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