The data used to analyze seasonality and detect epidemic periods concern confirmed malaria cases (across all groups) in Burundi. These cases are reported monthly over a 5-year period from 2019 to 2023. These data are collected by the National Integrated Malaria Control Program through the health districts. The raw data are gathered by the health district from the health facilities at the community level. Data are then stored in the District Health Information Software 2 (DHIS2), an information system used by the Ministry of Health and the Fight Against AIDS for data storage.

The data on confirmed malaria cases were initially stratified by health districts. To obtain the confirmed malaria cases at the national level, we summed the cases reported in all districts for each month of the observation period. A malaria case is considered confirmed if the result of the thick drop test or rapid test is positive. Missing data at the district level were not imputed. In consultation with the program staff, it was determined that these data are missing primarily because no malaria screenings were conducted during that period.

The study period was primarily selected due to its recency. The epidemiological dynamics of recent periods provide an updated understanding of malaria seasonality and transmission patterns, while also accounting for changes in key influencing factors such as climate particularly in an era marked by the imminent effects of climate change. This allows for a forward-looking perspective on malaria seasonality and epidemiology, using recent trends as a basis for anticipating future patterns. Furthermore, the choice of this study period was also guided by data availability within the DHIS2 database. Only data from the 2019-2023 period met the necessary standards of completeness, reliability, and quality required for rigorous analysis.

Burundi is a landlocked country in East Africa, located 1200 km from the Indian Ocean and 2200 km from the Atlantic Ocean [8] . It is bordered to the north by Rwanda, to the East and South by Tanzania, and to the west by the Democratic Republic of Congo [8] . It is a country in the Great Lakes region of Africa, located between the meridians 29˚00' - 30˚25' East and the parallels 2˚20' - 4˚25' South [8] . Its total area is 27,834 km² [8] . The population, which is predominantly rural with an urbanization rate around 10.4%, was estimated at 8,053,574 inhabitants with an average density of 310 inhabitants/km² in the 2008 national population census [8] . According to the preliminary results of the 2024 general population census, the population of Burundi was estimated at 12,332,788 [9] .

To describe the series of confirmed malaria cases in Burundi, we use various descriptive parameters, including the minimum, maximum, mean, and the confidence interval of the mean. These parameters will be used to describe each month of the year for the entire observation period.

The trend of our series was evaluated using LOESS (Locally Estimated Scatterplot Smoothing) adjustment. This is a non-parametric smoothing method for the series, based on the smoothing parameter known as the span, which controls the width of the smoothing. In our case, we did not specify a particular value for the span. The function used in R directly provides the optimal span enabling the generation of a relatively well-smoothed series.

The statistical analysis of confirmed malaria cases stratified monthly for the period from 2019 to 2023 aims to decompose the series and primarily extract the seasonal component. To achieve this, the method used is based on Haar wavelet decomposition. This is a non-parametric decomposition method.

The choice of a non-parametric model was based on the fact that the malaria case series does not meet the conditions for applying a parametric decomposition model. This primarily concerns the non-stationarity of the series. The stationarity test used was the Phillips-Perron test due to its robustness compared to other stationarity tests [10] . The null hypothesis of the test is that “the series is not stationary”. The p-value found is 0.3018, which suggests that there is evidence in favor of the null hypothesis. Therefore, the series is non-stationary. Secondly, the series is not periodic. This is evident when performing a simple parametric decomposition in R. The decomposition is impossible because the series has less than two periods. This provides little insight into the frequency of seasonal peaks in malaria cases. However, this may be due to the fact that the series of confirmed malaria cases is short, with only 60 observations.

The wavelet model is [11] :

${\left\{{\psi }_{u,s}\left(t\right)=\frac{1}{\sqrt{s}}\psi \left(\frac{t-u}{s}\right)\right\}}_{\left(u,s\right)\in IR\ast I{R}_0^{+}}$

with:

u the time parameter; s the scale parameter.

A wavelet is an oscillating function with a zero mean, called ψ possessing a certain degree of regularity and having finite support (which explains the term “wavelet,” meaning small wave) [11] . We choose the Haar wavelet to study the series with two components: the variation and the mean. The Haar model is given by its two components. The so-called Haar function ϕ and the Haar wavelet ψ given by [11] :

$\psi \left(x\right)=\{\begin{array}{l}-1if0\le x<1/2\\ 1if1/2\le x<1\\ 0ifnot\end{array}$, $\phi \left(x\right)=\{\begin{array}{l}1if0\le x<1/2\\ 1if1/2\le x<1\\ 0ifnot\end{array}$

In the definition of a wavelet basis, the function ϕ is called the scaling function, and the function ψ is called the mother wavelet function [11] . These components are respectively provided by the scaling coefficients and the wavelet coefficients. These two coefficients respectively measure the local means and the local variations of a series [12] . These coefficients can describe the series at different levels. The parameter related to this is the decomposition level parameter. We are interested in what happens at the finest scale of our series, that is, at the first decomposition level. To describe a signal Z at level j (at a scale of j∆t) we use coefficients Vj and Wj which respectively represent the scaling coefficients and the wavelet coefficients of length N/(2j) (if N is even) [12] . In the case where N is odd, the length of the coefficients will be N − 1/(2j). Thus, if we consider that 2 consecutive points are spaced by ∆t, the first decomposition level corresponds to the scale ∆t with wavelet coefficient vectors W1 which is proportional to the variations between two segments of width ∆t, and with scaling coefficient vectors V1 which is proportional to the means over two segments of width ∆t [12] .

