SpaceTime Modeling and Application to Emerging Infectious Diseases

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Space-Time Modeling and Application to Emerging
Infectious Diseases
National Health Research Institutes

• ???

July 26th, 2005
Division of Biostatistics and Bioinformatics
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Outline

• Introduction
• STARMA Models
• Methods for STARMA Modeling and Software IEAST
• Modeling Emerging Infectious Diseases using
  STARMA and IEAST
• Conclusion

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Introduction
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Introduction

• Toblers First Law of Geography
• Everything is related to everything else, but
  near things are more related than distant
  things.

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Introduction

• Biological and ecological processes are often
  organized and correlated in both space and time.
• Why use space-time data and space-time analyses?
• Various space-time models
• STKF, KKF, VARMA, STARMA, etc.
• Why STARMA models?
• Is emerging infectious diseases the only
  application?

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Scope of the Work

• An efficient and robust STARMA modeling method
• Space-time extensions of optimization algorithm
  and model fitness measures
• Refinement of the space-time modeling procedure
• Software development -- IEAST
• The first general-purpose STARMA modeling and
  analysis software
• Integrated Environment for Analyzing STARMA
  models
• Application to the spread of WNV in an epidemic
  in Detroit
• Modeling and analysis of Dead Crow Data
• Modeling and analysis of Human Case Data
• Cross analysis of Human Case Data and Dead Crow
  Data
• Statistical inferences from these space-time
  analyses

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STARMA Models
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Space-Time Variables Evolving over Time

• zt,x some ecological variable at spatial
  coordinates vector x at time t. zx forms a time
  series for location x.
• These time series are not independent, but
  influence each other via spatial proximity.

zt,(2,2)
random noise
zt,(1,2)
zt,(2,1)
zt,(0,0)
time
X
Y
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General STARMA Models

• The general STARMA model has the stochastic
  equation
• Model types
• STAR model (when ?k,b0)
• STMA model (when ?k,b0)
• Mixed model (when ?k,b ? 0 and ?k,b ? 0).

----- AR terms ----- ----- MA
terms ----- The strengths of the autoregressive
components is measured by ?k,b and the strengths
of shared moving average stochastic inputs are
?k,b.
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A Useful Form for STARMA Modeling

• By introducing the spatial weight matrices W(l),
  we can express the general STARMA model as the
  following form
• This is the equation actually used for the
  implementation of IEAST and applications.

where l spatial lag, k temporal lag zt is
the observation vector at time t W(l) is the
weight matrix for l-th order ?kl are the
parameters of autoregressive terms ?kl are the
parameters of moving average terms et is the
random noise vector at time t.
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Spatial Correlation Structure and Weight Matrices

• Spatial weight matrices are used to construct the
  spatial correlation structure among locations.
• The following ordering is an example of the
  definition of spatial correlation structure (up
  to 4th order neighbors) in 2D system.

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Some Limitations of STARMA Modeling

• Raster based
• Requires massive amount of space-time data
• Models generally may not be fully mechanistic
• Assumptions
• Stationarity
• Spatial Regularity
• Effects are constant
• Effects are linearly correlated

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Methods for STARMA Modeling and Software IEAST
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Box-Jenkins Modeling Method
Data
Model Identification
Parameter Estimation
Modify Model
Diagnostic Check
No
Good?
Yes
End
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Model Identification

• To determine the model type and orders.
• Conventionally, space-time autocorrelations (i.e.
  STACF/STPACF) are used (Pfeifer and Deutsch,
  1980).
• In this research, space-time extensions of model
  fitness measures (i.e. AIC, BIC) are used to
  assist identification when the method above does
  not work. These measures are more objective and
  computationally efficient.

