# Application of Functional Data Analysis to Identify Patterns of Malaria Incidence, to Guide Targeted Control Strategies

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Study Area and Dataset

#### 2.2. Statistical Analysis

#### 2.2.1. Estimating the Smooth Function (Functional Data) for Each Time Series

_{j}is the continuous time of week j, and ${\epsilon}_{ij}$ is an error term representing the difference between the function value and the observed data for village i at week j.

#### 2.2.2. Dissimilarity Measures and Hierarchical Ascending Clustering on Smooth Functions

^{2}. The connectivity indicates the degree of connectedness of the clusters, as determined by the k-nearest neighbors (in this work k = 10). The connectivity has a value between 0 and infinity and should be minimized. Both the silhouette width and the Dunn index combine measures of compactness and separation of the clusters. The silhouette width is the average of each curve’s silhouette value. The silhouette value measures the degree of confidence in a particular clustering assignment and lies in the interval [−1, 1] with well-clustered curves having values near 1 and poorly clustered curves having values near −1. The Dunn index is the ratio between the smallest distance between curves not in the same cluster to the largest intra-cluster distance. It has a value between 0 and infinity and should be maximized. To choose the final or optimal number of malaria incidence patterns, we performed a principal component analysis (PCA) [36] on assessed HAC results to look for the one which showed the best criteria of validity indices, i.e., with connectivity index close to 0, high Dunn index, and silhouette width and R

^{2}close to 1.

#### 2.2.3. Velocity and Acceleration

^{®}software (The R Foundation for Statistical Computing, Vienna, Austria) R 3.4.2 version. Maps were produced using QGIS

^{®}software (Open Source Geospatial Foundation, Boston, MH, USA) QGIS 3.10.1 version.

## 3. Results

#### 3.1. From Observed to Smoothed Malaria Incidence

#### 3.2. Identification of Malaria Incidence Patterns

^{2}(Figure 3, Panel A). The best classification should therefore be located in the upper right of the factorial plane of the dissimilarity measures and the number of patterns (Figure 3, Panel B). The DTWCORT1 dissimilarity measure took into account 46.2% of the temporal correlation between functional data and 53.7% of the geometric distance.

#### 3.3. Velocity and Acceleration of Malaria Incidence Patterns

## 4. Discussion

^{2}).

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Validity indices performed on each hierarchical ascending clustering’s results for 3 and 4 patterns: connectivity, Dunn, silhouette, and the percentage of inertia explained R

^{2}.

HAC Results (with 3 and 4 Clusters) and Dissimilarity Measures | Connectivity | Dunn | Silhouette | R^{2} |
---|---|---|---|---|

