A Space-Time Conditional Intensity Model for Invasive Meningococcal - PowerPoint PPT Presentation

Motivation Point Process Modelling Inference Data Analysis Summary A Space-Time Conditional Intensity Model for Invasive Meningococcal Disease Occurrence Sebastian Meyer 1 , 3 Johannes Elias 4 Michael Höhle 2 , 3 1 Division of Biostatistics, Institute for Social & Preventive Medicine, Univ. of Zürich 2 Department for Infectious Disease Epidemiology, Robert Koch Institute, Berlin 3 (previously) Department of Statistics, Ludwig-Maximilians-Universität, München 4 German Reference Centre for Meningococci, University of Würzburg, Würzburg QMUL – Institute of Zoology London, United Kingdom 7 September 2012 1 / 27

Motivation Point Process Modelling Inference Data Analysis Summary Outline Motivation 1 Space-Time Point Process Modelling 2 Inference 3 Data Analysis 4 Summary 5 2 / 27

Motivation Point Process Modelling Inference Data Analysis Summary Motivation and Aim Understanding the spread of an infectious disease is a step towards its control There is increased agreement that such dynamics are stochastic phenomena operating in a heterogeneous population The spatial and temporal resolution of infectious disease data is becoming better and better Aim Establish a regression framework for point referenced infectious disease surveillance data, where the transmission dynamics and its dependency on covariates can be quantified within the context of a spatio-temporal stochastic process . 3 / 27

Motivation Point Process Modelling Inference Data Analysis Summary Application: Invasive meningococcal disease (IMD) Description Life-threatening infectious disease triggered by the bacterium Neisseria meningitidis (aka meningococcus ) Involves meningitis (50%), septicemia (5–20%), pneumonia (5-15%) Transmission by mucous secretions, also airborne Epidemiology Yearly incidence (Germany, 2001–2008): 0.5–1 infections per 100 000 inhabitants Mainly affected are infants and adolescents Lethality: 8.4%, for meningococcal sepsis: ≈ 40% 4 / 27

Motivation Point Process Modelling Inference Data Analysis Summary Available IMD data T wo most common finetypes in Germany in 2002–2008: 336 cases of B:P1.7-2,4:F1-5 , 300 cases of C:P1.5,2:F3-3 Case variables: date, residence postcode, age, gender B:P1.7-2,4:F1-5 C:P1.5,2:F3-3 Number of cases of the serogroup C finetype Number of cases of the serogroup B finetype 16 16 14 14 12 12 10 10 8 8 6 6 4 4 2 2 0 0 2002 2003 2004 2005 2006 2007 2008 2009 2002 2003 2004 2005 2006 2007 2008 2009 Time (month) Time (month) 5 / 27

Motivation Point Process Modelling Inference Data Analysis Summary Spatial distribution B:P1.7-2,4:F1-5 C:P1.5,2:F3-3 4500 4500 ● ● ● ● ● ● 4000 4000 ● ● ● ● ● ● ● 54 ° N 54 ° N ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 3500 3500 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 3000 ● ● 3000 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 52 ° N 52 ° N ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2500 ● ● 2500 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2000 ● ● ● ● 2000 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1500 ● ● 1500 ● ● 50 ° N ● 50 ° N ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1000 ● ● ● ● 1000 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 500 500 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 48 ° N ● 48 ° N ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ● 0 ● ● ● ● ● 6 ° E 8 ° E 10 ° E 12 ° E 14 ° E 6 ° E 8 ° E 10 ° E 12 ° E 14 ° E Scientific question: Do the finetypes spread differently? My task: Quantify the transmission dynamics. 6 / 27

Motivation Point Process Modelling Inference Data Analysis Summary Relationship of IMD and influenza Weekly numbers of SurvNet influenza cases Weekly numbers of SurvNet IMD cases 4000 50 2002 2006 2002 2006 2003 2007 2003 2007 ● 2004 2008 2004 2008 2005 2005 ● 40 3000 ● Number of influenza cases Number of IMD cases 30 ● 2000 ● ● ● ● ● 20 ● ● 1000 10 ● ● ● 0 0 0 10 20 30 40 50 0 10 20 30 40 50 Week Week Scientific question: Do waves of influenza predispose to IMD accumulations? Statistical solution: Quantify and test the local effect of (lagged) numbers of influenza cases on occurrences of IMD 7 / 27

