trachoma is the leading infectious cause of blindness worldwide(resnikoff et al . , 2004 ) . mass , community - wide , 
 oral azithromycin treatments have proven effective in reducing the prevalence of the 
 causative agent , chlamydia trachomatis(schachter et al . , 1999 ; , 2006 ; house et al . , 
 2009 ; lietman et al . , 1999 ; gebre 
 et al . , 2011 ) . hygiene programs and latrine construction are also thought to reduce 
 transmission ( emerson et al . , 2000 ) . in some 
 areas , infection is disappearing even in the absence of a dedicated treatment 
 program ( chidambaram et al . infection may be going away because of mass 
 antibiotic distribution , improved hygiene , or secular changes . regardless of the 
 mechanism , public health stakeholders need to assess whether their programs are a 
 success or failure . progressive reduction of the mean prevalence of infection in a 
 region towards zero would certainly be an encouraging trend . however , monitoring 
 large numbers of villages longitudinally can take up scarce resources , and has not 
 been done consistently outside a few research programs . programs would benefit if 
 cross - sectional surveys alone could reveal whether infection was headed towards 
 elimination . mathematical models of infectious disease transmission suggest that as a 
 disease disappears , the only stationary distribution for the expected prevalence of 
 infection occurs when elimination has been achieved in all communities(brauer et al . , 2008 ) . however , the distribution 
 of the prevalence in those communities where infection still remains may approach a 
 quasi - stationary distribution ( cavender , 
 1978 ; nsell , 1999 , 1996 ) . in stochastic 
 susceptible - infectious - susceptible ( sis ) models , this quasi - stationary distribution 
 can be approximated by a geometric distribution ( cavender , 1978 ; nsell , 
 1999 , 1996 ) . specifically , if we 
 assume a reproduction number at invasion ( r0 ) distinctly 
 less than one , and a refuge of one permanently infected individual in each 
 community , then the number of infections in a set of uniform , fixed - size communities 
 approaches a geometric distribution as the population size of communities gets 
 larger ( nsell , 1996 ) . simulations 
 suggest that even with moderate population sizes , the geometric distribution or its 
 continuous analog , the exponential distribution , are excellent approximations ( nsell , 1999 , 1996 ; ray et al . , 
 2007 ) . this has been found for models of infection disappearing due to 
 low transmission or due to repeated mass antibiotic distributions ( nasell , 1999 ; ray et al . , 
 2007 ) . note that these models typically have assumed homogeneity within 
 and between communities ( blake et al . , 2009 ; 
 ray et al . , thus , the geometric 
 distribution may or may not fit well to field data , where these assumptions would 
 likely not strictly hold . recent trachoma surveys have assessed the prevalence of infection in multiple 
 communities biannually over several years . in each study , communities were 
 participating in an intensive treatment program , and infection was documented to be 
 decreasing over time . if the underlying conditions which promote transmission are 
 similar enough in an area , then each community could be regarded as a separate 
 sampling from a single distribution . here , we test the hypothesis that a geometric 
 distribution describes the prevalence of infection in different communities where 
 infection is disappearing . we considered results from the tef study in gurage , ethiopia , and the tana 
 study in amhara ethiopia ; in each , children from 24 communities were longitudinally 
 monitored for ocular chlamydial infection ( house et 
 al . , 2009 ; gebre et al . , 2011 ; , 2001 ; lakew et al . , 2009a ; 
 porco et al . , 2009 children are 
 considered a core - group for ocular chlamydia infection and are therefore the most 
 important population to monitor ( house et al . , 
 2009 ) . in the tef study , 15 year - old children were monitored 
 biannually from randomly selected communities treated biannually for 2 years ( lakew et al . , 2009a ) . in the tana study , 
 09 year - old children were monitored biannually from randomly selected 
 communities treated annually or biannually for 42 months ( gebre et al . , 2012 ) . in order to characterize the 
 distribution of infection when infection is disappearing , we included cross - sectional visits after at least two mass antibiotic distributions and where 
 the mean prevalence of infection was lower than seen 12 months previously ( chidambaram et al . , 2006 ; lakew et al . , 2009b ) . for display purposes only , we 
 produced an empirical distribution of each cross - sectional survey by assuming that 
 each community s contribution was the bayesian posterior derived from the 
 observed prevalence and a non - informative , uniform prior ( thus a beta distribution ) . 
