Modeling group dispersal of particles with a spatiotemporal point - PowerPoint PPT Presentation

Modeling group dispersal of particles with a spatiotemporal point process Samuel Soubeyrand INRA – Biostatistics and Spatial Processes Joint work with L. Roques, J. Coville and J. Fayard 9th SSIAB Workshop, May 10, 2012 Group Dispersal Project – Plant Health and Environment Dpt.

Spatiotemporal point processes in propagation models Object of interest: species spreading using small particles (spores, pollens, seeds...) Sources of particles generate a spatially structured rain of particles ◮ rain of particles → spatial point process ◮ spatial structure → inhomogeneous intensity of the process

Intensity of the spatial point process formed by the deposit locations of the particles The intensity is a convolution between ◮ the source process (spatial pattern and strengths) and ◮ a parametric dispersal kernel

Simulation of an epidemics

Dispersal kernel Dispersal kernel: probability density function of the deposit location of a particle released at the origin The shape of the kernel is a major topic in dispersal studies: it determines ◮ the propagation speed ◮ the spatial structure of the population ◮ the genetic structure of the population

Main characteristics of dispersal kernels: ◮ long distance dispersal (Minogue, 1989) ◮ non-monotonicity (Stoyan and Wagner, 2001) 200 Intensity 0.5 ◮ anisotropy 0.1 100 0.01 0.001 Ordinate (m) 0 − 100 − 200 − 300 − 200 − 100 0 50 100 Abscissa (m)

Observation of secondary foci (clusters) in real epidemics Epidemics of yellow rust of wheat in an experimental field (I. Sache) t = 1 t = 2 t = 3 t = 4

◮ Classical justifications for patterns with multiple foci: ◮ long distance dispersal ◮ spatial heterogeneity ◮ super-spreaders (a few individuals which infects many susceptible individuals)

◮ Classical justifications for patterns with multiple foci: ◮ long distance dispersal ◮ spatial heterogeneity ◮ super-spreaders (a few individuals which infects many susceptible individuals) ◮ An other justification to be investigated: Group dispersal ◮ Groups of particles are released due to wind gusts ◮ Particles of any group are transported in an expanding air volume ◮ At a given stopping time, particles of any group are projected to the ground

Group Dispersal Model (GDM): Spatial case Deposit equation for particles: A single point source of particles located at the origin of R 2 J : number of groups of particles released by the source N j : number of particles in group j ∈ { 1 , . . . , J } X jn : deposit location of the n th particle of group j satisfying X jn = X j + B jn ( ν || X j || ) , (1) where X j : final location of the center of group j , B jn : Brownian motion describing the relative movement of the n th particle in group j with respect to the group center ν : positive parameter

Assumptions about the deposit equation ◮ The random variables J , N j , X j and the random processes { B jn : n = 1 , . . . , N j } are mutually independent ◮ Number of groups: J ∼ Poisson ( λ ) ◮ Number of particles in group j : N j ∼ indep p µ, σ 2 ( · ) ◮ Group center location: X j ∼ indep f X j ( · ) (features of f X j : decrease at the origin is more or less steep, tail more or less heavy, shape more or less anisotropic...) ◮ The Brownian motions B jn are centered, independent and with independent components They are stopped at time t = ν || X j || . Then, B jn ( ν || X j || ) ∼ indep N (0 , ν || X j || I )

Dispersal from a single source ◮ Simulations: (Interpretation: Cox process or Neyman-Scott with double nonstationarity — in the center pattern and the o ff spring di ff usion) Ordinate D e n Ordinate s i t y Abscissa Abscissa 0.3 0.2 0.1 Ordinate 0.0 − 0.1 I n t Ordinate e n s i t y − 0.2 − 0.3 Abscissa − 0.3 − 0.2 − 0.1 0.0 0.1 0.2 0.3 0.4 Abscissa Marginal probability density function (dispersal kernel):

Dispersal from a single source ◮ Simulations: (Interpretation: Cox process or Neyman-Scott with double nonstationarity — in the center pattern and the o ff spring di ff usion) Ordinate D e n Ordinate s i t y Abscissa Abscissa ◮ Marginal probability density function (dispersal kernel): � � f X jn ( x ) = R 2 f X jn | X j ( x | y ) f X j ( y ) dy = R 2 φ ν , y ( x ) f X j ( y ) dy . The particles are n.i.i.d. from this p.d.f. while in the classical dispersal models the particles are i.i.d. from a dispersal kernel which may be of the form of f X j or f X jn

Discrepancies from independent dispersal The GDM is compared with two independent dispersal models (IDM) ◮ IDM1: the number of particles in each group is assumed to be one. Thus, particles are independently drawn under the p.d.f. f X jn . ◮ IDM2: the number of particles in each group is assumed to be one and the Brownian motions are deleted (i.e. ν = 0). Thus, particles are independently drawn under the p.d.f. f X j .

