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COVID-19 waves peaked at different times in different countries, but the extent to which national epidemics were synchronized, and how synchrony changed as variants, vaccination and policies changed, is rarely quantified with several complementary methods. To compare the timing, duration and magnitude of epidemic waves in ten countries and to evaluate their synchronization using cross-correlation, wavelet coherence, dynamic time warping (DTW), network analysis and time-varying dependence measures, with a random-effects meta-analytic synthesis across countries. A protocol-driven synthesis of a ten-country daily dataset (Australia, Brazil, France, Germany, India, Italy, Japan, South Africa, United Kingdom, United States; 16,751 country-day records, 239 complete weeks, 12 January 2020 to 4 August 2024) was combined with a narrative critical review of the methods. Weekly cases per million were used because the daily series were reported in weekly batches. Four meta-analyses (k = 10) used restricted maximum likelihood with Hartung-Knapp adjustment and prediction intervals. 45 waves were detected. The between-country standard deviation of peak timing (τ ) ranged from 3.4 weeks for the Omicron BA.1/BA.2 wave to 9.5 weeks for the Alpha/Beta/Delta period. Mean maximum cross-correlation was 0.72, with 22 of 45 pairs significant after surrogate testing and false-discovery-rate control, and 32 pairs peaking at zero lag. Mean wavelet coherence was 0.43-0.46 across bands, but only 12-18% of the usable time-frequency area was significant. Phase synchrony rose from 0.22 before vaccination to 0.59 in the late period (null mean 0.28). Omicron peaks were 3.6 times earlier peaks (95% CI 1.7-7.6), and the case-fatality ratio was 0.17 times the pre-Omicron ratio (95% CI 0.11-0.27). Synchrony with the other nine countries was high for levels (r = 0.84) and moderate for weekly growth (r = 0.43, I² = 78%). Synchrony was partial and episodic rather than constant: strongest for the Omicron wave and late-period dynamics, weakest during the first year and the Alpha/Beta/Delta period. Spatial proximity explained little of the pairwise synchrony. Case-based timing is affected by changes in testing and reporting, so the findings describe reported epidemics rather than true infection incidence.
Keywords: COVID-19; epidemic synchronization; wavelet coherence; dynamic time warping; network epidemiology; random-effects meta-analysis; phase shift; random-effectsmeta-analysis
epidemiology;random-effectsmeta-analysis;phaseshift 1.Introduction 2.1 Reviewdesignanditslimits Thestudyhastwocomponents.Thefirstisaprotocol- From2020to2024everycountryexperiencedseveral drivenquantitativesynthesisinwhichthetencountries COVID-19 waves, yet their timing differed widely: aretheunitsofsynthesis;itfollowsthereportinglogic somecountriespeakedwithinweeksofoneanother, ofPRISMA2020(Pageetal.,2021)whereapplicable whereas others were separated by many months. (flowofrecords,eligibility,appraisalofdataquality, Understanding why national epidemic waves occur synthesis,heterogeneity,andsensitivityanalysis),as at different times matters for forecasting, for the showninFigure1.Thesecondisanarrativecriticalre- design of coordinated surveillance, and for judging viewoftheanalyticalmethods.Themethodologicallit- whether policies in one country can be informed eratureinthisarticlewasidentifiednon-systematically, by experience in another. Most quantitative work fromkeytextsandfrompapersonmeasles,influenza has asked how many cases will occur, whereas the andCOVID-19synchrony;nodatabasesearchwitha timingandcoordinationofwavesacrosscountrieshas registeredprotocolwasrun.Section6thereforetreats receivedlesssystematicattention.
theliteraturecomponentasacriticalnarrativereview, notasasystematicliteraturesearch.
Ecological theory provides a vocabulary for this problem.Inmeasles,whoopingcoughandinfluenza, epidemicsinconnectedpopulationsshowwavesthat travelwithcharacteristicdelaysandsynchronythat dependsoncoupling,seasonalityandpopulationsize (Grenfelletal.,2001;Rohanietal.,1999;Viboudetal., 2006;Bjornstadetal.,1999).ForCOVID-19,manyfac- torscanshifttiming:thedateandsizeofintroductions, the arrival of variants with different transmissibility (Fariaetal.,2021;Tegallyetal.,2021;Vianaetal.,2022; Volzetal., 2021), non-pharmaceuticalinterventions (Flaxmanetal.,2020;Haleetal.,2021),mobilityand travelnetworks(Chinazzietal.,2020;Kraemeretal., 2020),seasonalforcing(Bakeretal.,2020;Kissleret al.,2020;Neheretal.,2020),vaccination(Mathieuet
(PRISMA-style,adaptedtoadatasynthesis). A range of methods can describe synchronization, 2.2 Datasourceandeligibility and each answers a different question. Cross- The dataset contains daily country-level records correlation quantifies linear association at a lag; for Australia, Brazil, France, Germany, India, Italy, waveletcoherenceresolvesassociationbytimeand Japan, South Africa, the United Kingdom and the period; dynamictimewarpingalignsseriesthatare United States, with 73 variables on cases, deaths, shifted or stretched in time; network epidemiology testing,hospitalization,vaccination,policystringency, treats countries as nodes with weighted links; and demographicsandexcessmortality.Recordsrunfrom time-varying dependence measures show whether 5 January 2020 to between 4 and 12 August 2024. couplingstrengthensorweakens.Thesemethodsare ThevariablestructurematchesthatoftheOurWorld rarely applied together to the same data, and their in Data COVID-19 compilation (Mathieu et al., 2020; assumptions and failure modes differ. This article Mathieuetal.,2021;Haleetal.,2021).Alltencountries thereforehasthreeaims.First,toreviewcriticallyhow mettheinclusionrule(completedailycoveragewith these methods have been used to study epidemic noduplicateddates)andnonewasexcluded. synchrony and what can mislead. Second, to apply themtoaten-countryCOVID-19datasetandcompare thetiming,durationandmagnitudeofwaves.Third,to 2.3 Data-qualityappraisalandpreprocessing synthesizecountry-leveleffectsquantitativelythrough Asananalogueofrisk-of-biasassessment,eachcoun- random-effectsmeta-analysis,withanassessmentof trywasappraisedforcoverage,shareofzero-valued heterogeneity.
