Modeling and inferring epidemic dynamics from: viral phylogenies and - - PowerPoint PPT Presentation

Introduction Main purposes Modeling epidemics Results Bibliography

Modeling and inferring epidemic dynamics from:

viral phylogenies and inference time series Miraine Dávila Felipe

Supervisors: Amaury Lambert and Bernard Cazelles

LPMA - UPMC & CIRB - CDF

Journée DIM RDM-IdF, September 2013

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Outline

Introduction Main purposes Modeling epidemics Results Bibliography

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Motivation

◮ The emergence of new pathogens and their distribution in the population have a

significant impact in terms of public health but also in terms of socio-economic development [MFF04].

• HIV, dengue, new variants of influenza, ...

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Motivation

◮ The emergence of new pathogens and their distribution in the population have a

significant impact in terms of public health but also in terms of socio-economic development [MFF04].

• HIV, dengue, new variants of influenza, ...

◮ Need to understand: interaction between epidemiological and evolutionary

mechanisms [WDD07].

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Motivation

◮ The emergence of new pathogens and their distribution in the population have a

significant impact in terms of public health but also in terms of socio-economic development [MFF04].

• HIV, dengue, new variants of influenza, ...

◮ Need to understand: interaction between epidemiological and evolutionary

mechanisms [WDD07].

◮ Recent works on modeling and inferring population dynamics from phylogenetic

data: [VPW+09], [Sta11], [RRK11]

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Motivation

◮ The emergence of new pathogens and their distribution in the population have a

significant impact in terms of public health but also in terms of socio-economic development [MFF04].

• HIV, dengue, new variants of influenza, ...

◮ Need to understand: interaction between epidemiological and evolutionary

mechanisms [WDD07].

◮ Recent works on modeling and inferring population dynamics from phylogenetic

data: [VPW+09], [Sta11], [RRK11]

◮ Few models exist linking both, epidemiological and evolutionary data Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Guidelines

• To develop stochastic models, called phylodynamics, combining evolutionary and

epidemic dynamics of different viral strains and the random sampling of viral sequences

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Guidelines

• To develop stochastic models, called phylodynamics, combining evolutionary and

epidemic dynamics of different viral strains and the random sampling of viral sequences

• To propose methods for the reconstruction of these multiscale stochastic

dynamics (by parameters estimation or model selection)

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Guidelines

• To develop stochastic models, called phylodynamics, combining evolutionary and

epidemic dynamics of different viral strains and the random sampling of viral sequences

• To propose methods for the reconstruction of these multiscale stochastic

dynamics (by parameters estimation or model selection)

• To compare the success of the different methods depending on the studied

pathogen in various phylodynamic scenarios of varying complexity

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Guidelines

• To develop stochastic models, called phylodynamics, combining evolutionary and

epidemic dynamics of different viral strains and the random sampling of viral sequences

• To propose methods for the reconstruction of these multiscale stochastic

dynamics (by parameters estimation or model selection)

• To compare the success of the different methods depending on the studied

pathogen in various phylodynamic scenarios of varying complexity

• To apply these methods to real data of dengue or influenza

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

SIR models

Standard SIR models

◮ In general there are 3 classes of individuals: Susceptible (S), Infected (I),

Recovered (R)

◮ There is homogeneity within each class and transition rates are modeled in

different ways (constant, time dependent, density dependent, etc...)

◮ Individuals behave independently from one another Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Model

If susceptible individuals are abundant ⇒ a simplified model without density dependence, a linear birth and death process (BD) ⇒ suitable for the infected population in outbreaks

Infected population

◮ It: infected population size at time t ≥ 0 ◮ I0 = 1 and is conditioned to survive until present time T0 (IT0 = 0) ◮ If we allow to general distribution for the infectious period: {It}t≥0 is a

Crump-Mode-Jagers (CMJ) process

Binomial sampling

Each infected individual at T0 is sampled independently with probability p

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Splitting tree, CMJ process

Individuals

◮ have i.i.d. infectious periods (with

general distribution)

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Splitting tree, CMJ process

Individuals

◮ have i.i.d. infectious periods (with

general distribution)

◮ transmit the disease at constant rate

during their infectious period

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Splitting tree, CMJ process

Individuals

◮ have i.i.d. infectious periods (with

general distribution)

◮ transmit the disease at constant rate

during their infectious period

◮ behave independently from one another Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Splitting tree, CMJ process

Individuals

◮ have i.i.d. infectious periods (with

general distribution)

◮ transmit the disease at constant rate

during their infectious period

◮ behave independently from one another

The process is stopped at present time T0

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Splitting tree, CMJ process

Individuals

◮ have i.i.d. infectious periods (with

general distribution)

◮ transmit the disease at constant rate

during their infectious period

◮ behave independently from one another

The process is stopped at present time T0

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Splitting tree, CMJ process

Individuals

◮ have i.i.d. infectious periods (with

general distribution)

◮ transmit the disease at constant rate

during their infectious period

◮ behave independently from one another

The process is stopped at present time T0

A splitting tree is characterized by a σ-finite measure Λ on (0, ∞) satisfying

• (0,∞)(1 ∧ r)Λ(dr) < ∞ (the lifespan measure).

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Splitting tree, CMJ process

Individuals

◮ have i.i.d. infectious periods (with

general distribution)

◮ transmit the disease at constant rate

during their infectious period

◮ behave independently from one another

The process is stopped at present time T0

A splitting tree is characterized by a σ-finite measure Λ on (0, ∞) satisfying

• (0,∞)(1 ∧ r)Λ(dr) < ∞ (the lifespan measure).

