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Does the early frog catch the worm? Disentangling potential drivers of a parasite age–intensity relationship in tadpoles

The manner in which parasite intensity and aggregation varies with host age can provide insights into parasite dynamics and help identify potential means of controlling infections in humans and wildlife. A significant challenge is to distinguish among competing mechanistic hypotheses for the relationship between age and parasite intensity or aggregation. Because different mechanisms can generate similar relationships, testing among competing hypotheses can be difficult, particularly in wildlife hosts, and often requires a combination of experimental and model fitting approaches. We used field data, experiments, and model fitting to distinguish among ten plausible drivers of a curvilinear age–intensity relationship and increasing aggregation with host age for echinostome trematode infections of green frogs. We found little support for most of these proposed drivers but did find that the parsimonious explanation for the observed age–intensity relationship was seasonal exposure to echinostomes. The parsimonious explanation for the aggregated distribution of parasites in this host population was heterogeneity in exposure. A predictive model incorporating seasonal exposure indicated that tadpoles hatching early or late in the breeding season should have lower trematode burdens at metamorphosis, particularly with simulated warmer climates. Application of this multi-pronged approach (field surveys, lab experiments, and modeling) to additional parasite–host systems could lead to discovery of general patterns in the drivers of parasite age–intensity and age–distribution relationships. Electronic supplementary material The online version of this article (doi:10.1007/s00442-010-1776-0) contains supplementary material, which is available to authorized users.

tadpoles can halt development (but not growth) during certain times of the year independent of temperature (Crawshaw et al. 1992), and because the growth rate model explained the observed dichotomous masses of the stage 36 tadpoles (due to growth without differentiation over winter) whereas the differentiation model did not (Fig. S-1). Parameter values were obtained from Berven et al. (1979). Because the total number of degree days in a season at our site was intermediate between Berven et al.'s (1979) montane and lowland populations (approximately 887 degree-days for our pond using Berven et al.'s (1979) formula), we used the mean of the parameter values from the two populations (mass model: α = -12.8, β = 2.42; differentiation model: α = -18.9, β = 5.29). Larval age estimates based on these average parameters were similar to those calculated using parameter values from either montane (mass model: α = -11.4, β = 2.00) or lowland (mass model: α = -2.84, β = 2.84) populations and indicated that tadpoles over-winter at least once in this population, as expected for a pond with approximately 900 degree days per season (Fig. 1, Fig. S-1, Berven et al. 1979). Because the density of tadpoles in Raffel et al. 2 terms of mass is approximately 1.0 g/mL (Skelly and Werner 1990), volume measurements of Berven et al. (1979) were assumed to be comparable to our mass measurements. Initial mass M i (mass at onset of stage 25) was estimated by regressing mass vs. stage for early stage (25)(26)(27)(28) tadpoles (N = 16, M i = 0.046 g).
The models of Berven et al. (1979) only apply to the green frog larval period, which begins at stage 25 when the embryo completes its transition from dependence upon its yolk sac into a feeding tadpole (Duellman and Trueb 1986). However, hatching in green frogs occurs at approximately Gosner stage 19 (Schalk et al. 2002). Thus, tadpoles should be exposed to trematode infections during their last 5 "embryonic" stages, as well as during their larval stages.
This post-hatching embryonic period was therefore added to the temperature-derived larval age (age since Gosner stage 25) of each tadpole. Data were unavailable for direct estimation of the length of the post-hatching embryonic period of green frogs, but leopard frogs take 6.9 days to complete this period at 18°C (Shumway 1940) and green frogs develop 19% slower as embryos than leopard frogs, at least during Gosner stages 1-19 (Moore 1939). This yielded an approximate post-hatching embryonic period of 8 days.
Average daily water temperatures were calculated using data collected every two hours from 6 March to 2 June of 2005 by a temperature logger (HOBO, Onset, Pocasset, MA) set 20 cm. below the water surface, and estimated for all other dates from 1 March 2004 to 1 November 2005 based on a strong correlation between air temperature (Bahrmann and Ayers 2005) and water temperature in State College PA, as described by Raffel et al. (2006).

