Data

Clinically suspected cases of dengue fever (DF) and dengue hemorrhagic fever (DHF) in Puerto Rico are reported to the surveillance system maintained by the Puerto Rico Department of Health and the Centers for Disease Control and Prevention. The data analyzed here include all reported cases from July 1986 through December 2006 by month. Reported cases, rather than laboratory confirmed cases, are used because approximately 60% of cases lack the samples necessary for a definitive laboratory diagnosis. Summaries of suspected dengue cases reported in Mexico in the years 1985–2006 were obtained from the Mexican Secretariat of Health (www.dgepi.salud.gob.mx/anuario). Monthly counts of reported DF and DHF were combined. In contrast, Thailand historically only included DHF in national surveillance. We analyzed Thai reported DHF cases for the years 1983–1996 [15]. Although, this represents only a portion of the cases actually occurring, it is a consistent measure through time in Thailand where DHF has long been established. Whether using confirmed or reported DF or DHF case counts, inaccurate estimation of the true burden of disease is inherent. However, for the purposes of this analysis, the absolute quantity is immaterial as long as the data accurately represent change over time. All included datasets represent relatively consistent measures of incidence over time. Although it is possible that DF and DHF exhibit different dynamics, they are expected to be highly correlated in settings with long-term endemicity such as Thailand. Each dengue time series was log-transformed and normalized prior to analysis to reduce skewing, remove the mean, and standardize the amplitude. As a result, change over time, the focus of this analysis, is more directly analyzed.

Weather data were obtained from the TYN CY 1.1 dataset of the Climate Research Unit at the University of East Anglia, United Kingdom [16]. This dataset is composed of 0.5° gridded, interpolated observations for the years 1901 through 2000 summarized to political boundaries [17]–[19]. The datasets used for each study area were normalized total monthly precipitation and minimum, maximum, and mean average monthly temperatures.

The ENSO index used is the normalized Oceanic Niño Index sea surface temperature anomaly index for Niño region 3.4 available from the Climate Prediction Center of the US National Weather Service (www.cpc.ncep.noaa.gov). This index is the difference between the 3-mo running average sea surface temperature for the area between 5°N, 5°S, 120°W, and 170°W, and the average of that value for the years 1971–2000 [20].

Wavelet Transformation

In the following we present the critical details of our analyses. For further depth, Torrence and Compo [21] is an excellent resource from which much of our methodology was derived. Wavelet analysis requires the selection of a basis function for the transformation. Here we use the Morlet wavelet, a complex sine wave localized by a Gaussian distribution,

where η is a scaled time unit and ω₀ describes the relative frequency of the sine wave ( here to satisfy admission criteria) [22]. Because it is a localized periodic function, it is ideal for analyzing periodic behavior such as multiyear ENSO or seasonal dengue variation.

The continuous wavelet transform is the convolution of the series x_n and the wavelet ψ₀ at time t and scale s, where x_n is a series of observations x₀, x₁, …, x_N−1 equally spaced in time by δ_t. This is defined as

where ψ* is the complex conjugate of the wavelet normalized by a factor of (δ_t/s)^1/2 to ensure unit energy, allowing comparability between scales and analyses [21]. δ_t is included to adjust the scales to a meaningful time frame for interpretation. Scales range from , the finest temporal resolution present in the dataset, to , the minimum temporal resolution, where . The scale resolution, , was selected on the basis of criteria detailed in the wavelet coherence significance section below and Figures S1 and S2. The wavelet transformation is complex and describes the time- and frequency-specific power and phase. The power, |W_n,s|², indicates the strength of the wavelet-like behavior at every point and is presented in the power spectrum of each transformation. Phase (θ) indicates the angular position of each point in its cyclical trajectory, from a trough at π radians to a peak at 0 or 2π radians. It is calculated as the inverse tangent of the imaginary component of the transform divided by the real component:

The wavelet itself extends both forwards and backwards in time. Consequently, in wavelet analysis the beginning and end of a time series are effectively joined in a loop so that there is prior and post information at every time point. To disconnect the beginning and end, we pad the time series with zeros. The zeros still have an effect on the transform at the extremities though, so, following Torrence and Compo [21], we shade the area of the transform where edge effects are significant.

The transform can be inverted to reconstruct the original time series,

where C_δ is an empirically defined wavelet basis-specific reconstruction factor ( for the Morlet [] wavelet) [21]. As written above, the reconstruction uses all scales (j). However, the scales over which the summation occurs in the second part of the equation can be limited to reconstruct scale-specific components of the original series. Reconstruction is imperfect due to the edge effects of the transformation, but information loss is minimal [21].

