International Journal for Parasitology 40 (2010) 1381–1388

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International Journal for Parasitology

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

Trickle or clumped infection process? A stochastic model for the infection processof 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 24 March 2010Received in revised form 30 June 2010Accepted 2 July 2010

Keywords:Ascaris lumbricoidesImmigration-death modelStochastic simulationClumped infectionIntraclass correlation coefficientSize distribution

0020-7519/$36.00 � 2010 Australian Society for Paradoi:10.1016/j.ijpara.2010.07.001

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

The importance of the mode of acquisition of infectious stages of directly-transmitted parasitic helminthshas been acknowledged in population dynamics models; hosts may acquire eggs/larvae singly in a‘‘trickle” type manner or in ‘‘clumps”. Such models have shown that the mode of acquisition influencesthe distribution and dynamics of parasite loads, the stability of host-parasite systems and the rate ofemergence of anthelmintic resistance, yet very few field studies have allowed these questions to beexplored with empirical data. We have analysed individual worm weight data for the parasitic round-worm of humans, Ascaris lumbricoides, collected from a three-round chemo-expulsion study in Dhaka,Bangladesh, with the aim of discerning whether a trickle or a clumped infection process predominates.We found that hosts tend to harbour female worms of a similar weight, indicative of a clumped infectionprocess, but acknowledged that unmeasured host heterogeneities (random effects) could not be com-pletely excluded as a cause. Here, we complement our previous statistical analyses using a stochasticinfection model to simulate sizes of individual A. lumbricoides infecting a population of humans. Weuse the intraclass correlation coefficient (ICC) as a quantitative measure of similarity among simulatedworm sizes and explore the behaviour of this statistic under assumptions corresponding to trickle orclumped infections and unmeasured host heterogeneities. We confirm that both mechanisms are capableof generating aggregates of similar-sized worms, but that the particular pattern of ICCs described pre-and post-anthelmintic treatment in the data is more consistent with aggregation generated by clumpedinfections than by host heterogeneities alone. This provides support to the notion that worms may beacquired in clumps. We discuss our results in terms of the population biology of A. lumbricoides and high-light the significance of our modelling approach for the study of the population dynamics of helminthparasites.

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

1. Introduction

The weights of female Ascaris lumbricoides infecting a popula-tion of human hosts have recently been shown to be statisticallysignificantly aggregated; hosts tend to harbour females of similarweight (Walker et al., 2010). A plausible mechanism for this aggre-gation (clustering) is the acquisition of infectious eggs from theenvironment in aggregates (rather than singly): the sizes of adultsacquired at the same time will be correlated. This is termed a‘‘clumped” infection or acquisition process (Tallis and Leyton,1969). The effect of clumped infections on the population biologyand transmission dynamics of directly-transmitted gastrointestinal(GI) nematodes of human and non-human hosts has been recogni-sed through inference from theoretical models. In particular, aclumped infection process is hypothesised to increase over-disper-sion (relative to the Poisson or random distribution) in per host

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worm loads (Tallis and Leyton, 1969), and enhance the spread ofanthelmintic resistance (Cornell et al., 2000; Cornell et al., 2003;Smith et al., 1999).

Aggregation among worm sizes is not exclusively the result of aclumped infection process. Heterogeneities among hosts can alsogenerate aggregates of similar-sized worms. For example, hostswho mount an effective immune response against infecting wormsmay increase the worms’ mortality rate and/or cause them to growless rapidly than worms within hosts whose response is weak orineffective. Host immune responses have been shown to limit thesize of Teladorsagia (=Ostertagia) circumcincta infecting sheep (Stearet al., 1997, 1999), Strongyloides ratti infecting rats (Viney et al.,2006; Wilkes et al., 2004), and hookworms infecting humans (Prit-chard et al., 1995). Similarly, intra-specific competition may causeworms to be smaller within hosts with large worm burdens com-pared with worms within hosts with small worm burdens. This ef-fect has been implicated in A. lumbricoides infections of humans(Monzon et al., 1990; Walker et al., 2009) and in other (non-hu-man) animal-GI nematode systems (Dezfuli et al., 2002; Michael

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1382 M. Walker et al. / International Journal for Parasitology 40 (2010) 1381–1388

and Bundy, 1989; Tompkins and Hudson, 1999). Irrespective of theparticular mechanism, heterogeneities among hosts may produceaggregates of similar-sized worms.