It is not Markovian in general

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Reconstructed tree and coalescent point process

For a fixed time T0 > 0, the reconstructed phylogeny from infected (sampled) individuals is a coalescent point process (CPP) [Lam10]:

◮ a sequence of i.i.d. random variables H1, H2, . . . killed at its first value greater

than T0

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Reconstructed tree and coalescent point process

Goal

To characterize the joint law of:

◮ Incidence data: (IT0, IT1 . . . , ITN ), at deterministic times T0 > T1 . . . > TN > 0 ◮ Coalescence times between sampled hosts at T0:

H1, H2 . . .

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Binomial sampling, present population size, general lifespans

From now on, let I ′

T0 be the number of sampled hosts at T0.

Theorem 1

The joint distribution of IT0 and the coalescence times between sampled individuals is characterized by the probability generating function (pgf) G: G(u) := Ex

• 1{

H1<t1,..., HK−1<tK−1}1{I ′

T0 =K}uIT0

• IT0 = 0
• which can be expressed in terms of the probability distribution of H and the sampling

probability p.

Remark:

This result is provided in the general framework, where individuals infectious period has arbitrary distribution (not necessarily exponential).

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Binomial sampling, present and past population sizes, exponential lifespans

Theorem 2

The likelihood of the variables is characterized by the N-dimensional pgf G : [0, 1]N+1 → R+, defined as: G(u0, u1, . . . , uN) = Ex

• 1{

H1<t1,..., HK−1<tK−1}1{I ′

T0 =K}u

IT0

u

IT1 1

. . . u

ITN N

• IT0 = 0
• which, again, can be expressed in terms of the probability distribution of H and the

sampling probability p.

Remark:

◮ This result relies in a path-wise decomposition of the contour process of a

splitting tree, and the fact that this contour is a Lévy process [Lam10]

◮ The population size process I is expressed as a sum of time inhomogeneous

branching processes (IBP) backward in time

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Contour process of a splitting tree

Splitting tree up to T0, the CPP The contour of a truncated splitting from sampled individuals tree is a Lévy process with measure Λ, together with IT0, IT1 drift −1, starting at x ∧ T0, reflected below T0 and killed at 0 [Lam10]

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Contour process of a splitting tree

Splitting tree up to T0, the CPP The contour of a truncated splitting from sampled individuals tree is a Lévy process with measure Λ, together with IT0, IT1 drift −1, starting at x ∧ T0, reflected below T0 and killed at 0 [Lam10] The contour process codes the splitting tree

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Some ideas of the proof: IBP

◮ The branching and immigration mechanisms are described through the excursions

• f the contour process from levels (Ti)0≤i≤N, starting from x and conditioned to

visit T0 before 0 (i.e. IT0 = 0):

◮ Immigrants: number of visits of level Tk∗, before it goes above Tk∗−1 (upper

level), without touching 0.

◮ Descendants of (⋆): number of visits of Tk∗, starting from Tk∗−1, before it goes

again above Tk∗−1, without touching 0.

◮ Sampled individuals: visits of level T0 which are sampled (with prob. p). ◮ and so on... Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Future direction

◮ The inversion of the probability generating function G, to allow for MLE or

bayesian estimation for the inference of the model parameters

◮ See if the inference is possible on simulated data ◮ Apply the inference methods on real data ◮ Conceive more suitable models for the specific characteristics of epidemics such

as dengue (several strains) Collaboration with E. Numminen from University of Helsinki in a study of the transmission dynamics of a pathogen in a population of children in Oslo, Norway. Application to dengue and influenza data from Southeast Asia:

• Thailand and Cambodia: DENFREE European project (FP7-HEALTH-2011)
• South Vietnam: National Institute of Hygiene and Epidemiology in Hanoi and

Oxford University Clinical Research Unit in Ho Chi Minh City

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

J R Gog and B T Grenfell. Dynamics and selection of many-strain pathogens. Proc Natl Acad Sci U S A, 99(26):17209–17214, December 2002. Amaury Lambert. The contour of splitting trees is a Lévy process.

• Ann. Probab., 38(1):348–395, 2010.

David M. Morens, Gregory K. Folkers, and Anthony S. Fauci. The challenge of emerging and re-emerging infectious diseases. Nature, 430(6996):242–249, July 2004. David A. Rasmussen, Oliver Ratmann, and Katia Koelle. Inference for nonlinear epidemiological models using genealogies and time series. PLoS Comput. Biol., 7(8):e1002136, 11, 2011. Tanja Stadler. Inferring speciation and extinction processes from extant species data. Proceedings of the National Academy of Sciences, 108(39):16145–16146, September 2011. Erik M. Volz, Sergei L. Kosakovsky Pond, M. J. Ward, Andrew J. Leigh Brown, and Simon D. W. Frost. Phylodynamics of infectious disease epidemics. Genetics, 183(4):1421–1430, 2009.

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Nathan D. Wolfe, Claire P. Dunavan, and Jared Diamond. Origins of major human infectious diseases. Nature, 447(7142):279–283, May 2007.

Miraine Dávila Felipe Modeling and inferring epidemic dynamics

Introduction Main purposes Modeling epidemics Results Bibliography

Thank You!

Miraine Dávila Felipe Modeling and inferring epidemic dynamics