Significance of the Wavelet Power Spectrum

The significance of a wavelet power spectrum is assessed by comparison with simulated or theoretical spectra representing a null hypothesis. Cazelles et al. [23] used the null hypothesis that “the variability of the observed time-series or the association between two time series is no different to that expected from a purely random process.” This definition implicitly assumes that sequential observations are independent. In fact, many geophysical [24] and ecological [25] processes exhibit significant memory, such that they are more accurately described as autocorrelated processes. In general terms, the observation x_n is related to the previous l observations of x by

where l, the order of autocorrelation, is particular to the system under study. For infectious diseases, such as dengue, intrinsic first-order autocorrelation results from the fact that transmission is dependent on a source of infection, such that for any given time point, current incidence is associated with incidence at the previous time point. Thus, current observations are correlated with past observations. Because autocorrelation limits instantaneous change in a variable, these time series are more likely to vary over longer periods of time, and power in the frequency domain shifts to longer periodicities simply due to observation-to-observation autocorrelation. All of the time series analyzed here exhibit first-order autocorrelation and are thus susceptible to appearing to vary over long time periods solely owing to stability over short time periods. To account for the potential influence of short-term autocorrelation on long-term characteristics, we employ a stricter null hypothesis: the variability of the observed time-series is equivalent to the expected variability of a random process with similar first-order autocorrelation. We estimate the first-order autocorrelation of the time series to be analyzed and create a theoretical Fourier power spectrum of a Gaussian process with equivalent first-order autocorrelation [24] and a χ² estimator [21] which allows us to establish 95% confidence bounds for the null hypothesis.

Wavelet Coherence

In the current study, our interest is the relationship between dengue incidence and climate on multiyear scales. Coherence measures time- and frequency-specific association between two wavelet transforms. High coherence indicates that two time series associate at a particular time and frequency, precisely the way that climate may influence dengue on multiyear scales. Here we calculate squared coherency (R²),

where s⁻¹ normalizes the energy, W^X is one wavelet transform, W^Y is the other, W^XY is the cross-wavelet spectrum (W^XW^Y*), and S(W) is the sequential smoothing function S_scale(S_time) [26]. S_time is the scale-specific convolution of W with a normalized Gaussian filter and S_scale is the time-specific convolution of the result with a normalized boxcar filter of width [27]. In the denominator, the power spectra are smoothed and in the numerator, the cross-wavelet spectrum is smoothed prior to finding the modulus and squaring. This ensures that the numerator and denominator are nonidentical.

The phase of coherency measures time- and frequency-specific differences in phase between the two time series:

The phase difference can be converted back into a time scale to measure the lag of coherence.

Significance of Wavelet Coherence

Statistical significance for coherence is determined by simulation. Pairs of time series representing the null hypothesis are generated and assessed for coherence to provide a measure of coherence that occurs by chance, as opposed to coherence due to a true association. A certain amount of coherence is intrinsic to the analysis because of the scale filtering process; variance on a particular frequency is likely to exhibit coherence with other time series on that same frequency solely because the frequency is roughly matched. As observed previously [28] and shown in Figure S2A, although coherence is theoretically sensitive to autocorrelation of individual time series, spectra of simulated time series with varying degrees of autocorrelation show that autocorrelation has no discernable effect on coherence. Thus, we generated the null series as simple random Gaussian variables. On the other hand, scale selection does affect coherence. Although the approximate minimum (s₀) and maximum (s_J) scales are dictated by the resolution and length of the dataset, the scales selected for analysis depend on the scale resolution, δ_j. At low δ_j random coherence varies markedly while at higher δ_j it stabilizes (Figure S2B). We selected δ_j to maximize computational efficiency while minimizing the random coherence associated with low δ_j.

Not only does coherence arise by chance, but, because it is random, it occurs in a nonstationary fashion. Random processes drift in and out of coherence transiently, mimicking nonstationary association. At small scales, random coherence is relatively brief, but as scales increase, it tends to occur over longer time periods. Because coherence occurs with or without a true relationship, we assess both the occurrence of transient coherence, as described above, as well as the duration of this coherence. For the latter test, we again generate random pairs of time series and assess them for significant coherence as described above. We then calculate the maximum duration of significant coherence on each scale over the area where edge effects are minimized. Using Monte Carlo simulations (10,000 here), we generate a scale specific distribution of the duration of maximum random coherence, which serves as a basis for measuring the probability of finding significant coherence of any given duration at any given scale.

Computing Environment

All analyses were performed using the statistical package R (version 2.6.0) [29]. Much of the code was adapted from MATLAB code by Torrence and Compo [21] and Grinsted [28]. The complete code is available from the corresponding author.

Wavelet Transforms

Figure 1B shows the wavelet power spectrum of dengue incidence in Puerto Rico. High power indicates frequency- and time-specific periodicity. The wavelet transform of dengue incidence in Puerto Rico showed significant periodicity on the 1-y scale. High power was also present in the 3–6-y period range, but did not reach significance compared to the autocorrelated null hypothesis. The temporal location of this power corresponds mainly to the large epidemics of 1994 and 1998. Temperature and precipitation in Puerto Rico also showed consistent significant power on the 1-y scale, but not at larger scales (Figure 1C and 1D). In contrast, ENSO exhibited little periodicity at the 1-y scale, and significant periodicity at 2–7 y (Figure 1E). Dengue incidence in Mexico and Thailand exhibited similar behavior to that in Puerto Rico (Figure S3). The yearly periodic was strong and significant through time and regions of higher (but not statistically significant) power occurred at a mode of approximately 8 y in Mexico (the detection limit for the length of the time series) and in the 1.5–3-y range in Thailand.

