Epidemics with Random Sampling

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Epidemics with Random Sampling

Amaury Lambert

MCEB 2013, Hameau de l’Étoile, 28 mai 2013

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Epidemics with sampling : 2 problems

Assumption :

Each infective is sampled/detected after some random time (symptoms, medical exam)

Goal :

Infer epidemiological dynamics from...

• Problem 1.Pathogen sequence data sampled atall detection times

= w/ T. Stadler and H. Alexander (ETH Zürich)

• Problem 2.Hospital data in antibiotic-resistant epidemics stoppedat the first detection time

= w/ P. Trapman (U. Stockholm)

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Epidemics with sampling : 2 problems

Assumption :

Goal :

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Epidemics with sampling : 2 problems

Assumption :

Goal :

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Problem 1 : Stoppingat all dots before timet

Problem 2 : Stoppingat first dot = timeT

r

r r

t

0

r r r

t

T

0

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General framework

• Continuous time

• Branching process assumption :Excess of susceptibles

• Age-dependentdeath rate :The duration of infectiousness can have anarbitrarydistribution

• Constantbirth rate(transmission)

• Constantdetection rate

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General framework

• Continuous time

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General framework

• Continuous time

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General framework

• Continuous time

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General framework

• Continuous time

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Splitting tree in forward time (Geiger & Kersting 97)

Thetransmission treewith samplingcan be described by an asexual population withmarkswhere

6 t

r r

r

r r

• individuals reproduce independently

• death rate may depend onage

• birth rate is constant

• detection rate is constant.

The population size process(Nt;t≥0)is anon-Markovian birth–death process.

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Splitting tree in forward time (Geiger & Kersting 97)

Thetransmission treewith samplingcan be described by an asexual population withmarkswhere

6 t

r r

r

r r

• individuals reproduce independently

• death rate may depend onage

• birth rate is constant

• detection rate is constant.

The population size process(Nt;t≥0)is anon-Markovian birth–death process.

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Without marks (1)

Thetransmission tree stopped attcan beorientedas follows...

6

?

t 0 1 2 3

H₁ H₂

H₃

b b

b b b

?H₄ 4

...where the timesH₁,H2,H3. . .are thenode depths.

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Without marks (2)

Theoriented,reconstructed tree attcan be represented...

Like this...

...Or like that

? ?

t

H₁ H₁

H₂ H₂

H₃ H₃

c

c c c c

? ?

H₄ ^c ^c ^c ^c H₄ ^c

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Coalescent point process

• Mathematical result :The node depthsH₁,H2,H3. . .of the tree form a sequence ofindependent and identically distributed positive random variables killed at its first value larger thant

(Lambert 2010, Lambert & Stadler 2013)

In particular, conditional onN_t6=0, the population sizeN_t follows ageometric distribution.

• Such a tree is called acoalescent point-process:

• fast simulationof reconstructed trees

• Easy computation of thelikelihood of a tree(product form).

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Coalescent point process

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Coalescent point process

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Coalescent point process

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Coalescent point process

6

t 1 2 3 4 5 6

?

H₁ H2

H₃ H₄

H₅ H₆

Likelihood of tree withnode depths(hi)1≤i≤n−1

L(T) =p(t)

n−1

∏

i=1

f(hi).

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Adding marks : 2 problems

• Problem 1.Likelihood of the treespanned by all the distinct detection times ?

;Previously known : case when death rate is constant(Stadler et al 2012)

• Problem 2.Likelihood of the treestopped at the first detection timeT?

;Previously known :NT is (still) geometric(Trapman & Bootsma 2009)

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Adding marks : 2 problems

• Problem 1.Likelihood of the treespanned by all the distinct detection times ?

;Previously known : case when death rate is constant(Stadler et al 2012)

• Problem 2.Likelihood of the treestopped at the first detection timeT?

;Previously known :NT is (still) geometric(Trapman & Bootsma 2009)

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Problem 1 : Assumptions and notation

• Asampled individualimmediatelyleavesthe infective population.

• Si:=sampling timeof individuali

• Ri:=coalescence timebetween individualsi−1 andi.

t

t t

t

0 1

2

3

? 6

?

S₁ 6

S₂

? 6

?

R₁ 6R₂

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Problem 1 : Result

Theorem (Lambert, Alexander & Stadler 2013) The pairs(Si,R_i)form akilled Markov chainwith semi-explicit transitions, where the transition probability only depends on the second component(Si).