3EUC | 113.57 | 0.04 | 0.38 | 0.1 |

3FDA | 73.44 | 0.08 | 0.52 | 0.1 |

3DTW | 80.47 | 0.03 | 0.55 | 0.19 |

3DWT | 112.88 | 0.04 | 0.37 | 0.1 |

3EUCCORT1 | 102.07 | 0.07 | 0.46 | 0.1 |

3EUCCORT2 | 117.83 | 0.03 | 0.36 | 0.1 |

3EUCCORT3 | 118.87 | 0.03 | 0.38 | 0.1 |

3EUCCORT5 | 122.23 | 0.03 | 0.38 | 0.11 |

3FDACORT1 | 121.69 | 0.04 | 0.36 | 0.1 |

3FDACORT2 | 124.79 | 0.03 | 0.36 | 0.1 |

3FDACORT3 | 118.93 | 0.03 | 0.36 | 0.11 |

3FDACORT5 | 121.7 | 0.03 | 0.38 | 0.11 |

3DWTCORT1 | 121.06 | 0.04 | 0.36 | 0.1 |

3DWTCORT2 | 123.79 | 0.03 | 0.36 | 0.11 |

3DWTCORT3 | 66.56 | 0.08 | 0.54 | 0.1 |

3DWTCORT5 | 121.55 | 0.03 | 0.38 | 0.11 |

3DTWCORT1 | 76.4 | 0.03 | 0.55 | 0.19 |

3DTWCORT2 | 84.01 | 0.02 | 0.53 | 0.19 |

3DTWCORT3 | 109.12 | 0.01 | 0.4 | 0.19 |

3DTWCORT5 | 89.88 | 0.02 | 0.52 | 0.19 |

4EUC | 149.3 | 0.04 | 0.35 | 0.12 |

4FDA | 158.82 | 0.02 | 0.2 | 0.12 |

4DTW | 174.53 | 0.01 | 0.33 | 0.21 |

4DWT | 149.92 | 0.04 | 0.34 | 0.12 |

4EUCCORT1 | 206.79 | 0.04 | 0.27 | 0.12 |

4EUCCORT2 | 171.17 | 0.03 | 0.32 | 0.12 |

4EUCCORT3 | 173.47 | 0.03 | 0.33 | 0.12 |

4EUCCORT5 | 156.54 | 0.03 | 0.34 | 0.12 |

4FDACORT1 | 188.3 | 0.04 | 0.3 | 0.12 |

4FDACORT2 | 181.55 | 0.03 | 0.32 | 0.12 |

4FDACORT3 | 157.83 | 0.03 | 0.34 | 0.12 |

4FDACORT5 | 156.44 | 0.03 | 0.35 | 0.12 |

4DWTCORT1 | 187.75 | 0.04 | 0.3 | 0.12 |

4DWTCORT2 | 180.55 | 0.03 | 0.32 | 0.12 |

4DWTCORT3 | 163.1 | 0.02 | 0.22 | 0.12 |

4DWTCORT5 | 156.27 | 0.03 | 0.35 | 0.12 |

4DTWCORT1 | 177.8 | 0.01 | 0.31 | 0.21 |

4DTWCORT2 | 126.73 | 0.02 | 0.44 | 0.21 |

4DTWCORT3 | 151.85 | 0.01 | 0.38 | 0.22 |

4DTWCORT5 | 166.42 | 0.01 | 0.23 | 0.21 |

**Figure A1.**Dendrogram resulting of hierarchical clustering on smooth function with DTWCORT1 dissimilarity measure: 12 villages with high-incidence pattern (red), 97 villages with intermediate-incidence pattern (blue border), and 466 with a low-incidence pattern (green border).

**Figure A2.**The velocity (

**Panel A**) and the acceleration (

**Panel B**) dynamics of malaria incidence patterns: high-incidence pattern in red line, intermediate-incidence pattern in blue line, and low-incidence pattern in green line.

**Table A2.**The PCA results on epidemiological indicator (EI) durations (Variables) and seasonal outbreaks of the patterns (Individuals): the PCA indicators (correlation between EI durations and dimensions representing also the coordinates of EI durations on dimensions, cosinus

^{2}measuring the quality of projection of EI durations or seasonal outbreaks on dimensions (or factorial axis), the percentage of contribution of EI durations or seasonal outbreaks on each dimensions (or factorial axis), and the coordinates of seasonal outbreaks on each dimension (or factorial axis); Dim is dimension or factorial axis resulting on PCA on which EI durations and seasonal outbreaks were projected.

PCA_Indicators | Dim.1 | Dim.2 | Dim.3 | Dim.4 | Dim.5 |
---|---|---|---|---|---|