Motivation Point Process Modelling Inference Data Analysis Summary 1 Motivation 2 Space-Time Point Process Modelling 3 Inference 4 Data Analysis 5 Summary 8 / 27

Motivation Point Process Modelling Inference Data Analysis Summary Conditional intensity function (CIF) A regular spatio-temporal point process N on ❘ + × ❘ 2 can be uniquely characterised by its left-continuous CIF λ ∗ ( t, s ) . Definition � � � N ([ t, t + Δ t ) × d s ) = 1 � H t − P λ ∗ ( t, s ) = lim Δ t | d s | Δ t → 0 , | d s |→ 0 Instantaneous event rate at ( t, s ) given all past events Key to modelling, likelihood analysis and simulation of evolutionary (“self-exciting”) point processes In application, N is only defined on a subset ( 0 , T ] × W ⊂ ❘ + × ❘ 2 (observation period and region) 9 / 27

Motivation Point Process Modelling Inference Data Analysis Summary Proposed additive-multiplicative continuous space-time intensity model ( twinstim ) λ ∗ ( t, s ) = h ( t, s ) + e ∗ ( t, s ) Inspiration Additive-multiplicative SIR (susceptible-infectious-recovered) compartmental model (Höhle, 2009) for a fixed population Spatio-temporal ETAS (epidemic-type aftershock-sequences) model (Ogata, 1998) 10 / 27

Motivation Point Process Modelling Inference Data Analysis Summary Proposed additive-multiplicative continuous space-time intensity model ( twinstim ) λ ∗ ( t, s ) = h ( t, s ) + e ∗ ( t, s ) Multiplicative endemic component � � o ξ ( s ) + β ′ z τ ( t ) ,ξ ( s ) h ( t, s ) = exp Piecewise constant function on a spatio-temporal grid { C 1 , . . . , C D } × { A 1 , . . . , A M } with time interval index τ ( t ) and region index ξ ( s ) Region-specific offset o ξ ( s ) , e.g., log-population density Endemic linear predictor β ′ z τ ( t ) ,ξ ( s ) includes discretised time trend and exogenous effects, e.g., influenza cases 10 / 27

Motivation Point Process Modelling Inference Data Analysis Summary Proposed additive-multiplicative continuous space-time intensity model ( twinstim ) λ ∗ ( t, s ) = h ( t, s ) + e ∗ ( t, s ) Additive epidemic (self-exciting) component � e ∗ ( t, s ) = e η j g α ( t − t j ) ƒ σ ( s − s j ) j ∈  ∗ ( t, s ; ϵ,δ ) Individual infectivity weighting through linear predictor η j = γ ′ m j based on the vector of unpredictable marks Positive parametric interaction functions, e.g., − � s � 2 � � and g α ( t ) = e − αt ƒ σ ( s ) = exp 2 σ 2 Set of active infectives depends on fixed maximum temporal and spatial interaction ranges ϵ and δ 10 / 27

Motivation Point Process Modelling Inference Data Analysis Summary Marked extension with event type Motivation: joint modelling of both finetypes of IMD Additional dimension K = {1 , . . . , K } for event type κ ∈ K Marked CIF � � λ ∗ ( t, s , κ ) = exp β 0 ,κ + o ξ ( s ) + β ′ z τ ( t ) ,ξ ( s ) � q κ j ,κ e η j g α ( t − t j | κ j ) ƒ σ ( s − s j | κ j ) + j ∈  ∗ ( t, s ,κ ; ϵ,δ ) T ype-specific endemic intercept ype-specific transmission, q k, ∈ {0 , 1} , k,  ∈ K T ype-specific infection pressure η j = γ ′ m j , κ j is part of m j T ype-specific interaction functions, e.g., variances σ 2 T κ 11 / 27