In a statistical analysis of worm weights (a proxy for worm size)of three A. lumbricoides population samples collected from humanparticipants in Dhaka, Bangladesh (one collected at endemic equi-librium and two after consecutive 6-month periods of re-infection(Hall et al., 1992)), we used the intraclass correlation coefficient(ICC) as a quantitative measure of the similarity among the sizesof worms within the same host (Walker et al., 2010). For each ofthe three population samples, the ICC was estimated adjustingfor host age, sex and total (male plus female) worm burden. Con-trolling for these variables reduced the influence of host heteroge-neities, permitting a more accurate estimation of the degree ofaggregation resulting from potentially clumped infections. Theanalysis resulted in significantly greater than zero ICC values equalto 0.21 or 0.22, 0.17 or 0.18, and 0.13 or 0.14 (depending on the ICCestimator used) for, respectively, the baseline, first re-infection andsecond re-infection populations. This indicates statistically signifi-cant aggregation among worm weights even after adjusting forknown host covariates, and a downward trend over the three pop-ulations (Walker et al., 2010). Clearly, controlling completely forheterogeneities among hosts is impossible and some variation,unaccounted for by the measured covariates (random effects), willinevitably contribute to the estimated ICCs. Disentangling aggrega-tion derived from heterogeneities among hosts from that arisingfrom clumped infections is not feasible using the available data.However, a theoretical modelling approach permits predictionsof the expected pattern of size aggregation over the three popula-tion datasets to be made under the two alternative aggregation-generating mechanisms. These predictions can be used to assesswhether the conclusions drawn from the analysis of the data pre-sented in Walker et al. (2010) are robust to alternativeexplanations.

In this paper, we present an extended stochastic immigration-death model (Tallis and Leyton, 1969) which permits simulationof the numbers of migratory larvae, and nominally ‘‘small” and‘‘large” adult A. lumbricoides infecting a population of human hosts.In order to mimic the situation from which the data were collected(Hall et al., 1992), we modelled three rounds of anthelmintic treat-ment with pyrantel pamoate, a drug capable of paralysing theworms, leading to their expulsion from the gut with a greater than80% ‘‘curative” efficacy (Keiser and Utzinger, 2008). From thesesimulated data we estimated the ICCs and explored their behaviourunder various assumptions regarding the mode of acquisition ofinfectious eggs from the environment and heterogeneities amonghosts. For the latter, we took a ‘‘black-box” approach, assumingthat there exists between-host variation but disregarding the spe-cific mechanism from which it arises. The aims of this approachwere: (i) to improve our understanding of how specific populationand infection processes affect aggregation among parasite sizesand the estimated ICC values, and (ii) to assess how likely it is thatthe pattern of ICCs reported in Walker et al. (2010), i.e., significantaggregation among worm weights and a downward trend from thebaseline to the second re-infection population, may be the out-come of a clumped infection process.

2. Materials and methods

2.1. Overview of modelling approach

The model is based on a stochastic immigration-death processdescribed by Tallis and Leyton (1969) which incorporates clumpedacquisitions of infectious stages from the environment (clumpedinfections). We modified this model to incorporate three worm

life-stages: pre-intestinal within-tissue migrating larval worms,‘‘small” adult worms and ‘‘large” adults who develop from thesmall ones. This immigration-death model considers only the partof the life-cycle of A. lumbricoides which occurs within the defini-tive host, assuming that the population of eggs within the environ-ment is constant and at temporal equilibrium. Consequently, thesexually reproductive aspects of A. lumbricoides biology and theco-infecting adult male population were ignored.

Heterogeneity among hosts was incorporated by defining asrandom variables the parameters determining the rate of growthfrom small to large worms and the mortality rate of worms. A sche-matic illustration of the deterministic representation of the modeldescribing the per host mean number of each A. lumbricoides life-stage is given in Fig. 1. Definitions of the parameters referred toin the subsequent description can be found in Table 1.

To mimic the setting from which the data were originally gath-ered (see Section 2.2), we modelled three rounds of drug treatment(6 months apart) with pyrantel pamoate and explored levels of sizeaggregation in simulated populations of worms at endemic equi-librium and in two re-infection populations. As done previously(Walker et al., 2009, 2010) we refer to these populations as the(simulated) baseline, first and second re-infection populations. Ineach population, the similarity among the sizes of worms infectingthe same host was quantified using the ICC. The ICC is a descriptivestatistic that measures how strongly units within the same groupresemble each other. In our context, it provides a suitable estimateof the correlation among the weights of worms within a host(Crowder, 1978; Ridout et al., 1999). An ICC that is statistically sig-nificantly greater than zero indicates a significant correlation (inour case an indication that hosts harbour similar-sized worms).

We modelled two distinct mechanisms by which aggregationamong worm sizes may be generated. In the first, hosts providedhomogenous environments for incoming worms and clusters ofsimilar-sized worms were driven solely by the mode of acquisitionof eggs from the environment. This scenario would correspond to ahypothetical situation in which heterogeneities among hosts,which may otherwise affect the estimated ICCs, are completely ad-justed for by the measured covariates, namely host age, sex and to-tal worm burden (Hall et al., 1992; Walker et al., 2010). In thesecond, worms were assumed to establish in a trickle mannerand variation among worm life-histories between hosts (mimick-ing unmeasured random effects) was incorporated to drive aggre-gation among worm sizes.