correlation_AB | 0.9 | −0.3 | −0.31 | −0.08 | −0.06 |

correlation _AD | 0.9 | −0.22 | −0.37 | 0.08 | 0.05 |

correlation _CE | 0.35 | 0.92 | 0 | 0.13 | −0.07 |

correlation _DG | 0.89 | 0.21 | 0.4 | −0.09 | −0.02 |

correlation _FG | 0.53 | −0.74 | 0.39 | 0.11 | 0.01 |

correlation _BF | 0.39 | 0.92 | 0.03 | −0.03 | 0.08 |

correlation _AG | 1 | 0 | 0.03 | −0.01 | 0.02 |

cosinus^{2}_AB | 0.8 | 0.09 | 0.1 | 0.01 | 0 |

cosinus^{2}_AD | 0.81 | 0.05 | 0.14 | 0.01 | 0 |

cosinus^{2}_CE | 0.13 | 0.85 | 0 | 0.02 | 0.01 |

cosinus^{2}_DG | 0.79 | 0.04 | 0.16 | 0.01 | 0 |

cosinus^{2}_FG | 0.28 | 0.55 | 0.15 | 0.01 | 0 |

cosinus^{2}_BF | 0.15 | 0.84 | 0 | 0 | 0.01 |

cosinus^{2}_AG | 1 | 0 | 0 | 0 | 0 |

contribution_AB | 20.32 | 3.71 | 17.59 | 13.75 | 19.18 |

contribution _AD | 20.38 | 2.03 | 24.91 | 13.5 | 12.86 |

contribution _CE | 3.16 | 35.1 | 0 | 31.09 | 30.65 |

contribution _DG | 19.98 | 1.75 | 29.12 | 17.03 | 1.6 |

contribution _FG | 7.15 | 22.74 | 27.98 | 23.24 | 0.42 |

contribution _BF | 3.77 | 34.68 | 0.2 | 1.22 | 33.84 |

contribution _AG | 25.24 | 0 | 0.2 | 0.17 | 1.43 |

2008H_ coordinates | 2.32 | −1.73 | 0.8 | −0.13 | 0.15 |

2008I_ coordinates | 1.08 | −1.28 | −1.15 | −0.01 | −0.16 |

2008L_ coordinates | −2.91 | 0.06 | −1.18 | −0.3 | 0.21 |

2009H_ coordinates | 2.14 | 3.73 | −0.21 | 0.11 | 0.13 |

2009I_ coordinates | 1.42 | −0.9 | 1.1 | −0.39 | −0.01 |

2009L_ coordinates | −3.89 | 1.06 | 1.05 | 0.1 | −0.13 |

2010H_ coordinates | 0.68 | 1.36 | −0.15 | −0.18 | −0.15 |

2010I_ coordinates | 0.94 | 1.61 | 0.28 | 0.08 | −0.03 |

2010L_ coordinates | −1.75 | −0.99 | 0.36 | 0.2 | −0.03 |

2011H_ coordinates | 1.06 | −0.84 | −0.74 | 0.01 | −0.18 |

2011I_ coordinates | 0.97 | −1.5 | −0.05 | 0.52 | 0.11 |

2011L_ coordinates | −2.06 | −0.58 | −0.1 | −0.02 | 0.09 |

2008H_ cosinus^{2} | 0.59 | 0.33 | 0.07 | 0 | 0 |

2008I_ cosinus^{2} | 0.28 | 0.4 | 0.32 | 0 | 0.01 |

2008L_ cosinus^{2} | 0.85 | 0 | 0.14 | 0.01 | 0 |

2009H_ cosinus^{2} | 0.25 | 0.75 | 0 | 0 | 0 |

2009I_ cosinus^{2} | 0.48 | 0.19 | 0.29 | 0.04 | 0 |

2009L_ cosinus^{2} | 0.87 | 0.06 | 0.06 | 0 | 0 |

2010H_ cosinus^{2} | 0.19 | 0.77 | 0.01 | 0.01 | 0.01 |

2010I_ cosinus^{2} | 0.25 | 0.73 | 0.02 | 0 | 0 |

2010L_ cosinus^{2} | 0.73 | 0.23 | 0.03 | 0.01 | 0 |

2011H_ cosinus^{2} | 0.46 | 0.29 | 0.23 | 0 | 0.01 |

2011I_ cosinus^{2} | 0.27 | 0.65 | 0 | 0.08 | 0 |

2011L_ cosinus^{2} | 0.92 | 0.07 | 0 | 0 | 0 |

2008H_ contribution | 11.38 | 10.3 | 9.79 | 2.57 | 11.82 |

2008I_ contribution | 2.48 | 5.64 | 20.16 | 0.01 | 12.25 |

2008L_ contribution | 17.84 | 0.01 | 21.49 | 14.11 | 21.09 |

2009H_ contribution | 9.66 | 47.62 | 0.69 | 1.96 | 7.92 |

2009I_ contribution | 4.24 | 2.79 | 18.69 | 24.31 | 0.1 |

2009L_ contribution | 31.88 | 3.84 | 16.96 | 1.68 | 8.4 |

2010H_ contribution | 0.98 | 6.32 | 0.34 | 5.15 | 10.77 |

2010I_ contribution | 1.85 | 8.91 | 1.22 | 1 | 0.52 |

2010L_ contribution | 6.47 | 3.33 | 1.94 | 6.11 | 0.5 |

2011H_ contribution | 2.35 | 2.42 | 8.51 | 0.01 | 16.01 |

2011I_ contribution | 1.97 | 7.68 | 0.04 | 43.06 | 6.39 |

2011L_ contribution | 8.92 | 1.13 | 0.17 | 0.04 | 4.23 |

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**Figure 1.**A graphical example for the seven epidemiological indicators: the beginning of seasonal outbreaks and the start acceleration of the growth phase (A); the beginning of the pre-slowdown of the growth phase (B); the deceleration’s beginning of growth phase (C); the peak (D) also corresponding after to the beginning of the acceleration of the decrease phase; the beginning of the deceleration of the decrease phase (E); the beginning of the tail (F); the end of the seasonal outbreaks (G); functional incidence in red line, functional velocity in black bold line (first derivative), and functional acceleration in black discontinuous line (second derivative).

