Rapid inﬂuenza reinfection · t−1, θ) the conditional transition density f (y t|x t, θ) the conditional distribution of the observation process f (x 0|θ) the initial density

Rapid influenza reinfection

Likely mechanisms and potential impacts during a pandemic

Anton Camacho, Sébastien Ballesteros, Andrea L. Graham, Fabrice Carrat & Bernard Cazelles

Department of BiologyUMR 7625, UPMC-CNRS-ENS

[email protected]

mailto:[email protected]

mailto:[email protected]

A two-wave epidemic on Tristan da Cunha (1971)

August 13

Landing

August 19

Max 1rst Wave

August 28

End 1rst Wave

September 13

Max 2nd Wave

October 10

End of 2nd Wave

0

10

20

30

40

50

Incid

en

ce

Phot

o: a

ll rig

hts r

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ved


August 13

Landing

August 19

Max 1rst Wave

August 28

End 1rst Wave

September 13

Max 2nd Wave

October 10

End of 2nd Wave

0

10

20

30

40

50

Incid

en

ce

Phot

o: a

ll rig

hts r

eser

ved


August 13

Landing

August 19

Max 1rst Wave

August 28

End 1rst Wave

September 13

Max 2nd Wave

October 10

End of 2nd Wave

0

10

20

30

40

50

Incid

en

ce

Phot

o: a

ll rig

hts r

eser

ved

Attack rates:• Infection : 95%• Reinfection : 30%

Objectives: Disentangling between 6 biological

mechanisms to explain rapid influenza reinfection

A simple mechanistic approach

S E I R Lλ ε ν γ

• λ = β I/N mass-action•1/ε : mean latent period•1/ν : mean infectious period•1/γ : mean removed period

Long-term immunity

A simple mechanistic approach


Flexible distribution of the removed period: quarantine, non-specific protection, etc.

I R1 ... Rk Lkγν kγkγ

H1: the virus mutated during the first epidemic-wave

S E1 I1 R1 L1λ1 ε ν γ



I2

Tmut



• σ ∈[0,1] cross-immunity• 2-strain history-based model (Rios-Doria & Chowell 2009)

L2 R2 I2 E2ενγ

σλ2

H2: two different viruses at the beginning


L2 R2 I2 E2ενγ

λ2

• no cross-immunity• 2-virus history-based model

H3: partially-protective immunity


σλ

σ ∈[0,1]: probability to be reinfected (Gomes et al. 2004)

H4: All-or-Nothing

S E I R Lλ ε ν αγ

α: the probability to develop long-term immunity (Mathews et al. 2007)

(1-α)γ

S E I R Lλ ε ν αγ

α: the probability to clear the viral load

(1-α)γ

H5: intra-host recrudescence of infection

H6: window-of-reinfection

S E I R Lλ ε ν

γ

1/τ: the mean duration of the window of susceptibility before developing immunity

Wλ

τ

Likelihood-based inference

For a given time series: y1:T = (y1, y2, ..., yT )and a state space model completely specified by:

M :

!""#

""$

f(xt|xt!1, !) the conditional transition densityf(yt|xt, !) the conditional distribution

of the observation processf(x0|!) the initial density

the likelihood is given by the identity:

f(y1:T |!) =T%

t=1

f(yt|y1:t!1, !)

where xt is the unobserved Markov process, ! is the unknown vectorof parameters and f(.|.) is a generic density specified by its arguments

Exploring the likelihood surface

Param 2

Para

m 1

LogLikelihood

Exploring the likelihood surface with MIF (Ionides et al. 2006)

5 10 15 20 25

5

10

15

20

25

Param 2

Para

m 1

0.95

0.9

5

0.95

0.9

0

.9

0.8

0.8

0.8

• 3 local maxima

• 1 global maximum

Global convergence

5 10 15 20 25

5

10

15

20

25

Param 2

Para

m 1

0.95

0.9

5

0.95

0.9

0

.9

0.8

0.8

0.8

• 3 local maxima

• 1 global maximum

Local trap:• initial θ• MIF para-metrization

Exploring the likelihood surface with MIF (Ionides et al. 2006)

Log-likelihood profile

5 10 15 20 25

5

10

15

20

25

Param 2

Para

m 1

0.95

0.9

5

0.95

0.9

0

.9

0.8

0.8

0.8

5 10 15 20 25

−120

−119

−118

−117

−116

−115

Log−likelihood

projection

95% confidence interval:

{!2 : 2[l(p)(!̂2)! l(p)(!2)] < "20.95(1)}

with !̂2 = argmax l(p)(!2)

Parameter identifiability

Structural non-

identifiability

Practical non-

identifiabilityIdentifiability

Raue et al. 2009

Structural non-identifiability (Mutation: H1)