days in daily new cases, and availability of testing, hospital, stringency, vaccination, excess-mortality andreproduction-numbervariables.Ineverycountry 86-90%ofthedaysin2020-2022hadzeroreportednew cases,whichshowsthatdailynewcaseswererecorded 1995).Correlationsofweeklygrowthwerecomputed onceperweek;dailysmoothingandday-levelanalyses asarobustnesscheck.
are therefore not valid, and weekly aggregation is Waveletcoherence. Squaredwaveletcoherencewas lossless. New cases and deaths were summed over computedwithaMorletwavelet(ω =6)followingTor- calendarweeksendingSunday,keepingonlycomplete renceandCompo(1998),TorrenceandWebster(1999) weeks present in all countries (12 January 2020 to andGrinstedetal. (2004),forperiodsof4-96weeks, 4 August 2024; 239 weeks), converted to incidence withtimeandscalesmoothing.Significance(5%)was permillion,andsmoothedwithathree-weekcentred setperscalefrom60pairsofsimulatedAR(1)series mean. For correlation-based and wavelet analyses, matchingthelag-1autocorrelationofeachseries.For log(1 + incidence) was linearly detrended and stan- threeperiodbands(8-16,16-32,32-64weeks)themean dardized;weeklygrowthwasthefirstdifferenceofthe coherenceoutsidetheconeofinfluence,thepropor- smoothedlogseries.
tionofsignificantarea,andthemeanphase-derived leadinweekswerecomputed.Phasewasinterpreted 2.4 Wavedetectionandwavecharacteristics asaleadonlywherecoherencewassignificant. Waveswereidentifiedaspeaksinsmoothedweekly incidencepermillionusingtopographicprominence Dynamic time warping. DTW (Berndt and Clifford, (peakprominenceofatleast8%ofthecountrymaxi- 1994) with a Sakoe-Chiba band of 26 weeks (Sakoe mumandatleast25permillion;minimumseparation andChiba,1978)wasappliedtostandardizedseries; six weeks) with the SciPy implementation (Virtanen the DTW distance, the mean signed warping offset et al., 2020). Timing was the peak week; duration (positive: the first country’s features occur earlier), was the width at half peak height; magnitude was andaverage-linkageclusteringwereobtained. DTW peakincidencepermillionandcumulativeincidence distancewascomparedwithEuclideandistance. between the prominence bases; growth and decay Network epidemiology. An undirected network were log-incidence slopes from base to peak and was formed from country pairs with FDR-significant peaktobase. Sensitivityanalysesusedprominence cross-correlation (edge weight = maximum |r|), and thresholds of 5% and 15%. For synchronization a directed lead-lag network from the sign of the and meta-analysis, seven a priori windows were optimal lag. Strength, degree, weighted clustering, definedfromdominantepidemiologicalperiods:first and greedy modularity communities (Clauset et al., wave (March-June 2020), autumn-winter 2020/21, 2004;Newman,2006)werecomputedwithNetworkX Alpha/Beta/Delta(March-September2021),Omicron (Hagberg et al., 2008). Mantel tests with 4,999 per- BA.1/BA.2 (28 November 2021-April 2022), summer mutations(Mantel,1967)relatedpairwisesynchrony 2022, winter 2022/23 and 2023/24. The highest to great-circledistance betweencapitals, difference smoothed week in each window was the window inmeanpolicystringency,differenceinmedianage, peak; peaks on a window boundary were treated difference in log population density, and opposite asindistinctandexcluded. Samplinguncertaintyof hemisphere.Geographicproximityisonlyaproxyfor eachwindowpeakdatewasestimatedby300Poisson mobility, which is not in the dataset (Balcan et al., parametricbootstrapresamplesoftheweeklycounts, 2009;BrockmannandHelbing,2013;Pastor-Satorras plusthevarianceofweeklyrounding(1/12week²). etal.,2015).