For our series, the coefficients that will be obtained will characterize the pairs January-February; March-April; May-June; July-August; September-October; and November-December of each year during the observation period. However, these coefficients have the drawback of not fitting the values of our series, whether in x or in y, and their length is N/(2j). To address this and obtain coefficients of length N, we will perform an inverse transformation of the wavelet and scaling coefficients in order to obtain, respectively, a detail series Dj and a smoothed series Sj. We can therefore consider that the details Dj and the smooth Sj obtained respectively as the corrected wavelet coefficients and the corrected scaling coefficients [12] .

We then obtain, for our series, the coefficients of variations and means for each month, but which still characterize the variations and means of consecutive month pairs. But in reality, we are unsure about the coefficients of the month transitions such as February-March; April-May; June-July; August-September; October-November; and December-January. To address this, we remove the first value of January 2019 from the original series. Thus, the decomposition process will start from February, and the obtained coefficients will correspond to the pairs February-March ; April-May; June-July; August-September; October-November; and December-January (of the following year). In order to maintain 60 observations and to obtain the coefficients for the December 2023-January 2024 month pair, the month of January 2024 was imputed by the average of the January months from the observation period, and the decomposition was performed.

Figure 1 shows a general downward trend in malaria cases from month 1 to month 20 (January 2019 to August 2020) and from month 39 to month 60 (March 2022 to December 2023), while an upward trend in malaria cases is observed from month 24 to month 38 (December 2020 to February 2022).

Regarding seasonality and epidemiology, this trend helps us to identify, in general terms, the time intervals during which local minima and maxima of malaria cases occur and the seasonal rhythm at which this pattern repeats.

According to Table 1 , a total of 31,777,626 malaria cases were recorded over the entire period. The year 2023 had the lowest number of malaria cases, with 4,320,032 cases, while 2019 recorded the highest number of cases throughout the period, with 8,913,210 malaria cases.

Table 2 indicates that August had the lowest average number of malaria cases, with 392,306 cases, whereas January had the highest, averaging 631,649 cases. This contrast offers an initial insight into the seasonal pattern of malaria, with January showing a peak in cases and August recording comparatively fewer cases than other months.

Similarly, all confidence intervals that do not contain zero show that these averages are significantly high. Continuing to discuss the epidemic nature of the months, January recorded the highest maximum number of observed cases, with 927,875 cases, while August has the second-lowest maximum number of cases with 520,680 cases, just behind September, which recorded 513,439 cases. At the same time, it is still the month of August that records the lowest minimum number of malaria cases, with 249,592 cases. In summary, the highest number of malaria cases is observed in the month of January, while the lowest number of malaria cases is observed in the month of August.

In Figure 2 , the means and variations obtained characterize the pairs of months: January-February; March-April; May-June; July-August; September-October; and November-December. The scale coefficients for certain months in our observation period show very high local means, while others have lower local means.

For illustration, the means for the pairs of the 1st and 2nd months; 3rd and 4th months; 5th and 6th months; 11th and 12th months; and 41st and 42nd months exceed 700,000 cases ( Figure 2A ). These correspond to January-February 2019, March -April 2019, May-June 2019, November-December 2019, as well as May-June 2022.

As for the variation in malaria cases, month pairs such as January-February 2019, July-August 2019, September-October 2019; January-February 2020, September-October 2020; January-February 2021, January-February 2022, July-August 2022; January-February 2023, and September-October 2023 have a variation in malaria cases greater than or equal to 100,000 ( Figure 2B ).

Figure 3 represents the coefficients for month pairs such as February-March; April-May; June-July; August-September; October-November, and December-January (of the following year).

For the month pairs of February-March , April-May, June-July 2019, as well as April-May and June-July 2022, the average number of cases exceeds the 700,000 mark ( Figure 3A ). As for the variations, month pairs such as April-May 2019, October-November 2020, October-November 2021, and June-July 2022 record variations in the order of 100,000 ( Figure 3B ).

The general period from February to June, as well as from August to December, are periods during which an increase in malaria cases is observed, with a more pronounced increase from August to December. In these same periods, the transitions from April-May and September-October show a significant increase in cases, with an increase of 80,888 and 108,444 cases respectively. At the same time, the periods from January to July and from October to December record malaria averages above 500,000 cases. The highest malaria figures are observed from May to June, due to its higher average. The transition from July to August is a turning point, with a notable average decrease of about 110,871 cases and an average of cases below 500,000 ( Table 3 ).