ISPM, Division of Biostatistics Modelling power-law spread of infectious diseases Sebastian Meyer and Leonhard Held Financially supported by the Swiss National Science Foundation (project 137919: Statistical methods for spatio-temporal modelling and prediction of infectious diseases ) SMIDDY 2013, 13 September 2013 Page 1
ISPM, Division of Biostatistics Epidemic Modelling – Prospective surveillance: outbreak detection (Farrington). – This talk is concerned with retrospective surveillance: – Explain the spread of epidemics through statistical modelling – Assess influential factors, e.g., seasonality, climate, concurrent epidemics of related pathogens, contact networks – Data basis: routine public health surveillance including temporal as well as spatial information – This talk deals with two types of surveillance data: – individual case reports – aggregated counts by week and administrative district Meyer & Held: Modelling power-law spread of infectious diseases Page 2
ISPM, Division of Biostatistics Mobility networks determine the spread of epidemics Source: Max Planck Institute for Dynamics and Self-Organization (http://www.mpg.de/4406928/) How to quantify spatial interaction between regions or individuals in the absence of network data? Meyer & Held: Modelling power-law spread of infectious diseases Page 3
ISPM, Division of Biostatistics Power law! Why? Brockmann et al., 2006: – Analysed trajectories of 464 670 dollar bills in the USA – Short-time travel behaviour follows a power law wrt distance Fig. 1c: Histogram of the distance r traversed within 4 days. Dashed line: P ( r ) ∝ r − 1 . 59 – “Starting point for the development of a new class of models for the spread of infectious diseases” Meyer & Held: Modelling power-law spread of infectious diseases Page 4
ISPM, Division of Biostatistics Power law! Why? Brockmann et al., 2006: – Analysed trajectories of 464 670 dollar bills in the USA – Short-time travel behaviour follows a power law wrt distance Fig. 1c: Histogram of the distance r traversed within 4 days. Dashed line: P ( r ) ∝ r − 1 . 59 – “Starting point for the development of a new class of models for the spread of infectious diseases” Let’s do it! We use this finding to improve upon two previously established model frameworks for infectious disease spread. Meyer & Held: Modelling power-law spread of infectious diseases Page 4
ISPM, Division of Biostatistics Two additive components (Held et al., 2005) Endemic: seasonality, population, socio-demography, climate, . . . ⊕ Epidemic: dependency on previously infected individuals Space-time point process model for individual case reports λ ∗ ( t , s ) = ν [ t ][ s ] ρ [ t ][ s ] + � (Meyer, Elias, η j · g ( t − t j ) · f ( � s − s j � ) and H¨ ohle, j : t j < t 2012) log( ν [ t ][ s ] ) = β 0 + β ⊤ z [ t ][ s ] , log( η j ) = γ 0 + γ ⊤ m j Multivariate time-series model for counts Y it | Y · , t − 1 ∼ NegBin( µ it , ψ ) (Held and Paul, � µ it = ν it e it + λ it Y i , t − 1 + φ it w ji Y j , t − 1 2012, and previous work) j � = i + β ( · ) ⊤ z ( · ) log( · it ) = β ( · ) + b ( · ) · ∈ { ν, λ, φ } 0 i it Meyer & Held: Modelling power-law spread of infectious diseases Page 5
ISPM, Division of Biostatistics Two additive components (Held et al., 2005) Endemic: seasonality, population, socio-demography, climate, . . . ⊕ Epidemic: dependency on previously infected individuals Space-time point process model for individual case reports λ ∗ ( t , s ) = ν [ t ][ s ] ρ [ t ][ s ] + � (Meyer, Elias, η j · g ( t − t j ) · f ( � s − s j � ) and H¨ ohle, j : t j < t 2012) log( ν [ t ][ s ] ) = β 0 + β ⊤ z [ t ][ s ] , log( η j ) = γ 0 + γ ⊤ m j Multivariate time-series model for counts Y it | Y · , t − 1 ∼ NegBin( µ it , ψ ) (Held and Paul, � µ it = ν it e it + λ it Y i , t − 1 + φ it w ji Y j , t − 1 2012, and previous work) j � = i + β ( · ) ⊤ z ( · ) log( · it ) = β ( · ) + b ( · ) · ∈ { ν, λ, φ } 0 i it Meyer & Held: Modelling power-law spread of infectious diseases Page 5
