International Journal for Parasitology 40 (2010) 1373–1380

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International Journal for Parasitology

journal homepage: www.elsevier .com/locate / i jpara

Trickle or clumped infection process? An analysis of aggregation in the weightsof the parasitic roundworm of humans, Ascaris lumbricoides

Martin Walker a,*, Andrew Hall b, María-Gloria Basáñez a

a Department of Infectious Disease Epidemiology, Faculty of Medicine (St. Mary’s Campus), Imperial College London, Norfolk Place, London W2 1PG, UKb Centre for Public Health Nutrition, School of Life Sciences, University of Westminster, 115 New Cavendish Street, London W1W 6UW, UK

a r t i c l e i n f o a b s t r a c t

Article history:Received 7 December 2009Received in revised form 10 March 2010Accepted 11 March 2010

Keywords:Ascaris lumbricoidesWeight distributionIntraclass correlation coefficientClumped infectionExtra-binomial variationGini coefficientLorenz curve

0020-7519/$36.00 � 2010 Australian Society for Paradoi:10.1016/j.ijpara.2010.03.009

* Corresponding author. Tel.: +44 (0)20 75943229;E-mail address: m.walker06@imperial.ac.uk (M. W

Studying the distribution of parasitic helminth body size across a population of definitive hosts canadvance our understanding of parasite population biology. Body size is typically correlated with egg pro-duction. Consequently, inequalities in body size have been frequently measured to infer variation inreproductive success (VRS). Body size is also related to parasite age (time since entering the definitivehost) and potentially provides valuable information on the mode of acquisition and establishment ofimmature (larval) parasites within the host: whether parasites tend to establish singly or in aggregates.The mode of acquisition of soil-transmitted helminths has been a theoretical consideration in the para-sitological literature but has eluded data-driven investigation. In this paper, we analyse individual Ascarislumbricoides weight data collected from a cohort of human hosts before and after re-infection followingcurative treatment, and explore its distribution within and among individuals in the population. Lorenzcurves and Gini coefficients indicate that levels of weight inequality (a proxy for VRS) in A. lumbricoidesare lower than other published estimates from animal-helminth systems. We explore levels of intra-hostweight aggregation using statistical models to estimate the intraclass correlation coefficient (ICC) whileadjusting for covariates using a flexible fractional polynomial transformation approach capable of han-dling non-linear functional relationships. The estimated ICCs indicate that weights are aggregated withinhosts both at equilibrium and after re-infection, suggesting that parasites may establish within the hostin clumps. The implications of a clumped infection process are discussed in terms of ascariasis transmis-sion dynamics, control and anthelmintic resistance.

� 2010 Australian Society for Parasitology Inc. Published by Elsevier Ltd. All rights reserved.

1. Introduction

The distribution of parasitic helminth body size within a popu-lation of definitive hosts offers a number of potential insights intothe population biology of worms. Body size is almost universallyrelated to egg production (fecundity) (Poulin, 1998), and sizeinequalities have been used frequently to approximate variabilityin reproduction success (VRS). VRS has important implicationsfor the population genetics of the parasite as high levels can purgegenetic diversity (Criscione and Blouin, 2005; Dobson, 1986). HighVRS has been detected in a number of parasitic helminths such asspecies of pseudophyllidean, tetraphyllidean and cyclophyllid(Hymenolepis) cestodes (Dobson, 1986; Shostak and Dick, 1987),acanthocephalans (Dobson, 1986), and anisakid nematodes (Szalaziand Dick, 1989).

Body size may also be used to estimate the time individual par-asites have been established within the host (parasite age). Fromthis, inference on the mode of acquisition by the host could be

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gained: aggregates of parasites of similar ages presumably havebeen acquired at the same or a similar moment in time. For soil-transmitted helminths (STHs), the mode of acquisition of infectivestages from the environment has long been a theoretical consider-ation in population dynamical models (initially considered byTallis and Leyton (1969)). Such models have demonstrated thatthe mode of acquisition may have a considerable impact on STHpopulation biology, not least because it could explain some ofthe over-dispersion (relative to the Poisson distribution) of thenumber of worms per host (Grenfell et al., 1995; Isham, 1995; Pug-liese et al., 1998; Tallis and Leyton, 1969), a characteristic featureof STH populations (Anderson and May, 1992). Despite this, therehave been very few attempts at analysing data with the purposeof discerning the predominant mode by which hosts acquireinfective stages from the environment (Heinzmann et al., 2009).