Time-varying dependence. Series were band-pass 2.5 Synchronizationmethods filtered(10-40weeks; third-orderButterworth,zero- Cross-correlation. The cross-correlation function of phase),instantaneousphaseswereobtainedwiththe standardizedserieswascomputedforlagsupto±26 Hilbert transform (Rosenblum et al., 1996), and the weeks(Eq. 1),andthemaximumabsolutevalueand Kuramoto order parameter R(t) = |N− exp(i j(t))| 1 Σ θ itslagwereretainedforeachofthe45pairs. measuredinstantaneousphasesynchrony(Kuramoto, r_xy(k)=(1/n) Σtzx(t)zy(t+k);positivek:xleadsy(1) 1984; Pikovsky et al., 2001); pairwise phase-locking Because trending and wave-like series produce valueswerealsocomputed(Lachauxetal.,1999).The spurious correlation (Yule, 1926; Granger and New- firstandlast13weekswerediscardedasfilter-edge bold, 1974), significance was assessed against 499 regions. R(t) was compared across four eras (pre– phase-randomized surrogate pairs for each country vaccine; vaccine rollout before Omicron; Omicron; pair(Theileretal.,1992;SchreiberandSchmitz,2000), late/endemic) with block-bootstrap intervals and a using the maximum absolute correlation over all Kruskal-Wallistest,andagainst200phase-randomized lags as the test statistic, with Benjamini-Hochberg surrogate datasets. Mean pairwise correlation in a false-discovery-ratecontrol(BenjaminiandHochberg, 52-weekslidingwindowgaveasecondview. 2.6 Meta-analysis areneededinadditiontoasinglecorrelation.Second, Effectsizeswerecomputedpercountryandpooled observedsynchronycanreflectsharedforcingrather byrandom-effectsmeta-analysiswithrestrictedmaxi- thancoupling:seasonality,agloballycirculatingvari- mum likelihood estimation of the between-country ant,orsimultaneouspolicyrelaxationcansynchronize variance ² (Viechtbauer,2005),Hartung-Knapp-Sidik- otherwiseunconnectedpopulations(Bjørnstadetal., τ Jonkman confidence intervals with t-distribution 1999; Pikovsky et al., 2001). Third, case reports are (Knapp and Hartung, 2003), and 95% prediction notincidence:variationintesting,casedefinitionsand intervals (IntHout et al., 2016). Heterogeneity was reportingcadencechangesbothamplitudeandappar- summarized by Cochran’s Q, ² and I² (Higgins and enttiming,anissuethatbecamesevereonceroutine τ Thompson,2002;Higginsetal.,2003).Foursyntheses testingdeclined.Fourth,nullmodelsmatter.Correla- were pre-specified: (MA1) log ratio of the Omicron- tion,coherenceandphase-lockingallhavenon-zero wavepeaktothemaximumearlierpeak,withvariance expectedvaluesforautocorrelatedseries,sosurrogate from Poisson counts in the peak window; (MA2) dataarerequiredtosaywhethersynchronyexceeds pooledpeaktimingandbetween-countrydispersion chance.
τ for each window, with the bootstrap variance above; (MA3) Fisher z of each country’s correlation withthemeanoftheotherninecountries,forlevels (MA3a) and weekly growth (MA3b), with variance 4.Results 1/(n_eff − 3) and the effective sample size n_eff 4.1 Includedcountriesanddataquality correctedforautocorrelation(BayleyandHammersley, Zero-casedaysofabout86%ineverycountryshow 1946); and (MA4) log ratio of the lagged (two-week) that daily new cases were recorded once per week; case-fatality ratio in the Omicron era (28 November from2023thezero-dayshareroseto93-100%inBrazil, 2021-27 November 2022) to that before Omicron (1 Germany,France,JapanandtheUnitedStates,indi- March 2020-21 November 2021). Subgroups were catingthatreportingmovedtoalowerfrequencyor SouthernandNorthernHemisphere.Single-covariate ceasedforsomedays.Availabilityoftesting,hospital meta-regressions(medianage,logpopulationdensity, andexcess-mortalitydatavariedwidely,soanalyses logGDPpercapita,humandevelopmentindex,mean reliedonreportedcasesanddeaths,whichwereavail- stringencyin2020-2021,hemisphere)wereexploratory ableforallcountries. Alltencountrieswerejudged because k = 10. Leave-one-out analyses, and Egger adequate for weekly analysis, with the main risk of regressionwheresamplingvariancesvaried,assessed biasbeingchangesintestingandcaseascertainment robustness and small-study effects. Analyses used overtime.
PythonwithNumPy,SciPy,statsmodelsandNetworkX; alltestsweretwo-sided.
