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Driving the model with actual temperature time series. The mechanistic model, driven by temperature with the mediation of temporary serotype cross-protection, was capable of producing qualitatively similar dynamics to those observed in the data (e.g., see S12 Fig in S1 Appendix), thereby supporting the hypothesis that temperature plays an important role. The ratios between reported cases and our modeled output (median of 0.11 across all provinces and points in time) are up to almost 30 times greater (for a cross-protection of 1 year) than those estimated by Reich and colleagues [19] (median values of between 0.0037 and 0.0120 across serotypes), suggesting that transmission rates in the model may be comparatively low. However, our results were robust to changes in transmission (see section “Robustness tests” in S1 Appendix). For simulation 1 (using temperature time series for the 72 provinces), the WMF for the model output had statistically significant synchrony at all times (Fig 2C). The Pearson and Spearman correlations between the modeled dengue output and the dengue data WMFs (Fig 2B and 2C) were 0.29 (P = 0.017) and 0.32 (P = 0.011), respectively. That these correlation coefficients were statistically significant, but those between dengue cases and the temperature were not, suggests that the nonlinearities in temperature encoded in the model, together with cross-protection, are important elements of dengue dynamics. These correlation coefficients were still significant for different assumptions on the mean duration of cross-protection (S1 Table in S1 Appendix). The phase angles between temperature and model output across provinces were, as expected, consistent and statistically significant over most time points (Fig 2E). WMFs and CWMFs for model outputs using different assumptions on cross-protection are shown in S13 Fig in S1 Appendix and show that with longer mean cross-protections (e.g., 2 years), the consistency in the phase angles between modeled dengue and temperature is lower than with shorter cross-protections. Both the WMF and CWMF for the model output contain a notable synchrony event in the early 1970s, reproducing that found in temperature; this feature is absent from the WMF and CWMF estimated using the data (Fig 2B and 2D). We also compared the correlations between the wavelet spectra of the modeled dengue output and dengue data for each province, for low- and high-synchrony periods separately. The correlations in high-synchrony conditions were also significantly greater than during low synchrony, but here, low-synchrony correlations were on average greater than the low-synchrony correlations between temperature and dengue data (see S2 Table and S16 Fig in S1 Appendix). These results show that the model (which includes dynamics of immunity) captures mechanisms at play in each province during low synchrony and explains some of the variations in multiannual dynamics of dengue that temperature alone cannot.

A temperature gradient can generate asynchronous dengue dynamics. For the idealized experiments of simulation 2, we found that with at least a mean of 1 year of temporary cross-protection between serotypes, the resulting dengue dynamics include multiannual cycles (Fig 4A– 4C), cycles that are completely attributable to the dynamics of immunity given that here there are no multiannual cycles in temperature. The periodicity of these cycles (and their amplitude), however, changed as a result of the temperature regime. As temperature conditions are optimized for transmission (i.e., as mean temperatures increase, approaching the higher temperatures that characterize temperature in, for instance, Bangkok; S42 Fig in S1 Appendix), the periodicity of multiannual components in dengue go down (i.e., multiannual cycles are more frequent), and the power at the multiannual timescales increases. At the highest mean temperature, with a mean of 2 years of cross-protection, seasonal cycles all but disappear (Fig 4C) due to 2 main trends. On the one hand, as cross-protection becomes longer, the power of the multiannual cycles (and, therefore, their amplitude) increases (Fig 4A–4C). On the other, the seasonal variation in the transmission is most limited at the highest mean temperatures (S44 Fig in S1 Appendix), thus producing a more limited seasonal variation in dengue cases. Taken together, these 2 trends mean that dynamics become increasingly dominated by the multiannual cycles produced through cross-protection and less so by seasonal variation in temperature. While the periodicity of the multiannual cycles for each temperature regime changes little between means of 1 and 2 years of cross-protection, their prominence does: With longer cross-protection, these cycles increase in mean wavelet power. The range of dynamics produced across locations under all assumptions of cross-protection is sufficiently diverse: No synchrony is detected (e.g., Fig 4D). Nevertheless, part of the asynchrony observed in dengue cases (Fig 2B) may also be due to asynchrony in the multiannual cycles in temperature across locations.

