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LFC designed the study, performed the research, analyzed the data, and wrote the paper. MP provided input on the methods and the design of the study and contributed to the writing of the paper.

The authors have declared that no competing interests exist.

Cutaneous leishmaniasis (CL) is one of the main emergent diseases in the Americas. As in other vector-transmitted diseases, its transmission is sensitive to the physical environment, but no study has addressed the nonstationary nature of such relationships or the interannual patterns of cycling of the disease.

We studied monthly data, spanning from 1991 to 2001, of CL incidence in Costa Rica using several approaches for nonstationary time series analysis in order to ensure robustness in the description of CL's cycles. Interannual cycles of the disease and the association of these cycles to climate variables were described using frequency and time-frequency techniques for time series analysis. We fitted linear models to the data using climatic predictors, and tested forecasting accuracy for several intervals of time. Forecasts were evaluated using “out of fit” data (i.e., data not used to fit the models). We showed that CL has cycles of approximately 3 y that are coherent with those of temperature and El Niño Southern Oscillation indices (Sea Surface Temperature 4 and Multivariate ENSO Index).

Linear models using temperature and MEI can predict satisfactorily CL incidence dynamics up to 12 mo ahead, with an accuracy that varies from 72% to 77% depending on prediction time. They clearly outperform simpler models with no climate predictors, a finding that further supports a dynamical link between the disease and climate.

Using mathematical models, the authors show that cutaneous leishmaniasis has cycles of approximately three years that are related to temperature cycles and indices of the El Niño Southern Oscillation.

Every year, 2 million people become infected with a pathogenic species of

It would be very useful to have early warning systems for leishmaniasis and other vector-transmitted diseases so that public health officials could prepare for epidemics—or spikes in the number of cases—of these diseases. Monitoring of climatic changes could form the basis of such systems. But although it is clear that the transmission of cutaneous leishmaniasis is affected by fluctuations in the climate, there have been no detailed studies into the dynamics of its seasonal or yearly variation. In this study, the researchers used climatic data and information about cutaneous leishmaniasis in Costa Rica to build statistical models that investigate how climate affects leishmaniasis transmission.

The researchers obtained the monthly records for cutaneous leishmaniasis in Costa Rica for 1991 to 2001. They then used several advanced statistical models to investigate how these data relate to climatic variables such as the sea surface temperature (a measure of El Niño, a large-scale warming of the sea), average temperature in Costa Rica, and the MEI (the Multivariate ENSO Index, a collection of temperature and air pressure measurements that predicts when the ENSO is going to occur). Their analyses show that cutaneous leishmaniasis cases usually peak in May and that the incidence of the disease (number of cases occurring in the population over a set time period) rises and falls in three-year cycles. These cycles, they report, match up with similar-length cycles in the climatic variables that they investigated. Furthermore, when the researchers tested the models they had constructed with data that had not been used to construct the models (“out of fit” data), the models predicted variations in the incidence of cutaneous leishmaniasis for up 12 months with an accuracy of about 75% (that is, the predictions were correct three times out of four).

The finding that interannual cycles of climate variables and cutaneous leishmaniasis coincide provides strong evidence that climate does indeed affect the transmission of this disease. This link is strengthened by the ability of the statistical models described by the researchers to predict outbreaks with high accuracy. The researchers' new insights into how climate affects the transmission of cutaneous leishmaniasis are important because they open the door to the possibility of setting up an early warning system for this increasingly common disease. The same statistical approach could be used to improve understanding of how climate affects the dynamics of other vector-transmitted diseases and to design early warning systems for them as well—the World Health Organization has identified 18 diseases for which climate-based early warning systems might be useful but no such systems are currently being used to plan disease control strategies. Finally, the improved understanding of the relationship between climate and disease transmission that the researchers have gained through their study is an important step towards being able to predict how the incidence and distribution of leishmaniasis and other vector-transmitted diseases will be affected by global warming.

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The leishmaniases are among the most important emerging and resurging vector-borne protozoal diseases, second only to malaria in terms of the number of affected people. Like malaria, the leishmaniases can be caused by infection with any of several species of parasites belonging to the genus

As with other vector-borne diseases, seasonal patterns in cases and vector abundance suggest that cutaneous leishmaniasis (CL) transmission is sensitive to climatic exogenous factors. More specifically, vector density is correlated to climate variables—producing seasonal patterns that have been widely described [

In the present paper, CL cycles and their relationship to climate variables are described, and linear statistical models that use climate variables as predictors are used to assess the accuracy of forecasts based on climatic variables.

Monthly records of CL cases, from January 1991 to December 2001, were obtained from the epidemic surveillance service, Vigilancia de la Salud, of Costa Rica. Data were normalized using a square root transformation. Sea Surface Temperature 4 (SST 4) (also known as the Niño 4 index;

(A) CL cases in Costa Rica.

(B) Mean temperature in Costa Rica.

(C) SST 4.

(D) MEI.

(E) Box plot with monthly square-root-transformed CL cases.

(F) The fits of (1) the Daubechies discrete wavelet (green lines), used to detrend the series so that the resulting data can then be analyzed for their dominant frequencies with a periodogram (a filter number 5 and eight levels of decomposition were used for this wavelet; the dashed line corresponds to periodic edges, and the dotted line to symmetric ones); (2) smoothing splines (blue solid line) and the first four reconstructed components of singular spectrum analysis (black dashed line; 60 orders). These methods were used to de-noise the signals so that dominant frequencies could be identified (with the maximum entropy spectral density method).