 to assess possible geographical clustering in tana , where gps coordinates were 
 available , we conducted a permutation test of moran s i defining neighboring 
 communities as those sharing a vertex in a delaunay triangulation . for each of the cross - sectional surveys of prevalence , we assessed the fit of 
 8 discrete distributions . three had a single continuous parameter : the geometric , 
 binomial , and poisson distributions . four had two continuous parameters , and with 
 the appropriate shape parameter included the geometric distribution as a special 
 case : discrete weibull , negative binomial , beta binomial ( at least asymptotically 
 approaching a geometric distribution as the population size increases ) , the 
 zero - inflated geometric , and the zero - inflated poisson distributions ( johnson et al . , 1993 ) . several cross - sectional 
 surveys were available for each of the two studies . in order to combine data from 
 the available visits , for each of the 2-parameter distributions we fitted a single 
 overall shape parameter per study , but a different scale parameter for each 
 available time point ( of which note that we 
 chose to parameterize the zero - inflated geometric and poisson allowing a different 
 proportion of zeros at each visit , but a single geometric scale parameter for all 
 visits ( see appendix ) . as a sensitivity analysis for the 2-parameter distributions , 
 we fitted each cross - sectional survey separately , allowing a unique shape and scale 
 parameter for each cross - sectional visit . since the 24 different communities in each 
 of the two studies had differing numbers of individuals , we parameterized such that 
 the fitted scale parameter for each cross - sectional survey represented the mean 
 proportion of the distribution ; note that this mean proportion of the distribution 
 was not necessarily the mean proportion of the data ( appendix a ) . since post - treatment distributions were right - tailed , the 
 first shape parameter of the beta binomial distribution was treated as the shape 
 parameter , and the second shape parameter as the scale parameter ( see 
 reparameterizations in the appendix ) . each distribution was assessed by determining 
 the parameter values that optimized the likelihood of the observed data , and ranked 
 by the sample size - corrected akaike information criterion ( aicc ) ( burnham and anderson , 1998 ) . the uncertainty of 
 the parameter estimates was assessed with bias - corrected 95% percentile 
 bootstrap confidence intervals with 999 resamples . we performed goodness - of - fit 
 testing with the chi - squared statistic , binning prevalence into 5% intervals 
 and comparing versus the geometric distribution with the optimum scale parameter . we 
 determined the p - value for the observed data by selecting 999 
 samples from the best - fit geometric distribution for each study visit , and treated 
 each sample in the same manner as the observed data . as sensitivity analyses , we fitted pre - treatment data from the two study 
 regions before a program was initiated ( time 0 months in table 1 and the black curves in fig . 1 and b ) ; infection was not thought to be 
 disappearing , and thus a geometric distribution was not expected . in addition , we 
 estimated community prevalence from a stochastic sis model , assuming 
 r0 = 0.5 ( infection disappearing ) , 
 r0 = 1.0 , r0 
 = 1.5 , or r0 = 2.0 ( infection not 
 disappearing ) , and creating a sample of 50 children from each of 24 communities 
 ( lietman et al . , 2011 ) . we determined the 
 quasi - stationary distribution of the prevalence of infection pre - treatment by 
 constructing an n  n matrix representing 
 the n kolmogorov forward equations without the zero state ; the 
 eigenvector associated with the largest eigenvalue of this matrix was taken as the 
 quasi - stationary distribution(brauer et al . , 
 2008 ; nasell , 1999 ; ray et al . the quasi - stationary 
 distribution after three annual mass distributions with 80% coverage was 
 estimated with 1000 simulations . in order to assess whether 24 communities were 
 sufficient to distinguish between different distributions , we fitted multiple 
 samples of 24 simulated communities from geometric and discrete weibull 
 distributions . all calculations were performed in mathematica 9.0 
 ( wolfram research , champaign , illinois ) . data were from studies registered with clinicaltrials.gov ( nct00221364 and 
 nct00322972 ) and approved by the national ethics review committee of the ethiopian 
