Forecasting Human African Trypanosomiasis Prevalences from Population Screening Data Using Continuous Time Models
[['Human African trypanosomiasis ( HAT ) , also known as sleeping sickness , is a parasitic disease that is caused by two sub-species of the protozoa Trypanosoma brucei: Trypanosoma brucei gambiense ( gambiense HAT ) and Trypanosoma brucei rhodesiense ( rhodiense HAT ) .', 'The infection causing the disease is transmitted from person to person through the tsetse fly .', 'It is estimated that there were 20000 cases in the year 2012 [1] and that 70 million people from 36 Sub-Saharan countries are at risk of HAT infection [2 , 3] .', 'Our work focuses on gambiense HAT , which represents 98% of all HAT cases [3] .', 'Gambiense HAT , which we will refer to as “HAT” from now on , is a slowly progressing disease and is fatal if left untreated .', 'In the first stage of the disease , symptoms are usually absent or non-specific [4] .', 'The median duration of this stage is about 1 . 5 years [5] .', 'By the time patients arrive at a healthcare provider , the disease has often progressed to the neurological phase , which causes severe health problems .', 'In addition , this treatment delay increases the rate of transmission , since an infected patient is a potential source of infection for the tsetse fly [4 , 6] .', 'Therefore , active case finding and early treatment are key to the success of gambiense HAT control [7 , 8] .', 'The current case finding strategy uses mobile teams that travel from village to village to conduct exhaustive population screening [4 , 8 , 9] .', 'For example , 35 mobile teams are active in the Democratic Republic of the Congo ( DRC ) .', 'Because this strategy has considerably reduced disease prevalence in several African countries [6 , 10–12] , the disease is no longer perceived as a major threat .', 'Consequently , donors are now scaling down their financial commitments [8] .', 'This , however , poses a serious risk to the control of HAT .', 'The disease tends to re-emerge when screening activities are scaled down , bringing about the risk of a serious outbreak , as shown by an epidemic in the 1990s [4 , 11 , 13] .', 'For example , the number of cases in 1998 is estimated to have exceeded 300000 [3] .', 'In order to minimize the risk of re-emergence when resources are scaled down , and in order to eliminate and eradicate the disease , maximizing the effectiveness of the control programs is crucial .', 'Mpanya et al . [9] suggest that the effectiveness of population screening is determined by ( among others ) the management and planning of the mobile teams .', 'Planning decisions—which determine which villages to screen , and at what time interval to screen them—have a direct impact on the risk and the magnitude of an outbreak .', 'Existing literature does not address these issues , as highlighted by the WHO [1] , and a wide variety of screening intervals have been applied in different control programs [12 , 14 , 15] .', 'To optimize the planning decisions , it is of key importance to be able to predict the evolution of the HAT prevalence level in the villages at risk .', 'This allows decision makers to assess the relative effectiveness of a screening round in these villages and to prioritize the screening rounds to be performed .', 'However , practical tools for predicting HAT prevalence appear to be lacking .', 'Existing models for HAT are mostly based on differential equations , describing the rate of change for the HAT prevalence level among humans and flies as a function of the prevalence levels among humans and flies ( some models also include an animal reservoir ) [16–22] .', 'As the information needed to use such models—e . g . , the number of tsetse flies in a village—is not available on the village level , using these models for prediction is impractical .', 'This paper therefore sets out to develop practical models describing and predicting the expected evolution of the HAT prevalence level in a given village , based on historical information on HAT cases and screening rounds in that village .', 'The main difference with the models mentioned in the previous paragraph is that our models make no assumptions about the causal factors underlying the observed prevalence levels: the “inflow” of newly infected persons and the “outflow” of infected persons by cure or death .', 'Instead , we just consider data on the net effect of these two processes—the evolution of the prevalence level—and fit five different models to this .', 'To analyze the predictive performance of these models , we make use of a dataset describing screening operations and HAT cases in the Kwamouth district in the DRC for the period 2004–2013 .', 'Furthermore , we use one of the models to analyze the fixed frequency screening policy , which assigns to each village a fixed time interval for consecutive screening rounds .', 'Specifically , we investigate screening frequency requirements for reaching elimination and eradication .', 'Here , eradication is defined as “letting the expected prevalence level go to zero in the long term” , and elimination is defined as “reaching an expected prevalence