A new way of analyzing malaria data: A non- stationary geostatistical modeling approach
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Authors Ejigu, Bedilu Alamirie;Moraga, Paula
Citation Ejigu, B. A., & Moraga, P. (2023). A new way of analyzing malaria data: A non-stationary geostatistical modeling approach. https://
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A new way of analyzing malaria data: A non- stationary geostatistical modeling approach
Bedilu Alamirie Ejigu ( [email protected] )
Department of Statistics, CNCS, Addis Ababa University https://orcid.org/0000-0003-1159-6308 Paula Moraga
Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia
Research Article
Keywords: Environmental-autocorrelation, Malaria, Non-stationarity, Model-based geostatistics, Third Law of Geography, Plasmodium falciparum
Posted Date: June 26th, 2023
DOI: https://doi.org/10.21203/rs.3.rs-3100450/v1
License: This work is licensed under a Creative Commons Attribution 4.0 International License.
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Computing interests: The authors declare no computing interests.
A new way of analyzing malaria data: A non-stationary geostatistical modeling approach
Bedilu A. Ejigu1, Paula Moraga2
Abstract
Geostatistical models are widely used to analyze malaria data, and obtain spatial predictions at un-sampled locations based on the First Law of Geography (close things in space are more similar than distant things). When environmental covariates affect not only the mean of the underlying process under investigation but also its covariance structure, stationary models for spatial prediction are questionable. In this paper, we illustrate how to incorporate spatially refer- enced environmental risk-factors into the covariance function to model non-stationary patterns of malaria risk. Specifically, we demonstrate the suggested modelling framework with a case study of malaria prevalence in Mozambique where we compare a non-stationary model with depen- dence structure governed by precipitation to the standard stationary model. Results reveal that non-stationary geostatistical modeling approaches are more useful to model the non-stationary patterns of malaria. Further the predication malaria risk map based on the non-stationary mod- eling approach show that in many part of the country (especially in the eastern part) malaria prevalence is above 30%. The demonstrated non-stationary modeling approach will play a great role for malaria elimination.
Keywords: Environmental-autocorrelation, Malaria, Non-stationarity, Model-based geostatistics, Third Law of Geography,Plasmodium falciparum.
1Department of Statistics, Addis Ababa University, Ethiopia.
2 Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia.
1. Introduction
Malaria remains a major public health problem in low-income countries. The Africa con- tinent and especially sub-Saharan African countries carry disproportionately high share of the global malaria burden. According to the 2022 world malaria report, the WHO Africa region was home to 95% of malaria cases and 96% of malaria deaths (WHO (2022)). A recent scoping re- view study on malaria prevalence among under five children (UN5) in sub-Saharan Africa shows that, despite the efforts done to protect people from malaria infection, the region accounted for 93% of all malaria deaths globally (Sarfo et al. (2023)). Even though policies and interventions towards protecting people against malaria infection accounted for a 47% reduction in mortality rates among UN5 between 2000 and 2019, nearly every minute, a child under-five dies of malaria (Sarfo et al. (2023); UNICEF (2022)). To identify the sub-national vulnerable group to malaria
Preprint submitted to Elsevier June 23, 2023
risk and to recommend possible intervention mechanisms, the High Burden High Impact (HBHI) strategy for malaria encourages countries to use multiple sources of available data. In order to draw valid recommendations by properly utilizing available empirical evidence to protect vul- nerable populations from malaria, much attention should be given on the use of robust statistical modeling approaches.
In order to guide the planning of geographically targeted malaria control programmes, na- tional malaria control programs need accurate malaria risk prediction maps that show affected areas by taking into account environmental covariates. Geostatistical modeling approaches have been increasingly used in low-resource settings where, due to the absence of disease registries, household surveys provide the main source of information for monitoring the burden of malaria (Schabenberger and Gotway (2005); Diggle and Giorgi (2016)). These approaches are employed to i) identify determinant factors that influence the transmission risk of malaria, ii) generate continuously smoothed malaria prevalence maps, and iii) identify hot-spot areas which need interventions to minimize or eradicate malaria risk (Julius Nyerere et al. (2020); Peter et al.