Note:samplingvariancesinMA1andMA4derivefrom 4.2 Timing,durationandmagnitudeofwaves Poissoncaseanddeathcountsandareverysmall;het- The detection rule found 45 waves across the ten erogeneity(I²)isthereforecloseto100%byconstruc- countries(3-6percountry; Figures2and3; Table4). tion, and τ, togetherwiththepredictioninterval, is Counts were sensitive to the prominence threshold: theinformativesummaryofbetween-countryvaria- with a 5% threshold 52 waves were detected and tion.Eggertestsandfunnelplotsareuninformativefor with15%34.WavesfromtheOmicronperiodonward thesetwosyntheses.
were taller (median peak 7,723 versus 1,969 cases permillionperweek;Mann-Whitneyp=<0.001)and 3.Criticalreviewofmethodsforepidemicsyn- narrower (median half-height width 5.2 versus 7.2 chronization weeks; p = 0.005; Figure 4). Waves rose faster than theydecayed(medianlog-growth0.160versus0.127 Table1summarizeswhateachmethodmeasures,its perweek),althoughthedifferencewasnotsignificant assumptions,andthefailuremodesmostrelevantto (Wilcoxon p = 0.160). Width did not differ between COVID-19reporteddata.
hemispheres(medians6.0and5.9weeks;p=0.581). Four lessons from the literature shape the analysis. Reporteddeathsfollowedreportedcaseswithalagof First,synchronyisscale-dependent:inmeasles,syn- 0-2weeksinallcountries(maximumcross-correlation chronyandtravellingwavesdifferbyperiodandbyspa- 0.69-0.88),whichsupportstheuseofcasetimingasa tialscale(Grenfelletal.,2001;Earnetal.,2000;Rohani proxyforepidemictimingwithinthelimitsdiscussed etal.,1999),whichiswhymethodsthatresolveperiod below.
(wavelets)andtime(rollingorphase-basedmeasures) Method Whatitcaptures Keyassumptions Principalpitfalls Keyreferences Spuriouscorrelationfor Yule(1926); trendingorwave-likeseries; Linearassociation Stationarity; GrangerandNewbold Cross-correlation asinglelaghidestimevariation; andbest-fittinglag linearrelation (1974); lagatthesearchboundary Theileretal.(1992) isuninformative TorrenceandCompo Association Edgeeffects(coneofinfluence); Adequatelength (1998); Wavelet resolvedin shortseries;smoothingchoices; relativetoperiod; Grinstedetal.(2004); coherence timeandperiod; phaseinterpretableonlywhen red-noisenull Cazellesetal.
phaselead coherenceissignificant (2007,2008) Canmatchunrelated Shapesimilarity Elasticalignment BerndtandClifford featuresifunconstrained; Dynamictime allowing ismeaningful; (1994); noinferencebydefault; warping shiftsand windowconstrains SakoeandChiba oftenclosetoEuclidean stretching warping (1978) distanceforsimilarseries Pastor-Satorrasetal.
Structureof Edgesaredefined Edgedefinitiondrivesstructure; (2015); pairwisecoupling; Networkanalysis byathreshold correlationisnottransmission; Brockmannand leadersand ortest confoundingbycommondrivers Helbing(2013); communities Balcanetal.(2009) Rosenblumetal.
Windowlengthtrade-off; (1996); Whethercoupling Time-varying Localstationarity phaseestimates Lachauxetal.
changesacross dependence withinwindows neednarrow-bandsignals; (1999); eras nullmodelsrequired Pikovskyetal.
(2001) Fewunits(k=10); Pooledeffect Exchangeable Borensteinetal.
samplingvariances Random-effects and countryeffects; (2009); ignoreascertainmentbias; meta-analysis between-country knownsampling IntHoutetal.
I² misleading heterogeneity variances (2016) whenvariancesaretiny
GDP Mean Population Median Density Cases Deaths Country percapita HDI Hemisphere stringency (millions) age (perkm²) permillion permillion (US$) 2020-21 Australia 26.2 37.9 3 44,649 0.940 South 453,107 964 63.0 Brazil 215.3 33.5 25 14,103 0.760 South 174,220 3,261 63.6 Germany 83.4 46.6 237 45,229 0.950 North 461,051 2,099 61.1 France 67.8 42.0 123 38,606 0.900 North 575,074 2,479 57.9 United 67.5 40.8 273 39,753 0.930 North 369,946 3,438 61.9 Kingdom India 1417.2 28.2 450 6,427 0.640 North 31,783 377 71.3 Italy 59.0 47.9 206 35,220 0.890 North 453,628 3,342 69.0 Japan 124.0 48.2 348 39,002 0.920 North 272,716 603 43.3 UnitedStates 338.3 38.3 36 54,225 0.930 North 305,764 3,527 61.5 SouthAfrica 59.9 27.3 47 12,295 0.710 South 68,000 1,713 58.2 4.3 Whydopeaksoccuratdifferenttimes?Window- (mid-December 2021) and Germany last, consistent leveltiming withrapidspreadfromsouthernAfrica(Vianaetal., Table5andFigure5givethepooledtimingofwindow 2022).Dispersionincreasedagaininlaterwindows(τ peaks. Dispersion of timing varied nearly threefold =4.2,6.6and6.4weeks). I² wasabove99%inevery across windows. The first wave (k = 6) peaked in window, asexpectedwhenbootstrapsamplingvari- mid-April 2020 with = 3.4 weeks; dispersion rose ancesaresmallcomparedwithrealtimingdifferences; τ inautumn-winter2020/21( =6.1weeks,withIndia thesubstantivequantityis τ .Hemisphereeffectswere τ first and Brazil last, a range of 18 weeks) and was inconsistent:SouthernHemispherecountriespeaked greatest for Alpha/Beta/Delta ( = 9.5 weeks; range later than Northern countries in autumn-winter τ 24weeks;Italyfirst,UnitedStateslast).TheOmicron 2020/21(6.1weeks)butearlierinsummer2022(-6.9 BA.1/BA.2 wave was the most synchronized ( = 3.5 weeks),whichindicatesvariantandimmunityeffects τ weeks,range13weeks),withSouthAfricapeakingfirst morethanafixedseasonallag.