ISPM, Division of Biostatistics Two additive components (Held et al., 2005) Endemic: seasonality, population, socio-demography, climate, . . . ⊕ Epidemic: dependency on previously infected individuals Space-time point process model for individual case reports λ ∗ ( t , s ) = ν [ t ][ s ] ρ [ t ][ s ] + � η j · g ( t − t j ) · f ( � s − s j � ) “ twinstim ” j : t j < t log( ν [ t ][ s ] ) = β 0 + β ⊤ z [ t ][ s ] , log( η j ) = γ 0 + γ ⊤ m j Multivariate time-series model for counts Y it | Y · , t − 1 ∼ NegBin( µ it , ψ ) � µ it = ν it e it + λ it Y i , t − 1 + φ it w ji Y j , t − 1 “ hhh4 ” j � = i + β ( · ) ⊤ z ( · ) log( · it ) = β ( · ) + b ( · ) · ∈ { ν, λ, φ } 0 i it Meyer & Held: Modelling power-law spread of infectious diseases Page 5
ISPM, Division of Biostatistics Power-law distance decay in twinstim f ( x ) = x − d not suitable: pole at x = 0 ⇒ not integrable. Meyer & Held: Modelling power-law spread of infectious diseases Page 6
ISPM, Division of Biostatistics Power-law distance decay in twinstim f ( x ) = x − d not suitable: pole at x = 0 ⇒ not integrable. 1.0 σ = 10 d=0.5 0.8 “Lagged”power law with uniform d=1 d=1.59 short-range dispersal: 0.6 f L ( x ) d=3 0.4 � 1 for x < σ, 0.2 f L ( x ) = � x � − d otherwise. 0.0 σ 0 20 40 60 80 100 Distance x Meyer & Held: Modelling power-law spread of infectious diseases Page 6
ISPM, Division of Biostatistics Power-law distance decay in twinstim f ( x ) = x − d not suitable: pole at x = 0 ⇒ not integrable. 1.0 σ = 1 d=0.5 0.8 d=1 Kernel of the density of the shifted d=1.59 0.6 f ( x ) d=3 Pareto distribution: 0.4 0.2 f ( x ) = ( x + σ ) − d 0.0 0 20 40 60 80 100 Distance x Meyer & Held: Modelling power-law spread of infectious diseases Page 6
ISPM, Division of Biostatistics Power-law distance decay in twinstim f ( x ) = x − d not suitable: pole at x = 0 ⇒ not integrable. 1.0 σ = 1 d=0.5 0.8 d=1 Kernel of the density of the shifted d=1.59 0.6 f ( x ) d=3 Pareto distribution: 0.4 0.2 f ( x ) = ( x + σ ) − d 0.0 0 20 40 60 80 100 Distance x – Joint ML-inference for all model parameters – Numerical cubature of f 2 D ( s ) = f ( � s � ) over polygonal domains in likelihood via product-Gauss cubature ( Sommariva and Vianello, 2007 ) Meyer & Held: Modelling power-law spread of infectious diseases Page 6
ISPM, Division of Biostatistics Power-law weights in hhh4 3 – On which distance measure between regions 3 2 2 should the power law act? 1 1 2 0 − → Order of neighbourhood o ji ! 1 1 2 3 Meyer & Held: Modelling power-law spread of infectious diseases Page 7
ISPM, Division of Biostatistics Power-law weights in hhh4 3 – On which distance measure between regions 3 2 2 should the power law act? 1 1 2 0 − → Order of neighbourhood o ji ! 1 1 2 3 – Generalisation of previously used 1.0 ● ● ● d first-order weights w ji : 0 0.8 0.5 ● ● 1 first-order power law ● 0.6 ● 1.59 ● o − d o − d ● ● ∞ 1 ( j ∼ i ) ● 0.4 ji ● ● ● ● ● ● ● ● ● ● ● – Normalisation: w ji / � ● k w jk 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● 2 4 6 8 10 12 14 Neighbourhood order o Meyer & Held: Modelling power-law spread of infectious diseases Page 7
ISPM, Division of Biostatistics Power-law weights in hhh4 3 – On which distance measure between regions 3 2 2 should the power law act? 1 1 2 0 − → Order of neighbourhood o ji ! 1 1 2 3 – Generalisation of previously used 1.0 ● ● ● d first-order weights w ji : 0 0.8 0.5 ● ● 1 first-order power law ● 0.6 ● 1.59 ● o − d o − d ● ● ∞ 1 ( j ∼ i ) ● 0.4 ji ● ● ● ● ● ● ● ● ● ● ● – Normalisation: w ji / � ● k w jk 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● – Estimate d within the penalised 2 4 6 8 10 12 14 likelihood framework simultaneously Neighbourhood order o with all other model parameters. Meyer & Held: Modelling power-law spread of infectious diseases Page 7
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