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QMRA for Drinking Water: 2. The Effect of Pathogen Clustering in Single‐Hit Dose‐Response Models

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  • Vegard Nilsen
  • John Wyller

Abstract

Spatial and/or temporal clustering of pathogens will invalidate the commonly used assumption of Poisson‐distributed pathogen counts (doses) in quantitative microbial risk assessment. In this work, the theoretically predicted effect of spatial clustering in conventional “single‐hit” dose‐response models is investigated by employing the stuttering Poisson distribution, a very general family of count distributions that naturally models pathogen clustering and contains the Poisson and negative binomial distributions as special cases. The analysis is facilitated by formulating the dose‐response models in terms of probability generating functions. It is shown formally that the theoretical single‐hit risk obtained with a stuttering Poisson distribution is lower than that obtained with a Poisson distribution, assuming identical mean doses. A similar result holds for mixed Poisson distributions. Numerical examples indicate that the theoretical single‐hit risk is fairly insensitive to moderate clustering, though the effect tends to be more pronounced for low mean doses. Furthermore, using Jensen's inequality, an upper bound on risk is derived that tends to better approximate the exact theoretical single‐hit risk for highly overdispersed dose distributions. The bound holds with any dose distribution (characterized by its mean and zero inflation index) and any conditional dose‐response model that is concave in the dose variable. Its application is exemplified with published data from Norovirus feeding trials, for which some of the administered doses were prepared from an inoculum of aggregated viruses. The potential implications of clustering for dose‐response assessment as well as practical risk characterization are discussed.

Suggested Citation

  • Vegard Nilsen & John Wyller, 2016. "QMRA for Drinking Water: 2. The Effect of Pathogen Clustering in Single‐Hit Dose‐Response Models," Risk Analysis, John Wiley & Sons, vol. 36(1), pages 163-181, January.
  • Handle: RePEc:wly:riskan:v:36:y:2016:i:1:p:163-181
    DOI: 10.1111/risa.12528
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    References listed on IDEAS

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    1. Vegard Nilsen & John Wyller, 2016. "QMRA for Drinking Water: 1. Revisiting the Mathematical Structure of Single‐Hit Dose‐Response Models," Risk Analysis, John Wiley & Sons, vol. 36(1), pages 145-162, January.
    2. Zhang, Huiming & Liu, Yunxiao & Li, Bo, 2014. "Notes on discrete compound Poisson model with applications to risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 325-336.
    3. James D. Englehardt & Ruochen Li, 2011. "The Discrete Weibull Distribution: An Alternative for Correlated Counts with Confirmation for Microbial Counts in Water," Risk Analysis, John Wiley & Sons, vol. 31(3), pages 370-381, March.
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    5. Hadar, Josef & Russell, William R, 1969. "Rules for Ordering Uncertain Prospects," American Economic Review, American Economic Association, vol. 59(1), pages 25-34, March.
    6. Bawa, Vijay S., 1975. "Optimal rules for ordering uncertain prospects," Journal of Financial Economics, Elsevier, vol. 2(1), pages 95-121, March.
    7. Rothschild, Michael & Stiglitz, Joseph E., 1970. "Increasing risk: I. A definition," Journal of Economic Theory, Elsevier, vol. 2(3), pages 225-243, September.
    8. James Englehardt & Jeff Swartout & Chad Loewenstine, 2009. "A New Theoretical Discrete Growth Distribution with Verification for Microbial Counts in Water," Risk Analysis, John Wiley & Sons, vol. 29(6), pages 841-856, June.
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