Much of the developing world, particularly sub-Saharan Africa, exhibits high levels of morbidity and mortality associated with infectious diseases. The incidence of Schistosoma sp. —which are neglected tropical diseases exposing and infecting more than 500 and 200 million individuals in 77 countries, respectively—is rising because of 1) numerous irrigation and hydro-electric projects, 2) steady shifts from nomadic to sedentary existence, and 3) ineffective control programs. Notwithstanding the colossal scope of these parasitic infections, less than 0. 5% of Schistosoma sp. investigations have attempted to predict their spatial and or temporal distributions. Undoubtedly, public health programs in developing countries could benefit from parsimonious forecasting and early warning systems to enhance management of these parasitic diseases. In this longitudinal retrospective (01/1996–06/2004) investigation, the Schistosoma haematobium time-series for the district of Niono, Mali, was fitted with general-purpose exponential smoothing methods to generate contemporaneous on-line forecasts. These methods, which are encapsulated within a state–space framework, accommodate seasonal and inter-annual time-series fluctuations. Mean absolute percentage error values were circa 25% for 1- to 5-month horizon forecasts. The exponential smoothing state–space framework employed herein produced reasonably accurate forecasts for this time-series, which reflects the incidence of S. haematobium–induced terminal hematuria. It obliquely captured prior non-linear interactions between disease dynamics and exogenous covariates (e. g. , climate, irrigation, and public health interventions), thus obviating the need for more complex forecasting methods in the district of Niono, Mali. Therefore, this framework could assist with managing and assessing S. haematobium transmission and intervention impact, respectively, in this district and potentially elsewhere in the Sahel.
Prevalent parasitic infectious diseases frequently evade the public health radar because infected individuals present with a clinical history that is characterized by a highly heterogeneous symptomatology. Schistosoma sp. , also known as bilharzias, expose and infect more than 500 and 200 million individuals in 77 countries, respectively [1], [2]; however, only those with severe symptoms seek available treatment. Though sub-clinical Schistosoma sp. infection detrimentally impacts the health of infected individuals, the enormous impact of seemingly asymptomatic and mildly symptomatic infection remains difficult to quantify. Furthermore, Schistosoma sp. incidence continues to rise because of 1) numerous irrigation and hydro-electric projects, 2) steady shifts from nomadic to sedentary existence, and 3) ineffective control programs unable to cope with population growth. With the mounting evidence that Schistosoma sp. impose an enormous burden on, as well as their control have paramount importance to improve public health in, developing countries, intervention programs therein could benefit from parsimonious forecasting and early warning systems to enhance management and hazard mitigation of these parasitic infections [1]–[8]. Most individuals at risk of Schistosoma sp. infection reside between latitudes 36° N and 34° S where average fresh water temperatures range from 25 to 30° C [1], placing African states among the most affected countries. Schistosoma mansoni and Schistosoma haematobium account for most Schistosoma sp. infection in Africa [1], 2. S. mansoni and S. haematobium cercarias enter the human circulation trans-cutaneously. Subsequently, adult forms mate, migrate, and lay eggs, which eventually lodge in the intestine (S. mansoni) or bladder (S. haematobium). Excreted eggs hatch under favorable aquatic conditions, releasing miracidia, which penetrate the intermediate snail host—in Africa, S. mansoni and S. haematobium infect Biomphalaria sp. and Bulinus sp. snails [8], [9]. Finally, mature cercarias emerge from their intermediate host to seek human reservoirs thus, perpetuating their life cycle [1]. Individuals infected with S. mansoni are usually asymptomatic or mildly symptomatic (rash, fever, aching, cough, diarrhea, and or gland enlargement). In serious infection, lodged S. mansoni eggs trigger a granulomatous immune response that may cause colonic obstruction, hemorrhages, portal