The size of a worm within the human gut will depend on its sizeat establishment, the time since establishment (for Ascaris lumbric-oides, we define the worm’s age to be this period plus an approxi-mately 2 week migratory phase (Crompton, 1989)) and on therelationship between size and age (determined by the growthrate). A worm’s rate of growth will depend on both host-parasite

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Table 1The number of female Ascaris lumbricoides collected before and after mass treatmentwith pyrantel pamoate.

Population Hostsstudied

Number of femaleworms collected andweighed

Hosts with atleast one femaleworm

Baseline 1765 16,302 1473First re-infection,

6 months aftertreatment

1257 8657 938

Second re-infection,6 months after re-treatment

1017 6194 753

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(such as immune responses) and parasite-parasite (such as re-source competition) interactions. Innate (genetically determined)factors will also affect growth. Variation in growth rates amongworms can be roughly separated into two types: that which is dri-ven by systematic differences between infrapopulations (the para-site population within a host (Esch et al., 1977)), and that resultingfrom differences between individual worms, independently of theinfrapopulation.

Systematic variation in the body size of worms within differentinfrapopulations may arise from a number of well-characterisedphenomena. For example, if body size is density dependent, thesize of all worms within a host will be affected by the total numberof worms present, although not necessarily uniformly or in a con-stant way through time. This results in variation in the averagebody size of worms among hosts harbouring different intensitiesof infection (worm burden). There are numerous published exam-ples of density dependence in the body size of gastro-intestinal (GI)nematode infections of animals (Michael and Bundy, 1989; Tomp-kins and Hudson, 1999; Dezfuli et al., 2002), and in A. lumbricoidesinfections of humans (Monzon et al., 1990; Walker et al., 2009).

Host-parasite interactions may also induce variability in thesize of worms among infrapopulations. The strength of these inter-actions may also be mediated by worm density (Paterson and Vi-ney, 2002). In sheep, acquired antibody-based responses to theGI nematode Teladorsagia (=Ostertagia) circumcincta reduces thesize of worms (Stear et al., 1997; Stear et al., 1999). Similar associ-ations have been documented in human hookworm infections(Pritchard et al., 1995). Experimental infections of rats with Stron-gyloides ratti have shown that worms are larger in immunosup-pressed hosts and smaller in immunized animals than worms incontrols (Wilkes et al., 2004). Heterogeneity in immunocompe-tence will depend on the nature of the immune response: if it ispredominantly acquired, it will depend on cumulative past expo-sure to parasite antigens and so, where worms are endemic, it islikely to be associated with the age of the host or, if principally in-nate, it will depend on host genetic factors. Most likely it will be acombination of both. Human immune responses to helminth infec-tions remain only partially understood (reviewed by Anthony et al.(2007)).

Variation in the size of worms within an infrapopulation willcrucially depend on the mode of parasite acquisition and establish-ment: whether immature stages are acquired singly and at randompoints in time in a ‘‘trickle” type manner (equivalent to a homoge-neous Poisson process) or in clusters (termed a ‘‘clumped” infec-tion process) (Tallis and Leyton, 1969; Grenfell et al., 1995;Isham, 1995). Assuming that the size of worms is related to theirage (Seo and Chai, 1980) and that innate variability in growth ratesis independent of the infrapopulation (which may be a less robustassumption in areas with highly focal transmission (Criscione et al.,2010)), under trickle infections, the size of worms within a hostwill be mutually independent. Conversely, clumped infections willresult in aggregates of similarly-sized worms.

In this analysis we had two aims: first, to assess the extent ofsize inequality in A. lumbricoides in order to estimate VRS. Toachieve this, we used Lorenz curves (Lorenz, 1905) and Gini coeffi-cients (Gini, 1921) to evaluate levels of weight inequality in wormscollected at endemic equilibrium and after two 6 month periods ofre-infection following chemo-expulsion therapy. These are tech-niques of choice for this type of analysis in the parasitological lit-erature (Dobson, 1986; Shostak and Dick, 1987; Szalazi and Dick1989; Hanelt, 2009; Poulin and Latham, 2002). Second, we aimedto contribute to the elucidation of the modality of the infectionprocess in A. lumbricoides: is it a trickle or clumped?. To achievethis, we used statistical models to evaluate the evidence for wormsof similar weights being aggregated within infrapopulations. Weadjusted for the effects of variables known to affect the size of

A. lumbricoides which, if ignored, would inflate the estimated de-gree of aggregation. Aggregation was measured by the intraclasscorrelation coefficient (ICC), which provides a quantitative mea-sure of similarity between the weights of worms infecting thesame host (Ridout et al., 1999).

