Relationship between the phase difference angles and the time series dynamics. When both time series are in phase, the phase difference angle is θ = 0 (upper left). When the red time series is ahead of the blue time series by a quarter of the period (3 units over a period of 12), the value of the phase difference angle is one quarter of the total circle, that is θ = π/2 (upper right). When both time series are anti-phase, the phase difference angle is θ = π (bottom left). When blue time series is behind by three quarter of period (here, 9 over 12 units), θ = 3π/2 (bottom right). We observe that we cannot distinguish mathematically with the situation where time series in blue is actually leading by one quarter. In our case, however, malaria incidence always follows the effect of climatic variables and the causality direction is clear. Hence, we avoid negative values when speaking about phase difference in months.

Time scales obtained after a maximal overlap discrete wavelet transform of depth 3 of an input time series.

Time series plots (2008-2019) of the different variables of our study. Malaria incidence (A) is computed in terms of cases per person-year, air temperature (B) and LST (C) are in °C, rainfall (D) units are in mm of precipitation, Niño 3.4 SST (E) is a sea surface temperature in °C, and bed net coverage (F) is a percentage of visited households where bed nets were effectively used.

Wavelet power spectra heat maps (left) and average wavelet spectra (right) of malaria incidence (A), air temperature (B), LST (C), rainfall (D) and Niño 3.4 SST (E). Colours represent the spectrum itself, while white contour lines delineate the regions where periodicity is assessed at the 90% confidence threshold, against red noise. Light blue areas on top of the maps represent the cone of influence, where periods are too high and time is too close to time series bounds to be able to assess seasonalities. Solid points in the average power spectrum are periods tested at the 95% (red) or 90% (blue) confidence threshold.

Cross-wavelet power spectra heat map (left) and average power spectra (right) of malaria incidence related to air temperature (A), LST (B), rainfall (C) and Niño 3.4 SST (D). Colours represent the spectrum itself, while white contour lines delineate the regions where periodicity is assessed at the 90% confidence threshold, against two simulated time series following red noise processes. Light blue areas on top of the maps represent the cone of influence, where periods are too high and time is too close to time series bounds to be able to assess seasonalities. Solid points in the average power spectrum are also periods tested at the 95% (red) or 90% (blue) confidence threshold. Arrows represent the phase differences between the climatic time series and the malaria incidence.

Results of the bivariate linear regressions between the malaria incidence time series and the predictors, i.e. the climatic time series at different time scales. The climatic subseries obtained through maximal overlap discrete wavelet transform have been standardized before computation of the model. The slope coefficient assesses the direction of the relationship. The numbers in brackets represent the 95% confidence interval bounds for the slope coefficient. The lags for the lowest time scale (D1, 1-to 4-month period) have been restricted to a maximum of 4, to avoid lags greater than the time scale. The lags for other time scales have been restricted to a maximum of 8.

Time series decomposition of the malaria incidence time series using the maximal overlap discrete wavelet transform. The global original time series is plotted at the bottom, while the different subseries are plotted above. The subseries D1 corresponds to time scales of 1- to 4-month, D2: 4-to 8-month, D3: 8-to 16-month and S3 retains seasonalities of periods above 16-month.

Evolution of the lags obtained through the phase differences. The phase differences of the climatic factors related to malaria incidence have been computed at a period of 6-month. Phase differences have been converted into lags by using the method described in the Methods section.

Results of the linear regressions of the lag evolution time series against the climate time series. The time series of the lag evolution of air temperature, LST and rainfall have been fitted with climatic and intervention variables time series: air temperature, LST, rainfall, Niño 3.4 SST and bed net coverage. Only the two models with the least AIC are displayed here.

Time series decomposition of the climatic factors, using the maximal overlap discrete wavelet transform. The global (original) time series is plotted at the bottom while the different subseries are plotted above. The subseries D1 corresponds to time scales of 1-to 4-month, D2: 4-to 8-month, D3: 8-to 16-month and S3 retains seasonalities of periods above 16-month.

Results of the linear regressions between the malaria incidence time series and the predictors, i.e. the climatic time series at different time scales, for different lags. Maximal overlap discrete wavelet transform-decomposed predictor subseries have been standardized before computation of the model. The first number is the slope coefficient assessing the direction of the relationship. The numbers in brackets represent the 95% confidence interval bounds for the slope coefficient. The last number in square brackets is the Akaike Information Criterion (AIC) of the model. The asterisk means the p-value of the slope coefficient is below the 1% threshold and the bold font highlights the best model for all lags according to minimal AIC. Lags for the lowest time scale (D1, 0-4-month) have been restricted to a maximum of 4, to avoid lags greater than the time scale.