2.2. Source of data for comparison with model predictions

The ICCs calculated from simulated data were compared withthose calculated from the ‘‘observed” data presented in Walkeret al. (2010). The parasitological data from which the ‘‘observed”ICCs were calculated were collected from a poor urban suburb ofDhaka, Bangladesh between 1988 and 1989 by Hall et al. (1992).Briefly, households were visited and all their occupants invited totake part in the study. Participants were given a dose of pyrantelpamoate and their stools were collected for a period of 48 hpost-treatment. The A. lumbricoides worms recovered from the fae-ces of each individual host were sexed, counted and weighed.Treatments and worm counts were repeated on two further occa-sions at 6-month intervals (+14 days maximum). These data thusformed the three observed population datasets (baseline, firstand second re-infection populations) analysed by Walker et al.(2010). In that analysis, the degree of similarity among dichoto-mised weights (‘‘small” and ‘‘large”) of female worms within thesame host was quantified by estimating the ICC, while controllingfor worm burden, host age and sex. Weight was dichotomised be-cause there was not a suitable choice of continuous probability dis-tribution with which to model the raw data on worm weights. (For

Fig. 1. A schematic illustration of the deterministic version of the stochastic immigration-death process used to model the number of migrating larvae, ‘‘small” and ‘‘large”adult female Ascaris lumbricoides. LiðtÞ, NS

i ðtÞ and NLi ðtÞ denote, respectively, the mean number of migrating larvae, small and large adult worms in host i at time t. The adult

worm mortality rate, li , and the growth rate, ci , vary among hosts.

Table 1Summary of the parameters and random variables referred to in the main text.

Parameter Description Value andunits

Random variablesC; c Number of eggs containing female L2 larvae acquired per infectious contact. C follows a zero-truncated negative binomial distribution. –1=li Life-expectancy of adult worms in host i. 1=li gamma-distributed among hosts (and so li is inversely gamma distributed). years1=ci Expected time spent as a ‘‘small” worm in host i. 1=ci is independently and identically distributed to 1=li . years

Varied constantsmC The mean of C. As mC ! 1, ðPðC ¼ 1Þ ! 1Þ, the infection process is trickle. –k The over-dispersion parameter for the zero-truncated negative binomial distribution of C. –u The rate of infectious contacts with clumps of eggs (larvae) in the environment. year�1

m Coefficient of variation of the gamma distribution used to model 1=li and 1=ci . m ¼ 1=ffiffiffiap

where a is the shape parameter of the gammaprobability density function.

–

Fixed constantscL=ðcL þ lLÞ The expected proportion of migrating larvae surviving migration. 1a

cL The Per capita rate at which larvae complete their pre-gut migratory phase. b2/52 year�1

a=b Mean life-expectancy of worms in the host population as a whole (b is the rate parameter of the gamma distribution describing 1=li and1=ci and since a=b ¼ 1, a ¼ b).

1 yearc

a Arbitrary value.b Crompton (1989).c Anderson and May (1992).

M. Walker et al. / International Journal for Parasitology 40 (2010) 1381–1388 1383

a description of how weight was dichotomised in each populationsee Section 2.3.3 below.) Male worms were excluded from theseanalyses because they are much smaller than females (Crompton,1989), which would have contravened one of the assumptions nec-essary for calculation of the ICC. That is, the probability that aworm is large is constant among the worms within a host (Ridoutet al., 1999; Walker et al., 2010). Also, we do not expect the infec-tion process to be different between male and female parasites.

2.3. Definition of the model

Throughout the following description of the model used to sim-ulate the weights of female worms infecting a population of hostsit is important to note that: (i) all rates are per capita rates, and (ii)because the model is completely stochastic, when reference ismade to a (per capita) rate it is implied that the probability of anevent occurring is given by the (per capita) rate multiplied by ashort time interval Dt.

2.3.1. Clumped infectionsHosts were assumed to make infectious contacts with eggs in

the environment at a rate u such that exposures occurred at timesof a homogeneous Poisson process. Upon each infectious contact, arandom number of eggs containing L2s of the female sex, C, wereingested. Following Tallis and Leyton (1969) and subsequent stud-

ies (Herbert and Isham, 2000; Isham, 1995), C was assumed to fol-low a zero-truncated negative binomial distribution withprobability mass,

PðC ¼ cÞ ¼ Cðc þ kÞCðkÞc!

ackk

ðaþ kÞcþk

11� 1k

: ð1Þ

Here 1 ¼ k=ðaþ kÞ, a is the mean of the non-truncated distribu-tion, and k is the over-dispersion parameter (as k!1 the distri-bution becomes a zero-truncated Poisson). The mean number ofeggs (containing female larvae) ingested per infectious contact,mC , is given by mC ¼ a=ð1� 1kÞ. For the limit mC ! 1, a singleegg containing a single female larva is acquired per infectious con-tact ðPðC ¼ 1Þ ! 1Þ and the acquisition process is ‘‘trickle”.