2. Materials and methods

2.1. Study area and data collection

Data were collected in a poor urban suburb of Dhaka, Bangla-desh between 1988 and 1989 by Hall et al. (1992). Briefly, house-holds were visited and all of their occupants invited to take part inthe study with the aim of recruiting as many individuals as possi-ble. All participants were asked to provide a faecal sample fromwhich the number of A. lumbricoides eggs were counted using aquantitative ether sedimentation technique (Hall, 1981) and theconcentration of eggs per gram of faeces (EPG) estimated. A doseof pyrantel pamoate was given to each subject and their stoolswere collected for a period of 48 h post-treatment. The wormsrecovered (A. lumbricoides) from the faeces of each individual weresexed, counted and weighed. Egg counts, treatment and wormcounts were repeated on two further occasions at 6 monthly inter-vals. Pyrantel pamoate paralyses A. lumbricoides so that they areexpelled from the gut by peristalsis (Abdi et al., 1995) with a curerate of approximately 88% (Keiser and Utzinger, 2008). Hence,these data provide a reliable and accurate measure of the numberand weight of worms per host. The population of worms recoveredafter the first round of treatment is termed the ‘‘baseline popula-tion” after the second round of treatment, the ‘‘first re-infectionpopulation” and after the third and final round, the ‘‘second re-infection population”. Only female worms were included in theseanalyses for two reasons. First, only females produce eggs and thusonly their weight is a useful measure of VRS and second, malestend to be much smaller than females, excessively complicatinganalyses aimed at assessing the degree to which hosts harbourworms (females) of similar weight. A summary of the data ana-lysed is given in Table 1.

2.2. The weight distribution of female worms across the hostpopulation

Lorenz curves and Gini coefficients were used to estimateinequality in the weight of female worms across the entire popula-tion. Lorenz curves were prepared by ranking worm weights in thesample and plotting the cumulative weights against the cumula-tive number of individuals. The more concave the curve, the great-er the degree of inequality. The Gini coefficient (denoted G)quantifies this inequality by determining the ratio of the area be-tween the line of perfect equality (when all worms are of the sameweight) and the Lorenz curve and the area under the Lorenz curve

M. Walker et al. / International Journal for Parasitology 40 (2010) 1373–1380 1375

(for full details and a history of these measures in the parasitolog-ical literature see Dobson (1986)). G was calculated from the meanof the absolute difference between every pair of weights in thepopulation (Damgaard and Weiner, 2000). A value of 0 indicatesperfect equality with inequality increasing as the coefficient in-creases to 1.

2.3. Modelling approach to analysing the distribution of female wormweights among hosts

The two contrasting infection processes under scrutiny, thehypotheses under investigation, their biological interpretation,and the choice of models to be investigated are summarised inTable 2. Briefly, for a trickle infection process the intraclass corre-lation coefficient, ICC (also referred to here by parameter q) will bezero, whereas for a clumped infection process the ICC will be sta-tistically significantly greater than zero. The reasons for analysingdichotomised female weight and choosing a family of binomialmodels are explained as follows. We assume that worm weightis a proxy for worm age (Seo and Chai, 1980). However, given thatboth the age of individual worms and the relationship between ageand weight is unknown, there was no natural choice of probabilitydensity function with which to model the individual weight data.Individual female worm weights were thus transformed into a bin-ary variable, Yi, (where the subscript i denotes the host from whichthe worm was collected) such that Yi = 1 for females with a weightgreater than a threshold T and Yi = 0 for worms with a weight lessthan or equal to T (we also refer to these females as being either‘‘large” or ‘‘small”, respectively). The total number of females abovethe threshold weight in host i is denoted Zi, where (Zi ¼

Pyi). Def-

initions of the variables and parameters used in this analysis maybe found in Supplementary Table S1. For the parasite populationcollected after each round of treatment, T was taken as the medianweight of all the female worms to ensure equal numbers of largeand small worms. In a simple binomial (BN) model, the probabilitythat a female worm is large is the same among hosts (sharing acovariate combination) and there is no correlation between theweights of worms within a particular host. In a beta-binomial(BB) model, the probability that a female is large is beta distributedamong hosts. This variation allows for the weights of females with-in hosts to be correlated, the magnitude of which is quantified bythe ICC estimated from the model (Crowder, 1978; Williams,1982). A detailed description of the binomial and beta-binomialmodels is given in the Supplementary Data S1, Section 1.

