a space-time conditional intensity model for invasive ... · pdf filea space-time conditional...

Motivation Point Process Modelling Inference Data Analysis Summary

A Space-Time Conditional Intensity Model forInvasive Meningococcal Disease Occurrence

Sebastian Meyer1,3 Johannes Elias4 Michael Höhle2,3

1Division of Biostatistics, Institute for Social & Preventive Medicine, Univ. of Zürich2Department for Infectious Disease Epidemiology, Robert Koch Institute, Berlin3(previously) Department of Statistics, Ludwig-Maximilians-Universität, München4German Reference Centre for Meningococci, University of Würzburg, Würzburg

QMUL – Institute of ZoologyLondon, United Kingdom

7 September 2012

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Outline

1 Motivation

2 Space-Time Point Process Modelling

3 Inference

4 Data Analysis

5 Summary

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Motivation and Aim

Understanding the spread of an infectious disease is astep towards its controlThere is increased agreement that such dynamics arestochastic phenomena operating in a heterogeneouspopulationThe spatial and temporal resolution of infectious diseasedata is becoming better and better

Aim

Establish a regression framework for point referencedinfectious disease surveillance data, where the transmissiondynamics and its dependency on covariates can bequantified within the context of a spatio-temporal stochasticprocess.

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Application:Invasive meningococcal disease (IMD)

Description

Life-threatening infectious disease triggered by thebacterium 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 inhabitantsMainly affected are infants and adolescentsLethality: 8.4%, for meningococcal sepsis: ≈ 40%

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Available IMD data

Two 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:F3-3Case variables: date, residence postcode, age, gender

B:P1.7-2,4:F1-5

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Spatial distributionB:P1.7-2,4:F1-5

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C:P1.5,2:F3-3

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Scientific question: Do the finetypes spread differently?

My task: Quantify the transmission dynamics.

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Relationship of IMD and influenzaWeekly numbers of SurvNet influenza cases

0 10 20 30 40 50

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Weekly numbers of SurvNet IMD cases

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200620072008

Scientific question: Do waves of influenza predispose to IMDaccumulations?

Statistical solution: Quantify and test the local effect of (lagged)numbers of influenza cases on occurrences of IMD

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1 Motivation


3 Inference

4 Data Analysis

5 Summary

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Conditional intensity function (CIF)

A regular spatio-temporal point process N on R+ ×R2 can beuniquely characterised by its left-continuous CIF λ∗(t,s).

Definition

λ∗(t,s) = limΔt→0, |ds|→0

P

�

N([t, t + Δt)× ds) = 1�

�Ht−�

Δt |ds|

Instantaneous event rate at (t,s) given all past eventsKey to modelling, likelihood analysis and simulation ofevolutionary (“self-exciting”) point processesIn application, N is only defined on a subset(0, T]×W ⊂ R+ ×R2 (observation period and region)

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Proposed additive-multiplicative continuousspace-time intensity model (twinstim)

λ∗(t,s) = h(t,s) + e∗(t,s)

Inspiration

Additive-multiplicative SIR(susceptible-infectious-recovered) compartmentalmodel (Höhle, 2009) for a fixed populationSpatio-temporal ETAS (epidemic-typeaftershock-sequences) model (Ogata, 1998)

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λ∗(t,s) = h(t,s) + e∗(t,s)

Multiplicative endemic component

h(t,s) = exp�

oξ(s) + β′zτ(t),ξ(s)�

Piecewise constant function on a spatio-temporal grid{C1, . . . , CD}× {A1, . . . , AM} with time interval index τ(t)and region index ξ(s)Region-specific offset oξ(s), e.g., log-population densityEndemic linear predictor β′zτ(t),ξ(s) includes discretisedtime trend and exogenous effects, e.g., influenza cases

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λ∗(t,s) = h(t,s) + e∗(t,s)

Additive epidemic (self-exciting) component

e∗(t,s) =∑

j∈∗(t,s;ϵ,δ)eηj gα(t − tj) ƒσ(s− sj)

Individual infectivity weighting through linear predictorηj = γ′mj based on the vector of unpredictable marksPositive parametric interaction functions, e.g.,

ƒσ(s) = exp�

− ‖s‖2

2σ2

�

and gα(t) = e−αt

Set of active infectives depends on fixed maximumtemporal and spatial interaction ranges ϵ and δ

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Marked extension with event type

Motivation: joint modelling of both finetypes of IMDAdditional dimension K = {1, . . . , K} for event type κ ∈ K

Marked CIF

λ∗(t,s, κ) = exp�

β0,κ + oξ(s) + β′zτ(t),ξ(s)�

+∑

j∈∗(t,s,κ;ϵ,δ)qκj,κ e

ηj gα(t − tj|κj) ƒσ(s− sj|κj)