 science and technology agency and the ucsf irb ( 10 - 02630 and 10 - 02576 ) . the sponsors of the study had no role in trial design , data collection , 
 data analysis , data interpretation , or writing of the report . tl had full access 
 to all the data in the study and had final responsibility for the decision to 
 submit for publication . the sponsors of the study had no role in trial design , data collection , 
 data analysis , data interpretation , or writing of the report . tl had full access 
 to all the data in the study and had final responsibility for the decision to 
 submit for publication . visits from the two studies were included if communities had received at 
 least two mass antibiotic distributions and if the infection was documented to have 
 decreased from 12 months earlier were included . the 2 and 6 month visits from tef 
 and the 6 and 12 month visits from tana were excluded because they were after a 
 single treatment . the post - treatment mean prevalence of infection in the two studies 
 varied from 1.5% to 10.9% ( table 
 1 ) , decreasing after mass treatment . density plots of the community - level 
 prevalence over time are displayed in fig . 1 . 
 in tana , where gps was available , we were unable to demonstrate statistically 
 significant geographical clustering for any post - treatment visit ( moran s i : 
 p = 0.32 for 18 months , p = 
 0.30 for 24 months , p = 0.14 for 30 months , 
 p = 0.64 for 36 months ) . we determined the parameter values which maximized the likelihood of 
 obtaining the observed data for the 7 distributions : geometric , binomial , poisson , 
 discrete weibull , negative binomial , beta binomial , zero - inflated geometric , and 
 zero - inflated poisson distributions ( table 
 2 ) . in each case , the geometric distribution had the lowest ( best ) 
 aicc . the binomial and poisson distributions were the other single 
 parameter distributions tested and offered far lower ( worse ) likelihoods . note that 
 the geometric distribution is a special case of the two - parameter distributions : 
 discrete weibull , negative binomial , and beta binomial ( at least asymptotically ) , 
 and the zero - inflated geometric distributions ; thus with appropriate parameter 
 choices , each can mimic a geometric distribution and achieve a likelihood of the 
 observed data as least as high as that found with the geometric ( with the exception 
 of the zero - inflated geometric since we chose to use a single geometric scale 
 parameter for each cross - sectional survey in a study , allowing the proportion that 
 were zero to vary at each time point ) . since none of these two - parameter 
 distributions offered a significantly better fit than the geometric , the aicc was 
 larger for these distributions . this suggests that these two - parameter distributions 
 are less parsimonious than the geometric . the 95% confidence intervals for 
 the estimation of the shape parameters for these two - parameter distributions 
 included one , which in these parameterizations were consistent with the geometric 
 ( table 2 ) . note that when we treated each 
 individual cross - sectional survey separately allowing the shape parameter to vary 
 with each visit , either the geometric or the zero - inflated geometric had the best 
 aicc ( for individual tana visits , see appendix 
 b ) ; the 95% cis of shape parameter of the other three 2-parameter 
 distributions included one for each separate visit . if the community - level 
 prevalence of infection were indeed from a geometric distribution , how often would 
 we have found results as close to a geometric as observed ? with goodness - of - fit 
 testing , we were unable to reject the hypothesis that the observed data from each 
 entire study , or from any single post - treatment visit within a study , came from a 
 geometric distribution ( table 1 ) . as sensitivity analyses , simulations of an sis model with mass action and 
 r0 = 0.5 were used to construct a dataset 
 similar to the ethiopian data with 24 communities evaluated at 4 different time 
 points . the geometric offered the optimal aicc , and in no case did the addition of a 
 second parameter result in a statistically superior fit ( likelihood ratio test , 
 p > 0.05 ) . similar sized sets of communities simulated 
 from discrete weibull distributions with shape parameters of 0.5 and 2.0 were easily 
 distinguishable from the geometric ( p < 0.05 ) , with the 
 weibull distribution having the optimal aicc . the model selection process correctly 