hypertension, ascites, and life-threatening esophageal varicose. S. haematobium produces similar unspecific symptoms whereas its fully symptomatic form manifests primarily as terminal hematuria. Moreau et al. [10] reported the pervasive endemicity of S. haematobium in West Africa, particularly in the Sahel (Figure 1) —i. e. the sub-Saharan region that spans the entire east-west African axis, bordering the Sahara desert to the north and the Savanna to the south [11]. Conversely, his collaboration demonstrated that the prevalence of S. mansoni is greater in Sudanese and Guinean savannas [10]. Along this line of investigation, several epidemiological studies have evaluated the Schistosoma sp. prevalence in Mali [12]–[15], which ranks among the poorest countries in the world, and which is transected by savannas, the Sahel, and the Sahara desert. Traore et al. [12] reported a 55% overall S. haematobium prevalence, with a case distribution orbiting the 7–14 age-category, in the district of Niono (Segou Region) and Dogon Plateau, Mali; circa 50 and 30% of infected individuals presented with clinical symptoms and pathologic lesions, respectively. The surveys conducted by Keita et al. [13] demonstrated that the Schistosoma sp. prevalence (7–14 age-category) in the community health center (CSCOM) service area of Molodo, in the district of Niono, was 72,68, and 51% for S. haematobium, S. mansoni, and co-infection, respectively. Finally, Medina et al. [11] reported that S. haematobium is the 5th most frequently diagnosed infectious disease, accounting for 2. 5% of total CSCOM consultations in the district of Niono. The high prevalence of Schistosoma sp. in this district may be attributed to an extensive irrigation system that supports predominantly rice monoculture. Unfortunately, district communities not only ingest water from the irrigation scheme but also wash their belongings, bathe, excrete, and amuse themselves in the canals (Figure 2), considerably increasing exposure to Schistosoma sp. infection. Notwithstanding the colossal scope of these parasitic infections in developing countries, only circa 0. 5% of Schistosoma sp. investigations have attempted to predict their spatial and or temporal transmission distributions e. g. [1], [2], [16]–[18]—meriting special attention, Yang et al. [18] modeled both the spatial and temporal S. japonicum transmission dimensions in Jiangsu province, China. [The number of reports investigating Schistosoma sp. spatial and or temporal distributions roughly obtain via keyword-searching “schistosomiasis”, “Schistosoma”, “bilharzias”, “forecast”, “forecasting”, “prediction”, and keyword combinations at www. pubmed. com (09/25/2007). A meta-analysis is beyond the scope of this manuscript. ] Regrettably, S. haematobium time-series (TS) forecasts are practically inexistent for Sahelian locations, such as Mali, where this neglected tropical disease tremendously deteriorate public health. Thus, the quest for robust S. haematobium TS forecasting methods to assist with preventing transmission, rapidly treating patients, as well as monitoring intervention impact must not be ignored. In this longitudinal retrospective (01/1996–06/2004) investigation, the S. haematobium consultation rate TS for the district of Niono, Mali (Fig. 1), was fitted with general-purpose exponential smoothing (ES) methods—encapsulated within a state-space framework—to produce contemporaneous on-line forecasts. On-line forecasts imply that historical records are continuously supplied to the execution program, which automatically revises external predictions. Although this state-space framework ignores direct effects from climate, public health intervention, and irrigation on S. haematobium TS evolution, it accommodates seasonal as well as inter-annual TS fluctuations. The ES methods within this framework may capture prior non-linear interactions between disease dynamics and the aforementioned covariates, potentially obviating the need for more complex predictive approaches in the district of Niono, Mali. [An intuitive overview of this ES state-space framework is conveyed by Figure 3. ] Therefore, not only does this analysis address the paucity of reported S. haematobium TS investigations but it also demonstrates that this state-space framework could assist with managing S. haematobium infection in this district and possibly elsewhere in the Sahel.