level of one case per 10000” .', 'Our paper thereby contributes to the branch of research on control strategies for HAT .', 'Next , we list several other papers that are highly related to our work .', 'The effectiveness of active case finding operations is analyzed by Robays et al . [23] , who define “effectiveness” as the expected fraction of cases in a village which will eventually get cured as a result of a screening round in that village .', 'The papers by Stone & Chitnis [16] , Chalvet-Monfray et al . [20] , and Artzrouni & Gouteux [24] introduce differential equation models to gain structural insights on the effectiveness of combinations of active case finding and vector control efforts and on the requirements for eradicating HAT .', 'The effect of active case finding activities is modeled through a continuous “flow” of infected individuals into the susceptible compartment .', 'Since we explicitly model the timing and the effects of a screening round , this is one of the main differences with our paper .', 'Finally , Rock et al . [10] study the effectiveness of screening and treatment programs and the time to elimination using a multi-host simulation model .', 'Their paper , however , considers the screening frequency as a given , whereas we consider the effects of changing this frequency .', 'Furthermore , we propose models for predicting prevalence on a village level , whereas their model implicitly assumes all villages to be homogeneous .'], ['Our dataset consists of information on screening operations in the period 2004–2013 in the health zone Kwamouth in the province Bandundu .', 'The raw data were cleaned up based on the rules described in S1 Text .', 'The number of villages in the dataset equals 2324 , and 143 of these villages were included in the data analysis based on three criteria: ( 1 ) the number of screening rounds recorded was at least two , ( 2 ) at least one case has been detected over the time horizon , and ( 3 ) at least one record of the number of people screened during the operation was available .', 'The first condition is necessary to enable modeling the prevalence level observed in a given screening round as a function of past observed prevalence levels , and the third condition is necessary for estimating prevalence itself .', 'We estimate the prevalence level in a village at the time of a screening round as the number of cases detected in that round over the number of people participating in that round .', 'Furthermore , lacking population size data , we estimate the population of a village as the maximum number of people participating in a screening round reported for that village .', 'Though our dataset also contains cases identified by the regular health system in between successive screening rounds , these do not yield ( direct ) estimates of prevalence levels in the corresponding villages , as required by the models proposed in the next section .', 'We therefore focus on the active case finding data only .', 'The total number of screening rounds reported for the 143 villages included equals 766 ( on average 5 . 4 per village ) .', 'Fig 1 shows cumulative distributions of the observed prevalence level in these screening rounds ( mean 0 . 0055 , median 0 . 0011 , standard deviation 0 . 0121 ) , the time interval between each pair of consecutive screening rounds ( mean 1 . 28 , median 1 . 00 , standard deviation 1 . 03 ) , the estimated population for each village ( mean 1073 , median 450 , standard deviation 2046 ) , and the participation level in the screening rounds ( mean 0 . 69 , median 0 . 72 , standard deviation 0 . 27 ) .', 'Note that the relatively large number of observations with a participation level of 100% is due to the method used to estimate the population sizes .', 'Before we propose our prediction methods , we introduce some notations .', 'A table of the most important notations used in this article can be found in S1 Table .', 'Let sv = {sv1 , sv2 , …} denote the vector of screening time intervals for village v , where sv1 denotes the time between the start of the time horizon and the first screening for this village , sv2 denotes the time between the first and the second screening , and so on .', 'The time at which the nth screening is performed is given by Svn = ∑m≤n svm and the participation fraction in this screening round is denoted by pvn .', 'Parameter Nv represents the population size of village v . Furthermore , let iv represent historical information on HAT cases in this village: the numbers of cases detected during past screening rounds .', 'We model the expected prevalence level at time t in village v as a function fv ( ⋅ ) of sv , iv , and some parameters β: fv ( t , sv , iv , β ) .', 'Note that the expected prevalence level is a latent , i . e . unobserved , variable , and that the observed prevalence level , xv ( t ) , generally deviates from the expected value .', 'We measure prevalence levels fv ( ⋅ ) and xv ( ⋅ ) as fractions and represent the difference between the expected and observed prevalence level in village v by the random variable εv:', 'x v ( t ) = f v ( t , s v , i v , β ) + ε v ( 1 ) Time series models