Common multiannual fluctuations in temperature can synchronize dengue dynamics. Because different temperature regimes can produce distinct dengue dynamics (Fig 4A–4C), one simple explanation for synchrony could be that temperatures across the country converge to a more similar temperature regime during those periods, resulting in more synchronous dengue dynamics. However, this has not been the case in Thailand (S38 Fig in S1 Appendix). The next hypothesis is that a common multiannual fluctuation in temperature can synchronize dengue dynamics. The result for simulation 3 was that when the amplitude of the multiannual fluctuation in temperature was at least 20% of the seasonal cycle, the 6 locations were temporarily synchronized (Fig 4E), under all assumptions on cross-protection. Synchrony in dengue was typically detected across a range of timescales. However, when the multiannual fluctuation had an amplitude of 10% of that of the seasonal cycle and the mean cross-protection was 2 years, synchrony was weaker (S32 Fig in S1 Appendix). Synchrony was also generally weaker when the multiannual fluctuation in temperature had a longer periodicity (e.g., 5 years) and a cross-protection of at least 1 year.

The roles played by immunity and temperature. The synchronization achieved in the simulation 3 experiments varied as a function of the mean duration of cross-protection: The degree of synchrony fell with a longer cross-protection (S24–S32 Figs in S1 Appendix). To explain why, we need to better understand how the multiannual timescales produced by intrinsic and extrinsic factors interact. Comparing simulations with (simulation 3) and without (simulation 2) a single multiannual fluctuation in temperature allows us to quantify how the multiannual timescales in dengue change specifically as a result of the multiannual fluctuation in temperature. We found that while the periodicity of the multiannual fluctuation in temperature is generally detectable in the resulting dengue dynamics, its importance relative to the timescales of the intrinsic dynamics (i.e., the dengue multiannual timescales attributable to cross-protection alone) varies as a function of mean temperature, seasonal amplitude in temperature, and mean cross-protection (Fig 5). Dengue appears to be more insensitive to multi-annual fluctuations in temperature with increasing mean temperatures, smaller seasonal variations in temperature, and longer cross-protections. Specifically, as the duration of cross-protection and/or mean temperatures increase, and as seasonality in temperature is reduced, the power of the multiannual intrinsic dynamics increases, such that the effect of the multiannual fluctuation in temperature can be overwhelming. In our simulations, this is particularly the case for higher mean temperatures and a mean cross-protection of 2 years (Fig 5).

It is also worth noting that the multiannual fluctuation in temperature is not necessarily straightforwardly reproduced in dengue dynamics. Instead, it elicits a transient response in dengue cases that can extend in time beyond the multiannual fluctuation in temperature, as can be clearly seen when plotting the phase angle between simulations with a multiannual fluctuation in temperature, and those without (S33 Fig in S1 Appendix). These show both that the effect of the multiannual fluctuation in temperature can be lasting and also confirm that the effect of the multiannual fluctuation in temperature is reduced with longer cross-protections and higher mean temperatures.

Discussion

There are several caveats to our analyses. We chose to focus on temperature because we have specific theories for how it might mechanistically impact dengue dynamics. However, we could not ascribe every pattern in synchrony in dengue to synchrony in temperature. For example, Thailand experienced a strongly synchronous 2-year fluctuation in temperature around 1972, and while this fluctuation was reproduced in our model output, it was conspicuously absent from the observed dengue dynamics. There are multiple reasons that could explain this discrepancy. During the earlier part of the records, dengue cases might have been detected less effectively, thus potentially not qualitatively capturing multiannual features of the underlying dynamics. On the other hand, other environmental variables (notably precipitation) and their interaction with temperature are likely important for the dynamics of the vector of dengue. For instance, the extent to which a synchronous event in temperature can synchronize multiannual cycles in dengue might depend on the amount of precipitation or the timing of the rainy seasons during those years. Additionally, other nonenvironmental mechanisms might also influence synchrony in dengue. Complex multiannual cycles, synchrony, and variations in the degree of synchrony over time can in theory be produced via, for example, the movement of hosts between locations [15]. There is also a range of concurrently ongoing changes in Thailand that may also influence multiannual dynamics and synchrony. For instance, the birth rates across the country have been declining [53], and the rates of decline may be spatially heterogeneous. Similarly, long-term trends in population densities across the country could also contribute to the observed spatiotemporal patterns. Exploring how these factors may interact with temperature and contribute to the observed spatiotemporal patterns deserves further attention. The mechanistic model we use here when compared to data, also leads to reporting ratios that are distinctly higher than those estimated in the literature [19], implying transmission rates in the model may be low. Lower transmission rates might be expected to affect the dynamics of immunity, as susceptibles would be depleted at a slower rate, and would lead to an increase in the periodicity of multiannual cycles produced by immunity. Nonetheless, our results on synchrony are robust to variations in transmission. Finally, the model we use necessarily makes simplifying assumptions (see section “Caveats on the dengue model” in S1 Appendix). Further work is required to understand which of these assumptions may significantly affect synchrony.