2.3.2. Homogenous hostsEach acquired egg hatches within the host into a female larva

and was assumed to develop into an adult worm at a rate cL suchthat the distribution of development times was exponential withmean 1=cL. The time taken before becoming a small adult femalerepresents the pre-intestinal migratory phase undergone by A.lumbricoides larvae before reaching their final site of establishmentwithin the host’s gut (Crompton, 1989). During their migratoryphase larvae died at a rate lL. The mean proportion of larvae sur-viving the migratory phase is given by cL=ðcL þ lLÞ (Supplementary

0 1 2 3 4 5

0.0

0.5

1.0

1.5

Prob

abili

ty d

ensi

ty

Life−expectancy of adult female worms (years)

Fig. 2. Probability density functions of gamma distributions used to modelheterogeneity among the life-expectancies of female Ascaris lumbricoides betweenhosts (and the expected time spent as a small worm, see text Section 2.3.3). Themeans of the distributions are set to 1 year. The solid line corresponds to acoefficient of variation (CV) of 0.25, the dashed line a CV of 0.5 and the dotted line aCV of 1.

1384 M. Walker et al. / International Journal for Parasitology 40 (2010) 1381–1388

material S1, Section 1). Parameter lL was arbitrarily assigned a va-lue of 0 so that 100% of larvae were expected to survive migrationðcL=ðcL þ lLÞ ¼ 1Þ. The mean duration of migration ð1=cLÞ was setto 2 weeks (Crompton, 1989). The rate of growth from a small toa large worm is denoted by c. Both small and large adults were as-sumed to have a mortality rate l with mean life-expectancy ð1=lÞset to 1 year (Anderson and May, 1992). Collectively, we refer to land c as life-history parameters.

2.3.3. Heterogeneous hostsTo incorporate heterogeneities among hosts, the life-history

parameters (l and c) were defined as random variables suchthat li and ci are, respectively, realisations of the adult wormmortality and growth rates in host i. We assumed that 1=ci

and 1=li were independently and identically gamma-distributedamong hosts with shape and rate parameters a and b, respec-tively. This assumption was made for simplicity in the absenceof any prior knowledge of a correlation between the life-historyparameters. The assumption also ensured that the expectednumbers of small and large adults were equal at equilibriumin accordance with Walker et al. (2010), who used the medianweight at equilibrium as a threshold for dichotomisation. Themean of the gamma distribution is a=b and the coefficient of var-iation (CV) is 1=

ffiffiffiap

. The CV is denoted m and provides a norma-lised measure of variability (m ¼ standard deviation=mean(Kirkwood and Sterne, 2003)) among worm life-history parame-ters between hosts. The average life-expectancy of femalesacross the host population as a whole was assumed to be 1 year(Anderson and May, 1992). Consequently, the gamma distribu-tion’s shape parameter a equals its rate parameter b. The distri-bution of worm life-expectancies, 1=li, among hosts (which isidentical to the distribution of expected time spent as smallworm, 1=ci) for various values of the CV is depicted in Fig. 2.The corresponding distribution for the rate parameters ci andli is inverse-gamma.

2.4. Force of infection

The force of infection is the average rate at which hosts acquireadult worms (Anderson and May, 1992). This is given by mCu (alllarvae establish). The degree of clustering of similar-sized wormsdepends on both components of this parameter. Naturally onewould expect such dependence on the mean number of larvae ac-quired per infectious contact (mC); worms acquired simulta-neously are more likely to be a similar size. Intuition also leadsto a similar conclusion regarding the rate of infectious contacts(u). Consider that for a host making contacts with clumps of infec-tious eggs (larvae) at rate u, the times between contiguous (con-secutive) acquisitions will be exponentially distributed withmean 1=u. Thus, as the rate of contacts increases and the time be-tween them decreases, it becomes increasingly difficult to discernbetween clumps of worms acquired at different times.

In the model, the force of infection was restricted to ensure amean of 10 female worms (small plus large) per host at equilib-rium in order to approximately reflect the corresponding value of9.23 per host in the observed baseline data. Values of u were as-signed to conform to this restriction for a chosen mC . In this way,the infection process was characterised either by the frequentacquisitions of small clumps of larvae or infrequent acquisitionsof large clumps.