Type-specific endemic interceptType-specific transmission, qk, ∈ {0,1}, k, ∈ KType-specific infection pressure ηj = γ′mj, κj is part of mj

Type-specific interaction functions, e.g., variances σ2κ

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1 Motivation


3 Inference

4 Data Analysis

5 Summary

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Log-likelihood of the proposed model

Observed spatio-temporal marked point pattern:

=n

(t,s,m) : = 1, . . . , no

Covariate information zτ,ξ on a spatio-temporal grid:

(θ) =

n∑

=1

logλ∗θ(t,s, κ)

−∫ T

0

∫

W

∑

κ∈Kλ∗θ(t,s, κ)dt ds

θ =�

β′0,β′,γ′,σ′,α′

�′

Integration of epidemic component e∗θ(t,s, κ) involves

∫min{T−tj;ϵ}0 gα(t|κj)dt and

∫

�

W∩b(sj;δ)�

−sjƒσ(s|κj)ds

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Numerical log-likelihood maximisation

For a polygonal region R, perform approximation∫

Rƒσ(s)ds ≈

n∑

j=1

j ƒσ(sj)

with fixed evaluation points s1, . . . ,snBenchmark experiment ⇒ two-dimensional midpoint rulewith adaptive bandwidth choice depending on the valueof σ as best trade-off between accuracy and speedRathbun (1996): existence, consistence and asymptoticnormality of a local maximum θ̂ML as T →∞ for fixed WNewton-algorithm using R’s nlminb function withanalytical score function and expected Fisher information

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Goodness-of-fit and simulation

Define residuals

Y = Λ̂∗(t)− Λ̂∗(t−1), = 2, . . . , n,

where Λ̂∗(t) is the cumulative intensity functionIf the estimated CIF describes the true CIF well in thetemporal dimension, then U = 1− exp(−Y)

iid∼ U(0,1)Use the Kolmogorov-Smirnov test and plot the empiricaldistribution function of the U’s to check for deviationsAlternative: compare the observed epidemic withsimulations from the model using Ogata’s modifiedthinning (Daley & Vere-Jones, 2003, Algorithm 7.5.V.)

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1 Motivation


3 Inference

4 Data Analysis

5 Summary

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Data representation: epidataCS classR> library("surveillance")R> # [... loads of data preparation ...]R> imdepi <- as.epidataCS(events, stgrid, W = germany, qmatrix = diag(2))R> print(imdepi, n=5, digits=5)

History of an epidemicObservation period: 0 -- 2562Observation window (bounding box): [4034.1, 4670.4] x [2686.7, 3543.2]Spatio-temporal grid (not shown): 366 time blocks, 413 tilesTypes of events: 'B' 'C'Overall number of events: 636

coordinates ID time tile type eps.t eps.s age sex BLOCK103 (4112.19, 3202.79) 1 0.99 05554 B 30 200 17 male 1402 (4122.51, 3076.97) 2 1.00 05382 C 30 200 3 male 1312 (4412.47, 2915.94) 3 6.00 09574 B 30 200 34 female 1314 (4202.64, 2879.7) 4 8.00 08212 B 30 200 15 female 2629 (4128.33, 3223.31) 5 23.00 05554 C 30 200 15 male 4

start popdensity influenza0 influenza1 influenza2 influenza3103 0 260.86 0 0 0 0402 0 519.36 0 0 0 0312 0 209.45 0 0 0 0314 7 1665.61 0 0 0 0629 21 260.86 0 0 0 0[....]

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IMD model selection by AIC

Joint analysis of the two finetypesTemporal interaction function g: constantϵ = 30 days, δ = 200 kmDistrict-specific population density as endemic offset

Compare all models composed by subsets of thefollowing terms:

Common or finetype-specific endemic interceptLinear time trend and sine-cosine time-of-year effectsLinear effect of weekly number of influenza casesregistered in the district of a point (lag 0 – lag 3)Epidemic predictor with gender, age (categorized as 0-2,3-18, ≥19), finetype and age-finetype interactionSpatial interaction function ƒ : Gaussian or constant

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Example code

R> fit <- twinstim(+ endemic = ~ 1 + offset(log(popdensity)) + I(start/365) ++ sin(start*2*pi/365) + cos(start*2*pi/365),+ epidemic = ~ 1 + type + agegrp,+ siaf = siaf.gaussian(1),+ tiaf = tiaf.constant(),+ data = imdepi, subset = !is.na(agegrp),+ nCub = 36, nCub.adaptive = TRUE,+ optim.args = list(par = startvalues), model = TRUE+ )

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Model summary (1)

R> toLatex(summary(fit), digits=2, withAIC=FALSE)

