cluster cross-immunity > 50%)3. σin ≤ σex
• Structural (model) and practical (data) identifiability analysis➽ 2 dimensional profile likelihood
Julian Beever
Purely punctuated antigenic drift model (σin = 0)
R0
1 2 3 4 5 6 7
2.0e+06
4.0e+06
6.0e+06
8.0e+06
1.0e+07
1.2e+07
−5600
−5400
−5200
−5000
−4800
−4600
Initi
al n
umbe
r of
sus
cept
ible
s
Purely punctuated antigenic drift model (σin = 0)
1 2 3 4 5 6 7
2.0e+06
4.0e+06
6.0e+06
8.0e+06
1.0e+07
1.2e+07
−5600
−5400
−5200
−5000
−4800
−4600
0 1 2 3 4 5
0
50
100
150
200
year
Incid
ence
/100
00 h
ab
datapredictionobservation
• R0 = 1±0.5 (seasonality)• Punctual immune escape σex = 20% to 30%• Initial proportion of susceptibles: 75%
Purely punctuated antigenic drift model (σin = 0)
0 2 4 6 8 10
0
50
100
150
200
250
time (years)
incid
ence
/100
00 h
ab
● ●
0.02 0.77 1.51 2.2 2.87 3.6 4.27 5
year
Prop
ortio
n
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Failed invasionReplacementCoexistenceTotal extinction
0.02 0.77 1.51 2.2 2.87 3.6 4.27 5
year
Prop
ortio
n
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Failed invasionReplacementCoexistenceTotal extinction
1 indwith
cluster 2
10 indwith
cluster 2
Gradual & punctuated antigenic drift model
R0
Initi
al n
umbe
r of
sus
cept
ible
s
1 2 3 4 5 6 7
2.0e+06
4.0e+06
6.0e+06
8.0e+06
1.0e+07
1.2e+07
−6000
−5500
−5000
−4500
−4000
0 1 2 3 4 5
0
50
100
150
200
year
Incid
ence
/100
00 h
ab
datapredictionobservation
1 2 3 4 5 6 7
2.0e+06
4.0e+06
6.0e+06
8.0e+06
1.0e+07
1.2e+07
−6000
−5500
−5000
−4500
−4000
Gradual & punctuated antigenic drift model
• R0 = 5±0.2 (seasonality)• Punctual immune escape: σex = 19%• Gradual immune escape: σin = 17% & 1/γ = 2 years• Initial proportion of fully susceptibles (S): 4%• Initial proportion of partially susceptibles (X): 94%
0.02 0.77 1.51 2.2 2.87 3.6 4.27 5
year
Prop
ortio
n
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Failed invasionReplacementCoexistenceTotal extinction
0.02 0.77 1.51 2.2 2.87 3.6 4.27 5
year
Prop
ortio
n
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Failed invasionReplacementCoexistenceTotal extinction
0 2 4 6 8 10
0
100
200
300
400
500
600
time (years)
incid
ence
/100
00 h
ab
● ●
Gradual & punctuated antigenic drift model
1 indwith
cluster 2
10 indwith
cluster 2
Δσ = 2% !
Gradual & punctuated antigenic drift model
0 2 4 6 8 10
0
50000
100000
150000
200000
time
infe
ctio
us h
istor
y
0 clusters1 clusters2 clusters
0 2 4 6 8 10
0
10000
20000
30000
40000
time
infe
ctio
us h
istor
y
0 clusters1 clusters2 clusters
1 2 3 4 5 6 7
2.0e+06
4.0e+06
6.0e+06
8.0e+06
1.0e+07
1.2e+07
−6000
−5500
−5000
−4500
−4000
1 2 3 4 5 6 7
2.0e+06
4.0e+06
6.0e+06
8.0e+06
1.0e+07
1.2e+07
−6000
−5500
−5000
−4500
−4000
Infection dominates Reinfection dominates
Summary• Punctual antigenic drift alone is sufficient to
reproduce cluster replacement but not the between-cluster dynamics. Parameter estimates are realistic.
• Adding gradual antigenic drift shift the MLE near the reinfection threshold and the between-cluster dynamics is reproduced. However, epidemics are too high at cluster transition whereas immune escape and initial conditions are biologically and epidemiologically unrealistic.
Complex model
Data
Profile likelihood (correlation, identifiability,
initial conditions)
Good fit?Biological/epidemiological
realism?
Systematic analysis
Systematic analysisComplex
modelData
Profile likelihood (correlation, identifiability,
initial conditions)
Good fit?Biological/epidemiological
realism? ✔✘
Modelcriticism
Discussion• Neither of these 2 simple model can reproduce the whole complex
dynamics of H3N2 for the considered period (1970 - 1975). Confidence in data could be assessed by fitting other data sets.
• For small R0 the gradual antigenic drift does not play a major role in the dynamics because σin is well below the reinfection threshold (1/R0). How to explain the high epidemics after the refractory periods? infection of susceptible or reinfection? maybe the SIRX model or temperate time series are not suitable to detect gradual antigenic drift?
• Gradual vs Punctuated antigenic drift is maybe a false debate, other mechanisms seem necessary to understand influenza dynamics in temperate area: environmental stochasticity on the transmission, more realistic seasonal forcing, variation in the introduction time of new clusters... work in progress!
Peak time
−5 0 5 10 15
20
40
60
80
100
120
140
Week
Incid
ence
per
10
000
hab
EN/72
VI/75
TX/77
SI/87
BE/89
BE/92
WU/95SY/97
1rst January
0 2 4 6 8 10
0
50
100
150
200
250
time (years)
incid
ence
/100
00 h
ab
● ●
0 2 4 6 8 10
0
50
100
150
200
250
time (years)
incid
ence
/100
00 h
ab
● ●
Influenza time series England
Reinfection threshold
Invasion outcomes