L2 R2 I2 E2ενγ

Structural non-identifiability between σ and β2 ⇒ β2 = β1

σλ2

Practical non-identifiability (In-Host: H5)

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10 20 30 40 50

−127

−126

−125

−124

−123

Log−

likel

ihoo

d

R0

●●

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4 6 8 10

20

30

40

50

Mean infectious period in days (1/ν)

R0

linear regression:transmission coefficient = 3.88 e.c./day

95%

IC R

0

1520253035404550

0 10 20 30 40 50 60

0

10

20

30

40

50

60

Day

Incid

ence

0 10 20 30 40 50 600

50

100

150

200

Day

Prev

alen

ce

Model selection: Akaike information criterion

Model Win AoN 2Vi Mut In-Host PPI

k 9 9 10 10 9 9

Log-Like -112.52 -112.78 -114.75 -115.20 -117.50 -118.44

ΔAICc 0 0.52 7.37 8.27 9.96 11.84

AICc = !2L(!MLE) + 2k +2k(k + 1)T ! k ! 1

with k = ||!||

Maximum likelihood parameters

Parameter Win AoN

R0 = β/ν 10.38 11.27

Mean latent period (days) 2.14 2.11

Mean infectious period (days) 2.01 2.43

Mean removed period (days) 13.58 (shape 5) 11.62 (shape 4)

Reinfection window (days) 4.86 -

Probability of long-term immunity

- 0.53

Immunological support: primary influenza infection

Time (days)Infection

0 21 4228

Innate + cellularimmune response

Antibody response(in 80% of the cases)

Win

AoN

18 (sd = 6)

16 (6)

5 (5)+ antibody response

+ antibody response (in 53% of the cases)

Large pop: N = 1 million hab.

R0 = 1.4 (2009 pH1N1)

Tristan da Cunha:N = 284 hab.

R0 ≈ 11

AoN

Win

0 10 20 30 40

0

200

400

600

800

AoN in large population (N=1 million hab., R0=1.4)

Incid

ence

per

10

000

hab Attack rates:

infection= 63.8 %reinfection= 9.1 %

Infected but stillsusceptible: 21.1 %

infection + reinfectionreinfection

0 10 20 30 40 50

0

10

20

30

40

50

AoN in Tristan da Cunha (N=284 hab., R0=11.27)

Incid

ence

Attack rates:infection= 95.4 %

reinfection= 42.6 %



0 10 20 30 40

0

200

400

600

800

Win in large population (N=1 million hab., R0=1.4)

Incid

ence

per

10

000

hab Attack rates:


Infected but stillsusceptible: 0 %


Week

0 10 20 30 40 50

0

10

20

30

40

50

Win in Tristan da Cunha (N=284 hab., R0=10.38)

Incid

ence


reinfection= 39.4 %



Day


R0 = 1.4 (2009 pH1N1)


R0 ≈ 11

AoN

Win

0 10 20 30 40

0

200

400

600

800


Incid

ence

per

10

000

hab Attack rates:




0 10 20 30 40 50

0

10

20

30

40

50


Incid

ence


reinfection= 42.6 %



0 10 20 30 40

0

200

400

600

800


Incid

ence

per

10

000

hab Attack rates:




Week

0 10 20 30 40 50

0

10

20

30

40

50


Incid

ence


reinfection= 39.4 %



Day


R0 = 1.4 (2009 pH1N1)


R0 ≈ 11

AoN

Win

0 10 20 30 40

0

200

400

600

800


Incid

ence

per

10

000

hab Attack rates:




0 10 20 30 40 50

0

10

20

30

40

50


Incid

ence


reinfection= 42.6 %



0 10 20 30 40

0

200

400

600

800


Incid

ence

per

10

000

hab Attack rates:




Week

0 10 20 30 40 50

0

10

20

30

40

50


Incid

ence


reinfection= 39.4 %



Day

Conclusion• Maximum likelihood via Iterated Filtering (MIF, Ionides et al. 2006) is a

rigorous statistical framework for parameter inference and selection based on AIC for non-linear stochastic models.

• Identifiability analysis and 95% CI via log-likelihood profile

• Rapid influenza reinfection is likely due to a combination of ecological and immunological factors:

1. Window-of-reinfection (Win) + high exposure

2. Lack of antibody response (AoN) + re-exposure

• During a pandemic, the lack of antibody response has a greater impact than the window-of-reinfection (break of herd immunity)

• Experimental validation and accurate predictions with a more realistic large population model (heterogeneous mixing, seasonal forcing, external reintroduction, pre-existing immunity, etc.)

Documents

Rapid inﬂuenza reinfection · t−1, θ) the conditional transition density f (y t|x t, θ) the conditional distribution of the observation process f (x 0|θ) the initial density