Estimate Std. Error z value P(|Z| > |z|)h.(Intercept) −20.365 0.087 −233.5 < 2 · 10−16h.I(start/365) −0.049 0.022 −2.2 0.03

h.sin(start*2*pi/365) 0.262 0.065 4.0 6 · 10−05h.cos(start*2*pi/365) 0.267 0.064 4.1 3 · 10−05

e.(Intercept) −12.575 0.313 −40.2 < 2 · 10−16e.typeC −0.850 0.257 −3.3 0.001

e.agegrp[3,19) 0.646 0.320 2.0 0.04e.agegrp[19,Inf) −0.187 0.432 −0.4 0.67

e.siaf.1 2.829 0.082

endemic: common intercept, no influenza effect

epidemic: no gender effect, no age-finetype interaction, Gaussian ƒ

Basic reproduction numbers

μ̂B = 0.25 (95% CI: 0.19− 0.34)μ̂C = 0.11 (95% CI: 0.07− 0.17)

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Model summary (2)

R> intensityplot(fit, which = "total intensity", aggregate = "time",+ types = 1, col = "orangered", ylim = c(0,0.3))

B:P1.7-2,4:F1-5

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roce

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total intensityendemic intensity

C:P1.5,2:F3-3

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Model summary (3)0.

40.

60.

81.

01.

21.

41.

6

Time

Mul

tiplic

ativ

e ef

fect

2002 2004 2006 2008

point estimate95% Wald CI

Typical IMD peak in late Februaryand minimum in August

0 50 100 150 200

0.0

0.2

0.4

0.6

0.8

1.0

Distance ||s − s j|| from hosteγ̂ C

I C(κ

j) f σ̂(||

s−

s j||)

point estimate type Bpoint estimate type C95% Wald CI for type B95% Wald CI for type C

Effective interaction range ≈ 50 km

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Goodness-of-fit (residual analysis)

R> checkResidualProcess(fit, plot=1)

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0.0

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0.4

0.6

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1.0

u(i)

Cum

ulat

ive

dist

ribut

ion

deterministic tie-breaking

0.0 0.2 0.4 0.6 0.8 1.0

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Cum

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ribut

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U(0,1)-scheme

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Goodness-of-fit (simulation)

R> simulate(fit, nsim = 100,+ data = imdepi,+ tiles = districts,+ W = germany)

Compare observed7-year incidences with(2.5%, 97.5%)quantiles from 100simulations from thefitted CIF model

Many excess districtsaround Aachen at theborder to theNetherlands

Edge effects hidepotentialtransmissions acrossthe border

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Summary

twinstim is a comprehensive framework for themodelling, inference and simulation of generalself-exciting spatio-temporal point processes, e.g.,epidemics, forest fires, residential burglaries, riots,. . . Details in Meyer, Elias & Höhle (2012)

. . . and most importantly . . .

The twinstim implementation is available in the popular Rpackage surveillance (Höhle, Meyer & Paul, 2012)

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Acknowledgements

Michael Höhle, Robert Koch Institute, for fruitfulcollaborationJohannes Elias and Ulrich Vogel, University of Würzburg,for supplying the IMD data and for discussions on themicrobiological aspectsLudwig Fahrmeir, Ludwig-Maximilians-UniversitätMünchen, for providing helpful suggestions andcommentsStephen Price for inviting me to this seminar and forproviding an interesting applicationThe Munich Center of Health Sciences and the SwissNational Science Foundation for financial support

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Literature

Daley, D. J. & Vere-Jones, D. (2003). An introduction to the theory of point processes(2nd ed., Vol. I: Elementary Theory and Methods). New York: Springer-Verlag.

Höhle, M. (2009, December). Additive-multiplicative regression models forspatio-temporal epidemics. Biometrical Journal, 51(6), 961–978. doi:10.1002/bimj.200900050

Höhle, M., Meyer, S. & Paul, M. (2012). surveillance: Temporal and spatio-temporalmodeling and monitoring of epidemic phenomena [Computer software manual].Retrieved from http://surveillance.r-forge.r-project.org/ (R packageversion 1.4-2)

Meyer, S., Elias, J. & Höhle, M. (2012). A space-time conditional intensity model forinvasive meningococcal disease occurrence. Biometrics, 68(2), 607–616. doi:10.1111/j.1541-0420.2011.01684.x

Ogata, Y. (1998, June). Space-time point-process models for earthquake occurrences.Annals of the Institute of Statistical Mathematics, 50(2), 379–402. doi:10.1023/A:1003403601725

Rathbun, S. L. (1996). Asymptotic properties of the maximum likelihood estimator forspatio-temporal point processes. Journal of Statistical Planning and Inference, 51(1),55–74. doi: 10.1016/0378-3758(95)00070-4

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http://surveillance.r-forge.r-project.org/

a space-time conditional intensity model for invasive ... · pdf filea space-time conditional...

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