 identified samples taken from discrete weibull distributions with shape parameters 
 0.5 and 2.0 . empirical , pre - treatment surveys from the two ethiopian areas studied 
 were not consistent with a geometric distribution ( table 2 ) . simulations of an sis model with mass action and 
 r0 = 1.0 , 1.5 , and 2.0 ( without treatment ) 
 were used to construct datasets similar to the ethiopian data with 24 communities 
 evaluated at 4 different time points . in all cases , the poisson and the binomial 
 distribution each had a far lower ( better ) aicc than the geometric distribution . similar sets with an estimated treated reproduction number of 
 rt = 1.0 , 1.5 , and 2.0 also were better fit 
 by the poisson and binomial than the geometric ( for rt 
 = 1.0 see appendix c ) . when assessing whether an infectious disease is on its way to elimination , 
 survey of infection in every community in a region over time may be infeasible due 
 to the costs of collection and processing . however , if the community - level 
 cross - sectional prevalence of infection in an area can be approximated by a 
 geometric distribution , then a single cross - sectional sampling of communities can 
 provide an accurate assessment of infection in an area . here , we assessed two large , 
 population - based surveys recently performed in multiple communities in controlled 
 settings where transmission conditions were similar and constant , and where trachoma 
 was known to be disappearing due to programmatic activity . in each case , the 
 geometric distribution fitted the observed data better than the other distributions 
 tested . furthermore , we were unable to reject the hypothesis that each prevalence 
 was drawn independently from a geometric distribution in these two settings where 
 trachoma was known to be disappearing . much of the distributional information 
 available from examining every community is captured by the single , sufficient 
 parameter of a geometric distribution . since a geometric distribution has a relatively heavy tail , outliers are to 
 be expected . while we can not discount the possibility that a high prevalence village 
 has inherently higher transmission than its neighbors , this need not be 
 hypothesized ; occasional high prevalence communities are to be expected even from a 
 set of communities with identically transmission characteristics , and do not 
 necessarily reflect a hot spot for transmission or a program failure . models predict 
 that infection will still disappear from the high prevalence communities in the tail 
 of this distribution ( ray et al . , 2007 , 2009 ) . reports 
 from nepal , tanzania , and the gambia have found that the prevalence of infection in 
 the most severely affected village in an otherwise hypo - endemic 
 area decreases in the absence of treatment ( jha et al . , 2002 ; gaynor 
 et al . , 2003 ; solomon et al . , 
 2004 ; burton et al . , 2010 ) . longitudinal studies that focused on the most severely affected community in an area 
 may have led researchers to interpret this regression to the mean as enhanced 
 efficacy of antibiotics and other interventions , relative to studies which included 
 a control group ( gebre et al . , 2011 ; gaynor et al . , 
 2003 ; solomon et al . , 2004 ; stoller et al . , 2011 ; west et al . recent models fitted to data from areas moderately endemic for trachoma 
 suggest the possibility of positive feedback ( lietman et al . , 2011 ) ; that is , additional infection in a community may 
 increase the per - susceptible risk of infection more than the proportional increase 
 assumed by the mass - action model . in meso - endemic areas this may result in a 
 bi - modal distribution of prevalence , with a peak at elimination , and a second peak 
 around an endemic equilibrium ( lietman et al . , 
 2011 ) . note though that even a positive ( or negative ) feedback model 
 should behave as a mass - action sis model at a low enough prevalence , and thus the 
 geometric distribution would still be expected . we would expect a mixture of two 
 distributions if a survey area included distinct areas of low and high transmission . if the unit of the survey was not a single community , but a cluster of communities , 
 we would expect the convolution of several geometric distributions ( i.e. a negative 
 binomial distribution with a shape parameter greater than one ) . these 24 years 
 longitudinal studies show decrease in prevalence , but do not follow to complete 
 elimination . the goodness - of - fit testing assumes independent sampling from an 