Coherence

We first analyzed the direct association between dengue incidence and ENSO in the three regions. In Puerto Rico, dengue incidence showed significantly coherence with ENSO on a 3.3- to 6-y scale from approximately 1995 to 2002 (Figure 2A). The significant local coherence between ENSO and dengue in Puerto Rico occurred over a maximum of 68 mo at a scale of approximately 3.6 y. In 10,000 simulations the probability of significant coherence of this duration or longer at this scale given randomly generated unrelated time series was 0.016. The phase difference between 1995 and 2002 shows increased dengue incidence followed increased ENSO by approximately 6 mo (Figure 2B). Figure 2C shows the reconstructed dengue and ENSO signals at the period of peak interannual coherence. In Mexico, no notable coherence occurred at time scales greater than 1 y (Figure 3A). Although there was coherence at multiyear scales in Thailand (in the 2–3-y mode), it did not reach statistical significance (Figure 3B).

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Figure 2. Coherence between ENSO and dengue in Puerto Rico.

(A) Squared coherence plot of dengue incidence in Puerto Rico and ENSO. Coherence increases from blue to red. Areas where coherence is significantly high (95% confidence level) are encircled by black lines. Shaded areas indicate the presence of edge effects. (B) Phase of ENSO (solid black) and dengue incidence (red) and phase difference (dashed black) at a periodicity of 3–4.5 y. (C) Reconstructed ENSO (black) and dengue incidence (red) at a periodicity of 3–4.5 y. A similarly scaled Morlet wavelet is superimposed (blue).

https://doi.org/10.1371/journal.pmed.1000168.g002

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Figure 3. Coherence between ENSO and dengue in Mexico and Thailand.

(A) Squared coherence plot of ENSO and dengue incidence in Mexico. (B) Squared coherence plot of ENSO and dengue incidence in Thailand. Features are as described in Figure 2A.

https://doi.org/10.1371/journal.pmed.1000168.g003

If ENSO has an effect on dengue transmission it is hypothesized that this will occur via changes in local temperature and precipitation. We assessed coherence between ENSO and local temperature and precipitation from 1950 to 2000. In both Puerto Rico (Figure 4A) and Thailand (Figure S5A), temperature cohered with ENSO on multiyear scales. In Puerto Rico, the longest period of significant coherence was approximately 197 mo at a frequency mode of approximately 2.5 y (Monte Carlo significance: p<0.001). The maximum duration in Thailand was 172 mo at a frequency of approximately 2 y (Monte Carlo significance: p<0.001). The association with temperature in both areas was positive; increased temperature followed a rise in the ENSO index by approximately 5 and 3 mo, for Puerto Rico and Thailand, respectively. There was no significant multiyear coherence between temperature and ENSO in Mexico (Figure S4A). In Thailand, precipitation was also positively associated with ENSO (Figure S5B, 108 mo at the 3-y mode with a 14-mo lag; Monte Carlo significance: 0.018). Both Puerto Rico (Figure 4B) and Mexico (Figure S4B) exhibited short-term coherence between ENSO and precipitation but they did not reach significance by the Monte Carlo test of duration.

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Figure 4. Coherence between ENSO, weather, and dengue in Puerto Rico.

(A) Squared coherence plot of ENSO and temperature. (B) Squared coherence plot of ENSO and precipitation. (C) Squared coherence plot of temperature and dengue incidence. (D) Squared coherence plot of precipitation and dengue incidence. Features are as described in Figure 2A.

https://doi.org/10.1371/journal.pmed.1000168.g004

Given the observed associations between ENSO and local weather, we analyzed coherence between local weather and local dengue incidence. In all three areas, temperature and precipitation cohered to dengue incidence on the annual scale (Figures 4C, 4D, S4C, S4D, S5C, S5D). At interannual scales the associations varied. Temperature did not cohere significantly with dengue incidence at multiyear scales in any of the areas. In Puerto Rico, precipitation cohered significantly with dengue incidence (60 mo at the 1.8-y mode; Monte Carlo significance: p = 0.006). The phase difference between precipitation and dengue incidence during the time period of significant coherence can be interpreted in two ways (Figure S6). The closest temporal association suggests that precipitation follows, rather than leads, variation in dengue transmission by approximately 4 mo. Alternatively, the phase difference observed could represent an inverse relationship where decreased precipitation leads to increased dengue 7 mo later. In Thailand, precipitation cohered significantly with dengue incidence (36 mo at the 2.5-y mode; Monte Carlo significance: p = 0.030). The phase difference suggests a negative association between precipitation and dengue incidence 2 mo later, similar to the results observed in Puerto Rico. In Mexico, precipitation did not cohere with dengue incidence on multiyear scales.