;Inference of model parameters from viral phylogenies (HIV, flu).

For any givenorientedtreeT withcoalescence times(xi)2≤i≤nand sampling times(yi)1≤i≤n,

L(T) =g(y₁)k(yn)

n

∏

i=2

f(yi−1;xi,yi).

Limitations :

1 Transitions are only semi-explicit ;

2 The Markov chain property depends on the orientation...

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Problem 1 : Result

L(T) =g(y₁)k(yn)

n

∏

i=2

f(yi−1;xi,yi).

Limitations :

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Problem 1 : Result

L(T) =g(y₁)k(yn)

n

∏

i=2

f(yi−1;xi,yi).

Limitations :

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Problem 2 : Assumptions

• Patients havei.i.d lengths of stayin the hospital, all distributed as some r.v.K

• Conditionalon infection, the length of stay of a patient is a size-biasedversion ofK

• Detection rate per patient=:δ

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Problem 2 : Notation

For individuali, set

• U_i:=time elapsed from entrance of the hospital up to infection

• Ai:=time elapsed from infection up toT

• Ri:=residual lifetime in the hospital afterT.

Setm:=E(K)and letφdenote the inverse of the convex function x7→x−λ

m Z

(0,∞]

(1−e^−xy)P(K>y)dy.

T 6

? 6

transmission

U A s R

? 6

J K

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Problem 2 : Result

Theorem (Lambert & Trapman 2013)

Conditional on NT=n, the triples(Ui,Ai,Ri)of the n infectives at time T areindependent and identically distributed, distributed as

E(f(U,A,R)) = λ

m

φ(δ) φ(δ)−δ

Z _∞

u=0

du Z _∞

a=0

da Z _∞

z=u+aP(K∈dz)e^−φ(δ)af(u,a,z−u−a),

In particular, the timesJ_i=U_i+A_ispent in the hospital up to time T areindependent and identically distributed, distributed as the r.v. J

P(J∈dy) = λ/m

φ(δ)−δ P(K>y) 1−e^−φ^(δ)y dy.

;Inference from hospital data (dates of entrance in the hospital).

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Problem 2 : Result

E(f(U,A,R)) = λ

m

φ(δ) φ(δ)−δ

Z _∞

u=0

du Z _∞

a=0

da Z _∞

P(J∈dy) = λ/m

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Problem 2 : Result

E(f(U,A,R)) = λ

m

φ(δ) φ(δ)−δ

Z _∞

u=0

du Z _∞

a=0

da Z _∞

P(J∈dy) = λ/m

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Jumping contour of a tree

a) Binary tree with edge lengths and b) Contour process of its truncation below timet.

6

@

@ @

@

@ a)

b) T

@

@@@@ @

@

- 6

T

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Jumping contour of a tree with marks

a) Binary tree with marks, b) its reconstructed tree and c) its contour process.

r r r

r r

t r

0 0

1 1

2 2

3 3

? 6

? ?

6 6

S₁ S₁

S2 S₃ S2 S₃

? 6

?

R₂ R3 R₂ R63

a) b)

r@ r r

@

@@

@

@@

@

t6

0 -

c)

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Co-authors

• Helen ALEXANDER(ETHZ) . . . .

• Tanja STADLER(ETHZ) . . . .

• Pieter TRAPMAN(U. Stockholm) . . . .

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SMILE : A cross-disciplinary group in CIRB

• CIRB =Center for Interdisciplinary Research in Biology (Collège de France)

• SMILE =Stochastic Models for the Inference of Life Evolution

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SMILE group in June 2012

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Institutions

• Stochastic Models for the Inference of Life Evolution(SMILE)

⊂Center for Interdisciplinary Research in Biology

⊂Collège de France

• Stochastics & Biology group

⊂Laboratoire de Probabilités et Modèles Aléatoires

⊂UPMC University Paris 06

• ANRModèles Aléatoires eN Écologie, Génétique, Évolution(MANEGE)

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Conference announcement Mathematics for an Evolving Biodiversity

Montréal, Canada September 16–20, 2013

Organizers : Jonathan Davies (McGill), Nicolas Lartillot (CNRS & U.

Montréal) and myself

http://www.crm.umontreal.ca/2013/Biodiversity13/