To set values of u given mC , it was necessary to derive anexpression for the former in terms of the other parameters in themodel. This was achieved by first considering the following set ofdifferential equations which describe the mean numbers of larvaeðLiðtÞÞ, small ðNS

i ðtÞÞ and large ðNLi ðtÞÞ adult females in host i at time

t,

dLiðtÞdt¼ mCu� ðcL þ lLÞLiðtÞ; ð2Þ

dNSi ðtÞ

dt¼ cLLiðtÞ � ðci þ liÞN

Si ðtÞ; ð3Þ

dNLi ðtÞ

dt¼ ciN

Si ðtÞ � liN

Li ðtÞ: ð4Þ

By setting the derivatives of Eqs. (2)–(4) to 0 the expected num-bers of each life-stage at equilibrium (denoted by *) can be found.Writing N�i ¼ NS�

i þ NL�i it follows that,

N�i ¼umCcL

ðcL þ lLÞ1li

� �: ð5Þ

The mean of N�i across the entire host population, denoted N�

(note the dropping of subscript i), is given by,

N� ¼ umCcL

ðcL þ lLÞab

� �; ð6Þ

where a=b is the mean of the gamma-distributed life-expectancies.Eq. (6) can be rearranged to give an expression for u,

u ¼ N�ðcL þ lLÞmCcL

ba

� �: ð7Þ

As previously described, we assumed that lL = 0.

2.5. Modelling treatment with pyrantel pamoate

The model was run for an initial period long enough to ensurethe worm population had reached equilibrium. All hosts were thentreated with pyramtel on three occasions, each 6 months apart.Pyramtel pamoate was assumed to kill each adult worm with prob-ability = 1 – drug efficacy. This probability (drug efficacy) was setto 0.88 (Keiser and Utzinger, 2008). It was assumed that migratinglarval worms were unaffected by drug treatment because pyrantelpamoate is poorly absorbed from the GI tract (Abdi et al., 1995).

M. Walker et al. / International Journal for Parasitology 40 (2010) 1381–1388 1385

2.6. Simulations and estimation of the intraclass correlation coefficient

The model was programmed in Berkeley Madonna (Macey, R.I.,Oster, G.F., 2000. Berkeley Madonna. Version 8.0.1 for Windows.1442-A Walnut Street #392-GO, Berkley, CA 94709-1405, USA)and simulations were run in discrete time steps of approximatelyhalf a day using a population of 1500 initially uninfected hosts.At each time step the ICC was estimated using the Fleiss–Cuzick(FC) method (Fleiss and Cuzick, 1979). This method of estimatingthe ICC is different from the maximum likelihood method usedin Walker et al. (2010). The FC estimator was not appropriate forour previous analysis because it does not generalise to situationswhere the probability of success (a female worm being large) de-pends on covariates. However, the FC estimator is among the bestperforming of the numerous ICC estimators for binary data interms of its mean square error and bias (Ridout et al., 1999). More-over, it is computationally advantageous over likelihood-basedmethods because it can be calculated directly from the simulateddata without using iterative procedures.

The ICC was not calculated at time steps in which any of the fol-lowing applied: (i) no large females were present in the population;(ii) all females in the population were large, and (iii) each host con-tained a single (small or large) worm (Ridout et al., 1999). Prentice(1986) showed that for binary data the ICC must satisfy a lowerbound which is a function of the maximum number of observationswithin a unit and the probability of success. In the context of thisanalysis, this condition is a function of the maximum number of fe-male worms per host and the proportion of worms that are large inthe population as a whole. At any time step, an ICC falling below thislower limit was replaced by its minimum permissible value (theequation for this lower bound may be found in Ridout et al.(1999) or in Prentice (1986)). Simulations were repeated to estimatethe mean and standard error (SE) of the ICCs at each time step.

3. Results

3.1. Aggregation generated by clumped infections

The temporal dynamics of aggregation among A. lumbricoidesbody sizes generated by clumped infections through three rounds

0.0 0.5 1.0 1.5 2.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Time post first treatment (years)

Mea

n IC

C

A

increasing mC

treatments

Fig. 3. The predicted temporal dynamics among aggregation among female Ascaris lumbof treatment with pyrantel pamoate. (A) Means of intraclass correlation coefficients (ICCsof the mean number of infectious eggs acquired per contact ðmCÞ: 6.125, 12.5, 25, 50, 1confidence intervals estimated from the ‘‘observed” data in our previous analysis (Walkethe ICCs estimated in the baseline, first and second re-infection populations, respectivelyindicated by the three downward arrows. (B) Mean of the change in the ICCs between thebetween the baseline and first re-infection populations is shown by the thick solid line anthe thick dashed line. The thin solid and thin dashed lines represent, respectively, the corrA and B, model simulations were repeated until the standard errors of the estimates (eithewider than the lines used to display them.

