Download Piecewise-deterministic Markov processes for spatio

Piecewise-deterministic Markov processes
for spatio-temporal population dynamics
Samuel Soubeyrand and Rachid Senoussi
Biostatistics and Spatial Processes research unit
Workshop “Statistique pour les PDMP”
Nancy – February 2-3, 2017
Spatio-temporal population dynamics
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Population dynamics: vast topic
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Examples:
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of particular interest in ecology and epidemiology
studied at various scales, from the microscopic scale to the
global scale
Dynamics of bluefin tuna in the Mediterranean see
Invasion of Europe by the Asian predatory wasp
Recurrence of the avian flu in Europe
Huge diversity of modeling approaches, e.g.:
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Diffusion
Trajectory
Branching process
Point process
Areal process
Regression
etc.
(Quasi-)mechanistic models for population dynamics
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Models based on reaction-diffusion equations
Aggregated
model
Figure from Soubeyrand and Roques (2014)
Models based on spatio-temporal point processes
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Intensity
Individualbased
model
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Figure from Mrkvička and Soubeyrand (in prep)
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Trade-off b/n model realism and estimation complexity
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Intensity
Extreme 1: models with
stochastic behavior and lots
of degrees of freedom
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Spatio-temporal PDMP: the missing link for modeling
population dynamics
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Extreme 2: models with
deterministic behavior and a
few degrees of freedom
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Need for intermediate
models to achieve rapid,
realistic and consistent
inference
→ Spatio-temporal
piecewise-deterministic Markov
processes can play this role
Contents of the presentation
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Coalescing Colony Model
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Metapopulation epidemic model
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Trajectory models from auto-regressive processes
A precursory example of PDMP in population dynamics:
the Coalescing Colony Model (Shigesada et al., 1995)
Modeling stratified diffusion in biological invasions
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Biological invasions may be driven by various modes of
dispersal
Ex.: Stratified dispersal process
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neighborhood diffusion
long-distance dispersal
Impact of long-distance dispersal: acceleration of range
expansion
Range expansion by neighborhood diffusion
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Diffusion equation with a Malthusian growth term (Skellam,
1951)
2
∂ n ∂2n
∂n
=D
+
+ n
∂t
∂x 2 ∂y 2
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n((x, y ), t): local population density at location (x, y ) and
time t
D: diffusion coefficient
: intrinsic growth rate of the population
Property
The rate of spread at the front
√ of the population range
asymptotically approaches 2 D when a small population is
initially introduced at the origin.
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Change with time in the population density:
→ Establishing phase followed by a constant rate spread
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The property above is robust to some modifications of the
growth term
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Ex.: Diffusion equation with a logistic growth term
(Fisher-KPP)
2
∂n
∂ n ∂2n
=D
+
+ (1 − n)n
∂t
∂x 2 ∂y 2
Invasion by stratified diffusion
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Homogeneous environment
Invading species expanding its range by both neighborhood
diffusion and long-distance dispersal
Simple approximation of Skellam or Fisher-KPP equations
augmented by long-distance dispersal:
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the establishing
phase is neglected
√
c = 2 D: constant rate expansion
λ(r ): rate of generation of new colonies by a colony with
radius r
Coalescing Colony Model
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Flow: a colony forms a disk of radius r
expanding in space at constant speed c
→ deterministic range expansion of
colonies
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Jumps: new colonies are generated by an
existing colony with rate λ(r ) and are
located at distance L from the mother
colony
→ stochastic generation of new colonies
⇒ One obtains a spatio-temporal PDMP
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Shigesada et al. (1995) characterized
the variation in the range expansion
r˜(t) of the total population
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λ(r ) = λ0 ⇒ r˜(t) constant
λ(r ) = λ0 r ⇒ r˜(t) bi-phasic
λ(r ) = λ0 r 2 ⇒ r˜(t) continually
accelerates
Bayesian inference for a PDMP modeling the dynamics of a
metapopulation
(Soubeyrand, Laine, Hanski and Penttinen, 2009)
A metapopulation epidemic model viewed as a PDMP