Zero-case Hospital Full Excess Country 2021 2022 2023 2024 Tests Stringency Rt days2020 patients vaccination mortality Australia 87 86 86 85 91 100 100 100 100 100 100 Brazil 88 86 86 95 100 100 100 100 100 100 100 Germany 86 86 86 93 100 100 100 100 100 100 100 France 90 86 86 93 100 100 100 100 100 100 100 United 87 86 86 85 86 100 100 100 100 100 100 Kingdom India 88 86 86 85 87 100 100 100 100 100 100 Italy 87 86 86 85 86 100 100 100 100 100 100 Japan 86 86 86 95 100 100 100 100 100 100 100 United 86 86 86 95 100 100 100 100 100 100 100 States South 88 86 86 94 91 100 100 100 100 100 100 Africa
Largestpeak First Dateof Medianhalf-height Country Waves (permillion detected largestpeak width(weeks) perweek) peak Australia 4 17,816 2022-01-16 6.2 2022-01-16 Brazil 5 5,471 2022-02-06 6.8 2020-08-09 Germany 5 18,333 2022-03-20 6.4 2020-12-20 France 6 32,975 2022-01-23 4.7 2020-11-01 United 4 17,450 2022-01-02 5.4 2021-01-10 Kingdom India 3 1,817 2021-05-09 6.6 2020-09-20 Italy 5 19,988 2022-01-23 5.9 2020-11-15 Japan 4 11,748 2022-08-14 7.5 2021-08-29 UnitedStates 4 15,151 2022-01-16 10.1 2021-01-10 SouthAfrica 5 2,156 2021-12-19 5.8 2020-07-19 4.4 Cross-correlation 4.5 Waveletcoherence Acrossthe45countrypairsthemeanmaximumcross- Mean wavelet coherence was 0.43, 0.46 and 0.40 in correlationwas0.72(median0.74);25pairsweresignif- the 8-16, 16-32 and 32-64-week bands, but the sig- icantatthe5%levelagainstphase-randomizedsurro- nificant fraction of usable time–frequency area was gatesand22afterfalse-discovery-ratecontrol(Figure only 18%, 17% and 12% (Figures 7 and 8; Table 7). 6;Table6).Theoptimallagwaszerofor32of45pairs Synchronywasthereforepatchy:pairssuchasBrazil- andatleast10weeksinabsolutevaluefor5pairs,all UnitedStatesshowedextendedsignificantcoherence ofwhichinvolvedIndiaorAustralia(forexampleIn- at8-32weeks,whereasFrance-SouthAfricashowedlit- dialedAustraliaby26weeks,thesearchboundary,so tle.Wherephasecouldbeinterpreted,leadsweresmall thatlagshouldbereadas"notestimablewithin±26 (forexampletheUnitedStatesledBrazilbyabout1.3 weeks").Weekly-growthcorrelationswerelowerbut weeksinthe16-32-weekband),andrankagreement followedthesamestructure(Table6footnote).India, betweenwavelet-basedandcross-correlationleader- AustraliaandSouthAfricahadthelowestmeancorre- shipwasweak(Spearman ρ =0.39).
lationwiththeothers(0.57,0.54and0.70),whereas Brazil,France,GermanyandtheUnitedStateshadthe 4.6 Dynamictimewarping highest(0.79,0.79,0.79and0.81).Australia’slowval- DTWdistanceswerestronglycorrelatedwithEuclidean uesareconsistentwithnear-absentcirculationuntil distances (Spearman ρ = 0.93), so elastic alignment theOmicronwave.
changedtheorderingofpairslittleinthissetofseries withsimilarwavecounts.Average-linkageclustering Meanmaximumrforweeklygrowthwas0.35.Positive (threeclusters)separated{India,SouthAfrica},{Brazil, lag:thefirstcountryleads.