of treatment with pyrantel pamoate are depicted in Fig. 3A. Eachline in the figure was constructed from the means of ICCs calcu-lated at each time step from repeated model simulations with aset value of the mean number of eggs acquired per infectious con-tact ðmCÞ. Overlaid are treatment times and the ICC values esti-mated from the observed data in Walker et al. (2010). The timepoints immediately prior to each treatment correspond to thebaseline, first and second re-infection populations, respectively.For each value of mC , the over-dispersion parameter ðkÞ was as-signed the same arbitrarily large value ðk!1Þ. This ensured thatthe distribution of eggs acquired per infectious contact ðCÞ wasPoisson. Simulations were also performed for ðk ¼ 1Þ so that Cwas negatively binomially distributed (Supplementary materialS1, Section 2). The temporal dynamics of the mean ICC calculatedfrom these simulated data were found to be very similar to thoseobtained when C was Poisson distributed and so are not reportedfurther in the main text.

The mean ICC at baseline depended critically on whether eggswere acquired rapidly and in small clumps or infrequently and inlarger ones. In Fig. 3A, the (model predicted) mean ICC at baseline,for each value of mC , is represented by a horizontal line prior to thefirst treatment. High degrees of aggregation among worm bodysizes occurred when larvae were ingested infrequently (low con-tact rate, u) and in large clumps (high mean, mC). By contrast,low degrees of aggregation arose with frequent acquisitions ofsmall clumps of larvae. The sensitivity of the mean ICC at baseline(and in the re-infection populations) to changes in mC (and so alsoin u) is reported in more detail in Section 2 of Supplementarymaterial S1.

Treatment of hosts with pyrantel pamoate, which removedapproximately 88% of the worm population, elicited complexdynamical behaviour of the mean ICC, the precise form of whichdepended on mC (Fig. 3A) and to a lesser extent on k (Supplemen-tary material S1, Section 2). By way of comparison with the ICCscalculated from the observed data in Walker et al. (2010), Fig. 3Bdepicts the (model predicted) mean change between the ICCs cal-culated at baseline and those calculated in the re-infection popula-tions for increasing values of mC . It is clear from Fig. 3B that formost values of mC the mean ICCs decreased over the three popula-tions, in accordance with the trend in the observed data (Walkeret al., 2010).

0 100 200 300 400 500

−0.

20−

0.10

0.00

0.05

0.10

B

Mea

n ch

ange

in I

CC

rel

ativ

e to

bas

elin

e

Mean number of eggs acquired per contact, mC

ricoides body sizes resulting from a clumped infection process through three rounds) estimated at each time step from repeated model simulations for increasing values00 and 200 (in the direction of the upward arrow). Overlaid are the ICCs and 95%r et al., 2010). These are plotted at 0, 0.5 and 1 year post initial treatment, denoting. Treatments with pyrantel pamoate of the entire host population occurred at timesbaseline and re-infection populations for increasing values of mC . The change in ICCsd the change between the baseline and second re-infection populations is shown byesponding changes estimated from the observed data in Walker et al. (2010). In bothr the mean of the ICCs at each time step (A) or the mean change in ICCs (B)) were no

1386 M. Walker et al. / International Journal for Parasitology 40 (2010) 1381–1388

3.2. Aggregation generated by heterogeneous worm life-histories

The temporal dynamics of aggregation among A. lumbricoidesbody sizes generated by between-host heterogeneities in wormlife-histories through three rounds of treatment with pyrantelpamoate are depicted in Fig. 4A. Each line in the figure was con-structed from the means of ICCs calculated at each time step fromrepeated model simulations with a set value of the CV of wormmortality ðlÞ and growth ðcÞ rates (life-history parameters). Over-laid are treatment times and the ICC values estimated from the ob-served data in Walker et al. (2010). The sensitivity of the mean ICCin the baseline and re-infection populations to changes in the CV ofthe life-history parameters is reported in Section 2 of the Supple-mentary material S1.

By comparing Fig. 3A with Fig. 4A, it is evident that the tempo-ral dynamics of the mean ICC are markedly different when aggre-gation among worm body sizes is generated by heterogeneousworm life-histories compared with when it is generated byclumped infections. For a low CV of worm life-history parameters,the ICC tended to be suppressed in the first re-infection populationrelative to baseline. In the second re-infection population, the ICCwas either equal to, or fractionally lower than, that in the firstre-infection population. For a moderate to high CV of worm life-history parameters, the ICC was greater in the first re-infectionpopulation compared with baseline and greater still in the secondre-infection population (Fig. 4B). This trend of increasing degrees ofaggregation among A. lumbricoides body sizes over the three popu-lations (baseline to first to second re-infection population) is theexact opposite to that seen in the observed data (Walker et al.,2010).