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Disks: host populations labelled by i
Colored disks: infected host populations
Points: contaminating particles released by infected hosts and
dispersed with kernel h (cluster point process)
Flow: deterministic growth t 7→ gi (t) = g (t − Ti ) of the
disease in infected populations (Ti : infection time for i)
Jump: particles deposited in healthy populations may
generate new infections (gi (Ti− ) = 0, gi (Ti ) > 0)
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Infections of populations (jumps) depend on a spatio-temporal
point process governed by the inhomogeneous intensity:
X
λ(t, x) =
cj g (t − Tj )h(x − xj )
j∈It
where
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t 7→ cj g (t − Tj ) gives the evolution of the infection strength of
j, which is deterministic after Tj (flow),
h is the spatial dispersal kernel
⇒ One obtains a spatio-temporal PDMP
Application: inference of the dynamics of Podosphaera
plantaginis in Åland archipelago
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Data:
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obs
Observation of sanitary states (Yn,i
: healthy / infected / NA)
of populations at the end of successive epidemic seasons
Covariates Zi
Application: inference of the dynamics of Podosphaera
plantaginis in Åland archipelago
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Data:
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obs
Observation of sanitary states (Yn,i
: healthy / infected / NA)
of populations at the end of successive epidemic seasons
Covariates Zi
Bayesian estimation
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Estimation of model parameters and latent variables, e.g.:
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parameters of the growth functions gi
parameters of the dispersal kernel h
infection times Ti
Joint posterior distribution:
obs
obs
obs
p(θ, T | Ynobs , Yn−1
, Z) ∝ p(Ynobs | T, θ, Yn−1
)p(T | θ, Yn−1
, Z)π(θ)
Y
obs
= p(Ynobs | T)π(θ)b(θ, Yn−1
, Z)
exp{−ai Λ(tend , xi )}
i healthy at tend
Y
×
exp{−ai Λ(Ti , xi )}λ(Ti , xi )
i infected
with Λ(t, x) =
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MCMC
Rt
t0
λ(t, x)dt
Bayesian estimation
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Estimation of model parameters and latent variables, e.g.:
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parameters of the growth functions gi
parameters of the dispersal kernel h
infection times Ti
Joint posterior distribution:
obs
obs
obs
p(θ, T | Ynobs , Yn−1
, Z) ∝ p(Ynobs | T, θ, Yn−1
)p(T | θ, Yn−1
, Z)π(θ)
Y
obs
= p(Ynobs | T)π(θ)b(θ, Yn−1
, Z)
exp{−ai Λ(tend , xi )}
i healthy at tend
Y
×
exp{−ai Λ(Ti , xi )}λ(Ti , xi )
i infected
with Λ(t, x) =
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MCMC
Rt
t0
λ(t, x)dt
Bayesian estimation
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Estimation of model parameters and latent variables, e.g.:
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parameters of the growth functions gi
parameters of the dispersal kernel h
infection times Ti
Joint posterior distribution:
obs
obs
obs
p(θ, T | Ynobs , Yn−1
, Z) ∝ p(Ynobs | T, θ, Yn−1
)p(T | θ, Yn−1
, Z)π(θ)
Y
obs
= p(Ynobs | T)π(θ)b(θ, Yn−1
, Z)
exp{−ai Λ(tend , xi )}
i healthy at tend
Y
×
exp{−ai Λ(Ti , xi )}λ(Ti , xi )
i infected
with Λ(t, x) =
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MCMC
Rt
t0
λ(t, x)dt
Bayesian estimation
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Estimation of model parameters and latent variables, e.g.:
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parameters of the growth functions gi
parameters of the dispersal kernel h
infection times Ti
Joint posterior distribution:
obs
obs
obs
p(θ, T | Ynobs , Yn−1
, Z) ∝ p(Ynobs | T, θ, Yn−1
)p(T | θ, Yn−1
, Z)π(θ)
Y
obs
= p(Ynobs | T)π(θ)b(θ, Yn−1
, Z)
exp{−ai Λ(tend , xi )}
i healthy at tend
Y
×
exp{−ai Λ(Ti , xi )}λ(Ti , xi )
i infected
with Λ(t, x) =
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MCMC
Rt
t0
λ(t, x)dt
Posterior distributions of infection times (i.e. jump times)
for a few populations
Perspective 1: Towards random jumps with spatial extents
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Random jumps in the epidemic model results from
independent population–to–population dispersal events
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However, the random jumps could be correlated in space and
time
Incorporating area–to–area dispersal
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Random jump: dispersal from a set of aggregated patches J
to a set of aggregated patches I
gi (t) = gi (t − ) + ∆Ji ({cj gj (t) : j ∈ J}),
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∀i ∈ I
Challenge: defining such a jump process yielding to a
tractable posterior
Perspective 2: Fitting a PDE-based PDMP to
epidemiological surveillance data
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Estimation of the introduction time and location of an
invasive species
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Handling multiple introductions modeled as a Markov process
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Example of objective: characterizing the inter-jump duration,
and its eventual non-stationarity