(2002); Samadoulougou et al. (2014); Macharia et al. (2018); Chipeta et al. (2019); Omumbo et al. (2005); Afolabi et al. (2022); Giorgi et al. (2018b); Jean Damascene and Ezra (2021);
Julius et al. (2017); Ejigu (2020); Moraga et al. (2021)).
Malaria is climate-sensitive disease (Ernst et al. (2009); Huang et al. (2011); Mordecai EA (2013); Marcia C. (2017)), significantly influenced by changes in temperature and precipita- tion. The World Health Organization estimates that climate change will lead to 60,000 additional deaths per year due to malaria between 2030 and 2050, an increase of nearly 15% in overall annual deaths from this preventable disease (WHO (2021)). Standard geostatistical models for malaria prediction rely on the assumption of stationarity and first-law-of geography by taking into account environmental factors only in the mean structure. However, since malaria is an environ- mentally mediated disease, in addition to the first-law-of geography (Tobler (1970)), modeling approaches should acknowledge the Third Law of Geography Zhu et al. (2018); Zhu (2022) with respect to environmental configurations.
To overcome the limitations inherent to the assumptions of stationarity and first law of ge- ography and to account for the third law of geography, in this paper we employ a non-stationary modeling approach by considering potential environmental risk factors both in the mean and co- variance structures of the underlying spatial process. The main contribution of this paper is to demonstrate the useful application of the non-stationary geostatistical model to analyze malaria data. The three main advantages of this approach are: i) in addition to spatial dependency range, the model gives an estimated value of the environmental dependency range, ii) accounts for het- erogeneity in geophysical and other environmental spatial processes, and iii) inference is com- putationally feasible by using a likelihood-based approach.
The remaining part of the paper is organized as follows. Section 2 presents a description of the malaria data, as well as the stationary geostatistical modeling approach. It also presents the non-stationary geostatistical modeling approach with its implementation in the statistical software R (R Core Team (2022)). Section 3 presents the results of the Mozambique malaria case study by using both the standard and non-stationary approaches. Discussion of the main findings of the case study and the use of non-stationary modeling over the standard geostatistical modeling approach to analyze malaria data are presented in Section 4. Finally, concluding remarks are presented.
2. Methods
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2.1. Malaria parasite survey and environmental data assembly
To demonstrate the use of the non-stationary geostatistical modeling approach, we use pub- licly available malaria prevalence survey data in Mozambique from year 2000, as well as demo- graphic and environmental covariates known to affect malaria transmission including altitude, temperature, precipitation, humidity, and distance from water. Figure 2 shows the malaria preva- lence information taken at 447 unique locations. Table 1 presents a summary of the covari- ate information. The summarized datasets (Table 1) on temperature, precipitation and altitude are taken from WorldClim database (www.worldclim.org), data for distance to inland water sources are taken from Worldpop database (www.worldpop.org) and data of humidity is taken from the University of East Anglia Climatic Research Unit database (UEACRU,www.cru.uea.
ac.uk). To download these covariates, The R package raster was used. Malaria prevalence data was obtained from the Malaria Atlas Project (www.malariaatlas.org). Temperature, precipi- tation and altitude were taken from WorldClim (www.worldclim.org). Distance to inland water was obtained from Worldpop (www.worldpop.org) and humidity data was taken from the Uni- versity of East Anglia Climatic Research Unit (UEACRU,www.cru.uea.ac.uk). This dataset was previously used to demonstrate how to obtain malaria prevalence predictions and assess its relationships with potential risk factors using a Bayesian spatial model implemented with the INLA and SPDE approaches Moraga et al. (2021).
Table 1: Covariates summary statistics from 447 unique locations
Covariate Minimum Maximum Mean St.deviation
Altitude 3 1447 294.8 341.75
Temperature 22.85 34.18 30.29 1.74
Precipitation 34.85 142.23 85.43 20.5
Humidity 60.93 79.72 73.15 4.12
Distance from water 0 55.48 8.82 10.96
To compare the predictive ability of the non-stationary modeling approach (mentioned under Subsection 2.4) with the standard geostatistical modeling approach (subsection 2.3), environ- mental covariates from 15,675 unique locations assembled. Table 2 presents the summary of the data used for prediction.