This longitudinal retrospective (01/1996–06/2004) S. haematobium TS investigation was conducted in the district of Niono, Mali (Fig. 1). Panel A in Fig. 1 is a satellite image that portrays Mali, with a projected population of 12 million in 2004 [19], along with its neighboring West African countries. Panel B—which corresponds approximately to an enlargement of the red demarcation in panel A—depicts the district of Niono (red rectangle), 330 km northwest of Bamako, 100 km north of the Niger River, in the Segou region. This district is a model location to test S. haematobium TS forecasting and early warning systems feasibility because its extensive irrigation network pervasively exposes communities to this neglected parasitic infection. Furthermore, the district of Niono shares epidemiological similarities with other regions in the Sahel where poverty- and disease-induced morbidity and mortality are rampant. The review of monthly clinical consultation records from the district of Niono, Mali, is part of a larger study on climate and health (“Putting climate in the service of public health”) that was approved by the “Columbia University Medical Center Institutional Review Board” (New York, U. S. A.) and the “Ethics Committee of the Mali National Medical School” (Bamako, Mali). Patient privacy was protected from inadvertent (or deliberate) violations because consultation records reflect monthly summaries that lack information with which individuals may be identified [11]. The assembled monthly data set (01/1996–06/2004) comprises consultation records for 20 diseases, which were tabulated by gender and age categories, from 17 CSCOM service areas within the district of Niono [11], [19], [20]. However, only the S. haematobium TS was analyzed here—diarrhea, acute respiratory infection of the lower tract (ARI), and malaria TS forecasts, as well as preliminary frequency description of all 20 diseases, have already been reported [11]. Of note, Schistosoma sp. consultation records reported by Medina et al. [11] and analyzed herein reflect cases of S. haematobium–induced terminal hematuria in over 99% of consultations, as discussed later, for which a single dose of 40 mg/kg of prazinquatel was prescribed in most cases. Monthly S. haematobium consultation records for the 17 CSCOM service areas, both genders, and all ages were amalgamated. Rather than interpolating missing observations with imputed CSCOM-specific monthly median values and excluding ineligible CSCOM service area TS [11], this amalgamated consultation rate TS, {yt}, was estimated by simultaneous adjustment of time-dependent nominator (cases) and denominator (population) observations, according to Equation 1 (1) where Ct, η is the monthly number of CSCOM-specific S. haematobium-induced terminal hematuria consultations for both genders and all ages; Pt, η is the time-dependent population of each CSCOM service area, which was adjusted for a national annual population growth rate of 3. 2% [11], [19], [20]; αt, η = 0 if Ct, η is missing for month t and CSCOM service area η, otherwise αt, η = 1; last, N = 17 is the total number of CSCOM service areas. The approximately random distribution of missing observations (∼17%) across months, years, and CSCOM service areas [11] ensures approximately unbiased {yt} estimation with Eq. 1, which holds as long as the denominator summation is positive. In simpler words, Eq. 1 estimates the monthly consultation rate for S. haematobium-induced terminal hematuria from CSCOM service areas for which records are available. Consultation rates and their forecasts are expressed as the monthly number of newly diagnosed S. haematobium–induced terminal hematuria cases per 1000 individuals in the district of Niono. Additional record details appear in Table 1, which was adapted from Medina et al. [11]. The amalgamated TS was fitted with ES methods, which are encapsulated within a state-space framework hereafter referred to as ETS for error (E), trend (T), and seasonal (S) components. The E component is either additive (A) or multiplicative (M); T and S components may be A, M, or inexistent (N); last, T may also be dampened additively (Ad) or multiplicatively (Md). For example, the ETS method MMN has E (M), T (M), and S (N) structures. Therefore, there are 30 possible ES combinations within this forecasting framework, comprising linear and non-linear ones. However, only the 15 ES methods with multiplicative error structures (heteroskedastic) were herein considered for TS analysis [21]–[32]. Not only do multiplicative error structures are conservative but they also yield more realistic 95% prediction interval (PI) values. Furthermore, a reduction in the number of ES methods evaluated also diminishes the expensive computational time. The versatile and fully automatic ETS framework requires neither stationarity nor “strict” linearity to produce contemporaneous TS forecasts for variable time horizons (h) [21]–[32]. Consequently, it is extensively employed in, e. g. , econometrics and inventory control where automatic forecasts are required for a large number of diverse TS. This forecasting framework, whose performance was recently and favorably compared to those of several forecasting techniques across thousands of TS [32], adapts to underlying alterations in disease dynamics and automatically revises forecasts on-line as new observations accumulate (Fig. 3). This adaptability is essential for epidemiological forecasting methods because interventions (e. g. medical and prophylactic treatment) almost ubiquitously perturb disease TS dynamics. An intuitive description of the ETS framework appears in Fig. 3; it is only succinctly described below because it has been meticulously derived elsewhere [21]–[31]. In the ETS framework, the expected mean of a forecasted observation, E[F (yt+h|It) ], is conditioned on the information set (It) available at time t—i. e. these are external predictions. The information set It contains unobserved level (lt), trend (rt), and or seasonal (sm|t: month, m = [1,12]) components, whichever pertinent, depending on the underlying ES method. Possible lower-frequency “harmonics”, i. e. inter-annual fluctuations, are handled by lt and rt components in the ETS framework because the limited temporal window (01/1996–06/2004) considered in this investigation precludes stable estimation of periodicity much longer than 12 months. The observed amalgamated