# Expanded Fermi Solutions in Pathogens' Dose-Response Curves

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A foodborne or waterborne pathogen's dose-response curve is a plot of the probability of acute infection or death as a function of its ingested number. This Demonstration generates hypothetical plots based on the expanded Fermi solution concept, that is, that the probability of infection is determined by the number ingested and a set of probabilities that a given number of the pathogens will survive during their passage through the digestive system. Unlike in the "standard Fermi solution" where the probabilities are specified by single values, in the expanded version the probabilities are specified by their ranges. These allow analytical calculation of the lognormal distribution's logarithmic mean and standard deviation whose cumulative form represents the dose-response curve. The Demonstration also calculates and displays the hypothetical pathogen's dose that will result in 5, 50, and 95 percent probability of infection or death.

Contributed by: Mark D. Normand and Micha Peleg (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1: dose-response curve of a relatively mild pathogen plotted on linear coordinates (5 probabilities)

Snapshot 2: dose-response curve of a virulent pathogen plotted on linear coordinates (6 probabilities)

Snapshot 3: dose-response curve of a pathogen plotted on log-linear coordinates (6 probabilities)

Snapshot 4: dose-response curve of an extremely virulent pathogen plotted on log-linear coordinates (6 probabilities)

This Demonstration generates dose-response curves of hypothetical pathogens using the expanded Fermi solution method. The number, , of viable pathogens needed to cause an infection after reaching and settling in the gut is entered with a slider. (The lower is, the more virulent the pathogen.) The probabilities that determine the pathogen's survival in the digestive tract, to , are specified by their ranges whose lower and upper limits, minimum and maximum, are also entered with sliders. They can be, for example, the probabilities that the pathogens survive the stomach acids and enzymes, pancreatic juices, bile, competition from resident microbiota, etc.

The Demonstration calculates the lognormal distribution's logarithmic mean, , and standard deviation, , whose cumulative form (CDF) represents the dose-response curve that corresponds to the entered value and probability ranges. The calculation method is described in the appendix of [2] and has been the basis of the analytical solution in the three Demonstrations listed in the related links. The Demonstration then plots this curve accompanied by the numeric values of and and of the infective dose (ID) at three levels, 5, 50, and 95 percent. It also displays the value of and of , the number of ingested pathogens, entered with a slider, and its corresponding probability of infection in percent. Since the number of ingested pathogens, , and the number that causes infection, , can vary dramatically, a setter bar is used to enter their order of magnitude.

The generated dose-response curve may be plotted on either linear or log-linear coordinates as chosen by the user with a setter bar. The upper limit of the plot's axis is specified with a matching slider.

Not all the plots that can be generated by the Demonstration correspond to the dose-response curves of actual pathogens.

References:

[1] H. C. von Baeyer, *The Fermi Solution: Reflections on the Meaning of Physics*, London: Penguin, 1993.

[2] M. Peleg, M. D. Normand, J. Horowitz, and M. G. Corradini, "An Expanded Fermi Solution for Microbial Risk Assessment," *International Journal of Food Microbiology*, 113(1), 2007 pp. 92–101.

## Permanent Citation