such as discrete time ARMA , ARIMA or ARIMAX models seem to be the most popular methods for predicting prevalence ( or incidence ) ( see e . g . [25 , 26] ) .', 'These models describe the prevalence level at time t as a linear function of the prevalence levels at time t − 1 , t − 2 , … and ( optionally ) some other variables .', 'Their applicability in our context is however limited .', 'Discrete time models require estimates of the prevalence level at each time unit ( e . g . , each month ) , whereas information to estimate the HAT prevalence level is available only at moments at which a screening round is performed .', 'Namely , many HAT patients are not detected by the regular health system , particularly if they are in the first stage of the disease [8] .', 'The class of continuous time models is much more suitable for analyzing data observed at irregularly spaced times .', 'These models assume that the variable of interest , fv ( t , sv , iv , β ) , follows a continuous process , defining its value at each t > 0 .', 'The next subsections propose five continuous time models for predicting HAT prevalence levels .', 'We again note that models describing the causal processes determining the observed prevalence levels in detail ( e . g . , by explicitly modelling disease incidence , passive case finding , death and cure ) may be most intuitive , but require data that are not available on a village level .', 'Therefore , to safeguard their relevance for practical application , the variables we include are only those that are available on a large scale .', 'This does not imply that our models neglect the causal processes .', 'Instead , they are to some extent accounted for in an implicit way by fitting the models to the observed prevalence levels .', 'Data that are typically available at village level are numbers of HAT cases found during screening rounds and the times of these screening rounds .', 'For a given village , the first yields estimates of past prevalence levels , and the latter yield the time intervals between past screening rounds .', 'We hypothesize that the current expected prevalence level at time t is related to past prevalence levels , past screening intervals , and in particular the time since the last screening round , which we denote by δ v - ( t ) = min n { t - S v n | S v n ≤ t } .', 'Hence , we include ( functions of ) these variables in our models .', 'Linear regression models are very widely used in the world of forecasting ( see e . g . [27] ) .', 'Major advantages of these models are that they are easy to understand , to implement , to fit , and to analyze .', 'Therefore , the first model we introduce is a linear model ( model 1 ) , which also serves as a benchmark for our more advanced models .', 'This model describes the expected HAT prevalence in a given village as a function of the time since the last screening and past prevalence levels .', 'Such linear model is , however , very vulnerable to a typical structure present in active case finding datasets .', 'High past prevalence levels tend to increase the priority of screening a village , causing the time intervals between screening rounds to decrease .', 'As a result , δ v - ( t ) is a highly “endogenous” variable .', 'More formally , external variables ( past prevalence levels ) are correlated with both the dependent variable ( fv ( t ) ) and the independent variable ( δ v - ( t ) ) , which makes it hard to quantify the ( causal ) relation between them .', 'In response to this , we present four alternative models .', 'Model 2 is a fixed effects model , which adds a dummy-variable for each village to the initial model .', 'Model 3 is a ( non-linear ) exponential growth and decay model which is inspired by the SIS epidemic model .', 'This model is being used extensively for modeling epidemics that are characterized by an initial phase in which the number of infected individuals grows exponentially , and a second phase in which this number levels off to a time-invariant carrying capacity .', 'We refer to model 3 as the logistic model with a constant carrying capacity .', 'Finally , model 4 is a less data dependent version of model 3 and model 5 is variant of model 3 in which the carrying capacity is allowed to vary over time .', 'As HAT prevalence levels are very low , the variance of these levels is high , which enhances the chance that there are significant outliers among the observations .', 'For example , no cases were detected in three out of four screening rounds performed in a village of 122 people , whereas five cases were detected in the 4th round .', 'This implies two things .', 'First , observed prevalence levels will generally deviate significantly from expected prevalence levels .', 'Second , we need to choose a technique for estimating the model coefficients that is robust with respect to outliers .', 'Instead of Least Squares ( LS ) regression , one of the most commonly applied model fitting methods , we therefore use Least Absolute Deviations ( LAD ) regression to fit the model parameters , which is known to be relatively insensitive to outlying observations [32] .', 'An alternative technique