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Model Identificationusing Space-Time
Autocorrelation Functions

• Example 1 STAR (MaxT2, MaxS1)
• STACF tails-off
• STPACF cuts-off at T-lag2 S-lag1
• Example 2 STMA (MaxT1, MaxS1)
• STACF cuts-off at T-lag1 S-lag1
• STPACF tails-off

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Model Identification using Space-Time
Autocorrelation Functions
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Model Identification using Model Fitness
Measures
Accuracies (number in red) of model type
selection using (1)Variance of residuals, (2)AIC,
(3)BIC, and (4)AICBIC based on 150 Monte Carlo
simulated datasets
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Parameter Estimation

• To calculate coefficients of a candidate model
  for given model type and orders.
• Two methods needed for two kinds of models
• Linear models (i.e. STAR) Linear ML estimator.
• Non-linear models (i.e. STMA and Mixed)
  Multi-variate nonlinear optimization.
• The multi-variate and non-linear nature raises
  problems while in optimization
• Converge to local optima
• Very time-consuming
• A good starting point is crucial for optimization
• Extra step Pre-estimation
• Space-time extended Hannan-Rissanen Algorithm is
  used.

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Diagnostic Check

• To decide the adequacy of a candidate model for
  representing the given data.
• Methods
• Variance of residuals
• Space-time autocorrelations of residuals
• Significance testing of parameters
• Space-time extension of AIC/BIC

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Modeling Procedures
Data
Model Identification
Parameter Estimation
Modify Model
Diagnostic Check
No
Good?
Yes
End
Box-Jenkins method
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Software for STARMA Modeling -- IEAST

• Developed using GNU Octave v2.1.40 and able to be
  used under various popular OS, e.g. MS Windows,
  Mac OS, Unix.
• Two interfaces menu-driven mode and programming
  mode.
• Features
• True spatio-temporal analysis software
• Analyzing 2D lattice space-time datasets
• Full configurability
• Programming environment
• Improved estimation algorithms
• Improved diagnostic measures
• Estimation of spatial correlation structure
• Cross correlation analysis
• 2D/3D plotting abilities

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IEAST Menu-Driven Mode vs Programming Mode
In menu-driven mode, users can conduct the
modeling procedure by selecting a series of
commands/options from the menu hierarchy.
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IEAST Menu-Driven Mode vs Programming Mode
In programming mode, a set of sophisticated
instructions can be used to compose programs to
control the modeling flow and to conduct
statistical analyses.
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Modeling Emerging Infectious Diseases using
STARMA and IEAST
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State of Art for Statistical Analyses of Emerging
Infectious Diseases

• As far as we know, no true spatial-temporal
  statistical models and methods have been used.
• Space-time cluster analysis available
  (Theophilides et al, 2003 Mostashari et al,
  2003 Hoebe et al, 2004)
• Spatial models available (Watson et al, 2004).
• Temporal models available.

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Limitations of Simply Observing How a Spatial
Distribution Changes over Time

• For example, expansion of the leading edge of a
  disease range.
• Is the disease spreading directly over long
  distances but infrequently, or over short
  distances frequently?
• This is important for projecting the future
  spread.

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STARMA Has Potential for the Early
Characterization of Infectious Diseases.

• STARMA acts as a prism. Can filter the
  spatial-temporal correlations into direct effects
  with known magnitude and spatial and temporal
  lags.
• Not generally a complete, mechanistic model, but
  puts critical constraints on models.

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West Nile Virus

• The West Nile Virus (WNV) was first detected
  in a woman with a mild fever in the West Nile
  District of Uganda in 1937. Since then WNV has
  been spreading to North Africa, Europe, West and
  Central Asia, and the Middle East.

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West Nile Virus in the United States

• Outbreak in NYC in Sep 1999. Vector is Culex
  mosquitoes.
• Wild birds (89 are American crows) are the
  principal hosts. Humans, horses, etc. are
  incidental hosts.
• The incidence rate among crows is high. Infected
  crow almost always die (68).

• Surveillance of Dead crows has been used as an
  indicator of WNV epidemic.