Germany,France,UnitedKingdom,Italy,Japan,United States}and{Australia}(Figure9).Theclosestpairswere Southern Pooledpeak τ Range minus Window k I² (%) First Last date (weeks) (weeks) Northern (weeks) W1Firstwave 6 2020-04-16 3.4 99 9 AUS FRA – W2Autumn–winter 9 2020-12-09 6.1 100 18 IND BRA 6.1 2020/21 W3Alpha/Beta/ 9 2021-05-27 9.5 100 24 ITA USA -2.6 Delta W4Omicron 10 2022-01-25 3.5 99 13 ZAF DEU -2.3 BA.1/BA.2 W5Summer2022 10 2022-07-06 4.2 100 13 AUS JPN -6.9 (BA.4/5) W6Winter 9 2023-01-09 6.6 100 19 FRA IND 1.0 2022/23 W72023/24 4 2023-11-19 6.4 100 13 GBR IND –
Pair Maxr Lag(weeks) Zero-lagr p(surrogate) p(FDR) BRA–USA 0.94 0 0.94 0.002 0.006 DEU–FRA 0.92 0 0.92 0.002 0.006 DEU–USA 0.91 0 0.91 0.002 0.006 DEU–JPN 0.89 0 0.89 0.002 0.006 BRA–FRA 0.89 0 0.89 0.002 0.006 FRA–USA 0.89 0 0.89 0.002 0.006 JPN–USA 0.87 0 0.87 0.002 0.006 BRA–JPN 0.86 0 0.86 0.002 0.006 FRA–JPN 0.85 0 0.85 0.008 0.020 DEU–ITA 0.85 0 0.85 0.002 0.006 BRA-USA(1.9),DEU-FRA(2.2),DEU-JPN(3.0);themost 4.8 Time-varyingdependence distant involved Australia (AUS-BRA (11.9), AUS-ZAF Phasesynchronychangedmarkedlyovertime(Figure (12.5),AUS-IND(13.4)).LeadershiprankingsfromDTW 11; Table 9). Mean R(t) was 0.22 before vaccination, agreedwiththosefromcross-correlation(ρ =0.91)but 0.52duringvaccinerolloutbeforeOmicron,0.46inthe notwithwaveletphase(ρ =0.35). DTWdistancewas Omicroneraand0.59inthelateperiod(Kruskal-Wallis relatedtogeographicdistance(Mantel ρ =-0.42,p= p<0.001).Theoverallmean(0.47)wasabovethemean 0.043;notadjustedformultipletests).
ofphase-randomizedsurrogates(0.28;95thpercentile 0.35),andthelate-periodintervallayabovethesurro- 4.7 Networkepidemiology gate95thpercentile,sosynchronyexceededchance TheFDR-significantsynchronynetworkhad22edges especiallyafter2021.Themeanphase-lockingvalue (density0.49)andthreecommunities(modularity0.11): acrosspairswas0.31. Beforevaccination,meanR(t) {Germany,France,Italy,Japan,UnitedStates},{Brazil, wasbelowthesurrogatemean; theintervalsforthe UnitedKingdom,India,SouthAfrica},{Australia}(Fig- vaccine-rolloutandlateeraslayabovethesurrogate ure10;Table8).Australiawasisolated.Edgestructure 95thpercentile,andthelowerlimitfortheOmicronera didnotfollowgeography: synchronywasnotsignifi- (0.33)wasjustbelowit. Sliding-windowcorrelations cantlyassociatedwithgreat-circledistance(Mantel ρ oflevelsfluctuatedmorewidely(about0to0.7)than =-0.28,p=0.220),hemisphere(ρ =-0.12,p=0.538), correlationsofweeklygrowth(about0.1-0.3),which differenceinpolicystringency(ρ =-0.09,p=0.750),dif- showsthattheapparentstrengthofdependencede- ferenceinlogpopulationdensity(ρ =-0.33,p=0.126) pendsonwhetherlevelsorchangesareused. ordifferenceinmedianage(ρ =-0.30,p=0.066). Di- rectedlead–laglinkswerefew,becausemostpairshad zerooptimallag,andnetleadershipscoreswerenear 4.9 Meta-analysis zeroforallcountriesexceptFrance(-0.77)andSouth Table11andFigure12givethepooledresults.Magni- Africa(0.77);leadershipisthereforenotastablefeature tude(MA1). Omicron-wavepeakswere3.6timesthe ofthisnetwork.
largestearlierpeak(95%CI1.7to7.6;p=0.003),but positive:firstcountryleads).