4. Discussion

The aim of this study was to explore patterns of aggregationamong A. lumbricoides body sizes under two alternative generativemechanisms: clumped infections and heterogeneities among hosts.This was achieved using a stochastic immigration-death modelwhich: (i) permitted hosts to ingest multiple eggs (larvae) peracquisition (clumped infections), and (ii) allowed worm life-histo-ries (growth and mortality rates) to vary among hosts. The motiva-

0.0 0.5 1.0 1.5 2.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time post first treatment (years)

Mea

n IC

C

increasing ν

treatmentsA

Fig. 4. The predicted temporal dynamics among aggregation among female Ascaris lumbrounds of treatment with pyrantel pamoate. (A) Means of intraclass correlation coefficienvalues of the coefficient of variation (CV) of worm life-history parameters ðmÞ: 0.25, 0confidence intervals estimated from the ‘‘observed” data in our previous analysis (Walkethe ICCs estimated in the baseline, first and second re-infection populations, respectivelyindicated by the three downward arrows. (B) Mean of the change in the ICCs between thebetween the baseline and first re-infection populations is shown by the thick solid line anthe thick dashed line. The thin solid and thin dashed lines represent, respectively, the corrA and B, model simulations were repeated until the standard errors of the estimates (eithwider than the lines used to display them.

tion behind this approach arose from a need to better understandhow the ICCs estimated from the parasitological data analysed byWalker et al. (2010) relate to A. lumbricoides transmission and pop-ulation processes and, in particular, to determine whether thoseestimates are congruent with the conjecture of a clumped infectionprocess.

There are two main findings of our work. First, we confirm thataggregates of similar-sized worms can arise from clumped infec-tions and/or heterogeneities among hosts. Second, and moreimportantly, we show that clumped infections more readily gener-ate a similar pattern of size aggregation to that seen in the dataanalysed by Walker et al. (2010). That is, a decrease in aggregationover three population samples, one collected at baseline (the base-line population) and two collected after consecutive 6-month peri-ods of re-infection following treatment with pyrantel pamoate (thefirst and second re-infection populations, respectively).

Reductions in aggregation among worm body sizes from thebaseline to the first re-infection population arose from many sim-ulations. For clumped infections, aggregation in the first re-infec-tion population tended to be lower than aggregation at baselinewith the exception of when the mean number of larvae acquiredper clump was very small. Aggregation also tended to be lowerin the first re-infection population than at baseline for low to mod-erate degrees of heterogeneity among worm life-histories. Theopposite was true for high degrees of heterogeneity. A reductionin aggregation from the first to the second re-infection populationoccurred in all instances when aggregation was generated byclumped infections.

Aggregation generated by heterogeneities among worm life-histories was fractionally lower in the second re-infection popula-tion compared with the first for a small number of simulations cor-responding to coefficients of variation of worm life-historiesbetween 0.125 and 0.55 (see Fig. 4B). These predicted patterns ofthe aggregation among worm sizes lead to the conclusion thatacquisitions of clumped infections are more consistent with theobserved data. Clumping is more likely than heterogeneous wormlife-histories to drive aggregation in the second re-infection popu-lation lower than aggregation in the first re-infection population.This conclusion does not preclude heterogeneous worm life-histo-ries; indeed it is most likely that both mechanisms operate. How-

0 1 2 3 4

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CV adult worm life−history parameters, ν

ricoides body sizes resulting from heterogeneous worm life-histories through threets (ICCs) estimated at each time step from repeated model simulations for increasing.5, 1 and 2 (in the direction of the upward arrow). Overlaid are the ICCs and 95%r et al., 2010). These are plotted at 0, 0.5 and 1 year post initial treatment, denoting. Treatments with pyrantel pamoate of the entire host population occurred at timesbaseline and re-infection populations for increasing values of m. The change in ICCs

d the change between the baseline and second re-infection populations is shown byesponding changes estimated from the observed data in Walker et al. (2010). In bother the mean of the ICCs at each time step (A) or the mean change in ICCs (B)) were no

M. Walker et al. / International Journal for Parasitology 40 (2010) 1381–1388 1387

ever, the simulations lend support to the conjecture of clumpedinfections and suggest that this process may predominate in gener-ating aggregates of similar-sized A. lumbricoides within hosts.