**Figure 2.**Weekly evolution of malaria incidence for each village from January 2008 to December 2012: observed time series (

**Panel A**) and at the square root scale (

**Panel B**), smoothed time series at the square root scale (

**Panel C**) and at the scale of untransformed observations (

**Panel D**).

**Figure 3.**Principal component analysis on validity indices and dissimilarity measures for 3 and 4 number patterns: validity indices map (Variables,

**Panel A**), dissimilarity measures map (Individuals,

**Panel B**). 4DTWCORT3 is the assessed hierarchical ascending clustering (HAC) result with the potential number of patterns chosen as 4, and performed with DTWCORT3 dissimilarity measure (${d}_{\mathrm{DTWCORT}3}$) taking into account the 9.4% of ${d}_{DTW}$ (data value) and 90.5% CORT (data behavior) (Table 1, $\xi $ = 3), 3FDA is the assessed HAC result with the potential number of patterns chosen as 3, and performed with ${d}_{FDA}$ dissimilarity measure taking into account 100% of data value, etc.

**Figure 4.**The spatial distribution of malaria incidence pattern villages in the study area: Senegal map and the location of the study area pointed by the arrow, high-incidence pattern villages in red dot, intermediate-incidence pattern villages in blue dot, and low-incidence pattern villages in green dot.

**Figure 5.**The smoothed functions (functional data) for each malaria incidence pattern between January 2008 to December 2012: high-incidence pattern in red line, intermediate-incidence pattern in blue line, and low-incidence pattern in green line.

**Figure 6.**Weekly observed malaria incidence in black solid line, smoothed malaria incidence in color solid line, and smooth 95% point-wise confidence intervals in discontinuous color line: high-incidence pattern in red (

**Panel A**), intermediate-incidence pattern in blue (

**Panel B**), and low-incidence pattern in green (

**Panel C**).

**Figure 7.**Smoothed incidence in color solid line, their velocity in black bold solid line, their acceleration in black discontinuous line, and the epidemiological indicator of their seasonal outbreaks (A: onset, B: near slowdown of growth, C: beginning slowdown of growth, D: peak, E: beginning acceleration of decline, F: beginning of tail, G: end): high-incidence pattern in red (

**Panel A**), intermediate-incidence pattern in blue (

**Panel B**), and low-incidence pattern in green (

**Panel C**).

**Figure 8.**Principal component analysis on duration epidemiological indicators and seasonal outbreaks of the patterns: epidemiological indicator map (Variables,

**Panel A**) (the duration of strict growth’s acceleration phase (AB); the duration between start and peak (AD); the duration between slowdown of growth and decline (CE) indicating the width of the peak area; the duration between peak and the end (DG); the duration between the tail and the end of seasonal outbreaks (FG); the duration between pre-slowdown and the tail (BF) indicating the intermediate width of epidemics; the duration between the start and the end of epidemic episodes (AG) indicating the duration of the seasonal outbreaks); seasonal outbreaks of the patterns map (Individuals,

**Panel B**) (L=Low, I=Intermediate, H=High, 2009L is the seasonal outbreak starting in year 2009 in the malaria low-incidence pattern).

**Table 1.**The percentage of contribution in ${d}_{CORT}$ dissimilarity measure according to the parameter $\xi $.

$\mathit{\xi}$ | Behavior Contribution (%) | Values Contribution (%) |
---|---|---|

0 | 0 | 100 |

1 | 46.2 | 53.7 |

2 | 76.2 | 23.8 |

3 | 90.5 | 9.4 |

$\ge $5 | ~100 | ~0 |

**Table 2.**The description of epidemiological indicators and the determination of their corresponding date for a functional data ${C}_{q}$ of pattern q.