2.4. Treatment of covariates

A number of variables may affect the weight of female A. lumb-ricoides, including the number of co-infecting worms (worm

Table 2Summary of the two contrasting infection processes under scrutiny, the hypothesesunder investigation, their biological interpretation, and the resulting choice ofmodels.

Infectionprocess

Hypothesis being tested andbiological interpretation

Model

Single ortrickleinfection

The weights of worms co-infectinga host are not correlated, as wormsarrive and establish singly andindependently

Binomial (BN).Intraclass correlationcoefficient ðqÞ � 0

Clumpedinfectionprocess

The weights of worms co-infectinga host are positively andsignificantly correlated, as wormsare acquired in clumps andestablish together in their final sitewithin the definitive host

Modified binomialmodel (beta-binomial) to allow forq > 0:

burden) (Monzon et al., 1990; Walker et al., 2009) and the age ofthe host (Elkins and Haswell-Elkins, 1989; Walker et al., 2009).Failure to account for these and other host-specific covariates of fe-male weight would inflate our estimated ICC and so make rejectionof the null hypothesis of trickle infections more likely (Type I er-ror). Therefore, BN and BB models were adjusted for the effectsof the number of co-infecting worms, host age and sex by includingthem as covariates of the probability that a worm is large ðpiÞ in alinear model using a logit ln½pi=ð1� piÞ�ð Þ link function (McCullaghand Nelder, 1989). Fractional polynomial (FP) covariate transfor-mations (Royston and Altman, 1994; Sauerbrei and Royston,1999) were applied to the continuous explanatory variables (hostage and worm burden) in order to account for potential non-linear-ities in the relationship between these variables and the response.FP transformations exploit the array of functional forms that poly-nomial expressions can take in order to approximately linearise therelationship between response and explanatory variables. A de-tailed description of this approach is given in the SupplementaryData S1, Section 2.

2.5. Model fitting and fractional polynomial selection

A selection strategy first proposed by Royston and Altman(1994) and subsequently modified by Sauerbrei and Royston(1999) and Ambler and Royston (2001) was employed to find thebest-fit FP transformations of host age and worm burden (sex isa categorical variable and so was not subjected to FP transforma-tion). In essence, this procedure uses a likelihood ratio test (LRT)-based strategy to select best-fit FP transformations (full detailscan be found in the Supplementary Data S1, Section 3). For eachLRT a nominal p-value of 0.05 (5%) was chosen to determinewhether a particular FP transformation was accepted or rejected(Sauerbrei et al., 2007). The best-fit FP transformations were deter-mined separately for BN and BB models in each of the three popu-lations (baseline, first and second re-infection). BN models werefitted to the data using standard techniques appropriate for gener-alised linear models (GLMs) (McCullagh and Nelder, 1989). BBmodels were fitted by direct maximisation of the beta-binomiallog-likelihood (Crowder, 1978).

2.6. Model comparisons

Akaike’s information criterion (AIC) (Akaike, 1974) was used tocompare the goodness-of-fit of the BN and BB models chosen fromthe fractional polynomial selection procedure in each population.In certain circumstances, BB model assumptions can be restrictiveand lead to inaccurate estimates of the ICC (Collett, 2002). To scru-tinise this, additional and more flexible quasi-likelihood based ICCestimates were made (Williams, 1982). These were found to be al-most identical to the BB model estimates (unpublished observa-tions) and are not reported further. Confidence intervals for theBB ICC estimators were calculated from standard errors (SEs) esti-mated from the information matrix resulting from the model fit-ting procedure, and by making the assumption that the ICCswere asymptotically normally distributed.

3. Results

3.1. Average weight, variability and inequality

The Lorenz curves constructed using the data from the threepopulations show a modest degree of concavity (Fig. 1). The esti-mated Gini coefficients indicate that the levels of weight inequalityare fairly low compared with similar estimates in other parasitichelminths of vertebrates (Hanelt, 2009), suggesting that the female

Table 3Summary statistics for the distributions of female Ascaris lumbricoides (in grams) inthe baseline, first and second re-infection populations before and after pyrantelpamoate mass treatment in Dhaka, Bangladesh.

