How to extract the basic components of epidemiological relevance

from a time-series?

Wladimir J. AlonsoDirector of Origem Scientifica (Brazil)

Contractor and Research Fellow atFogarty International Center / NIH (US)

www.epipoi.info

Brazilian dataset of deaths coded as pneumonia and influenza

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

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

Series to be analyzed

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

Average

Many times this information is

all we need!

mortality at time t

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

Linear trend

• Better now!

Trend (linear)

We can use this information (e.g. is the disease increasing/decreasing? -

but then the data needs to be incidence)

Mortality at time t

Linear trendsMean Mortality

Trend (with quadratic term too)

Mortality at time t

Quadratic trends

2

210ttY

t

• 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

Getting rid of the trend

Blue line: “detrended series”

But let’s keep the graphic of the original series for illustrative purposes

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

Mortality at time t

Linear and quadratic trends

2

210ttY

t

Mean Mortality

We can see some rhythm…

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

Linear and quadratic trends

2

210ttY

t

Mean Mortality

Mortality at time t

Jean Baptiste Joseph Fourier(1768 –1830)

Fourier series

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

It is a way to represent a wave-like function as a combination of simple sine waves

Before modeling cycles:

…so, remembering, these are the residuals before Fourier

Linear and quadratic trends

2

210ttY

t

Mean Mortality

Mortality at time t

… and now with the incorporation of the annual harmonic

Mortality at time t

trends

Annual harmonicMean

Mortality

or with the semi-annual harmonic only?

Mortality at time t

trends

semiannual harmonicMean

Mortality

Much better when the annual + semi-annual harmonics are considered together!

Mortality at time t

trends

Annual and semi-annual harmonicsMean

Mortality

Although not much difference when the quarterly harmonic is added…

Mortality at time t

trends

Periodic (seasonal) componentsMean

Mortality

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, let’s extract some interest parameters (remember, we always need a “number” to compare, for instance, across different sites)

Timing and Amplitude

average seasonal signature of the original

series

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

Latit

ude

(deg

rees

)

5

0

-5

-10

-15

-20

-25

-30

-35

Amplitude of the major peak (%)0 10 20 30 40 50 60 70 80 90

(p < 0.001)

The seasonal component was found to be most intense in southern states, gradually attenuating towards central states (15oS) and remained low near the Equator

Latit

ude

(deg

rees

)

5

0

-5

-10

-15

-20

-25

-30

-35

Amplitude of the major peak (%)0 10 20 30 40 50 60 70 80 90

(p < 0.001)

5

0

-5

-10

-15

-20

-25

-30

-35

Phase of the major peak (months of the year)J F M A M J J A S O N D

Latit

ude

(deg

rees

)

(p < 0.001)

Variations in peak timing of influenza with latitude

5

0

-5

-10

-15

-20

-25

-30

-35

Phase of the major peak (months of the year)J F M A M J J A S O N D

Latit

ude

(deg

rees

)

(p < 0.001)

Peak timing was found to be structured spatio-temporally: annual peaks were earlier in the north, and gradually later

towards the south of Brazil

5

0

-5

-10

-15

-20

-25

-30

-35

Phase of the major peak (months of the year)J F M A M J J A S O N D

Latit

ude

(deg

rees

)

(p < 0.001)

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.

But can we still improve the model?

Mortality at time t

trends

Periodic (seasonal) componentsMean

Mortality

Yes, and in some cases we should,Mostly to model excess estimates

e.g. pandemic year

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

Ok, so now we can count what was the impact of the pandemic here right?

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

Residuals after modeling year to year variance

(1.96 SD above model)

Mortality at time t

trends

Periodic (seasonal) components error term

Mean Mortality

)()3

2sin()

3

2cos()

6

2sin()

6

2cos()

12

2sin()

12

2cos( 332211

2210 t

ttttttttYt

This is a measure of excess that is much closer to the real impact of the pandemic

Geographical patterns in the severity of pandemic mortality in a large latitudinal range

Schuck-Paim et al 2012 PLoS One

You can perform all these analyses in epipoi software. If you do, please cite the following reference:

Alonso  &  McCormick (2012) A user friendly analytical tool for extraction of temporal and spatial parameters from epidemiological time-series.  BMC Public Health 12:982

www.epipoi.info

Example from diarrhea mortality in Mexico (1979-1988)

Alonso WJ et al Spatio-temporal patterns of diarrhoeal mortality in Mexico. Epidemiol Infect 2011 Apr;1-9.

quantitative and qualitative change of diarrhea in Mexico 1917-2001

Winter peaks

Summer 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

State-specific rates, sorted by the latitude of their capitals, from north to south (y axis)

Timing of annual peaks (1979-1988)

First peak in the Mexican capital !

Major Annual Peaks of diarrhea of the period 1979-1988 in Mexican states, sorted by their latitude

Monthly climatic data were obtained from worldwide climate maps generated by the interpolation of climatic information from ground-based meteorological stations

Climatologic factors

Mitchell TD, Jones PD. An improved method of constructing a database of monthly climate observations and associated high-resolution grids. International Journal of Climatology 2005;25:693-712. (data at: http://www.cru.uea.ac.uk/cru/data/hrg/)

Early peaks in spring in the central region of Mexico (where most of the people lives) followed by a decrease in summer

Early peaks in the monthly average maximum temperature in the central region of Mexico followed by a decrease in summer too !

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

Mild summers - with average maximum temperatures

below 24 oC

Thanks! wladimir.alonso@nih.gov