The patterns in synchrony in temperature we have observed arise as a result of broader climatological patterns. Specifically, the synchronous events in temperature (Fig 2A) coincide with strong El Niño and La Niña events (1972 to 1973, 1982 to 1983, 1997 to 1998, and 2014 to 2016), during which respectively higher and lower than average temperatures were recorded across the country [54]. The multiannual timescale at which synchrony is detected might then depend on the extent to which El Niño events are followed by La Niña events and their respective strengths. Because of the broad geographical reach of the effects of the El Niño Southern Oscillation (ENSO), we might therefore expect the temperature to synchronize dengue dynamics in other countries where ENSO has a similar manifestation in local climate conditions, as shown, for example, in van Panhuis and colleagues [40]. Indeed, several studies have sought links between dengue dynamics and ENSO (e.g., [35,38,40,55,56]). Here, our aim was to test specific hypotheses on how temperature, an environmental variable that we know affects the vector of dengue, can produce asynchrony and synchrony in dengue cases. Care should be taken when drawing the link between ENSO and dengue dynamics, because ENSO affects the environment beyond just temperature, and the physical manifestations of ENSO in any location are complex and can change over time [57]. For example, precipitation, which likely also plays a role in dengue dynamics, can also vary with ENSO (drier conditions associated with El Niño, wetter with La Niña; [58]), but in Thailand, the correlation between precipitation and ENSO has increased over time, while in India, the opposite trend is true [59].

Any disease that is primarily driven by extrinsic (environmental) factors might exhibit a similar pattern of coinciding and related synchronizing events in the environment and in the disease, as we observed in this study. The feature that makes dengue a complex system in the context of our study is that intrinsic dynamics can have variation at similar (multiannual) periodicities as the extrinsic drivers (temperature), leading to difficulty in disentangling the role of each in this system. This feature is common to other disease systems, and disentangling extrinsic and intrinsic factors has been a major question in previous work [29–34]. Here, we have applied and developed a series of methods (CWMFs, correlations at low and high synchrony) and used simulations to better understand the relative role of each. Our approach might therefore be useful to other systems where there is a similar intermingling of timescales and for which intrinsic dynamics might constitute an important filter to consider (as might be the case in, e.g., Zika, chikungunya, respiratory syncytial virus, influenza).

Most locations’ temperatures appear to be increasing in mean and decreasing in seasonal variation (S39 and S42 Figs in S1 Appendix). If these trends were to be maintained, our results suggest that dengue dynamics could slowly shift, too, and approach a regime characterized by higher average incidence (S19–S21 Figs in S1 Appendix), and dynamics increasingly dominated by multiannual, rather than seasonal, cycles. Multiannual cycles may also be, to greater extents, dictated by intrinsic factors over the multiannual timescales present in temperature, meaning that in the future, the temperature may be less able to synchronize dengue dynamics. However, these hypotheses are contingent on a more precise understanding of the interactions between serotypes and contributions from other extrinsic (e.g., rainfall) and intrinsic factors (such as changes in demography and movement of hosts between locations). 

Materials and methods

Data

Temperature time series were obtained from the GHCN CAMS gridded (0.5˚ by 0.5˚ resolution) monthly mean temperature data set (provided by NOAA/OAR/ESRL PSD, Boulder, Colorado, USA, downloaded on 18 June 2019). This dataset combines station observations from the Global Historical Climatology Network v2 and the Climate Anomaly Monitoring System [60] (S37 and S38 Figs in S1 Appendix). The time series start in 1948. We downloaded shapefiles for Thai provinces on 12 February 2019, and estimated province centroids using the function “calcCentroid” in the R package “PBSmapping” v2.72.1. A time series for each province was then obtained by using the grid point nearest to the centroid of each province.

For simulations using real temperature time series (simulation 1; Table 1), we use the 2020 population sizes of each province (rounded to the nearest 1*105 ) downloaded from the Thai National Statistical Office, on 24 November 2021. As with the numbers of cases, we combine populations for provinces created from 1982 onwards.