Mean (g) T = median (g) Variance (g2) Skewnessa

Baseline population3.26 3.3 2.77 0.15

First re-infection population2.90 3.0 1.59 �0.039

Second re-infection population2.84 2.9 1.93 0.064

a Skewness is defined asP½weight-meanðweightÞ�3

sample size�½SDðweightÞ�3; SD denoting the standard deviation.

1376 M. Walker et al. / International Journal for Parasitology 40 (2010) 1373–1380

worms in this population have a low variability in reproductivesuccess. Modest levels of inequality are also indicated by the rela-tively small skew of the three weight distributions (Table 3 andFig. 2.). The average weight (and variance) of females at baselineis slightly higher than after 6 months re-infection. The skew ofthe distribution at baseline is also greater than after re-infection,reflecting a relatively higher proportion of small worms beforetreatment (Table 3).

3.2. Aggregation of worm weights within hosts

In each population the BB model was a better fit to the data thanthe BN model indicating the presence of statistically significantclustering of (dichotomised) worm weights (Table 4). The esti-mated ICCs were statistically significantly greater than 0 for eachof the parasite populations, and a downward trend in the valueswas observed, from the baseline population (highest ICC) to thesecond re-infection population (lowest ICC).

3.3. Covariate functional forms

For the BB models, FP transformations of both host age ðxi1Þ andworm burden ðxi2Þ were preferred over the models with untrans-formed covariates in the baseline population. In the first and sec-ond re-infection populations, transformations of worm burdenwere selected and host age was left untransformed (Supplemen-tary Table S2). The functional forms generated by these transfor-mations in the baseline and first re-infection populations aredepicted in Figs. 3 and 4, respectively. From these figures, a satu-rating relationship between the fitted probability that a worm islarge ðp̂iÞ and host age ðxi1Þ is evident at baseline (Fig. 3A). An in-verse transformation of xi1 captured this relationship (Supplemen-tary Table S2). In the first and second re-infection populations hostage was included as an untransformed covariate (SupplementaryTable S2) and did not have a marked impact on p̂i (Fig. 4A, resultsnot shown for second re-infection population).

The relationship between p̂i and worm burden ðxi2Þ initially in-creased with increasing ðxi2Þ before decreasing. This pattern wasconsistent across the three populations (Figs. 3B and 4B; resultsnot shown for the second re-infection population) and in accor-dance with initial facilitation followed by limitation in wormweight (Walker et al., 2009). Identical transformations were iden-tified as best capturing this relationship in the baseline and first re-infection populations (a linear combination of a square root and

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

A

G = 0.29

Cum

ulat

ive

prop

ortio

n of

fem

ale

wei

ghts

0.0 0.2 0.4

0.0

0.2

0.4

0.6

0.8

1.0

G = 0.25

B

Cumulative propor

Fig. 1. Lorenz curves of the weights of all females collected from the baseline (A), fiappropriate plots and represent the ratio of the shaded areas between the Lorenz curve ax-axis.

logarithmic transformation, see Supplementary Table S2) whilein the second re-infection population a square transformationwas selected (Supplementary Table S2).

4. Discussion

There are two main findings of this study. First, we showed thatthe degree of inequality in the weight of female A. lumbricoides isfairly low and is similar at endemic equilibrium and after two6 month periods of re-infection. This suggests that levels of VRSare not particularly high in A. lumbricoides. Second, we demon-strated that female worms of similar weights are statistically sig-nificantly aggregated within hosts suggesting that larval wormsestablish in clumps rather than singly.

The higher average weight and variability in weight of femalesat baseline compared with after 6 months re-infection is expectedsince, on average, worms are younger in the re-infection popula-tions and, excluding worms which may have survived the drugtreatment, maximum age is restricted by the 6 month re-infectionperiod. The higher (and positive) estimated skewness at baselinereflects a relatively higher proportion of smaller worms than inthe re-infection populations. A previous study which also lookedat the weight distributions of A. lumbricoides collected at endemicequilibrium and after an 11-month period of re-infection reportedsimilar results, but found a more homogenous distribution withfewer small worms after re-infection (Elkins and Haswell-Elkins,1989). These findings indicate that the regulatory mechanisms act-ing on the size of females are more pronounced at endemic equilib-rium and, as discussed by Elkins and Haswell-Elkins (1989) andoriginally proposed by Jung (1954), this may result from an

0.6 0.8 1.0

tion of female worms

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

C

G = 0.28

rst (B) and second (C) re-infection populations. Gini coefficients, G, are given onnd the line of complete equality (dashed line) and between the Lorenz curve and the

0 2 4 6 8

050

010

0015

00

Freq

uenc

yA

0 2 4 6 8

050

010

0015

00

B

Weight (g)0 2 4 6 8

050

010

0015

00

C

Fig. 2. Histograms of the weights of female Ascaris lumbricoides collected from the baseline (A), first (B) and second re-infection (C) populations.