would be to use a maximum likelihood estimation ( MLE ) approach based on a heavy-tailed probability distribution for the observed prevalence levels .', 'In S2 Table , we show the results obtained when assuming a Poisson , Beta-Binomial , or Negative Binomial distribution .', 'Each of the MLE approaches is , however , clearly outperformed by the LAD regression approach .', 'The variance of the observed prevalence level strongly depends on the sample size .', 'For example , under the assumption of an independent infection probability for each person , the variance is inversely proportional to the sample size .', 'We therefore weight the fitting deviation evn = fv ( Svn ) − xv ( Svn ) for observation n for village v by weight w v n = N v · p v n , yielding the following weighted LAD regression problem:', 'min β S a b s ( β ) = ∑ ( v , n ) w v n | e v n | ( 13 ) To deal with the risk of overfitting , we select the variables to be included in the models by means of a backward elimination method .', 'This method initially includes all variables in the model and iteratively removes the least significant variable ( if its p-value > 0 . 10 ) and estimates the model with the remaining variables .', 'The algorithm stops as soon as all remaining variables are significant or if only one variable is left .', 'We enforce that αv and κ cannot be removed by the backward elimination method so as to preserve essential elements of the corresponding models .', 'Hence , only parameters β1-β8 in models 1 and 2 , and parameters β1-β2 in models 3–5 could be removed .', 'Finally , to test the predictive performance of the models , we split the data in an estimation sample ( which we use for fitting the model ) and a prediction sample .', 'Specifically , for each of the 143 villages , we include the last screening round in the prediction sample , and include the others in the estimation sample .', 'Next , we measure performance based on the mean of the prediction errors M E = ∑ v e v n ^ | V | , indicating whether the predictions obtained by the model are biased , and based on two indicators for the amount of explained variation in the prevalence levels: the mean absolute error , M A E = ∑ v | e v n ^ | | V | and the mean relative error , M R E = ∑ v | e v n ^ | ∑ v x v ( S v n ^ ) .', 'Here , the index combination v n ^ indicates the last screening round for village v . The intuition behind the measures of explained variation is that they equal 0 if the predicted prevalence levels are exactly equal to the observed prevalence levels ( i . e . , the model perfectly explains the variation in the observed prevalence levels ) and that their value increases when the absolute difference between predicted and observed levels increases ( i . e . when the model explains less variation in observed prevalence levels ) .', 'We use Matlab R2015b for the implementation of our methods .'], ['Table 1 presents the coefficient estimates for the variables of the five presented models .', 'The results for models 1 and 2 are very similar .', 'Seven of the eight variables are identified as being non-significant by the backward elimination algorithm: the interaction terms , the long term prevalence level , the time since the last screening round , and the square root of the time since the last screening round .', 'The resulting model provides a clear prediction method: the expected prevalence equals 24 . 5% of the prevalence level observed at the previous screening round ( note , if this level was 0 . 0% , the estimated expected prevalence remains 0 . 0% ) according to model 1 , and equals 14 . 7% of this prevalence level plus a constant fraction αv according to model 2 .', 'Hence , this model predicts that , in the absence of screening activities , the expected prevalence remains the same over time .', 'The fitted models 3 , 4 , and 5 reveal a clear and intuitive relationship between screening frequency , prevalence , and carrying capacity: a larger historical prevalence indicates a higher carrying capacity , and facing an equal historical prevalence for a higher historical screening frequency indicates a higher carrying capacity .', 'The constant term has been identified as non-significant for models 3 and 4 and as significant for model 5 .', 'To illustrate the typical output of models 3 and 4 , Fig 2 shows the development of the expected prevalence levels for two villages over time ( the lines ) , as well as the observed prevalence levels ( stars and circles ) .', 'Furthermore , Fig 3 depicts the carrying capacities for the 143 villages in Kwamouth , as estimated by the LMCCC model .', 'Though data to validate these estimates are lacking , we note that they are in the same order of magnitude as prevalence levels found during screening rounds .', 'The latter are usually between 1% and 5% in high or very high transmission areas , and exceed 10% in some extreme cases [33 , 34] .', 'As mentioned in Section Model Fitting , we measure the predictive performance of the different models in terms of the prediction bias and in terms of the amount of variation explained .', 'Table 2 contains the values of the different indicators for each of the models , and Fig 4 and S1 Fig . compare the prediction errors produced by the different models .', 'These prompt several interesting observations .', 'First , the prediction bias ranges from 0 . 