Table 2: Covariates summary statistics, from 15,675 unique prediction locations
Covariate Minimum Maximum Mean St.deviation
Altitude -0.5 1736.7 352.5 298.24
Temperature 20.32 35.08 30.59 1.61
Precipitation 33.62 187.26 83.43 22.71
Humidity 60.93 79.79 72.01 4.15
Distance from water 0 82.33 14.42 12.43
2.2. Method of Data Analysis
2.3. Standard stationary geostatistical model
Let{Y1, . . . ,Yn}represent a georeferenced dataset collected over a spatially discrete set of locations{x1, . . . ,xn}within a region of interestA⊂R2. Model-based geostatistics helps us to
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Figure 1: Malaria prevalence by survey locations
make predictive inference on a spatially continuous space based on the first law-of-geography and a principled likelihood-based estimation paradigm. The standard linear geostatistical model takes the following form.
Yi=d(xi)⊤β+S(xi)+Zi, (1) whered(x) represents a vector of covariates, the spatially structured residuals{S(x) :x∈R2}is a stationary and isotropic Gaussian process with zero mean, varianceσ2and correlation function Corr{S(x),S(x′)}=ρ(x,x′). Assuming stationarity and isotropy, the correlation function ofS(x) is purely a function of geographical proximity, specifically the Euclidean distance betweenxand x′, hence we writeρ(x,x′)=ρ(kx−x′k). TheZiare assumed to be i.i.d. Gaussian variables with mean zero and varianceτ2. The unstructured random effectZiaccounts both spatial variation on a scale smaller than the minimum observed distance between survey locations and unexplained unstructured variation at survey location level.
2.4. Non-stationary geostatistical model
Standard geostatistical modeling approaches include spatially referenced risk factors only in the mean structure of the model to aid spatial prediction of malaria risk at unobserved locations.
Further, the standard modeling approach solely rely on the first law of geography, which does not take into account geographical configuration (third law of geography).
To take into account the third law of geography via the non-stationary modeling approach as suggested by Ejigu et al. (2019), we replace the stationary Gaussian processS(xi) in Eqn (1) with another Gaussian processS(xi,ei) which is a function of distances over both space and environmental configuration. The non-stationary model forYitakes the form:
Yi=d(xi)⊤β+S(xi,ei)+Zi, i=1,· · ·,n. (2) 4
In the above non-stationary modeling approach, the distance in an environmental factorei can be used to model both the mean ofYiby includingeias one of the components of the covariate vectord(xi), as well as the covariance function of the Gaussian random field. We then assume thatS(xi,ei) is a Gaussian process with mean zero and covariance function
Cov{S(x,e),S(x′,e′)}=σ2ρ(x,x′;e,e′).
Under the assumption of separable covariance functions (Genton (2007)), the correlation ρ(x,x′;e,e′) in the above equation can be written as:
ρ(x,x′;e,e′)=ρ1(kx−x′k)ρ2(|e−e′|). (3) Our choice of eitherρ1(·) orρ2(·) in (3) is to use a Mate´rn correlation function Mate´rn (1960) which is given by:
ρ(u)= 1
Γ(κ)2κ−1(u/φ)κKκ(u/φ), κ >0,u≥0, (4) whereKκ(.) denotes the modified Bessel function of the third kind of orderκ, andφis the scale parameter which controls the rate at which the correlation gets close to zero with increasing separation distanceu = kx−x′k. Underk = 1/2, the Mate´rn covariance function gives an exponential correlation function, given by
Cov{S(x,e),S(x′,e′)}=σ2exp{−us/φs}expn
−ue/φp
o, (5)
whereus=kx−x′kis the distance in space between the location of any two survey locations and up=ke−e′kis environmental covariate difference in two survey locations.
The choice of includingeionly in the covariance structure or both in the mean and covariance structure depends on the fit of the model to the data. Further details on the estimation of the model parameters inEqn2 are described in Ejigu et al. (2019). Further, among a set of environmental covariates/risk-factors which may affect both the mean and covariance structure, the data modeler may select the influential risk factor by investigating the variogram and correlogram plots.