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Dead Crow Data (DCD) Human Case Datasets (HCD)
in 2002

• Time Summer in 2002 (AprilOctober)
• Place Detroit metro area (Oakland, Macomb, and
  Wayne)
• DCD were collected systematically before and
  during an outbreak among humans. Data mainly
  consisted of locations and dates of reported
  public sightings.
• HCD were obtained from clinicians in Michigan.
  Data on address of residence and date of onset of
  disease were obtained from the case-patient or
  attending physician through telephone interviews.

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Two Datasets Collected in 2002
Human Cases
Interview
GIS - ArcMap
Toll-free
Dead Crows
Longitude/Latitude
Data Cleaning Geocoding
WWW pages
From www.rci.rutgers.edu/ insects/crowid.htm
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Space-Time Analysis for Dead Crow Data
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The Dead Crow Data

• Totally, 1817 dead crow sightings scattered
  within the three counties (red lines), spanning
  28 weeks.
• Covered area (after truncation) a rectangular
  area of 31.6x25.8 mi
• Divide the covered area into 10x10 cells. Cell
  size 3.16x2.58mi

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Spatial Correlation Structure and Trends

• Spatial correlation structure (uniform weighting)
• Preprocessing
• Remove spatio-temporal trend
• Spatial trend 4th order polynomial regression
  trend surface
• Temporal trend averaging over space.
• Remove mean

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Model Identification STACF
Tail-off
STACF tails-off
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Model Identification STPACF
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Parameter Estimation
The parameters (?ts) of this STAR model can be
estimated in IEAST by linear maximum likelihood
estimator.

• Values in dark blue are nominally significant at
  the 0.001 level.
• Values in light blue are nominally significant
  at the 0.01 level.

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Diagnostic Check

• Statistical significance of parameters
• The probabilities P that ?ts are not significant
  are
• Residuals autocorrelations

STACF
STPACF
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Interpretations for the DCD Analysis

• STAR(3,4) model is the best-fitted one.
• The max. of spatial and temporal lags that are
  important are still smaller. S2 (or 6.4 km) and
  T2 weeks.
• Compare S1 to S2. Value for S1 is much
  largercell boundary length effects.
• The virus is not spreading very far very fast.
  Crows are not much spreading the virus spatially,
  though they probably are amplifying it locally.
• Negative Autoregressive Effect At S1, and T2,3.

• Appears to be a real effect.
• May be due to crow population depletion.
• Suggests there is a mixture of two STAR
  processes, the dominant one reflecting
  probability of infection, the other an echo
  effect from depletion.

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Additional Analyses and Results

• Additional Analyses
• Using 20x20 and other cell configurations
• Using different lag structures Pfeiffers vs.
  Ring structure
• Using various polynomials for Spatial de-trending
• Using sub-sample of the data
• Results
• Consistent over various methods of spatial
  de-trending, except high order polynomials
  resulted in smaller AR.
• Consistent AR values using different lag
  structures and cell sizes.
• Consistent implied spatial and temporal scales
  over which there are significant or substantial
  AR effects

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Distances for Which There Are Significant Spatial
Correlation

• Based on different cell configurations 10x10,
  16x16, and 20x20
• The effective correlated area in the modeling
  result is consistently about 10.75 km regardless
  of cell sizes.

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Alternative Spatial Correlation Structures
Ring structure
Pfeifers
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Space-Time Analysis for Human Case Data
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Human Case Data

• Over 500 human cases spanning 13 weeks
• Date of onset-converted to week
• Home addresses (names stripped)-converted to
  cell, same as for DCD.
• Used same arrays of cell sizes and spatial
  correlation structures as for DCD.
• Same spatial and temporal de-trending method

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Model Identification STACF
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Model Identification STPACF
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Parameter Estimation
Spatial lags
Temporal lags (weeks)

• Values in dark blue are nominally significant at
  the 0.001 level.
• Values in light blue are nominally significant
  at the 0.01 level.