TS is symbolized by {yt}, as previously defined, whereas unobserved TS components enter the vector xt, according to the general state and transition Equations 2 & 3, respectively: (2) (3) where, For ES methods with multiplicative error structures, w (xt-1) and r (xt-1) have both the form of the expected mean of a forecasted observation, E[F (yt|It-1) ]. Otherwise, w (xt-1) = E[F (yt|It-1) ] and r (xt-1) = 1 for ES methods with additive error structures (not discussed hereafter). All ES methods rely on the adjustment of lt, rt, and or sm|t TS components with their corresponding smoothing control α, β, and γ values; furthermore, φ controls smoothing of rt-dampening if present. Basically, contemporaneous forecasts obtain via TS extrapolations whereby previous deviations between forecasts and their corresponding observations are exponentially adjusted with α, β, γ, and or φ. Large smoothing control values confer greater weights to recent information and effectively shorten the smoothing “memory”, i. e. the recent-past has a more pronounced influence on estimated components than does the distant-past [11], [21]–[31]. Three important remarks: 1) a single or multiple smoothing control values may be required depending on which TS components are present in the selected ES method; 2) although smoothing controls are symbolized with the same notation across distinct ES methods, their function may vary from one ES method to another because the relationship between TS components may also differ (e. g. multiplicative vs. additive rt); last, 3) the function of smoothing control values approximately parallels that of the bandwidth in a one-side Nadaraya-Watson exponential kernel. Smoothing controls plus unobserved components are estimated for all ES methods within the ETS state-space framework using a maximum likelihood function analog [31]. Here, the general ETS constraints are: 0<α, φ<1; 0<β<α; and, 0<γ<1−α; strictly multiplicative error structures; multiplicative sm|t values add annually to 12 because m = [1,12]; and, 36 months [≡3p] for initial training, the possible specification of longer intervals notwithstanding. Defaulted ETS constraints are specified for several reasons [21]–[31] among them to prevent the forecast execution program from producing unrealistic results. Once each ES method within the ETS framework is optimized at time t, that which minimizes the Akaike' s Information Criterion (AIC) is selected to generate the h-month horizon forecast path distribution. The h-month horizon forecast path distribution, F (yt+h|It), obtains via recursive iterations (Eqs. 2 & 3) of B = 1000 ordinary {εt} bootstrap-generated pseudo-TS [11], [31], [33]. With the accumulation of each new observation, ES methods within the ETS framework are re-optimized and the best-performing ES method is re-selected based on the AIC. Subsequently, F (yt+h|It) is again recursively generated from B = 1000 ordinary {εt} bootstrap-generated pseudo-TS. For example, observations from January 1996 to December 1998 initialize the ETS execution program (Fig. 3) that predicts consultation rates for January 1999 to May 1999, assuming h = [1,5]. Once the January 1999 forecast plus the available TS (including the most contemporaneous observation of January 1999) become available to the on-line system (Fig. 3), the execution program cycles again and optimizes all considered ES methods, re-selecting the best-performing one (which may or may not be the same one employed prior to the arrival of the new observation). As a result, revised consultation rate predictions for February 1999 to June 1999 ensue. This process repeats ad infinitum (Fig. 3). The 95% PI values for the simulated F (yt+h|It) paths are estimated from distribution percentiles. Although a full portrayal of the ETS framework (Eqs. 2 & 3) encapsulating the 15 considered ES methods [21]–[31] is beyond the scope of this investigation, those ES methods which have been selected at least once during this TS analysis are described in terms of E[F (yt|It-1) ] and xt recursions (Table 2). [Table 2 caption also provides an ES method example explicitly written in matrix notation. ] As discussed afterwards in the Results section, none of the selected ES methods (Table 2) is seasonal, reflecting the endemicity of the TS analyzed herein. For further details concerning the ETS framework, refer to, e. g. , Hyndman et al. [27], [29], [31]. Standard accuracy and dispersion measures were employed in this analysis. Accuracy—which measures the forecasting competence—is defined here as the mean absolute percentage error (MAPE) between observed and forecasted TS values whilst infrequently reported PI values reflect the dispersion of forecast distributions; the dispersion of simulated F (yt+h|It) probability density functions were also summarized as the average coefficient of variance (). MAPE and values are calculated with Equations 4 & 5, respectively: (4) (5) and MAPE (external) values are expressed in percentage (%) as a function of the h-month horizon forecast. In Eqs. 4 & 5, T is the TS length and f = 3p−1+h reflects the actual time when the h-month horizon forecast begins. Large MAPE and values imply low accuracy and large dispersion, respectively, and vice-versa. The distinction between MAPE and PI (or) values is an important one. The first assesses the competence, i. e. the skill, of the h-month horizon forecast; the latter only measures the dispersion of the h-month horizon forecast path distribution. Thus, PI (or) values have paramount importance for calculating, e. g. , the probability that a future observation will be smaller or greater than the expected forecast distribution mean by a certain margin. Likewise, the number of individuals at risk may be calculated for a specified probability. This TS has not undergone Box-Cox transformations. Notice however, that TS frequently undergo such transformations prior to the forecasting analysis. Regardless, contemporaneous forecasts and standard accuracy measures (e. g. MAPE) must be (and were) superimposed onto and computed for, respectively, the originally observed TS because accuracy may be severely distorted in the transformed dimension—i. e. occasionally, forecasts may be simultaneously accurate and inaccurate in the transformed and original dimensions, respectively. All calculations were performed in R: A language and environment for statistical computing [30], [31].