Band(weeks) Pair Meancoherence Significantarea Phaselead(weeks) 8-16 JPN-USA 0.78 70% -0.2 8-16 DEU-FRA 0.66 42% -0.5 8-16 BRA-USA 0.62 46% -0.0 8-16 FRA-ITA 0.56 38% 0.3 8-16 GBR-ITA 0.55 27% 0.4 16-32 BRA-USA 0.78 72% -1.3 16-32 IND-USA 0.73 65% -1.6 16-32 FRA-ITA 0.66 44% 0.2 16-32 JPN-USA 0.66 48% -1.5 16-32 BRA-JPN 0.64 43% 1.1 32-64 BRA-ZAF 0.73 51% -1.0 32-64 FRA-USA 0.73 40% -5.0 32-64 DEU-GBR 0.67 46% -5.3 32-64 DEU-USA 0.64 42% -4.1 32-64 DEU-FRA 0.63 43% 1.3
Weighted Lead–lagscore MeanDTW Country Degree Strength Community clustering (cross-correlation) distance Australia 0 0.00 0.00 3 -4 10.8 Brazil 7 5.87 0.47 2 -1 5.5 Germany 5 4.42 0.75 1 -1 5.0 France 6 5.09 0.61 1 -2 5.4 United 4 3.22 0.58 2 3 5.9 Kingdom India 2 1.43 0.80 2 4 8.4 Italy 4 3.23 0.59 1 2 6.7 Japan 4 3.47 0.93 1 -4 6.6 UnitedStates 7 6.02 0.56 1 0 5.6 SouthAfrica 5 3.90 0.52 2 3 6.4 between-countryvariationwaslarge(τ =1.03;predic- Eighteenmeta-regressionswererunwithk=10coun- tioninterval0.30to44.2):theratiowasbelowonein tries;noneoftheassociationsshouldbetreatedascon- India, where the 2021 wave was larger, and 30-fold firmatory,andthep-valuesarenotadjustedformul- inAustralia. Synchrony(MA3). Countrylevelscorre- tiplicity. Associationswiththehumandevelopment latedwiththemeanoftheotherninecountriesatr= indexandGDPinMA1,andwithmedianageinMA4, 0.84(95%CI0.74to0.91;I² =0%),butweeklygrowth werenominallysignificantandplausiblyreflectdiffer- ratescorrelatedatonlyr=0.43(95%CI0.29to0.56; encesintestingcapacityandintheagestructureof I² =78%;predictioninterval-0.03to0.74),withSouth pre-Omicronmortality;theycannotseparatetheseex- Africalowest(r=0.07)andtheUnitedStateshighest planations.
(r=0.70). Severity(MA4). Thelaggedcase-fatalityra- tiointheOmicronerawas0.17timesthepre-Omicron 4.10 Sensitivityanalyses ratio(95%CI0.11to0.27;predictioninterval0.03to Resultswerestabletothewave-detectionthresholdfor 0.85),consistentwithlowerseverityperreportedcase qualitativeconclusions(wavespercountrychangedby intheOmicronperiod(Nybergetal.,2022)butalsowith 0-5acrossthresholds;Section4.2).Excludinganysin- greatercaseascertainmentofmildinfectionsandvac- glecountrydidnotchangethedirectionorsignificance cination.Timing(MA2)isgiveninTable5.Hemisphere of MA1, MA3 or MA4. Growth-based synchrony was subgroupsdidnotdifferforanysynthesis(Q-betweenp lowerthanlevel-basedsynchronybecausetrending =0.849,0.623,0.262and0.406).Leave-one-outpooled levelssharecommonwavestructurethatisremoved estimatesstayedwithin2.9-4.3(MA1),0.39-0.47(MA3b) bydifferencing;conclusionsabout"highsynchrony" and0.15-0.19(MA4),andEggertestsforMA3werenot thereforedependonthemeasureused.DTWandEu- significant(p=0.318and0.371).
clideanorderingsagreedclosely(ρ =0.93). levels.
Era Weeks MeanR(t) 95%interval Meansliding-windowr Pre-vaccine 38 0.22 0.14-0.25 0.45 Vaccinerollout,pre-Omicron 47 0.52 0.37-0.66 0.21 Omicron 57 0.46 0.33-0.60 0.32 Late/endemic 71 0.59 0.55-0.69 0.47
Covariate(pairwise) Spearman ρ p(4,999permutations) Great-circledistance -0.28 0.220 Differenceinstringency -0.09 0.750 Differenceinmedianage -0.30 0.066 Oppositehemisphere -0.12 0.538 Differenceinlogdensity -0.33 0.126
(blue,pre-vaccine;yellow,vaccinerolloutbefore Omicron;red,Omicron;green,late/endemic). 5.Discussion Wavesoccurredatdifferenttimesbecausenationalepi- demicsweredrivenbyprocessesthatwereonlypartly shared.Thetiminganalysisshowsthatsynchronywas episodic.Beforewidespreadvaccination,andduring theAlpha/Beta/Deltaperiod,peakswereseparatedby up to 18-24 weeks and between-country dispersion
Tegallyetal., 2021; Volzetal., 2021), whennational policiesdivergedmost(Flaxmanetal.,2020;Haleet al.,2020;Kraemeretal.,2020).TheOmicronBA.1/BA.2 al.,2021),andwhentravelwasrestricted(Chinazziet wavewasthemostsynchronized.Itshightransmissibil- 95% 2 Hemisphere Synthesis(effect) Pooled 95%CI prediction τ I² (%) Qp Q-betweenp interval MA1:Omicron peak/earlierpeak 3.63 1.73to7.60 0.30to44.20 1.069 100 <0.001 0.849 (ratio) MA3a:synchrony, 0.84 0.74to0.91 0.73to0.91 0.000 0 0.493 0.623 levels(r) MA3b:synchrony, 0.43 0.29to0.56 -0.03to0.74 0.041 78 <0.001 0.262 growth(r) MA4:CFR Omicron/ 0.17 0.11to0.27 0.03to0.85 0.441 100 <0.001 0.406 pre-Omicron (ratio)