Table 4Comparison of the goodness-of-fit of the binomial (BN) and beta-binomial (BB)models for the number of ‘‘large” Ascaris lumbricoides per host.

Modela ICCb, q (95% CIc) Log-likelihood DFd AICe

Baseline populationBN q � 0 �3881 6 7774BB 0.21 (0.19,0.23) �2997 6 6006

First re-infection populationBN q � 0 �2103 6 4219BB 0.17 (0.14,0.19) �1782 6 3569

Second re-infection populationBN q � 0 �1520 6 3052BB 0.13 (0.11,0.16) �1352 5 2713

a Model equations including fractional polynomial covariate transformations canbe found in Supplementary Table S2 (BB models) and Supplementary Table S3 (BNmodels).

b Intraclass correlation coefficient.c 95% confidence interval, only applicable for BB model (see text).d Degrees of freedom.e Akaike’s information criterion; a lower AIC value indicates a better fit.

M. Walker et al. / International Journal for Parasitology 40 (2010) 1373–1380 1377

increasing ability of larger and more established worms to inhibitthe development of those more recently acquired.

The estimated levels of weight inequality (measured by the Gcoefficient) are similar in the three populations (G � 0.3) but arelower than estimates obtained for other parasitic helminths ofvertebrates. For example, Shostak and Dick (1987) calculated G

Fig. 3. Observed (open squares) and fitted probabilities that a female Ascaris lumbricoipopulation. In (A), worm burden is adjusted to its median value. Host age is similarly adcalculated from estimated standard errors assuming a normal sampling distribution.

to be 0.9 in the pike cestode, Triaenophorus crassus and Dobson(1986) published a value of 0.67 for the cestode, Chimaerocestosprudhoei parasitising chimaeroid fish. There have been no previ-ously published analyses of inequalities in the size of helminthinfections of humans (a list of all previously published estimatescan be found in Table 1 of Hanelt (2009)).

Inequalities in the weights of female worms may arise from var-iation in the age of worms, heterogeneities in host environmentsand individual genetic variability (Shostak and Dick, 1987). Dobson(1986) described two distinct (but not exclusive) patterns ofgrowth in parasitic helminths: determinant growth, whereby indi-viduals tend to grow at a decreasing rate with age and indetermi-nant growth, where the rate of growth is less disposed to declineand is more dependent on the host environment and levels of com-petition such as the number of co-infectors or the availability of re-sources. The populations of hosts studied here harbour highlyheterogeneous worm burdens (from 1 to 187), are of a wide varietyof ages (from 1 to 98 years) and are likely to be heterogeneous inmany other less immediately quantifiable ways. Furthermore, inthese analyses, and in those presented in Walker et al. (2009), itis shown that the weight of a female A. lumbricoides is dependenton the total worm burden and host age. Despite this, inequalityin female weight is low, suggesting that either their growth is rel-atively determinant and resilient to heterogeneous environments,or levels of competition are lower than compared with, for example,the cestode species studied by Dobson (1986) and Shostak andDick (1987). Interestingly, Dobson also showed that size inequali-

des is ‘‘large” (lines) estimated from the beta-binomial (BB) model in the baselinejusted in (B). The grey shaded areas represent 95% confidence intervals which were

Fig. 4. Observed (open squares) and fitted probabilities that a female Ascaris lumbricoides is ‘‘large” (lines) estimated from the beta-binomial (BB) model in the first re-infection population. In (A), worm burden is adjusted to its median value. Host age is similarly adjusted in (B). The grey shaded areas represent 95% confidence intervalswhich were calculated from estimated standard errors assuming a normal sampling distribution.

1378 M. Walker et al. / International Journal for Parasitology 40 (2010) 1373–1380

ties in the cestode C. prudhoei could be reduced by experimentallydecreasing levels of intra-specific competition.