This longitudinal retrospective (01/1996–06/2004) investigation analyzed the S. haematobium consultation rate TS for the district of Niono, Mali. In Figure 4, the observed amalgamated S. haematobium consultation rate TS is symbolized by black lines. The TS is excessively noisy from 1996 to 1999 when a sharp rise in consultation rates clearly ensues. From 2001 onwards, consultation rates decline because of large-scale prophylactic de-parasitation programs. Regardless, 2- to 5-month horizon forecasts clearly captured these inter-annual tendencies (Fig. 4) —red traces correspond to contemporaneous on-line 2-, 3-, 4-, and 5-month horizon forecasts (panels A, B, C, and D, respectively) whilst their 95% PI values are depicted in dots of the same color. Abscissa TS projections span 102 months (01/1996–06/2004) while ordinate scales represent the number of newly diagnosed (or forecasted) S. haematobium–induced terminal hematuria cases per 1000 individuals. TS observations were continuously submitted to a family of general-purpose ES methods—encapsulated within the ETS state-space framework—to produce contemporaneous on-line forecasts (i. e. external predictions). Predictions were superimposed onto the original TS to allow visual evaluation of prediction accuracy. While this superimposition is absolutely essential, it might convey the false impression that forecasts represent internal predictions—this is not the case. Fig. 4 should be considered dynamically. As observations and forecasts become available to and from the on-line forecast execution program (Fig. 3), respectively, the actual graphing of forecasts (red traces) precede that of observations (black lines) by exactly h-month horizon. Generally, the ETS framework accommodates seasonal and inter-annual fluctuations, producing reasonably accurate TS forecasts. Here, inter-annual fluctuations dominate the S. haematobium TS while seasonal oscillations are practically inexistent (Fig. 4). These fluctuations are intuited from the observed consultation rate TS (black lines), as well as implied by the absence of {st|m} vis-à-vis the presence of {lt} and or {rt} components in automatically selected ES methods (Table 2). Only 3 ES methods were automatically selected with the AIC during this S. haematobium TS forecasting analysis. These selected ES methods, which have been described in terms of E[F (yt|It-1) ] and xt recursions (Table 2), are: the multiplicative error/ trendless/ aseasonal (MNN), multiplicative error/ damped additive trend/ aseasonal (MAdN), and multiplicative error/ damped multiplicative trend/ aseasonal (MMdN) ES methods. None of them are seasonal and hence exogenous forcing (e. g. climate covariates) was not invoked to improve predictions. Table 3 lists the frequency (n) with which these ES methods were re-selected during the forecasted investigational period plus the method-specific median (and IQR: inter-quartile range) of pertinent smoothing control values. Smoothing control values are time-dependent because they are continuously re-estimated as new observations accumulate. Yet, their magnitude drifts little in this investigation. Hence, they were reported as median and IQR values. The MNN smoothing control α is obviously large since this method only has a level {lt} component, i. e. the MNN ES method lacks {rt} and {st|m} components as well as their corresponding smoothing control β, φ and γ values. For MAdN and MMdN methods, β≤α≪φ due to large dampening of minute rt components. As new observations accumulated, the automatic and criterial re-selection of ES methods conferred an additional layer of flexibility to the ETS framework and consequent TS forecasts. [Smoothing control values may differ in functional form across ES methods despite the retained notation (Methods). ] MAPE and values for 1- to 5-month horizon forecasts were circa 25 and 45%, respectively (Figure 5). values reflect the average dispersion of simulated F (yt+h|It) probability density functions whilst MAPE values measure the mean absolute percentage error between TS observations and their forecasts. Accuracy (MAPE; panel A) becomes approximately asymptotic as the h-month horizon forecast increases beyond 6 months because of a minute {rt} component irrespectively of the selected ES method, significant inter-annual {lt} fluctuations notwithstanding. As expected, dispersion (; panel B) increases as innovations propagate through longer stochastic h-month horizon forecast paths.