Appendix 2: Probabilistic modeling of the age intensity curve
The cyst count for the environmental echinostomes represented cumulative exposure over the lifespan of each individual tadpole in the wild. For each individual, i, infection was assumed Raffel et al. 3 to occur as an exponential process with a time-varying rate parameter, λ i (t). In our most general model, where ε(t) and σ i (t) are the relative seasonal exposure rate and relative susceptibility, respectively, of individual i on day t, and c is an unknown constant. The cumulative exposure over an individual's lifetime in the wild, from the date of emergence to the date of collection, is: Determination of the seasonal exposure rate ε(t) was based on published data from surveys of echinostome-infected snail populations. No data were available on the seasonal abundance of echinostome cercariae for the pond from which these tadpoles were collected.
However, the seasonal dynamics of relative echinostome abundance are similar across geographic regions (Sapp and Esch 1994;Schmidt and Fried 1997), and the relative level of exposure should be proportional to the abundance of infected snails, the first intermediate host of these echinostomes. Thus, we used the well-described seasonal infection dynamics of P. trivolvis snails (Sapp and Esch 1994;Wetzel and Esch 1996) as an index of the seasonal exposure of green frogs to echinostome cercariae at our site ( Fig. 2A). Monthly estimates of snail abundance and infection prevalence were available except for the winter months, during which we assumed no exposure occurred since cold temperatures drive snails into the deep water or the substratum (Sapp and Esch 1994) and inhibit cercarial production (Lo and Lee 1996). For our modeling analysis, we required daily values of the relative exposure intensity, ε(t), from the estimated emergence date of the oldest tadpole, 19-Jul-2004, to the date of capture, 14-Oct-2005.
To estimate ε(t), the monthly values were assumed to hold for the 15 th day of each month, and Raffel et al. 4 daily values were estimated by linear interpolation. The relative exposure intensity was assumed to be proportional to the product of snail abundance and the prevalence of cercarial shedding.
The relative susceptibility of tadpole i as a function of date, σ i (t), was derived from the relative susceptibility values for each developmental stage, σ(s). σ(s) was calculated using the quadratic fit of developmental stage to the arc-sine square-root transformed proportion of successfully encysted experimental trematodes (i.e., sin -1 (√(σ(s i ))) = β 0 + β 1 (s i ) + β 2 (s i 2 )) and the estimates of the developmental stage of individual i as a function of calendar date, s i (t), using the temperature-dependent differentiation rate model of Berven et al. (1979) as described above.
Tadpoles at stage 36 included the youngest second-year tadpoles and the oldest first-year tadpoles. In order to separate the animals which did over-winter from those that did not, the Gosner (1960)  We tested five models for this infection process, which include increasing degrees of covariate information. In the null model of constant exposure rate, exposure and susceptibility were fixed through time so values of λ i,tot scale linearly with tadpole age at capture. More complex models incorporated stage-dependent susceptibility σ i (t), seasonal variation in exposure ε(t), or both. In models for which susceptibility or exposure did not vary through time, ε and σ in Raffel et al. 5 Equation (3) were set to the mean values of ε(t) and σ(s) respectively. Because ε(t) and σ i (t) are completely specified by independent data, all models for λ i,tot have one free parameter, c.
These models were fit to the observed cyst counts using maximum likelihood methods, i.e., for each model the free parameter(s) θ were varied to maximize the log-likelihood where y i is the count of environmental cysts, N = 92 is the number of tadpoles for which environmental cyst counts were available, f(y) is the probability density function of the probability distribution, and θ represents the free parameters of the model. Because the parasite intensities were overdispersed, the data were modeled using a negative binomial distribution with intensity λ i,tot and free dispersion parameter, k. Optimization was conducted using the multi- Raffel et al. 6 The robustness of these results to our limited sample size was investigated using a bootstrapping analysis. The observed dataset was sampled 10,000 times with replacement and the Akaike weight calculated for each of the four candidate models, to determine the frequency with which each of the four models was selected. The mean and median Akaike weights mirrored the weights reported in Table 2. The results indicate clearly that our results are not influenced by the specific composition of the data set.

Appendix 3: Modeling the effect of hatching date on trematode burden at metamorphosis
To predict how hatching date influences the total trematode burden of a frog at metamorphosis, the larval period (days to metamorphosis) for a tadpole hatched on a given date in Beaver 1 pond was estimated using the degree-day model of Berven et al. (1979) and cumulative exposure through metamorphosis was calculated assuming seasonal exposure as shown in Fig. 2A. The degree day model of Berven et al. (1979) is: where L is the larval period in days, T is temperature in degrees Celsius, α is the temperature below which development does not take place and K is the total number of degree-days required for metamorphosis (degree day = one degree of temperature above the developmental zero for a period of one day). Parameters α and K were again estimated based on the average Berven et Daily exposure to trematodes was estimated using interpolated seasonal exposure rates as described above, and cumulative exposure to trematodes was estimated by summing daily exposure rates across the larval period. Stage-dependent susceptibility was excluded from this model because it was found to be unimportant in the age-intensity analysis. To simulate climatic warming/cooling or latitudinal variation in temperature, the entire annual pond temperature profile was uniformly raised or lowered in 1-degree increments up 6°C and down 3°C. This approximated the expected temperature range across the geographic range of R. clamitans (35-45°N and 70-95°W, Conant and Collins 1998), based on average temperatures in this region for the years 1993-2002 (the most recent 10 years available at the time of the analyses). Regional temperature data were obtained from the Climate Research Unit (CRU TS 2.1) dataset, University of East Anglia, available at http://www.cru.uea.ac.uk/ (Mitchell and Jones 2005). An increase of 6°C also approximates the predicted change in temperature for North America over the next century, given a worst-case scenario of high population growth, slow economic development and slow technological change (IPCC 2007). This model makes the simplifying assumptions that seasonal exposure rates are invariant with respect to average temperature and that the number of degree days needed to metamorphose (K) is constant across green frog populations.

