Ecological Archives M081-006-A4

John F. Bruno, Stephen P. Ellner, Ivana Vu, Kiho Kim, and C. Drew Harvell. 2011. Impacts of aspergillosis on sea fan coral demography: modeling a moving target. Ecological Monographs 81:123–139.

Appendix D. Estimating the size dependence of infection risk during the early epizootic.

Here we describe how we estimated the size dependence of infection risk during the early epizootic using the Florida Keys permanent transect survey data (Kim and Harvell 2002, 2004). We used all data collected from August 1997 through January 1999 (four survey periods), to represent the peak-outbreak period.

The fitted model was a modification of our population model that incorporated size-dependent infection risk. Because survey data have consistently shown a linear relationship between size and disease prevalence in larger colonies (20 cm height or taller), we assumed a linear relationship between the height of a disease-free larger colony and its risk of becoming infected, with a maximum risk of 100% for extremely large colonies. For colonies below 20 cm, we assumed a size-independent risk of infection, as in the main text. The infection risk model is thus piecewise linear and determined by 3 parameters: the small-colony infection risk, the infection risk at 20 cm, and the slope of the linear increase in infection risk for heights above 20 cm.

We also modified the population model in two ways to represent differences between the later endemic and earlier outbreak periods. First, we treated the probability that an infected colony recovers from the disease as a free parameter to be fitted, rather than assuming the value estimated from the endemic-phase demographic monitoring data. Second, we added a parameter to represent elevated disease-related mortality during the early epizootic. The endemic-phase data show very low disease-related mortality in large colonies, which is at odds with the observations during the epizootic peak. For example, Kim et al. (2004) observed mortality rates as high as 90% in size ranges where the endemic-phase data exhibit very low disease-related mortality. We therefore assumed a constant size-independent risk of disease-related mortality in addition to the size-related risk already present in the model. The mortality rate in the fitted model was therefore the sum of the size- and infection-dependent risk estimated from the demographic monitoring data (Fig. 3), the hurricane-related mortality risk of larger colonies (Appendix B), and an additional mortality risk for any infected fan regardless of size.

Because the relationship between size and disease prevalence in the population model converges quickly to its steady state, we estimated the five free parameters by fitting the model's steady-state size-prevalence relationship to the survey data. Specifically, let nH(x,θ) and  nI(x,θ) be the steady-state size distributions of healthy and infected colonies; then the predicted disease prevalence as a function of size is p(x,θ)=nI(x,θ)/(nH(x,θ)+nI(x,θ)). The likelihood function for observed disease states is thus the product over all surveyed colonies of

colonies of where xj is the size of colony j and Ij  is 1 if colony j is infected and 0 if it is uninfected. Model parameters and standard errors were estimated by numerically maximizing the likelihood using the optim and mle functions in R. To test the adequacy of the linear model for size-dependent infection we also fitted an infection-risk function with quadratic dependence on size in 20 cm or higher colonies; addition of the quadratic term was not significant (P = 0.76 based on the asymptotic chisquaredf1 distribution of the likelihood ratio statistic).

The estimated parameters and their asymptotic standard errors are:

ln (infection risk at 20cm):  
-3.0 ± 0.38
infection risk slope:  
0.0083 ± 0.0006
ln (disease-related mortality):  
-0.48 ± 0.12
recovery probability:   
0.10 ± 0.07 .

Although the confidence limits are large for some parameters, these estimates depict a very different host–pathogen interaction than we observed in the demographic monitoring plots; for example the lower limit of the approximate 95% confidence interval on disease-related mortality is 0.49/year. This is far greater than the disease-related mortality of larger colonies in the demographic monitoring plots, but is consistent with observations of high disease-related mortality during the initial outbreak period. Kim et al. (2006) report mortality rates of 5% to 95% per year in infected colonies during outbreaks in the Florida Keys between 1996 and 1998, and a 15% recovery rate over a 2-year period, corresponding to an annual recovery probability of 8%, in a small (n = 20 colonies) monitoring study of infected colonies. The correspondence between our indirect estimates and these direct observations support the approach that we have used to infer disease-process parameters during the early outbreak period.

A plausible a priori hypothesis was that the observed size-dependent prevalence was driven by size-dependent disease-associated mortality, i.e., with smaller colonies dying more quickly, and by size-independence infection rates. However, we rejected this alternative explanation. By tagging and following the fate of individual colonies we were able to directly measure size-dependent infection rate as well as size-dependent mortality of both infected and uninfected colonies. Additionally, our extensive field surveys and monitoring data do not support this hypothesis, which predicts a decrease in the number of small colonies during the epizootic. Even early in the outbreak, there was no decrease in the frequency of small fans (Fig. D1). For example, at the two Florida Keys sites with the greatest disease severity early in the epizootic, the relative frequency of small colonies increased from September 1997 to May 1998.

FigD1
 
   FIG. D1. Size-class distribution of sea fan colonies in the first three transect surveys during the rise of the epizootic. For each sampling period, the plotted curve gives the total number of colonies, classified by height (0–10 cm, 11–20 cm, etc.) in all permanent transects at two Florida Keys sites (WSAM = Western Sambo, WDRK = Western Dry Rocks) where disease severity and the number of dead colonies was the highest during the early outbreak period (Kim and Harvell 2002).

 

LITERATURE CITED

Kim, K., and C. D. Harvell. 2002. Aspergillosis of sea fan corals: disease dynamics in the Florida Keys. Pages 813–824 in J. W. Porter and K. G. Porter, editors. The Everglades, Florida Bay, and coral reefs of the Florida Keys: an ecosystem sourcebook. CRC Press, Boca Raton, Florida, USA.

Kim, K., and C. D. Harvell. 2004. The rise and fall of a six-year coral–fungal epizootic. American Naturalist 164:S52–S63.

Kim, K., A. P. Alker, K. Shuster, C. Quirolo, and C. Harvell. 2006. Longitudinal study of aspergillosis in sea fan corals. Diseases of Aquatic Organisms 69:95–99.


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