The seasonality of CL cases was assessed by using a box diagram (see

A time series is stationary if it has a constant mean and variance [_{t},

The second method consisted of the maximun entropy spectral density _{k}

The above characterizations of the cycles consider the whole temporal extent of the data and therefore provide, as such, only an average picture of dominant frequencies in the data. More recently, the importance of localizing these frequencies in time has been emphasized, particularly for nonstationary data. The wavelet power spectrum (WPS_{y}

Besides allowing us to determine how the variability of the data is allocated to different frequencies at different times, the wavelet transform can be used to study the patterns of association between two nonstationary time series [

Seasonal autoregressive models were fitted to the data using the Kalman recursions for their state space representation ([

On the basis of the cross-correlation functions, a full model was fitted that included as predictors all the statistically significant lagged climate variables [

Forecasts were obtained for time intervals 1, 3, 6, and 12 mo ahead using a total of 24 mo for each time interval. The model was refitted before computing the next prediction in two different ways, by including (i) all the previous months in the series or (ii) only the months from the previous 9 y.

The accuracy of the forecast was measured using the predictive ^{2}, which has an interpretation similar to the ^{2} of a linear regression as defined in [

CL shows, on average, a seasonal peak during May, though epidemic outbreaks happen around the year, as demonstrated by the existence of outliers in February, July, September, October, November, and December (

(A and B) Smoothed periodograms for (A) the detrended series (using Daubechies discrete wavelet with filter number 5 and periodic edges) and (B) the detrended series (using Daubechies discrete wavelet with filter number 5 and symmetric edges). In the periodograms, the blue lines are the 95% point confidence intervals [

(C and D) Maximum entropy spectral density for (C) the de-noised series (with smoothing splines) and (D) the de-noised series (with singular spectrum analysis). For the periodograms and the maximum entropy spectral density, frequencies are in cycles per year.

(E) Wavelet power spectrum. The solid line is the cone of influence indicating the region of time and frequency where the results are not influenced by the edges of the data and are therefore reliable. The dashed line corresponds to the 95% confidence interval for white noise based on the variance of the square-root-transformed incidence series. The intervals were obtained using a chi-squared distributed statistic with one degree of freedom (see [

In all analyses, the cases are square-root-transformed. Maximum entropy spectral densities were computed using the software described in [

The coherency scale is from zero (blue) to one (red). Thus, red regions indicate frequencies and times for which the two series share variability. The cone of influence (within which results are not influenced by the edges of the data) and the significant (

(A–C) Cross-correlation functions (CCF) with (A) SST 4, (B) MEI, (C) Temperature. The blue dashed lines are the 95% point confidence intervals for the cross-correlation between two series that are white noise [

(D) Predictive ^{2} measuring the accuracy of the predictions. Blue is for predictions with only 9 y of training data (used to fit the model) and black for predictions generated with all months preceding the prediction. (The value for 12-mo predictions with temperature is not shown, because it was negative.)

The forecasting accuracy is higher for the model selected by the likelihood ratio tests and Akaike information criterion values (_{1}_{a}_{t}^{2} for the temperature model is negative for predictions 12 mo ahead, and the mean squared error is two orders of magnitude higher than the original variance of the series (

Model Selection and Parameter Values

The description of cycles in any natural phenomenon is relevant to predict its dynamic behavior. The finding of a period close to 3 y for the interannual cycles of CL is robust, as the four applied methods indicate consistently the presence of cycles between 2.7 and 3.2 y in this series. The robustness of this result indicates that its statistical significance is not an artifact of any particular methodology. However, the description of cycles by itself is not sufficient to assess the effects of exogenous drivers or gain insights into the processes driving such oscillations.

A deeper understanding of the relationship between climate and disease dynamics is key for anticipating the potential effects that trends in a changing climate would have on the incidence and distribution of the disease [

A strong association between climate and CL incidence is further supported here by the finding that linear models can forecast satisfactorily the incidence of this disease, with an accuracy between 72% and 77%. In particular, MEI and temperature are identified as useful variables sustaining predictability for a window of 1 y. Interestingly, MEI is defined as the first principal component of several climate variables that predict ENSO [

The finding that the model with MEI as the only predictor outperforms the model with just temperature supports the recent proposal that large-scale climate indices may be more useful for forecasting than local climate variables [

For CL there are several plausible ways in which climate could affect transmission dynamics. As already pointed out, vector density is sensitive to climate variability, with vector densities varying seasonally [

Future work should compare the forecasting ability of nonlinear models and more mechanistic formulations. While mechanistic models are necessary to propose and evaluate methods of control [

This series is nonstationary, because the autocorrelation function is statistically significant for lags different from zero and decays over time, a pattern that is superimposed on that resulting from seasonality, which produces a significant autocorrelation at a lag of 1 y.

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Blue is for predictions with only 9 y of training data, and black, for all previous months preceding the prediction interval.

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cutaneous leishmaniasis

El Niño Southern Oscillation

Multivariate ENSO Index

Sea Surface Temperature 4