47/1000 ( rLMCCC model ) to -1 . 86/1000 ( LM model ) .', 'Given that the average observed prevalence in the 766 screening rounds in our dataset equals 5 . 5/1000 , we consider the biases of the LM model and the LMVCC model as quite substantial .', 'Yet , this may be very well explained by the highly variable character of the HAT epidemic .', 'A small number of outbreaks may substantially shift the average observed prevalence level .', 'For example , without the four most negative prediction errors , the prediction bias for the LM model would be only -0 . 69/1000 .', 'Second , the LM model performs relatively well in terms of explained variation .', 'Yet , we see two vulnerabilities of this model: ( 1 ) as discussed before , this model is likely to be hampered by endogeneity , inducing a potential bias in the coefficient estimates , and ( 2 ) the variation in the screening intervals is relatively small for the villages with the highest endemicity levels in our sample , as these villages are screened almost every year .', 'When there is little variation in δ v - ( t ) , the true effects of variations might not become visible .', 'These two fundamental vulnerabilities may very well explain why ( a function of ) δ v - ( t ) has not been identified as significant for the LM model .', 'As a result , this model unrealistically predicts that the value of the expected prevalence level remains the same over time in the absence of screening activities , contrasting with vast historical evidence .', 'The same vulnerabilities apply to the FEM model , which also provides a counter-intuitive relation between the expected prevalence and δ v - ( t ) .', 'On top of that , its predictive power is relatively low , which could be explained by the fact that , for many villages , there is insufficient data to estimate the fixed effect accurately .', 'As variants of the logistic model already fix the structure of the relationship between δ v - ( t ) and f v ( S v n + δ v - ( t ) ) based on epidemiological insights , these models do not suffer from the vulnerabilities mentioned above .', 'We therefore consider these models to have most potential for accurately predicting HAT prevalence levels in general ( i . e . , in any region and for any time horizon ) .', 'Among the three logistic model variants , model 3 ( LMCCC ) performs reasonably well in terms of both criteria .', 'Model 5 ( LMVCC ) has a substantial prediction bias , but performs best in terms of explained variation , as can be seen in Fig 4 ( its performance is closest to the “perfect fit” ) .', 'Though model 4 ( rLMCCC ) performs best in terms of prediction bias , it performs very weakly in terms of explained variation .', 'Hence , among the logistic model variants , there is no clear winner when both criteria are assigned equal importance .', 'For planning decisions , however , we consider model 5 to be most suitable , followed by model 3 .', 'The reason is that , in contrast with prediction bias , explained variation indicates the ability to identify differences in expected prevalence levels between villages , as required for effective planning decisions .', 'Hence , identifying an effective prioritization of the different villages will be more important than obtaining unbiased estimates of the resulting prevalence levels .', 'The sensitivity level s is known to differ between regions [35] .', 'Furthermore , the population size of a village had to be estimated , which induces a potential bias in the participation level estimates .', 'These issues beg to question the robustness of our results on the logistic model variants ( note , models 1 and 2 are not affected by this as these do not use these parameters ) .', 'S3 Table shows the results of a sensitivity analysis , which largely confirm our findings .', 'In all scenario’s analyzed , model 5 remains best in terms of explained variation , followed by model 3 , and models 3 and 4 outperform model 5 in terms of prediction bias .', 'Another assumption that questions the robustness of our results is the one about the expected prevalence level at the beginning of the time horizon ( i . e . , at 01-01-2004 ) .', 'S4 Table provides the results of a sensitivity analysis on this assumption .', 'Again our main findings remain the same .', 'In the previous section we argue that , among the models analyzed in this paper , variants of the logistic model have most potential for accurately predicting HAT prevalence levels in general .', 'In this section we demonstrate the applicability of one of these model variants to analyze the effectiveness of screening operations .', 'In particular , since information on the development of the carrying capacities is lacking , as required by the LMVCC model , and since we consider the predictive performance of model 3 superior to that of model 4 , we choose to use the LMCCC model as a basis for this analysis .', 'We do note that the theoretical results presented here also hold for model 