Synthesis Covariate Coefficient 95%CI p MA1(peakratio) Medianage 0.56 -0.10to1.22 0.086 MA1(peakratio) Logpopulationdensity -0.51 -1.19to0.17 0.125 MA1(peakratio) LogGDPpercapita 0.75 0.23to1.27 0.010 MA1(peakratio) Humandevelopmentindex 0.77 0.27to1.27 0.007 MA1(peakratio) Meanstringency2020–21 -0.18 -0.96to0.61 0.617 MA1(peakratio) SouthernHemisphere 0.20 -1.54to1.94 0.797 MA3b(growthsynchrony) Medianage 0.10 -0.06to0.26 0.195 MA3b(growthsynchrony) Logpopulationdensity 0.02 -0.15to0.20 0.757 MA3b(growthsynchrony) LogGDPpercapita 0.10 -0.06to0.26 0.199 MA3b(growthsynchrony) Humandevelopmentindex 0.10 -0.07to0.26 0.212 MA3b(growthsynchrony) Meanstringency2020–21 -0.02 -0.20to0.16 0.826 MA3b(growthsynchrony) SouthernHemisphere -0.20 -0.55to0.16 0.234 MA4(CFRratio) Medianage -0.52 -0.81to-0.23 0.003 MA4(CFRratio) Logpopulationdensity -0.05 -0.56to0.46 0.819 MA4(CFRratio) LogGDPpercapita -0.41 -0.80to-0.02 0.040 MA4(CFRratio) Humandevelopmentindex -0.45 -0.81to-0.08 0.022 MA4(CFRratio) Meanstringency2020–21 0.14 -0.36to0.64 0.545 MA4(CFRratio) SouthernHemisphere 0.35 -0.74to1.43 0.483
variant,seasonalforcingandimmunity)aswellaswith coupling;thesecannotbeseparatedwithoutmobility ityandimmuneescape,togetherwithconnectedpop- orgenomicdata(Pikovskyetal.,2001;Bjørnstadetal., ulationsthathadlargelyrelaxedcontrols,canreduce 1999).
differencesintiming,andtherapidglobalspreadfrom southernAfricaisdocumentedgenomically(Vianaet The results do not support a simple geographic al.,2022).Thephase-basedmeasureshowsthatsyn- explanation. Proximity, hemisphere, policy- chrony stayed above the surrogate level after 2021, stringency similarity, density and age similarity
Comparing methods is itself an outcome. Cross- correlation gave a clear overall picture (high cor- relations, mostly zero lag) but cannot show how synchronychanges;waveletcoherenceshowedthat significantcoherenceoccupiedonlyaminorityofthe time–frequencyplane;andthephase-basedmeasure showed when synchrony rose. DTW added little be- yondEuclideandistanceforseriesofsimilarstructure, anditswarpingoffsetsagreedwithcross-correlation leadership but not with wavelet phase. Leadership rankingsthereforedependonmethod,andthedata donotsupportstable"leader"countriesotherthana
(10–40-weekband;shadedwhiteareasarefilter-edge Practitionersshouldusemorethanonemethodand regionsexcludedfromsummaries).Bottom:mean testagainstsurrogatenulls(Theileretal.,1992). pairwisecorrelationina52-weekwindow. Themeta-analyticresultsshowthatmostquantities werenotsignificantlyrelatedtopairwisesynchrony. ofinterestvarysubstantiallybetweencountries.Peak The DTW–distance association (ρ ≈ −0.42,p = sizesduringOmicronrelativetoearlierpeaks,andthe 0 043) . wasweakevidenceandunadjustedformultipletests.Thisagreeswiththeviewthatlong changeincase-fatalityratio,hadwidepredictionin- − distanceconnectivity,notonlyadjacency,determinesthespreadofcontagion BrockmannandHelbing, Balcanetal., ,butwithtencountriesthepowertodetectsuchanassociationislow. tervals, which means that a pooled estimate alone ( 2013; 2009) shouldnotbeappliedtoanewcountry. InMA1and BA.1/BA.2 wave (τ= 3.5 weeks) and largest for the MA4 the near-100% I² follows from very small Pois- Alpha/Beta/Delta period (τ = 9.5 weeks); phase sonsamplingvariances;thepredictionintervalsand τ synchrony increased after 2020 and exceeded the aretheinformativesummaries.Synchronyofweekly surrogate null in later eras; and pairwise synchrony growth(MA3b)hadmoderateheterogeneity(I² ≈ 78%), wasnotexplainedbydistance,hemisphere,stringency, whereassynchronyoflevelswashomogeneous,which age or density. Omicron-era waves were taller and isexpectedbecausesharedmulti-wavestructuredom- narrower than earlier waves, and the case-fatality inateslevels.
ratiofelltoabout0.17ofitspre-Omicronvalue. For forecasting and coordination, the findings imply that cross-country lead times are unstable, so a 6.Limitations country’speakisaweakpredictorofanother’soutside Reportedcasesarenotincidence.Changesintesting periods dominated by a common variant, and that policy,casedefinitionsandreportingfrequency,and multi-method analyses with surrogate nulls are thedeclineinroutinetestingfrom2022,alterampli- neededbeforeclaimingsynchronyorleadership. tudeandmayshiftapparenttiming.TheOmicron-era peakratio(MA1)andthecase-fatalityratio(MA4)are particularlysensitivetothis.Deathsfollowedcasesat