Epidemiological Indicators (EI) | Determination of EI’s Dates |
---|---|

Beginning of seasonal outbreaks and the start acceleration of the growth phase (A) | ${t}_{A}=\{\begin{array}{c}firsttsuch{C}_{q}{}^{\prime}\left(t\right)0\\ {C}_{q}{}^{\u2033}\left(t\right)0\\ on3weeks\end{array}$ |

Beginning of the pre-slowdown of the growth phase (B) | ${t}_{B}=\{\begin{array}{c}\underset{tsuch{C}_{q}{}^{\prime}\left(t\right)0}{\mathrm{argmax}}\left({C}_{q}{}^{\u2033}\left(t\right)\right)\end{array}$ |

Deceleration’s beginning of growth phase (C) | ${t}_{C}=\{\begin{array}{c}\underset{tsuch{C}_{q}{}^{\u2033}\left(t\right)=0}{\mathrm{argmax}}\left({C}_{q}{}^{\prime}\left(t\right)\right)\end{array}$ |

Peak of seasonal outbreaks and beginning of the acceleration of the decrease phase (D) | ${t}_{D}=\{\begin{array}{c}{C}_{q}{}^{\prime}\left(t\right)=0\\ {C}_{q}{}^{\u2033}\left(t\right)0\end{array}$ |

Beginning of the deceleration of the decrease phase (E) | ${t}_{E}=\{\begin{array}{c}\underset{tsuch{C}_{q}{}^{\u2033}\left(t\right)=0}{\mathrm{argmin}}\left({C}_{q}{}^{\prime}\left(t\right)\right)\end{array}$ |

Beginning of the tail of seasonal outbreaks (F) | ${t}_{F}=\{\begin{array}{c}\underset{tsuch{C}_{q}{}^{\prime}\left(t\right)0}{\mathrm{argmax}}\left({C}_{q}{}^{\u2033}\left(t\right)\right)\end{array}$ |

End of seasonal outbreaks (G) | ${t}_{G}=\{\begin{array}{c}firsttsuch{C}_{q}{}^{\prime}\left(t\right)=0\\ on3weeks\end{array}$ |

**Table 3.**Incidence description of malaria incidence patterns: the type of pattern, their number of villages, and their ranges peaks of smoothed seasonal outbreaks with 95% CI and their observed cumulative incidence over the five years of the study period.

Malaria Incidence Patterns | Number of Villages | Range Peaks of Smoothed Seasonal Outbreaks (Cases/100,000 Person-Weeks) with [95% CI] | Observed Cumulative Incidence over the Five Year-Study Period (Cases/1000 Person-Years) |
---|---|---|---|

High | 12 | 227 [65, 487]–884 [420, 1518] | 114 |

Intermediate | 97 | 26 [7, 56]–131 [51, 248] | 13 |

Low | 466 | 7 [2, 16]–34 [7, 81] | 3 |

Start Year Seasonal Outbreak | EI | DateHigh | DateInter | DateLow | WeekHigh | WeekInter | WeekLow |
---|---|---|---|---|---|---|---|