4 and , if the carrying capacity remains constant , for model 5 also .', 'Our analysis will concentrate on the fixed frequency screening policy .', 'This policy assigns to each village a fixed time interval for consecutive screening rounds based on the village’s characteristics .', 'As the policy is relatively easy to understand and implement , it has been the basis for guideline documents for HAT control .', 'For example , the WHO recommends a screening interval of one year for villages reporting at least one case in the past three years , and an interval of 3 years for villages that did not report a case in the last three years , but did report at least one case during the past five years [1] .', 'In the first part of this section , we mathematically analyze the impact of a fixed screening policy for a given village and investigate the screening frequency required to eradicate HAT in that village .', 'As mentioned in the introduction , we define that HAT is eradicated in the long term if the expected prevalence level goes to zero in the long term .', 'A shorter term objective is to eliminate HAT , where elimination is defined as having at most one new case per 10000 persons per year [1 , 7] .', 'For example , the WHO’s roadmap towards elimination of HAT states the aim to eliminate ( gambiense ) HAT as a public health problem by 2020—which is defined as having less than one new case per 10000 inhabitants in at least 90% of the disease foci [1]—and to reach worldwide elimination by 2030 .', 'The second part of this section presents analytical results about the time needed to reach elimination and about the screening frequency requirements for reaching elimination within a given time frame .', 'As our models consider expected prevalence instead of incidence , we redefine elimination as “reaching an expected prevalence level of one case per 10000” .', 'We argue that the times and efforts required to reach this elimination target are practically suitable lower bounds on the times and efforts needed to reach the WHO’s targets .', 'First , incidence and prevalence levels are argued to be “comparable” for HAT if mobile units visit afflicted areas infrequently [16] .', 'If mobile teams visit the areas more frequently , incidence will only become larger compared to prevalence and the prevalence level target will be easier to achieve than the incidence level target ( e . g . , under the assumption that the fraction of flies infected is proportional to the fraction of humans infected , this follows directly from the epidemic model presented by Rogers [22] ) .', 'Second , even if the expected prevalence level is below the defined threshold level , the intrinsic variability of the HAT epidemic may induce an actual prevalence level that exceeds this threshold .', 'Throughout this section , we consider an imaginary village with a constant carrying capacity K . ( For sake of conciseness we omit the subscript v in this section ) .', 'Furthermore , we assume a constant participation level pvn = p , 0 < p < 1 , and a fixed screening interval τ .', 'The expected prevalence level at the beginning of the time horizon is denoted by f ( 0 ) , f ( 0 ) > 0 .', 'Finally , recall that s , 0 < s < 1 , denotes the sensitivity level .'], ['This paper introduces and analyzes five models for predicting HAT prevalence in a given village based on past observed prevalence levels and past screening activities in that village .', 'Based on the quality of prevalence level predictions in 143 villages in Kwamouth ( DRC ) , and based on the theoretical foundation underlying the models , we conclude that variants of the logistic model—a model inspired by the SIS model—are most practically suitable for predicting HAT prevalence levels .', 'Sensitivity analyses show that this conclusion is very robust with respect to assumptions about participation levels , the sensitivity of the diagnostic test , or the initialization value of the prevalence curves are violated .', 'Second , we demonstrate the applicability of one variant of the logistic model to analyze the effectiveness of the fixed frequency screening policy , which assigns to each village a fixed time interval for consecutive screening rounds .', 'Due to the intrinsic variability of the HAT epidemic , observed prevalence levels will generally deviate significantly from predicted prevalence levels .', 'We strongly believe , however , that this does not render predictions worthless in the context of planning decisions .', 'In contrast , a major contribution of our models is that they indicate the expected disease burden in different villages and can hence be applied to develop planning policies that aim to minimize the total expected disease burden for the villages considered .', 'Our analysis of the fixed frequency screening policy reveals that eradication of HAT is to be expected in the long term when the screening interval is smaller than a given threshold .', 'This threshold strongly depends on the case detection fraction: the fraction of cases who participate in the screening rounds and are detected by the diagnostic tests .', 'Under current conditions , we estimate the