Appendix 4: Simulation analysis to rule out small sample size effects
Small sample sizes can lead to underestimation of mean intensity for aggregated distributions of parasites (Gregory and Woolhouse 1993). This means that the small sample size of second-year tadpoles could have led to underestimation of parasite intensity in older tadpoles and a false indication of a curvilinear age-intensity relationship (i.e., the significant quadratic effect of exposure time on intensity). We therefore conducted a simulation analysis to confirm Raffel et al. 8 that the observed parasite intensities for second-year tadpoles were indeed lower than expected under the null model of a linear age-intensity relationship. Our sample size for first-year animals was much larger than for second-year animals and thus should provide more reliable intensity estimates. Hence, we used the null model of a linear fit for the relationship between exposure time and intensity for the first-year tadpoles to estimate the null expectation for the intensity of larval echinostomes in the second-year tadpoles, using a negative binomial generalized linear model (glm) with a log link. To ensure that our analyses were conservative, we used the lower 95% confidence limit around this intensity estimate because it was closer to the observed secondyear tadpole parasite intensities. In our simulation, we replaced the parasite intensity data for the eight second-year tadpoles with eight parasite intensities selected randomly from a negative binomial distribution with a mean equivalent to the lower 95% confidence limit just described and theta equal to theta of the glm for the first-year tadpoles. We then fit a quadratic negative binomial glm on the full age-intensity dataset (including the original data for first-year tadpoles and the simulated data for second-year tadpoles). We repeated this procedure 10,000 times to determine the proportion of simulations that produced quadratic coefficients that were at least as negative as the quadratic coefficient for the observed relationship. Only 2.01% of simulations yielded quadratic coefficients as negative as that observed for the real data, even under these conservative assumptions.

Appendix 5: Analyzing residuals from probabilistic models
Model selection using the Akaike information criterion only yields conclusions about the relative support for models in the candidate set of models under consideration. Just because one of the probabilistic models (i.e., seasonal exposure) fits the data better than the others does not necessarily indicate that it fully explains the curvature of the age-intensity relationship. To Raffel et al. 9 determine whether curvature remained after fitting probabilistic models to the data, we tested for significant curvature and linear trends in the residuals from these models. The residuals were log-transformed to improve normality, following addition of a constant to ensure that all values were positive. The adjusted and transformed residuals were given by where r min < 0 is the minimum value of all residuals for a given model. The adjusted residuals were regressed against age using linear and quadratic functions. Because of the complexity of our model and the non-conventional transformation applied to the residuals, we evaluated the results of these regressions using a third bootstrap analysis (Efron and Tibshirani 1994). For each candidate model, 1000 replicate datasets were simulated by generating negative binomial random variates with means equal to λ i,tot for the model in question, and dispersion parameters k equal to the maximum likelihood estimate derived from fitting the true data. By fitting the negative binomial model to each of these simulated data sets, and analyzing the residuals as described above, the sampling distributions and p-values for the regression parameters could be determined.
Log-residuals from all models exhibited significant negative slopes when regressed against age. We hypothesized that this pattern arose because negative binomial models generate predominantly negative residuals from aggregated data, and the degree of aggregation increased with age. To test this hypothesis, and to ensure that these trends did not arise because the model failed to predict the intensity of infection in older tadpoles, we applied a fourth bootstrap analysis to assess the distribution of slopes derived from data sets simulated from the negative binomial models. From this analysis, the linear slopes of these residuals were lower than expected for the . This result confirmed that the negative residual slopes were caused by properties of the negative binomial distribution, indicating that there was no significant signal in the data beyond that explained by seasonal exposure. It is worth noting that there was no correlation between the residuals from these best-fit models and numbers of experimental cysts (P > 0.3), further confirming the lack of interaction between the two echinostome species.  were log-transformed to improve normality (see Methods). The best-fit linear (thick solid line), quadratic (thick dashed line) fits to these residuals, and the expected trend based on bootstrap analysis (narrow dashed line), are shown on each graph. All 4 models result in negative trends in the residuals, but this is to be expected since aggregated data yields predominantly negative residuals in negative binomial models and the degree of aggregation increased with age. tadpole hatching on a given date, based on the degree day model of Berven et al. (1979). The model output using the Beaver 1 pond temperature profile (0°C) is indicated by a bold solid line, and narrower (solid, dashed or dotted) lines indicate model outputs given simulated warming (+1 to +6°C) or cooling (-1 to -3°C). Gray shading indicates the period outside the known green frog breeding season