2.5. Model Implementation
In this section, we fit a non-stationary model for the malaria data in Mozambique by including risk factors as well as a spatial random effect whose covariance structure depends not only on the geographical distance, but on the difference in precipitation. We also compare the performance of the non-stationary model with three alternative models that incorporate precipitation in the fixed or random effects by using the Akaike information criterion (AIC) Akaike (1973).
Among the set of environmental factors, we choose precipitation to model the covariance function as the variogram investigation reveals considering precipitation into the covariance structure better fits the data. The study by Victor A. et al. (2021) also shows precipitation is the best fit covariate among a set of environmental variables considered to estimatePalsmodium falciparumparasite prevalence in East Africa.
The models fitted are as follows. The first two models are specified under the framework of standard stationary geostatistical model by including and excluding precipitation in the mean structure of the model. Specifically,Model1 completely ignores the effect of precipitation on malaria prevalence, whileModel2 considers precipitation in the mean structure but ignores its effect on the covariance structure. Model3 andModel4 are specified under the framework of non-stationary geostatistical modeling approach. Model3 considers precipitation only in the covariance structure.Model4 considers precipitation both in the mean and covariance structure.
The models are specified as follows, whereeidenotes the precipitation at locationxi. 5
• Model 1,Yi=β0+β1alt+β2temp+β3hum+β4water+S(xi)+Zi;
• Model 2,Yi=β0+β1alt+β2temp+β3prec+β4hum+β5water+S(xi)+Zi;
• Model 3,Yi=β0+β1alt+β2temp+β3hum+β4water+S(xi,ei)+Zi;
• Model 4,Yi=β0+β1alt+β2temp+β3prec+β4hum+β5water+S(xi,ei)+Zi.
The implementation of the proposed non-stationary geostatistical model, as well as the pre- sented figures and analysis results were done using R statistical software R Core Team (2022).
The code to reproduce the analyses can be found in the GitHub repositoryhttps://github.
com/BediluEjigu/NS-MBG-Malaria.
3. Results
In this study a total of 447 unique data locations in Mozambique were identified during the data assembly process. The locations of each survey with their respective malaria prevalence are presented in Figure 1. The overall malaria prevalence is 39.86%. In spatial data analysis, variogram plots took the lion-share to assess the existence of spatial autocorrelation. Figure 2 presents the semivariogram and correlogram plot based on geographical separation distance and precipitation difference. The result reveals that, in addition to geographical separation distance, malaria prevalence correlated with precipitation difference over space.
3.1. Modeling Results
Table 3 presents the parameter estimates and the AIC values of the four models used to fit the data. Because of its lowest AIC value,Model3 that includes precipitation in the covariance function is considered as the best model. The results show that both the point estimates and the 95% CI of the exploratory variables in common to all models are comparable. As compared with non-stationary geostatistical models, the 95% CI for stationary geostatistical model parameter estimates are narrower. The results presented in Table 3 show that after incorporating precipita- tion into the covariance structure of the model, the estimated value of the spatial scale parameter increases. Further, we observe that the inclusion of precipitation only in the covariance structure inModel3 leads to a reduction both in the varianceσ2and the spatial scale parameterφsof the Gaussian process.
We observe that conditionally upon fixed and location-specific random effects, the log odds of malaria increases by 0.386 for one unit change in temperature. In general, conditionally on survey location-specific random effects, covariates temperature, humidity and altitude are positively associated with malaria prevalence.
Figure 3 shows the impact of three different values of precipitation difference on the spatial correlation based on the Euclidean distance. The difference between the correlation functions clearly indicate that non-stationary effects play an important role in the stochastic variation of malaria prevalence. Further, this figure demonstrates how the correlation on the Euclidean dis- tance domain changes for different values of precipitation.
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Figure 2: Empirical semivariogram (upper panel) based on separation distance (left) and precipitation difference (right panel), and respective correlogram plots (lower panel)
3.2. Model Validation
The validity of the adopted spatial correlation structure for the considered dataset was as- sessed by simulating 1000 empirical variograms under the fitted modelModel2 andModel3 for the stationary and non stationary models, respectively. Using the simulated variogram we com- puted 95% confidence interval at any given spatial distance of the variogram (Figure 4). Since the empirical variogram obtained from the data falls within the 95% tolerance bandwidth, the adopted spatial correlation function is compatible with the data (Figure 4).
