Slide 1

Wavelets and excess disease models for analysis of time series dataDan WeinbergerFogarty International CenterNational Institutes of Health

Analyzing time series dataWavelets: evaluate timing of peaks and dominant frequencyRegression models: estimate seasonal baseline and calculate excess incidence

Part 1: WaveletsMotivating example: Measles

Figures from Grenfell et al, Nature 2001Does the frequency of the measles epidemics change after vaccination?

Shift in dominant frequencyTimeWavelets: a powerful solutionIdentification of the dominant frequencies in a series (ie annual, monthly) at each specific timeDetermine the phase of these cycles and compare seriesFilter to smooth a series (remove high frequency noise)And many other applicationsBasic concepts

Higher frequency

Sample time series (wave with 10 unit cycle=sin(2*/10*t)-Wavelets:little waves of a specific shape-slide wavelet along timeseries to determine strength of correlation-repeat, while shrinking and expanding the wavelet -Can use different shapes of wavelets for different situations

Wavelet spectrum of a sine wave10 year cycle dominates

Wavelet spectrum of a sine waveGlobal wavelet=Average across entire time periodGrey area=Cone of influence:Less confidence in this regionColor=power of spectra:Red=higher amplitude at thatfrequency and timeSignificance tested by a permutation testMultiple frequencies

Wavelet with changing frequenciesInterpretation: Wavelength increases from ~0.25 to ~0.5 (from 4 cycles/year to 2 cycles/year) Example: epidemic timing

Example: Using wavelets to extract phase (timing) information

1.5-3 year componentApplying wavelets to real dataStep 1: remove any long term trends from the data (calculate baseline using spline of summer months and then divide by baseline)Step 2: Square root or log-transform the dataStep 3: Use transformed data in wavelet transform, evaluate spectra, extract phase dataNote: it is important to have a complete time series without missing data for the wavelets. Need to have relatively long time series since accuracy of wavelets is poor at the beginning and end of the time seriesPart 2: Excess Disease modelsPart 2: Calculation of excess mortality for seasonal diseasesMotivating question: How can we quantify the annual disease burden attributable to specific pathogens for diseases that show a strong seasonal pattern?Answer: Count excess above a typical seasonal baseline

Thanks to Ccile Viboud and Vivek Charu for contributing slides to this sectionSerfling RegressionStep 1: Define influenza, non-influenza periodStep 2: Set a baseline and threshold (95% confidence interval) for pneumonia during non-influenza periodStep 3: Calculate excess mortality for each yearSum of observed mortality subtracted from the model baseline during epidemic months (when flu deaths cross threshold)

USA P&I Deaths per 100,000 (1972-2006), All Ages 18To illustrate this process: In this graph we have mortality data due to P&I for the US from 1972-2006. The black dots are the actual data points (monthly), and the blue line simply connects them.

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USA P&I Deaths per 100,000 (1972-2006), All Ages 1919While fitting the seasonal baseline, we are interested in periods of time with no influenza activity. We obtain this from viral surveillance data. We know from these data that in the US, the period of May-Nov is associated with little influenza activity. We explicitly account for non-influenza periods by limiting our baseline model to months in which influenza viruses do not circulate. Here, in black, Ive highlighted those months so that you get an idea of how this looks.

USA May-Nov P&I Deaths per 100,000 (1972-2006), All Ages OBTAIN A BASELINE MODEL FROM THESE DATA2020

USA May-Nov P&I Deaths per 100,000 (1972-2006), with Model Baseline, All Ages Full Model: E(Yi) = + 1cos(2 ti/12) + 2sin(2 ti/12) + 3 ti + 4ti2 + 5ti3 + 6ti4 + 7ti5 + iBest-fit Model: E(Yi) = + 1cos(2 ti/12) + 2sin(2 ti/12) + 3ti + 4ti2 + 5ti3 + 6ti4 + i12-month sine and cosine waveto account for baseline seasonal variationsLinear time trendPolynomial time trendTo account for long-term fluctuations 21To give you an idea of the types of terms we include, we have:

An intercept term Two harmonic terms (cos and sin), which capture the sinusoidal nature of a seasonal baseline. Note that if we were working with weekly data, wed use 52 instead of 12. Linear, quadtratic, other polynomial time trends because the time series isnt necessary flat throughout the period. Error term.