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Diagnostic Check

• Residuals STACF and STPACF

STACF
STPACF
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Interpretations for the HCD Analysis

• Most people are getting infected at or near their
  homes.
• The incidences are highly autocorrelated in space
  and time.
• The distribution or probability of infection is
  highly localized.
• The WNV load and probability of human infection
  is spreading slowly, in the sense of not
  spreading very far very fast.
• Suggests localized spraying could reduce cases.
• Without depletion effect, the human case data
  show positive and significant above zero for
  T-lag2 and S-laggt1, esp. at S-lag1.

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Space-Time Cross Analysis for HCD and DCD
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Space-Time Data HCD and DCD

• The areas for cross analysis are same for both
  datasets.
• The configuration is again 10x10 and spanning 28
  weeks.
• Cell size is 6.31x6.31 km.

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Both Temporal Epidemic Curves

• Dead crow reported is leading human cases in time.

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Space-Time Cross Correlations
-3
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Interpretations for Space-Time Cross Correlations

• Drop smoothly to zero spatially and temporally.
• Very large (as high as 0.7).
• Across all spatial lags, the max. cross
  correlations are aligned at 3 weeks.
• The cross correlations at spatial lag 1 is
  slightly greater than at spatial lag 0.
• When temporal lag decreases to 8 or below, the
  correlations between these two datasets are
  negligible (lt0.1).
• When spatial lag increases up to 10, the cross
  correlations are reduced to as low as 0.2.

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Is the Cross Correlations Spurious?
The autocorrelation of the DCD can spuriously
contribute to cross correlations. To eliminate
this effect, both datasets were pre-whitened
before calculating cross correlations.

• The result shows that the real cross
  correlations are much larger than the spurious
  components.

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Summary for Modeling the Spread of WNV

• Crows are not spreading the disease spatially
  very far very fast.
• Spread is very localized, perhaps other animals
  or the mosquitoes themselves are spreading it
  spatially.
• Humans are being infected largely at or near
  their homes.
• Both crows and humans appear to be responding to
  local viral loads.
• Dead crow findings precede human cases by two to
  three weeks. Dead crows can be a good indicator
  of human epidemics.

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Conclusion

• It appears that STARMA modeling could be an
  important tool of the early characterization of
  many emerging and re-emerging infectious disease
  epidemics.
• During the course of an epidemic, it could be
  used (in principle) for forecasting, under
  existing conditions or under potential courses of
  action.
• While not generally a mechanistic model, STARMA
  does inform spatial and temporal scales of
  spread, hence places constraints on mechanistic
  models (which otherwise may have too many
  parameters).

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Funding Acknowledgements

• Michigan Agricultural Experiment Station,
  Michigan State University.
• Center for Emerging Infectious Diseases, Michigan
  State University.
• Centers for Disease Control and Prevention, USA.

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• Thanks for your attention!
• Questions?

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References

• C.J.P.A. Hoebe, H. de Melker, L. Spanjaard, J.
  Dankert, and N. Nagelkerke. Space-time cluster
  analysis of invasive meningococcal disease,
  Emerging Infectious Disease, Vol.10, No. 9,
  p1621-1626, 2004.
• C.N. Theophilides, S.C. Ahearn, S. Grady, and M.
  Merlino. Identifying West Nile virus risk areas
  The dynamic continuous-area space-time system.
  American Journal of Epidemiology, 157843-854,
  2003.
• J. Watson, R. Jones, K. Gibbs, and W. Paul. Dead
  crow reports and location of human West Nile
  virus cases, Chicago, 2002. Emerging Infectious
  Diseases, 10(5)938-940, 2004.
• F. Mostashari, M. Kulldorff, J.J. Hartman, J.R.
  Miller, V. Kulasekera. Dead bird clustering A
  potential early warning system for West Nile
  virus activity. Emerging Infectious Diseases,
  9641-646, 2003.