Changes in multiple dimensions (e. g. human behavior, population, disease TS, climate, and environment) will confer an ever-increasing role to infectious diseases forecasting and early warning systems. These predictive systems are based upon a single dimension or a combination thereof, whichever alternative best compromises among realism, feasibility, robustness, and parsimony. With the mounting evidence that S. haematobium—a neglected tropical disease—imposes an enormous burden on developing countries, public health programs therein could benefit from parsimonious forecasting and early warning systems to enhance management and control of this parasitic infection. Not only does this report address the paucity of S. haematobium TS forecasting investigations but it also advocates the usage of parsimonious state-space frameworks to forecast neglected tropical diseases. The ETS state-space forecasting framework employed herein generated reasonable 1- to 5-month horizon S. haematobium TS forecasts, obliquely capturing prior non-linear interactions between disease dynamics and exogenous covariates (e. g. climate) and hence, obviating the need for more complex predictive methods in the district of Niono, Mali. Thus, this and other e. g. [11], [21]–[32] results suggest that the remarkable performance of state-space forecasting methods since the 1960s may be capitalized by the public health sector, providing a basis for local re-organization and strengthening of intervention programs in this and potentially other Sahelian districts. The operational simplicity, generality, and flexibility of state-space frameworks, such as the one employed here, conveniently allow for: 1) unsupervised model selection without disease-specific methodological tailoring; 2) on-line adaptation to fluctuations in partially- and fully-stable disease TS; and, 3) automatic switches between distinct forecasting methods as new TS perturbations dictate. Generally, state-space approaches are malleable to the dynamic incorporation of covariates (e. g. climate), expert opinion, and even a spatial dimension as needed. Therefore, fully automatic and user-friendly state-space forecasting frameworks, incorporating myriad (e. g. univariate, multivariate, and spatial-temporal) options, could considerably enhance disease control and hazard mitigation in regions where vulnerability to neglected tropical diseases is pervasive and statistical expertise is scarce.
Adequate forecasting and early warning systems are based upon observations of human behavior, population, disease time-series, climate, environment, and/or a combination thereof, whichever option best compromises among realism, feasibility, robustness, and parsimony. Fully automatic and user-friendly state–space forecasting frameworks, incorporating myriad options (e. g. , expert opinion, univariate, multivariate, and spatial-temporal), could considerably enhance disease control and hazard mitigation efforts in regions where vulnerability to neglected tropical diseases is pervasive and statistical expertise is scarce. The operational simplicity, generality, and flexibility of state–space frameworks, encapsulating multiple methods, could conveniently allow for 1) unsupervised model selection without disease-specific methodological tailoring, 2) on-line adaptation to disease time-series fluctuations, and 3) automatic switches between distinct forecasting methods as new time-series perturbations dictate. In this investigation, a univariate state–space framework with the aforementioned properties was successfully applied to the Schistosoma haematobium time-series for the district of Niono, Mali, to automatically generate contemporaneous on-line forecasts and hence, providing a basis for local re-organization and strengthening public health programs in this and potentially other Sahelian districts.
mathematics/nonlinear dynamics
infectious diseases
infectious diseases/neglected tropical diseases
public health and epidemiology/epidemiology
infectious diseases/helminth infections
public health and epidemiology/infectious diseases
mathematics/statistics
infectious diseases/tropical and travel-associated diseases
infectious diseases/epidemiology and control of infectious diseases
State–Space Forecasting of Schistosoma haematobium Time-Series in Niono, Mali