There are many methods to fit a best-fit line in this case Ive use backwards selection via F-tests (nested models), we test the null hypothesis that one of the coefficients is zero, and minimize AIC.

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USA P&I Deaths per 100,000 (1972-2006), Model Baseline and Upper 95% Confidence Band, All Ages 22In order to define epidemic periods, we examine those instances where the observed P&I mortality exceeds the 95% CI of the seasonal baseline.

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USA P&I Deaths per 100,000 (1972-2006), Model Baseline and Upper 95% Confidence Band, All Ages Epidemic Months Highlighted in Grey23Here, Ive highlighted those months in greywe term these epidemic months.

To reiterate, these are months in which the observed P&I mortality for the entire population exceeds the 95CI of a seasonal baseline that captures expected P&I mortality during periods of non or low influenza virus activity.

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USA P&I Deaths per 100,000 (1972-2006) and Model Baseline 65-89 Year OldsEpidemic Months (Grey) Defined by All-Ages P&I ModelExcess24Now that we have defined epidemic months, we can repeat the procedure for our specific age groups.

Here we see the data from 65-89 year olds.

Note that we first fit a SEPERATE seasonal baseline to the May-Nov P&I mortality of this age group! We then sum deviations from the observed P&I mortality from this age group, from the seasonal base line of this age group, in epidemic months, highlighted in grey. Recall that epidemic months were defined by the all-ages data.

You can see in red that this is simply the area between the blue and black curves. 24Calculating Excess MortalityMonthly Excess Mortality: For epidemic months (months in which the observed P&I mortality exceeds the upper 95% CI of the model baseline for all-ages):Observed P&I mortality model baseline predicted P&I mortality Seasonal Excess Mortality: For each influenza season (defined as Nov.-May in the US): Monthly excess mortality25There are two types excess mortality we can consider.

Monthly excess mortality is calculated

25Seasonal US Excess Mortality TableSeasonAge GroupNo. of Epidemic MonthsExcess P&I Deaths per 100,000Excess A-C Deaths per 100,0001977/197865-89430.41137.051978/197965-8910.757.401979/198065-89315.6866.851980/198165-89328.45136.711981/198265-8933.6727.351982/198365-89412.6847.831983/198465-8949.5133.901984/198565-89421.38122.61AVERAGES- -3 (median = 3, range = 1-5)41.3478.3726To give you an idea of how seasonal excess mortality table looks like, we see the first eight seasons in the US.

Note that there is some variability in excess P&I deaths, and excess A-C deaths.

Also note that P&I would be considered a less sensitive measure of influenza-related mortality than is A-C. 26Pros and Cons of the Serfling Approach Very flexible: can be used without virological dataespecially useful for data on past pandemicsHowever, need at least three years of data Only works if disease is seasonalNeeds clear periods with no viral activity that can be used to create the baseline Cannot be used as is for tropical countries that have year-round influenza circulation There are techniques to adapt Serfling models for these purposes 27We need at least three years of data in order to fit a seasonal basleine. The method, as described here, only works if the disease is seasonal--27An alternative/complementary approachWhat proportion of pneumonia and influenza hospitalizations can be attributed to influenza? Use regression models with terms for seasonal variation, influenza, RSV (can be viral surveillance data, viral-specific hospitalization codes)A quick review of regressionLinear regression:Y=1x1+ 2x2 + a 1: 1 unit increase in x1 results in 1 increase in YPoisson regressionUsed when Y is a count variable/incidence rate rather than continuousUsually has a skewed distributionMultiplicative Poisson model: Y=e(1x1+ 2x2 + a)If data are not Poisson distributed, use an alternative model, such as negative binomial

Estimation of influenza hospitalization burdenOutcome = weekly pneumonia and influenza hospitalization rate