threshold to be approximately 15 months .', 'This suggests that annual screening , as recommended by the WHO for endemic areas , will eventually lead to eradication .', 'More specifically , our model predicts that annual screening will lead to eradication if the case detection fraction exceeds 55% .', 'The logistic model also reveals expressions for the time needed to reach the more short term target of eliminating HAT and for the screening interval required to eliminate HAT within a given time frame .', 'These suggest that it takes 10 years to eliminate HAT in a village or focus with a prevalence of 5/1000 ( under current conditions and annual screening ) .', 'Furthermore , we estimate that it is only feasible to reach elimination within five years if the case detection fraction is very high—roughly above 75%—or if the current prevalence level is very low—roughly below 1/1000 .', 'We argue that these figures are practically suitable lower bounds on the time or efforts needed to reach the WHO’s targets for elimination .', 'Our results on requirements for eradication or elimination are based on a deterministic model , which begs to question their validity for reality , where events are stochastic .', 'We note , however , that we model the expected behavior of a stochastic system , and hence that our results also hold in expectation for the stochastic system .', 'On the other hand , we acknowledge that our models are not perfect .', 'For example , we neglect interaction effects between neighboring villages .', 'It would therefore be interesting and relevant to investigate whether our results can be reproduced by a validated simulation model .', 'A necessary condition for the applicability of our prediction models is that data about possibl
['Introduction', 'Materials and Methods', 'Results', 'Discussion']
['To eliminate and eradicate gambiense human African trypanosomiasis ( HAT ) , maximizing the effectiveness of active case finding is of key importance .', 'The progression of the epidemic is largely influenced by the planning of these operations .', 'This paper introduces and analyzes five models for predicting HAT prevalence in a given village based on past observed prevalence levels and past screening activities in that village .', 'Based on the quality of prevalence level predictions in 143 villages in Kwamouth ( DRC ) , and based on the theoretical foundation underlying the models , we consider variants of the Logistic Model—a model inspired by the SIS epidemic model—to be most suitable for predicting HAT prevalence levels .', 'Furthermore , we demonstrate the applicability of this model to predict the effects of planning policies for screening operations .', 'Our analysis yields an analytical expression for the screening frequency required to reach eradication ( zero prevalence ) and a simple approach for determining the frequency required to reach elimination within a given time frame ( one case per 10000 ) .', 'Furthermore , the model predictions suggest that annual screening is only expected to lead to eradication if at least half of the cases are detected during the screening rounds .', 'This paper extends knowledge on control strategies for HAT and serves as a basis for further modeling and optimization studies .']
['The primary strategy to fight gambiense human African trypanosomiasis ( HAT ) is to perform extensive population screening operations among endemic villages .', 'Since the progression of the epidemic is largely influenced by the planning of these operations , it is crucial to develop adequate models on this relation and to employ these for the development of effective planning policies .', 'We introduce and test five models that describe the expected development of the HAT prevalence in a given village based on historical information .', 'Next , we demonstrate the applicability of one of these models to evaluate planning policies , presenting mathematical expressions for the relationship between participation in screening rounds , sensitivity of the diagnostic test , endemicity level in the village considered , and the screening frequency required to reach eradication ( zero prevalence ) or elimination ( one case per 10000 ) within a given time-frame .', 'Applying these expressions to the Kwamouth health zone ( DRC ) yields estimates of the maximum screening interval that leads to eradication , the expected time to elimination , and the case detection fraction needed to reach elimination within five years .', 'This paper serves as a basis for further modeling and optimization studies .']
['medicine and health sciences', 'infectious disease epidemiology', 'african trypanosomiasis', 'tropical diseases', 'parasitic diseases', 'health care', 'mathematics', 'forecasting', 'statistics (mathematics)', 'screening guidelines', 'neglected tropical diseases', 'infectious disease control', 'research and analysis methods', 'public and occupational health', 'infectious diseases', 'zoonoses', 'epidemiology', 'mathematical and statistical techniques', 'protozoan infections', 'trypanosomiasis', 'differential equations', 'health care policy', 'physical sciences', 'statistical methods']