How to extract the basic components of epidemiological relevance from a time-series?

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How to extract the basic components of
epidemiological relevance from a time-series?
Wladimir J. Alonso Director of Origem Scientifica
(Brazil) Contractor and Research Fellow
at Fogarty International Center / NIH (US)
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www.epipoi.info
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Brazilian dataset of deaths coded as pneumonia
and influenza

• We are going to extract as much information as
  possible from this series

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Brazilian dataset of deaths coded as pneumonia
and influenza

• Example of analyses performed in Schuck-Paim et
  al 2012 Were equatorial regions less affected by
  the 2009 influenza pandemic? The Brazilian
  experience. PLoS One.
• Data source Department of Vital Statistics from
  the Brazilian Ministry of Health

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Series to be analyzed

• Typical epidemiological time series from where to
  obtain as many meaningful and useful parameters
  as possible

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Average
mortality at time t

• Many times this information is
• all we need!

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Average

• But, it still leaves much of the variation
  (residuals) of the series unexplained
• the first of which seems to be an unbalanced
  between the extremities

mortality at time t
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Linear trend

• Better now!

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Trend (linear)
Mortality at time t

• We can use this information
• (e.g. is the disease increasing/decreasing? -
  but then the data needs to be incidence)

Mean Mortality
Linear trends
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Trend (with quadratic term too)

• Better definition
• It gets more complicated as a parameter to be
  compared across time-series
• But better if our purpose is eliminate the
  temporal trend

Mortality at time t
Quadratic trends
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Getting rid of the trend

• Blue line detrended series

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But lets keep the graphic of the original series
for illustrative purposes

• Clearly, there are still other interesting
  epidemiological patterns to describe

Mortality at time t
Mean Mortality
Linear and quadratic trends
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We can see some rhythm

• The block of residuals alternates cyclically
• Therefore this is something that can be
  quantified using few parameters

Mortality at time t
Mean Mortality
Linear and quadratic trends
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Jean Baptiste Joseph Fourier(1768 1830)
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Fourier series
It is a way to represent a wave-like function as
a combination of simple sine waves

• Some real world applications
• Noise cancelation
• Cell phone network technology
• MP3
• JPEG
• "lining up" DNA sequences
• etc etc

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Before modeling cycles
Mortality at time t

• so, remembering, these are the residuals before
  Fourier

Mean Mortality
Linear and quadratic trends
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and now with the incorporation of the annual
harmonic
Mortality at time t
Annual harmonic
Mean Mortality
trends
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or with the semi-annual harmonic only?
Mortality at time t
semiannual harmonic
Mean Mortality
trends
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Much better when the annual semi-annual
harmonics are considered together!
Mortality at time t
Annual and semi-annual harmonics
Mean Mortality
trends
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Although not much difference when the quarterly
harmonic is added
Mortality at time t
Periodic (seasonal) components
Mean Mortality
trends
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average seasonal signature of the original series

• We obtained therefore the average seasonal
  signature of the original series (where
  year-to-year variations are removed but seasonal
  variations within the year are preserved)
• Now, lets extract some interest parameters
  (remember, we always need a number to compare,
  for instance, across different sites)

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Timing and Amplitude
average seasonal signature of the original series
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Variations in relative peak amplitude of
pneumonia and influenza coded deaths with
latitude Alonso et al 2007 Seasonality of
influenza in Brazil a traveling wave from the
Amazon to the subtropics. Am J Epidemiol
Latitude (degrees)
(p lt 0.001)
0 10 20 30 40 50 60
70 80 90
Amplitude of the major peak ()
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The seasonal component was found to be most
intense in southern states, gradually attenuating
towards central states (15oS) and remained low
near the Equator
Latitude (degrees)
(p lt 0.001)
0 10 20 30 40 50 60
70 80 90
Amplitude of the major peak ()
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Variations in peak timing of influenza with
latitude
(p lt 0.001)
Latitude (degrees)
J F M A M J J A S
O N D
Phase of the major peak (months of the year)
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Peak timing was found to be structured
spatio-temporally annual peaks were earlier in
the north, and gradually later towards the south
of Brazil
(p lt 0.001)
Latitude (degrees)
J F M A M J J A S
O N D
Phase of the major peak (months of the year)
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Such results suggest southward waves of influenza
across Brazil, originating from equatorial and
low population regions and moving towards
temperate and highly populous regions in 3
months.
(p lt 0.001)
Latitude (degrees)
J F M A M J J A S
O N D
Phase of the major peak (months of the year)
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But can we still improve the model?
Yes, and in some cases we should, Mostly to model
excess estimates
e.g. pandemic year
Mortality at time t
Periodic (seasonal) components
Mean Mortality
trends
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Residuals after excluding atypical (i.e.
pandemic) years from the model

• To define what is normal it is necessary to
  exclude the year that we suspect might be
  abnormal from the model

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Ok, so now we can count what was the impact of
the pandemic here right?
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No! (unless you consider all the other anomalies
pandemics (and anti-pandemics)

• That is why we need to include usual residual
  variance in the model, and calculate excess
  BEYOND usual variation

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Residuals after modeling year to year variance
(1.96 SD above model)
Mortality at time t
Periodic (seasonal) components
error term
Mean Mortality
trends
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This is a measure of excess that is much closer
to the real impact of the pandemic
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Geographical patterns in the severity of pandemic
mortality in a large latitudinal range
Schuck-Paim et al 2012 PLoS One
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You can perform all these analyses in epipoi
software. If you do, please cite the following
referenceAlonso   McCormick (2012) A user
friendly analytical tool for extraction of
temporal and spatial parameters from
epidemiological time-series.  BMC Public
Health 12982
www.epipoi.info
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Example from diarrhea mortality in Mexico
(1979-1988)
Alonso WJ et al Spatio-temporal patterns of
diarrhoeal mortality in Mexico. Epidemiol Infect
2011 Apr1-9.
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quantitative and qualitative change of diarrhea
in Mexico 1917-2001
Summer peaks
Winter peaks
Gutierrez et al. Impact of oral rehydration and
selected public health interventions on reduction
of mortality from childhood diarrhoeal diseases
in Mexico. Bulletin of the WHO 1996 Velazquez et
al. Diarrhea morbidity and mortality in Mexican
children impact of rotavirus disease. Pediatric
Infectious Disease Journal 2004 Villa et al.
Seasonal diarrhoeal mortality among Mexican
children. Bulletin of the WHO 1999
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State-specific rates, sorted by the latitude of
their capitals, from north to south (y axis)
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Timing of annual peaks (1979-1988)
First peak in the Mexican capital !
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Major Annual Peaks of diarrhea of the period
1979-1988 in Mexican states, sorted by their
latitude
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Climatologic factors
Monthly climatic data were obtained from
worldwide climate maps generated by the
interpolation of climatic information from
ground-based meteorological stations
Mitchell TD, Jones PD. An improved method of
constructing a database of monthly climate
observations and associated high-resolution
grids. International Journal of Climatology
200525693-712. (data at http//www.cru.uea.ac.
uk/cru/data/hrg/)
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Early peaks in spring in the central region of
Mexico (where most of the people lives) followed
by a decrease in summer
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Early peaks in the monthly average maximum
temperature in the central region of Mexico
followed by a decrease in summer too !
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Mild summers - with average maximum temperatures
below 24 oC
The same climatic factors that enabled a dense
and ancient human occupation in the central part
of Mexico prevent a strong presence of bacterial
diarrhea and the observed early peaks
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Thanks! wladimir.alonso_at_nih.gov