Explanatory variables=influenza-specific and RSV-specific hospitalizations (proxies of viral activity),

Seasonal estimates of influenza-related hospitalization rates obtained as sum of predicted rates minus baseline rates (influenza covariate set to 0)

30Example: Estimation of influenza hospitalization burden in California in seniors over 65 yrs

31Comparison between Serfling and Poisson regression

Newall, Viboud, and Wood, 2009 Epidemiolo Infect.32Alternative Models for Estimating Influenza BurdenPeri-season modelsUse months surrounding influenza epidemics as baselineARIMA models Estimate seasonal baseline by adjusting for serial autocorrelation. Serfling-Poisson combined modelsSerfling seasonal excess mortality estimates are regressed against seasonal virus prevalence. Takes care of random variations in virus prevalence at small time scales Iterative Serfling models (for non-seasonal data)

33Estimates of influenza-associated mortality were of similar magnitude. Poisson regression models permit the estimation of deaths associated with influenza A and B, but require robust viral surveillance data. By contrast, simple peri-season rate-difference models may prove useful for estimating mortality in countries with sparse viral surveillance data or complex influenza seasonality.33Validity tests for influenza disease burden modelsRegression diagnosticsChecks based on the epidemiology of influenzaA/H3N2 vs A/H1N1/B dominant seasons (2-3 ratio)RSV vs influenza (age!)Higher rates in 65 yrs and overMultiple years: seasons will little influenza circulation very precious! Difficult to estimate disease burden with precisionMild seasonsYoung childrenMiddle age groups

34AcknowledgementCecile Viboud for providing some R program samplesCcile Viboud and Vivek Charu for slides on Serfling regressionhttp://www.ecs.syr.edu/faculty/lewalle/ (Jacques Lewalle) for some ideas on presentationChart10360.1564343343.20791151786.3090167420.3090167423.40673631996.58778482290.45399014513.58778482296.80901652650.58778482293.7431443526.95105618830.70710631213.866024961570.80901652653.95105618836.95105700830.89100610253.99452176596.80901808620.95105618833.99452204336.58778696970.98768815383.95105700836.309019265713.86602628836.00000265360.98768856893.74314612765.69098578170.95105700833.58778696975.41221732390.89100730723.4067387445.19098503330.80901808623.20791411345.04894463170.70710818853.000002653650.58778696972.79209107785.04894217170.45399250942.59326610435.19098035410.30901926572.41221732395.41221088350.15643695492.25685742365.69097821060.00000265362.13397636535.9999946928-0.15643171312.0489446317-0.30901421832.0054785115-0.45398778072.0054776793-0.58778267612.0489421717-0.70710443572.1339723849-0.80901496672.2568520968-0.89100489782.4122108835-0.95105536832.5932588318-0.98768773872.792083291-12.9999946928-0.987688984-0.9510578283-0.8910085119-0.8090196459-0.7071100648-0.5877891165-0.4539948738-0.3090217894-0.1564395758-0.0000053072

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Excess P&I hospitalizations associated with InfluenzaExpected P&I HospitalizationsObserved P&I HospitalizationsModeled BaselineDateHospitalizations per 100,000 persons