 identical distribution ; while the samples were exchangeable , there may be 
 geographical correlations . none of the sentinel communities in either tef or tana 
 were contiguous , and in tana , where gps was available , we were unable to demonstrate 
 any significant geographical clustering . note that any correlation would have made 
 it more difficult to exclude a geometric distribution , hence in that sense our 
 analysis was conservative . likewise , the prevalence of infection in treatment arms 
 in tana was not significantly different ; separation by treatment arm would have 
 resulted in smaller datasets and more difficult exclusion of the geometric 
 distribution . the endgame for disease control while districts 
 or subdistricts may meet thresholds for control , concern for missing higher 
 prevalence communities has led some to argue that programs need to examine every 
 community . knowledge that community - level prevalence approximates a geometric 
 distribution , and that the scale of the distribution can be estimated from a survey 
 sample , may allow programs the confidence they need to conserve resources . population - based surveys of all communities in trachoma - endemic areas would be an 
 enormous undertaking , and appears to be unnecessary . programs can assess whether 
 they are on track towards elimination by a single cross - sectional survey . if the 
 distribution of the community prevalence of infection in a particular area is 
 consistent with a geometric distribution , this may be a good sign , since this would 
 not be expected in areas where infection is not disappearing ( nsell , 1996 ; lietman et al . , 2011 ) . note that elimination would only be expected to 
 eventually occur if conditions remain the same . an increase in transmission due to 
 discontinuation of programmatic activity of deterioration of socioeconomic status 
 could prevent elimination . additional studies will be necessary to determine whether 
 the prevalence of the clinical signs of infection on which programs rely so heavily 
 can also be approximated by a geometric distribution . while 
 the who recommends hygiene and latrine programs , no non - antibiotic measure has yet 
 been proven to decrease infection ( emerson et al . , 
 2000 mass 
 antibiotics distributions have proven effective in cluster - randomized trials , and 
 have so far been well tolerated ( schachter et al . , 
 1999 ; chidambaram et al . trachoma has been 
 disappearing in some areas , even in the absence of a dedicated trachoma program 
 ( hoechsmann et al . , 2001 ; this report suggests that if efforts are continued , 
 infection will eventually disappear in previously hyper - endemic areas of ethiopia . 
 if these are representative of other severely affected regions , if trachoma programs 
 can be maintained , and if antibiotic resistance does not evolve , then global 
 eradication of trachoma may indeed be feasible .
mathematical models predict that the prevalence of infection in different 
 communities where an infectious disease is disappearing should approach a 
 geometric distribution . 
 trachoma programs offer an opportunity to test this 
 hypothesis , as the world health organization ( who ) has targeted trachoma to be 
 eliminated as a public health concern by the year 2020 . 
 we assess the 
 distribution of the community prevalence of childhood ocular chlamydia infection 
 from periodic , cross - sectional surveys in two areas of ethiopia . 
 these surveys 
 were taken in a controlled setting , where infection was documented to be 
 disappearing over time . 
 for both sets of surveys , the geometric distribution had 
 the most parsimonious fit of the distributions tested , and goodness - of - fit 
 testing was consistent with the prevalence of each community being drawn from a 
 geometric distribution . when infection is disappearing , the single sufficient 
 parameter describing a geometric distribution captures much of the 
 distributional information found from examining every community . 
 the relatively 
 heavy tail of the geometric suggests that the presence of an occasional 
 high - prevalence community is to be expected , and does not necessarily reflect a 
 transmission hot spot or program failure . 
 a single cross - sectional survey can 
 reveal which direction a program is heading . 
 a geometric distribution of the 
 prevalence of infection across communities may be an encouraging sign , 
 consistent with a disease on its way to eradication .