The effect of size inequalities on VRS will depend strongly onthe relationship between the measure of size and lifetime repro-ductive output. Non-linearities in this relationship will renderinequality estimates less intuitive and their relevance to heteroge-neities in the contribution of transmission stages harder to inter-pret. Sinniah and Subramaniam (1991) recorded egg counts fromfemale A. lumbricoides by dissection of their uteruses and describeda linear relationship between these data and the weights of theworms. It is not clear however, whether they compared this withthe fit of any non-linear functions.

The approach of dichotomising worm weight into ‘‘small” or‘‘large” provided a useful simplification to the analysis albeit witha loss of statistical power compared to modelling weight as a con-tinuous variable. We accepted this loss of power because the alter-native would have added considerable methodologicalcomplexities to the detriment of clarity. For example, the familiarBox–Cox techniques (Box and Cox, 1964) for determining a suit-able transformation would have been inappropriate because ofthe necessary random effects structure of the model required forestimating the degree of weight clustering (Gurka, 2006; Gurkaet al., 2007). Moreover, simultaneous assessment of the adequacyof both response and (FP) covariate transformations would requirenovel selection procedures far beyond the scope and aims of thisanalysis.

The FP transformation of covariates provided a robust and flex-ible approach to the statistical modelling of worm weights. Thebest-fitting models in each population provided an excellentdescription of the functional dependence of the probability that afemale A. lumbricoides is large with host age and worm burden.The latter relationship is exactly in accordance with previous anal-yses of these data exploring density dependence of worm weightand its impact on patterns of egg production (Walker et al.,2009) and is not be discussed here further. In a broader context,our results exemplify the usefulness of FP transformations in pro-viding a convenient and easily implementable technique to ac-count for non-linear functional relationships which are commonin epidemiological and parasitological data (Royston et al., 1999).

The functional form of the relationship between the probabilitythat a female worm is large and host age (as a continuous variable)extends the analyses of Walker et al. (2009) in which host age wastreated as a two-level categorical variable. The saturating relation-ship found at baseline may represent a number of possible phe-nomena. Since the age of a worm within a host cannot exceed

the time since the host’s initial exposure to infectious stages, themaximum age of worms in an ageing cohort of hosts will be trun-cated by the time since the cohort was initially exposed. Thus,assuming hosts are continually exposed from birth (which is likelyin this setting), the average age of worms will increase withincreasing host age. It is unlikely, however, that this accounts en-tirely for this effect. The probability that a female is large increasesmarkedly over approximately the first 20 years of host age but gi-ven that the life expectancy of A. lumbricoides is thought to be1–2 years (Anderson and May, 1992), one would expect saturationafter a much shorter duration. An alternative but not exclusiveexplanation is that that the size of worms is restricted in youngerhosts who have a smaller gut volume than adults. To reconcile thiswith the lack of obvious age dependence in the re-infection popu-lations we presume that worms are able to grow unrestricted foran initial period after establishment before gut volume becomesa limiting factor.

Elkins and Haswell-Elkins (1989), working in southern India, re-ported at endemic equilibrium a decreased proportion of small fe-male A. lumbricoides in older hosts while after an 11 month periodof re-infection, children (aged 1–9 years) harboured consistentlyheavier females than at baseline. These results are in accordancewith those presented here for the baseline population but are atodds with the results after re-infection. Despite the difference inre-infection periods between the current study and that ofHaswell-Elkins et al. (6 and 11 months, respectively), it is notewor-thy that the best-fit FP transformation of host age ðxi1Þ for the BNmodel fitted to the first re-infection data produced a functionalrelationship describing an increased probability that a worm isheavy in young children relative to adults (SupplementaryTable S3). This highlights the added uncertainty introduced by ex-tra-binomial variation and the importance of distributionalassumptions in drawing robust conclusions.

The beta-binomial model provided a better description of thedata than the binomial model, reflecting the presence of consider-able extra-binomial variation and in particular, of intraclass (intra-individual host) correlation of (dichotomised) worm weight. Extra-binomial variation can result from a clumped infection process inwhich worms’ ages are not mutually independent within a hostand from which aggregated weight distributions arise. Factors thatcan be assumed to affect all hosts approximately equally, such asseasonal changes in the force of infection, will not generate aggre-gation. However, it is a general limitation of our approach thatextra-binomial variation can also stem from unmeasured host het-erogeneities. We have minimised the influence of intensity-, host