Chart337.361115926537.361115926537.602965626936.494595700536.637092889836.637092889835.402283160236.068394992336.460658072836.460658072836.1996318835.575246741235.954529384235.954529384233.616222027835.277964794336.017955495236.017955495234.062737310935.143294761736.502903933436.502903933437.156450343834.985335787535.821734914735.821734914734.700616286835.069205849835.656206175835.656206175834.987661825935.141981272236.143424696336.143424696335.43417710935.423699542636.753680711336.753680711338.0494809135.78109753837.307649603637.307649603637.985693012536.279843891237.628715829237.628715829238.655465937136.838226412638.632821309938.632821309937.539177729337.568507828739.639827810539.639827810539.29334491338.290186767541.004623905541.004623905541.7491789739.212168994741.435156347641.435156347640.569102864740.158788703343.02195048343.02195048340.186375479241.233360135544.347730708344.347730708343.120618768142.45655322346.027655542646.027655542647.490089752843.672674904647.381860491947.381860491948.98910534645.158992572349.456751477449.456751477447.713347394346.717041129450.132416803150.132416803152.656909457248.317423619654.330386490254.330386490256.516077261250.637114054355.754862285855.754862285857.47289572551.848751202762.171165978962.171165978969.720172061554.369221021169.232772835169.232772835194.342300529756.051879628475.837167467775.8371674677103.081242498956.77924532476.657733516176.657733516198.488513872856.819288903372.865712702472.865712702477.087674232757.496285822372.804535106572.804535106575.49297679357.25575016572.040871905872.040871905874.057749097357.420005099171.516891118271.516891118274.823203868457.066099941566.956603407666.956603407672.71820324856.750427288264.082308628764.082308628769.752066010355.977944650960.084982888460.084982888462.416457787955.688509930858.817945031258.817945031253.518046074654.819617223257.624560524457.624560524452.943954996454.250215850155.562645163755.562645163752.050924430253.4912492754.072024779454.072024779447.936605035952.34786288853.42021388153.42021388148.893423499751.371252101751.75533214251.75533214247.649559496750.329500534150.323798136650.323798136650.328651195349.211593501949.190649454949.190649454949.467514577947.888956753248.000494698448.000494698444.651528310246.834849749346.368210507346.368210507343.248194563345.546556541145.491426277345.491426277344.587740412644.436338922444.141428498144.141428498144.874785951743.359233554643.342726510743.342726510741.653497123642.29019089542.130148237142.130148237146.788422879341.245094480541.250629341541.250629341540.728572608640.338950516940.343114488540.343114488542.610315587439.539760779739.461298872539.461298872543.726603795238.805375325339.124464268539.124464268540.98372419938.131731176338.452913085138.452913085138.514885972237.561070694638.00194877438.00194877440.044245301737.120565645437.513313343837.513313343834.519824866536.766309554337.438185890337.438185890336.205241270536.569878172637.466483556137.466483556137.984291919136.475038052837.360912408137.360912408134.270133547436.494396902337.284041136137.284041136133.396213930536.5824588237.765866910537.765866910535.643435802536.807606483438.179622444938.179622444938.233983238237.169305719238.450921296438.450921296437.953080504237.643159307439.429522899639.429522899635.300110238738.214936743939.663836696739.663836696737.204006546938.917473224240.401524466440.401524466439.919399642239.685601370541.445782599341.445782599340.855742088840.575105729742.724356364742.724356364743.102963960841.593775990543.809147663243.809147663240.418782280442.602229170546.247087539446.247087539442.291467173743.73402297246.479097888846.479097888846.036836960344.896596106648.804369980448.804369980449.438881183146.203944227249.371697620849.371697620844.975648854147.424994584250.901887213350.901887213352.591234086948.731212377852.922233691152.922233691155.088147277950.046441994954.528074969954.528074969954.838455958851.335052451454.6284427754.6284427753.68363360852.59187428656.461673320556.461673320568.977226903253.812835959358.364788354458.364788354481.212101539555.254958255759.866256470559.866256470574.564070168356.297699288260.616585904660.616585904663.889766276457.51517036563.360588098963.360588098959.613802436758.593340289264.203809351364.203809351358.302923011460.040298898466.671922761866.671922761867.91603879760.360294897565.596989931965.596989931964.045823350960.797198395465.211306603865.211306603863.73370920260.170186970962.743237060862.743237060863.203115148959.533012540362.860038115762.860038115756.367815288359.113025128460.20340331860.20340331856.617506607557.958998249758.693927862758.693927862759.676225266556.885921400557.890081203957.890081203959.051996968755.919062814256.46359339156.46359339161.548910159854.602124154555.388517959355.388517959357.772328958353.442919020554.276745237454.276745237454.089382001552.136638458351.861970813751.861970813752.716079746450.489589077550.503152524550.503152524552.029428618949.166728106849.617183625449.617183625449.938263821447.9814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