Moments X : Deposit location of a particle Q ( x + dx ): Count of points in x + dx Criterion Model Value ( 0 E ( X ) GDM 0 ) ( 0 IDM1 0 ) ( 0 IDM2 0 ) V ( X ) GDM V ( X j ) + ν E ( || X j || ) I IDM1 V ( X j ) + ν E ( || X j || ) I IDM2 V ( X j ) E ( || X || 2 ) E ( || X j || 2 ) + 2 ν E ( || X j || ) GDM E ( || X j || 2 ) + 2 ν E ( || X j || ) IDM1 E ( || X j || 2 ) IDM2 E { Q ( x + dx ) } GDM λ µ f X jn ( x ) dx IDM1 λ f X jn ( x ) dx IDM2 λ f X j ( x ) dx λ [ µ f X jn ( x ) dx + ( σ 2 + µ 2 − µ ) E { φ ν , X j ( x ) 2 } ( dx ) 2 ] V { Q ( x + dx ) } GDM IDM1 λ f X jn ( x ) dx IDM2 λ f X j ( x ) dx λ ( σ 2 + µ 2 − µ ) E { φ ν , X j ( x 1 ) φ ν , X j ( x 2 ) } ( dx ) 2 cov { Q ( x 1 + dx ) GDM , Q ( x 2 + dx ) } IDM1 0 IDM2 0

GDM: larger variance of Q ( x + dx ) and positive covariance (decreasing with distance) → clusters (even with µ = 1) We expect multiple foci in the spatio-temporel case

Group dispersal model: Spatio-temporal case → multiple foci under the GDM Ordinate Ordinate − 1.0 − 0.5 0.0 0.5 1.0 − 1.0 − 0.5 0.0 0.5 1.0 GDM − 1.0 − 1.0 − 0.5 − 0.5 0.0 Abscissa Abscissa 0.0 0.5 0.5 1.0 1.0 0 10000 20000 30000 40000 50000 60000 Ordinate Ordinate − 1.0 − 0.5 0.0 0.5 1.0 − 1.0 − 0.5 0.0 0.5 1.0 − 1.0 IDM1 − 1.0 − 0.5 − 0.5 0.0 Abscissa Abscissa 0.0 0.5 0.5 1.0 1.0 0 10000 20000 30000 40000 Ordinate Ordinate − 1.0 − 0.5 0.0 0.5 1.0 − 1.0 − 0.5 0.0 0.5 1.0 − 1.0 − 1.0 IDM2 − 0.5 − 0.5 0.0 Abscissa Abscissa 0.0 0.5 0.5 1.0 1.0 0 5000 10000 15000 20000 25000

Simulation study of the number of foci: Definition A δ -focus is a set of cells (from a regular grid) which are connected and whose intensity of points is larger than δ model ν 2 model ν 2 σ σ 8 8 IDM2 0 0 GDM 0.001 50 IDM1 0.005 0 GDM 0.005 50 GDM 0.005 10 GDM 0.01 50 GDM 0.005 50 GDM 0.1 50 6 6 Number of − foci Number of − foci GDM 0.005 100 δ δ 4 4 2 2 0 0 0 50000 100000 150000 0 50000 100000 150000 Threshold δ Threshold δ

Farthest particle (link with propagation speed) Definition The maximum dispersal distance during one generation is R max = max { R jn : j ∈ J , n ∈ N j } where R jn = || X jn || J = { 1 , . . . , J } if J > 0 and the empty set otherwise N j = { 1 , . . . , N j } if N j > 0 and the empty set otherwise By convention, if no particle is released ( J = 0 or N j = 0 for all j ), then R max = 0 Ordinate Abscissa

R max = max { R jn : j ∈ J , n ∈ N j } Under the GDM and IDMs, the distribution of the distance between the origin and the furthest deposited propagule is zero-inflated and satisfies: P ( R max = 0) = exp � � λ { p µ, σ 2 (0) − 1 } f R max ( r ) = λ f R max ( r ) exp { λ ( F R max ( r ) − 1) } , ∀ r > 0 , j j is the p.d.f. of the distance R max where f R max = max { R jn : n ∈ N j } j j between the origin and the furthest deposited propagule of group j , and F R max is the corresponding cumulative distribution function j � r ( r ) = P ( R max ( F R max = 0) + 0 f R max ( u ) du ). j j j → Distribution of R max ? j

Under the IDMs, N j = 1 for all j ∈ J and, consequently, p µ, σ 2 (0) = 0 and f R max ( r ) = f R jn ( r ) j �� 2 π rf X jn (( r cos θ , r sin θ )) d θ for the IDM1 0 = � 2 π rf X j (( r cos θ , r sin θ )) d θ for the IDM2 . 0

Under the GDM, the distribution of R max is zero-inflated and j satisfies: P ( R max = 0) = p µ, σ 2 (0) j � f R max ( r ) = R 2 f R max | X j ( r | x ) f X j ( x ) dx j j + ∞ � � R 2 f R jn | X j ( r | x ) F R jn | X j ( r | x ) q − 1 f X j ( x ) dx , = qp µ, σ 2 ( q ) q =1 where f R jn | X j is the conditional distribution of R jn given X j satisfying � r 2 h 1 ( u , x ) h 2 ( r 2 − u , x ) du , f R jn | X j ( r | x ) = 2 r 0 h i ( u , x ) = f i ( √ u , x ) + f i ( −√ u , x ) ∀ i ∈ { 1 , 2 } , 2 √ u , � � − ( v − x ( i ) ) 2 1 f i ( v , x ) = exp ∀ i ∈ { 1 , 2 } , , � 2 ν || x || 2 πν || x || � r x = ( x (1) , x (2) ) and F R jn | X j ( r | x ) = 0 f R jn | X j ( s | x ) ds .