M. Walker et al. / International Journal for Parasitology 40 (2010) 1373–1380 1379

age- and sex-dependent heterogeneities as best as possible by con-trolling for these variables. Other potentially influential variablescould not be controlled for. For example, there was no way to ac-count for genetic variation among hosts which could potentiallydrive heterogeneities in their ability to limit the growth of adultworms. Regardless of the number of measured covariates, struc-tured data (worms within hosts) of the sort analysed here are al-ways liable to over-dispersion; indeed McCullagh and Nelder(1989) recommend assuming extra-binomial variation unless it isotherwise proved to be absent. The estimated ICC for models notincluding covariates is approximately 20% higher at baseline com-pared with the ICC estimated from the model presented in the Re-sults (Supplementary Table S4), roughly indicating that around80% of the extra-binomial variation is not due to host age-, sexor worm burden-associated effects in this population.

A stochastic implementation of an immigration-death model,extended to include multiple worm life-stages to simulate num-bers of ‘‘small” and ‘‘large” adult females, has been used to esti-mate model-derived ICCs and explore their behaviour in apopulation of hosts at equilibrium (baseline) and after two simu-lated 6 month periods of re-infection post- (pyrantel pamoate)treatment (Walker et al., unpublished data). This modelling ap-proach confirmed the capacity for both a clumped acquisition pro-cess and host-level random effects (represented in the model bybetween-host variability in adult female worm mortality andgrowth rates) to generate aggregations of similarly-sized worms.We also demonstrated that the dynamics of aggregation generatedby the two processes are distinct. In particular, clumped acquisi-tions tend to cause the ICC to sequentially decrease over the threepopulations, a trend suggested from the results of the analyses pre-sented here (Table 4).

The distributional structure of parasite weight investigated inthese analyses may help explain some of the characteristic featuresof A. lumbricoides population biology. Since female egg productionis associated with body size (Sinniah and Subramaniam, 1991), andthe total egg output from a host is the sum of individual worms’egg outputs, the aggregation of female weights may contribute tothe high variability in the per host net egg output which is a char-acteristic feature of A. lumbricoides and many other STHs (Andersonand May, 1992). Walker et al. (2009) demonstrated, from the dataanalysed in this study, that the total per host net egg output isassociated with the per host mean weight of females but that thisexplained only a small fraction of the overall variability. Some ofthis unexplained variation may be due to the aggregation of weightwhich would greatly inflate the variance compared to the case ofweight being independently and identically distributed acrossthe host population.

The mode of acquisition and establishment has been shown tohave important implications for the population biology and trans-mission dynamics of STHs. Clumping generates over-dispersion inthe distribution of the number of worms among hosts (Tallis andLeyton, 1969; Grenfell et al., 1995; Isham, 1995; Pugliese et al.,1998) a characteristic feature of STH infections (Anderson andMay, 1992). It can also affect the dynamics of this distribution(Quinnell et al., 1995) and the stability of fluctuating host-parasitesystems (Rosà and Pugliese, 2002). Furthermore, clumping may aidthe rate of emergence of anthelmintic drug resistance by promot-ing the spread of rare alleles by non-random mating and increasedrates of inbreeding (Smith et al., 1999; Cornell et al., 2000, 2003;Churcher et al., 2008). The possibility of anthelmintic drug resis-tance emerging in helminth infections of humans is receivingincreasing attention (Churcher and Basáñez, 2008, 2009; Churcheret al., 2009) due to the widespread implementation of chemother-apy-based helminth control programmes which require prolongedduration and extensive coverage (Basáñez et al., 2006; Hotez et al.,2007).

The dataset analysed in this study provided a unique opportu-nity to answer a question which, until now, has proved elusive:by what mode are STH infective stages acquired from the environ-ment? From the results of our analyses, we have argued thatA. lumbricoides tend to establish in clumps rather than singly. Wehave also discussed the advantages and limitations of our statisti-cal modelling approach and the inherent difficulties in addressingthe problem. In addition, we have measured levels of inequality inthe weights of female A. lumbricoides which suggest that this spe-cies has a far lower variability in reproductive success than otherhelminth parasites of non-human vertebrates studied to date.

Acknowledgements

We are extremely grateful to Dr. Melissa Haswell-Elkins forsharing her data with us and providing an invaluable means ofcomparison between populations of Ascaris in India and Bangla-desh. We thank the Medical Research Council (MRC, UK) for fund-ing this work through a Doctoral Training Account (MW) and aCareer Establishment Grant (M-GB).

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.ijpara.2010.03.009.

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