3.3. Malaria Prediction
Using environmental covariates considered in the model fitting, prediction was carried out at 15,675 unique locations using parameter estimates from the geostatistical model. Figure 5 presents maps of the predictive probability that malaria prevalence lies below a threshold of 30%. The Figure reveals, in many part of the country (especially in the eastern part) malaria prevalence is above 30%.
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Table 3: Maximum likelihood estimates and 95% confidence intervals for the parameters of the fitted models
Factor
Stationary geostatistical Model Non-stationary geostatistical model
Model 1 Model 2 Model 3 Model 4
Estimates 95% CI Estimates 95% CI Estimates 95% CI Estimates 95% CI
Intercept -23.807 (-33.937,-13.677) -23.004 (-32.96,-13.04) -23.114 (-33.210,-13.026) -22.242 (-32.401,-12.082) Temperature 0.399 (0.218,0.581) 0.388 (0.211,0.566) 0.386 (0.205,0.568) 0.374 (0.193,0.556)
Altitude 0.002 (0.001,0.003) 0.002 (0.001,0.003) 0.002 (0.001,0.003) 0.002 (0.001,0.003)
Precipitation 0.005 (-0.005,0.014) 0.004 (-0.005,0.013)
Humidity 0.141 (0.707,0.211) 0.13 (0.058,0.202) 0.137 (0.067, 0.206) 0.126 (0.053,0.198)
Distance from water 0.013 (-0.0001,0.026) 0.013 (0.0001,0.026) 0.013 (-0.002,0.025) 0.013 (-0.039,0.944)
σ2 0.438 0.455 0.449 0.452
φs 0.398 0.332 0.687 0.651
φp 9.879 9.976
ω2 0.928 0.903 0.9 0.894
AIC 543.153 542.238 107.041 108.23
Figure 3: The black line corresponds to the functionexp(−u/φs)∗exp(−9.16/φp), the red line corresponds toexp(−u/φs)
×exp(−18.96/φp) and the green line corresponds toexp(−u/φs)×exp(−65.30/φp), whereexp(·) is an exponential correlation function, (9.16, 18.96, 65.30) are first, second and third quartiles of precipitation, andφs,φpare parameter estimates fromModel3 in Table 3.
4. Discussion
Malaria remains one of the major causes of mortality in Africa and Asia. In recent decades spatial models have been established as an important tool to identify malaria hotspot areas, pre- dicting the prevalence of malaria at unsampled locations and hence guiding the development of strategies to control it (Macharia et al. (2018); Giorgi et al. (2021)). Malaria prevalence maps are useful, not only for enabling a more precise malaria risk stratification, but also for guiding the planning of more reliable spatial control programmes by identifying affected areas (?Gething
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Figure 4: Results from variogram diagnostic check for the presence of residual spatial correlation (left panel) and for compatibility of the adopted spatial correlation function for malaria prevalence data under the geostatistics modeling framework (right panel)
Figure 5: Predicted prevalence of malaria
et al. (2012).) To generate prevalence maps, geostatistical modelling is increasingly used to com- bine multiple malaria surveys from multiple locations into a detailed model of local prevalence or incidence (Giorgi et al. (2018a); Daniel J et al. (2017); Diggle and Giorgi (2016)). However, when standard geostatistical modelling is used to analyze malaria data, the non-stationarity of malaria risk due to environmental factors is ignored. The presented non-stationary geostatisti-
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cal modeling approach offers a means to use location dependent risk-factors into the covariance structure of the model. By considering both the first and third law of geography into the analyt- ical paradigm, the resulting malaria risk maps by such modeling approach are more reliable on the spatial heterogeneity of malaria endemicity.