Wavelet transforms

We applied continuous WTs to explore how the oscillatory behavior of dengue cases has changed over time. WTs have now been applied extensively in the study of infectious disease dynamics [37,40,56,61,62] and more broadly in ecology [50,51]. WTs decompose time series into frequency components, but, as opposed to Fourier transforms, WTs are also localized in time. Thus, WTs can be used to characterize nonstationary time series and how the relative importance of different frequencies changes over time. The basis for the WT is a “mother” wavelet, a wave localized in both frequency and time (i.e., of limited duration), which is then scaled or stretched (for the frequency component) and shifted along the time axis (for the temporal component) to derive a set of“daughter” wavelets. The WT can then be understood as a correlation between (or more specifically the convolution of) the time series and the set of daughter wavelets [63].

We use the WT as implemented in the R package “WaveletComp” v1.1—details are provided in the package documentation [64] and code, and a summary is given here. The package uses a Morlet mother wavelet, ψ(t) = π−1/4 exp(i ω t) exp(−t 2 /2), where t is time (t = 1,. . ., T) and ω is the angular frequency, set to 6 radians t −1 [64]. The WT, W, at a point in time τ and scale s (proportional to period, the inverse of frequency) is:

where x(t) is the original time series, and *denotes the complex conjugate. The modulus of W (τ,s), |W(τ,s)|, gives the local amplitude A at time point τ and scale s. For any plots of the WT or calculations performed directly on it, we correct A so that A(τ,s) = s −1/2 |W(τ,s)| [65]. A(τ,s) 2 is the wavelet energy density. W(τ,s) also yields the instantaneous local wavelet phase. Although the WT contains significant redundancy in time and scale, it is possible to reconstruct the original time series (or time series containing specific scales only) on the basis of summing over the real part of the WT [63].

For these analyses, prior to performing the WT, a value of 1 was added to the time series of dengue cases prior to ln-transforming and normalizing to a mean of 0 and a standard deviation of 1. The time series of dengue cases were not detrended prior to calculating wavelet power because detrending a time series with zeroes produced artificially odd patterns. Time series of temperature were not transformed; long-term nonlinear trends were removed by taking the residuals of a local polynomial regression (using function “loess” with a span of 0.75), and the detrended time series were subsequently normalized, as above. Due to the finite length of the time series, whenever the wavelets extend beyond the edges of the time series, estimates of transform coefficients become less accurate. This issue is exacerbated as scales increase, due to wavelets extending further in time [63,64]. Regions, where these edge effects are present (often referred to as the “cone of influence”), are clearly indicated in wavelet plots by thick dashed lines.

While the dynamics of dengue are seasonal, previous studies have also identified distinct multiannual timescales [27,35,37,40,56]. We are specifically interested in these multiannual timescales; for this reason, we focus on timescales greater than 1.5 years. This lower bound prevents leakage from the annual signal into the multiannual timescales. As the timescales increase, edge effects increasingly dominate and the length of time we can reliably analyze becomes shorter. Furthermore, previous studies [35,40] typically identified multiannual components at 2 to 3 years, albeit using shorter time series. For these reasons, we here focus on the 1.5- to 5-year timescales (henceforth, “multiannual” components or timescales refer to this range). See S43 Fig in S1 Appendix for the time series of dengue and temperature and their respective WT for 4 representative Thai provinces characterized by different thermal regimes.

Wavelet mean fields as a measure of synchrony

WMFs provide a measure of synchrony as a function of both scale and point in time (further details can be found in Sheppard and colleagues [50,51]; only a summary is provided here). More intuitively, WMFs indicate scales and time points at which both phases and magnitude of oscillations are consistent (or more synchronous) across provinces. WMFs are a mean of the power-normalized WTs of each location n across all N locations (n = 1,. . ., N), where we normalized each WT by

To test whether synchrony is statistically significant in the multiannual range against a null hypothesis of no synchrony, we focus on the consistency of phases across locations (see section “A note on estimating the significance of (cross-)wavelet mean fields” in S1 Appendix for an explanation as to why the significance of WMFs focusses on the consistency of phases). We do so by estimating the wavelet phasor mean field (WPMF), defined as

which retains information on the complex phases of the transforms [51]. When different locations have a similar phase, the value of the WPMF will be large, while when the phase in each location is independent of all other locations, the value of the WPMF will be small. We test significance by generating surrogate datasets where the autocorrelation structure within each time series is preserved. We do this by generating time series with the same Fourier spectrum as the original time series [51]. Following this approach, we produced 1,000 surrogate time series, using the function “surrogate” in R package “wsyn” v1.0.2 [66]. For each surrogate dataset, we produced a WPMF, and then, for each point in time, and across the multiannual scales, we

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