2008 | A | 13/05/2008 | 29/04/2008 | 17/06/2008 | 20 | 18 | 25 |

2009 | A | 19/05/2009 | 26/05/2009 | 28/07/2009 | 21 | 22 | 31 |

2010 | A | 08/06/2010 | 01/06/2010 | 29/06/2010 | 24 | 23 | 27 |

2011 | A | 26/04/2011 | 10/05/2011 | 14/06/2011 | 18 | 20 | 25 |

2012 | A | 03/04/2012 | 24/04/2012 | 29/05/2012 | 15 | 18 | 23 |

2008 | B | 09/09/2008 | 02/09/2008 | 26/08/2008 | 37 | 36 | 35 |

2009 | B | 25/08/2009 | 08/09/2009 | 25/08/2009 | 35 | 37 | 35 |

2010 | B | 14/09/2010 | 31/08/2010 | 07/09/2010 | 38 | 36 | 37 |

2011 | B | 23/08/2011 | 23/08/2011 | 23/08/2011 | 35 | 35 | 35 |

2012 | B | 28/08/2012 | 28/08/2012 | 28/08/2012 | 36 | 36 | 36 |

2008 | C | 30/09/2008 | 23/09/2008 | 23/09/2008 | 40 | 39 | 39 |

2009 | C | 22/09/2009 | 22/09/2009 | 15/09/2009 | 39 | 39 | 38 |

2010 | C | 05/10/2010 | 28/09/2010 | 28/09/2010 | 41 | 40 | 40 |

2011 | C | 13/09/2011 | 20/09/2011 | 20/09/2011 | 38 | 39 | 39 |

2012 | C | 18/09/2012 | 18/09/2012 | 25/09/2012 | 39 | 39 | 40 |

2008 | D | 28/10/2008 | 21/10/2008 | 21/10/2008 | 44 | 43 | 43 |

2009 | D | 27/10/2009 | 20/10/2009 | 20/10/2009 | 44 | 43 | 43 |

2010 | D | 02/11/2010 | 26/10/2010 | 02/11/2010 | 45 | 44 | 45 |

2011 | D | 11/10/2011 | 25/10/2011 | 18/10/2011 | 42 | 44 | 43 |

2012 | D | 23/10/2012 | 23/10/2012 | 30/10/2012 | 44 | 44 | 45 |

2008 | E | 02/12/2008 | 02/12/2008 | 25/11/2008 | 49 | 49 | 48 |

2009 | E | 16/02/2010 | 01/12/2009 | 08/12/2009 | 8 | 49 | 50 |

2010 | E | 18/01/2011 | 18/01/2011 | 30/11/2010 | 4 | 4 | 49 |

2011 | E | 29/11/2011 | 29/11/2011 | 22/11/2011 | 49 | 49 | 48 |

2012 | E | 27/11/2012 | 20/11/2012 | 27/11/2012 | 49 | 48 | 49 |

2008 | F | 06/01/2009 | 23/12/2008 | 23/12/2008 | 2 | 52 | 52 |

2009 | F | 16/03/2010 | 12/01/2010 | 29/12/2009 | 12 | 3 | 1 |

2010 | F | 15/02/2011 | 08/02/2011 | 21/12/2010 | 8 | 7 | 52 |

2011 | F | 20/12/2011 | 13/12/2011 | 13/12/2011 | 52 | 51 | 51 |

2008 | G | 12/05/2009 | 24/03/2009 | 10/02/2009 | 20 | 13 | 7 |

2009 | G | 11/05/2010 | 04/05/2010 | 02/03/2010 | 20 | 19 | 10 |

2010 | G | 26/04/2011 | 26/04/2011 | 22/03/2011 | 18 | 18 | 13 |

2011 | G | 20/03/2012 | 03/04/2012 | 28/02/2012 | 13 | 15 | 10 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Dieng, S.; Michel, P.; Guindo, A.; Sallah, K.; Ba, E.-H.; Cissé, B.; Carrieri, M.P.; Sokhna, C.; Milligan, P.; Gaudart, J.
Application of Functional Data Analysis to Identify Patterns of Malaria Incidence, to Guide Targeted Control Strategies. *Int. J. Environ. Res. Public Health* **2020**, *17*, 4168.
https://doi.org/10.3390/ijerph17114168

**AMA Style**

Dieng S, Michel P, Guindo A, Sallah K, Ba E-H, Cissé B, Carrieri MP, Sokhna C, Milligan P, Gaudart J.
Application of Functional Data Analysis to Identify Patterns of Malaria Incidence, to Guide Targeted Control Strategies. *International Journal of Environmental Research and Public Health*. 2020; 17(11):4168.
https://doi.org/10.3390/ijerph17114168

**Chicago/Turabian Style**

Dieng, Sokhna, Pierre Michel, Abdoulaye Guindo, Kankoe Sallah, El-Hadj Ba, Badara Cissé, Maria Patrizia Carrieri, Cheikh Sokhna, Paul Milligan, and Jean Gaudart.
2020. "Application of Functional Data Analysis to Identify Patterns of Malaria Incidence, to Guide Targeted Control Strategies" *International Journal of Environmental Research and Public Health* 17, no. 11: 4168.
https://doi.org/10.3390/ijerph17114168