Conclusions drawn from any theoretical approach are highlydependent on the assumptions made. In particular, it is easy toconceive alternative and more complex ways to model heterogene-ity among worm life-histories. For example, it might be reasonableto assume that the mortality and growth rates of female A. lumbric-oides are negatively correlated: a host mounting an effective im-mune response may kill worms more rapidly in addition toreducing their growth rate. A positive correlation may also beenvisaged: worms that grow slowly may attain a larger size andexperience less mortality than worms that grow rapidly. Life-his-tory traits have been studied from an evolutionary perspective, of-ten with the aim of predicting the consequences of selectivepressures induced by mass chemotherapy on the fecundity of nem-atodes (Gemmill et al., 1999; Lynch et al., 2008; Morand and Pou-lin, 2000; Skorping and Read, 1998). However, we are not aware ofany work that has examined variability in life-history traits amongindividual worms within a host population. Thus, in the absence ofany such empirical evidence, we used a simplified approach in ouranalysis, modelling the two life-history parameters asindependent.

The true nature of the distribution of eggs acquired per infec-tious contact is likely to depend on a number of factors. If eggs be-come randomly displaced from their initial site of deposition thentheir spatial distribution will be homogenised, potentially makingtrickle infections more likely. Alternatively, if there is little dis-persal, acquisitions may be rarer but larger in terms of the num-bers of eggs (larvae) ingested. This may occur when the faeces ofmany hosts are deposited in the same local environment or at apreferential site of defecation. More importantly, A. lumbricoideseggs are sticky (Crompton, 1989; O’Lorcain and Holland, 2000)and may adhere together in clumps, regardless of other factors.

The efficiency of the ICC estimator in accurately portraying the‘‘true” degree of aggregation is difficult to assess. However, a largesimulation study (Ridout et al., 1999) found that the Fleiss andCuzick estimator performed generally well on data simulated usinga variety of ICCs, success probabilities, and group sizes drawn froma truncated negative binomial distribution. This suggests that thisestimator should perform well on the data simulated in this anal-ysis which were invariably over-dispersed and the probability thata female worm was large was temporally dynamic.

A number of previous theoretical studies have modelled theacquisition of eggs in discrete clumps rather than in a continuoustrickle. These studies have mostly been concerned with the effectof clumped infections on the distribution of worms among hosts.Tallis and Leyton (1969) were among the first to consider thisquestion, determining properties of this distribution by analyti-cally exploring a simple (single life-stage) immigration-deathmodel. Isham (1995) also took an analytically tractable approachto explore the effect of clumped infections, parasite-induced hostmortality and heterogeneous contact rates on the distribution ofworms. This methodology was extended by Herbert and Isham(2000) to include multiple parasite life-stages (larvae, mature par-asites and offspring). More complex, full transmission models,were presented by Grenfell et al. (1995a, 1995b). The effect ofclumped infections on parasite inbreeding and the rate of emer-gence of anthelmintic resistance in nematode infections of live-stock has also been explored (Smith et al., 1999; Cornell et al.,2000, 2003).

Non-specific (random) host heterogeneities have been includedin infection models mostly by modelling host contact rates as het-erogeneous. Generally, the effect of such heterogeneity on the dis-tribution of worm burdens has been of primary interest (Andersonand Gordon, 1982; Pacala and Dobson, 1988) (for examples of het-

erogeneities incorporated into clumped infection models seeIsham (1995) and Herbert and Isham (2000)). Rates of other dy-namic processes have most often been assumed constant or deter-ministically dependent on another modelled variable. For example,acquired immunity has been modelled by assuming that parasite(either adult or larval) mortality is dependent on the host’s cumu-lative past exposure to infection, modelled as a deterministic func-tion of host age (Anderson and May, 1985; Berding et al., 1986;Woolhouse, 1992).

Heterogeneous host contact rates do not generally alter the pre-dicted mean parasite load per host. This is useful because simpledeterministic models can be employed to explore the dynamicsof the mean when other distributional properties are not of inter-est. The heterogeneous worm mortality and growth rates appliedin the model described in this paper do not preserve the mean inthis way. To illustrate this, in Section 3 of Supplementary materialS1, we use a simplified version of our model to show that, for pop-ulations with identical forces of infection, the rate of increase inthe mean number of worms per host depends on the precise natureof the distribution of worm mortality rates among hosts. This hasimmediate practical implications for the assessment of chemother-apy-based control initiatives: a slower or more rapid than expectedrise in the average intensity of infection following anthelmintictreatment could be too hastily attributed to changes in the forceof infection if the potential for heterogeneities among hosts werenot fully appreciated.

Acknowledgements

We thank Dr. Nicholas Grassly (Imperial College London, UK)for providing plug-in functions for Berkeley Madonna to enableefficient random number sampling. We also thank the Medical Re-search Council (MRC) and the Faculty of Medicine of Imperial Col-lege London, UK for funding this work through a Doctoral TrainingAccount (MW) and the MRC for a Career Establishment Grant (M-GB). Two anonymous referees made valuable comments on an ear-lier version of the manuscript.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.ijpara.2010.07.001.

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