In the past few decades, stationary model-based geostatistics have been considered as a golden standard for interpolating malaria risk maps within geographical areas of interest using different malaria survey datasets and employed in Gambia by Peter et al. (2002), Burkina Faso (Samadoulougou et al. (2014)), Kenya (Macharia et al. (2018)), Uganda (Julius et al. (2017)), Rwanda (Jean Damascene and Ezra (2021)), Somalia (Abdisalan M et al. (2008); Giorgi et al.
(2018b)), Malawi (Kazembe et al. (2006); Chipeta et al. (2019)), Mozambique Ejigu (2020), Ghana (Yankson et al. (2019)), and East Africa (Omumbo et al. (2005); Victor A. et al. (2021)).
Similarly, to identify determinant factors and to estimate malaria prevalence stationary Bayesian geostatistical modeling approaches also employed to estimate malaria prevalence in Rwanda (Jean Damascene and Ezra (2021); Muhammed et al. (2020)), Nigeria (Adigun et al. (2015)), Angola (Laura et al. (2010)), Tanzania (Laura et al. (2012)) and Mozambique (Moraga et al.
(2021)).
While in the previous studies, environmental variables were accounted only on the mean structure of the model, studies reveal that environmental variables play a key role not only on the occurrence malaria but also in its variability over space Ferrao et al. (2021); Anne Caroline et al. (2011). The World Health Organization estimates that climate change will lead to 60,000 additional deaths per year due to malaria between 2030 and 2050, an increase of nearly 15% in overall annual deaths from this preventable disease (WHO (2021)). As demonstrated by simula- tion and case-studies, the study by Ejigu et al. (2019) shows that ignoring non-stationarity using standard geostatistical models can provide wrong predictive inferences which ultimately affect malaria intervention efforts.
The present analysis extends that of Moraga et al. (2021) by explicitly including precipitation in the covariance structure of the model. Moraga et al. (2021) were concerned in the analysis of geostatistical data using INLA and SPDE methods under the assumption of stationarity. The study by Anne Caroline et al. (2011); Ferrao et al. (2021) shows that in a highly endemic area, high-resolution precipitation data can directly predict malaria incidence. But, this study and others (Moraga et al. (2021)) considered precipitation only on the mean structure of the model did not demonstrate significant relationship with precipitation with malaria.
The fitted model was used to generate maps that shows areas with low (<5%) and high-risk (>30%) malaria prevalence. The study done by Ejigu (2020) shows the malaria risk were high (>30%) in Niassa, Cabo Delgado, Nampula, Zambezia, Manica and Inhambane provinces. Es- timated malaria prevalence in un-sampled location using geosatistical models is one of the key metrics in stratification and intervention process. To maximize its utility in decision making, the non-stationarity of environmental variables that trigger the existence of malaria parasite over- space should be taken into account in order perform reliable malaria predictions. As presented in Figure 5, malaria risk is high in the eastern part of Mozambique.
There are several ways to extend the presented non-stationary geostatistical model. To men- tion few, i) by considering more than one environmental variable in the covariance structure of the model, ii) considering non-separable covariance structure of the model (Gneiting (2002)), and iii) developing test-statistic to choose the better covariance structure of the model.
One of the main limitations of this study is that, the implementation of non-stationary geo- statistical modeling fitting function is not included in the standard spatial data modeling R pack- ages, such as inPrevMap Giorgi and Diggle (2017). An additional limitation is the method
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demonstrated by considering malaria data from one country. However, despite these limita- tions, researchers can use our R-function from GitHub repository (https://github.com/
BediluEjigu/NS-MBG-Malaria) to model the nonsationary patterns of malaria in other coun- tries.
5. Conclusion
The non-stationary geostatistical modeling approach presented is a useful method to analyze malaria data by taking into account environmental factors which trigger the existence of malaria in the covariance structure of the model. To make prediction to un-sampled location, the method takes into account both first- and third-law of geography.
Author statement file
Bedilu Alamirie Ejigu: Conceptualization; Data curation; Formal analysis; Investigation;
Methodology; Visualization; writing the original draft, review and editing. Pauala Morga: Data curation; validation; writing review and editing.
Acknowledgments
The author thanks Dr Kefyalew Alene from Curtin School of Population Health,Curtin Uni- versity for his valuable comments.
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