## Introduction

In the context of gastrointestinal infection, a dose–response curve usually refers to a plot of the relationship between the number of pathogens’ units ingested (x-axis) and the probability or frequency that it will have a clinical manifestation as illness or death (y-axis). The pathogen can be a food or waterborne infectious microorganism (including protozoa), a virus, and, less frequently, a bacterial spore. Dose–response curves have been an important tool in microbial risk assessment (MRA). They have been amply investigated and documented, as well as mathematically characterized (see Cassin and others 1998; Holcomb and others 1999; Haas and others 2000; Buchanan and others 2000, Kothary and Babu 2001; Strachan and others 2005, for example). The most commonly used 2 parameter mathematical model used to describe dose–response curves is the Beta Poisson (BP) model

where *P*_{inf}(*dose*) is the probability of infection after ingesting a *dose*, the number of pathogen units, and α and β constants, characteristic to the particular pathogen and affected by environmental circumstances (for example, Holcomb and others 1999; Teunis and others 1999; Haas and others 2000). Alternative models include lognormal, log logistic, simple exponential, a “flexible” exponential, and Weibull–Gamma models (Holcomb and others 1999; Haas and others 2000), expanded versions of the BP and other models (for example, Bartrand and others 2008) and a version of the BP model into which the Time Post Incubation (TPI) has been incorporated (Huang and Haas 2009). The mathematical properties of these functions, their fit to simulated and experimental data, and relation to the infectious mechanism have also been amply studied and compared (for example, Teunis and others 1996; Holcomb and others 1999; Teunis and Havelaar 2000; Xie Yang and others 2000; Latimer and others 2001). In many cases, experimental dose–response data on a particular pathogen have been limited for logistic, technical and ethical considerations, especially when human volunteers have been involved. The problem is further complicated by that the infectivity of pathogens can vary dramatically from a few units in some of the most virulent (Alam and Zurek 2006) to many thousands or even millions in the less infectious (Teunis and others 1996). Thus while the ordinate of the dose–response curves’ plots is almost always linear, that of the abscissa is frequently logarithmic. Also, data obtained from epidemiological and laboratory studies frequently do not cover the entire dose–response curve and are notoriously scattered. For these reasons, comparison of the fit of the various dose–response models has not always rendered a clear winner.

The scatter in dose–response data is not unexpected. As has been long recognized, the clinical manifestation of an infection, and sometimes even its reporting, is the result of a stochastic process, involving probabilities that can rarely be accurately assessed, let alone determined exactly—see subsequently. Therefore, an element of uncertainty is inherent in the construction of dose–response curve, regardless of the mathematical model used for its characterization and whether the independent variable is the actual number of pathogens ingested or their logarithm. Since infection by a food or water borne pathogen is largely but not exclusively affected by the conditions of the infective agent, the particulars of the medium in which it has been transmitted, and the immunological state of the humans that have ingested it, the observed scatter in recorded and reported dose–response curves is not at all surprising.

Construction of a dose–response curve, in contrast with merely fitting epidemiological or experimental laboratory data, is based on certain underlying assumptions. The most well known of these, is the “single-hit” hypothesis and its relation to the Beta Poison model, which was critically evaluated by Teunis and Havelaar (2000). The “stochastic approach” is based on identifying the chain of events that leads to infection or death, assigning them with probabilities and multiplying these probabilities to obtain the occurrence's probability (Teunis and others 1996). The concept has been the foundation of several risk assessment methods—see Cassin and others 1998, for example. The starting point of the methods based on it is the tacit admission that the details of the events leading to the infection or death are rarely, if ever, known with certainty and hence the appeal to probabilities. The “Fermi Solution” is a general method of estimation based on the multiplication and division of several factors whose values, which are unknown, can nevertheless be “reasonably assumed.” It is named after the great physicist Enrico Fermi (1901 to 1954) who made it a high art (von Baeyer 1993). We will explain the term “reasonably assumed” and describe the method and its expansion in the following sections. The objectives of this study were to use an expanded version of the Fermi Solution, which has been originally developed for microbial risk assessment (Peleg and others 2007), to generate food and water pathogens’ dose–response curves when hard information on what happens to them prior to and after their ingestion is scant or nonexistent, and to develop a user-friendly interactive program to do the calculations and post it on the Internet as freely downloadable software.