## Scientific Basis And Implications Processes in Onset of Infection

A dose-response model, if it is to have a mechanistic basis, should take into account several features of the process of infection. First, especially at low average doses administered to a population, there is heterogeneity in the actual number of organisms received by individual members. In other words, not all members receive actually identical doses. One or more organisms having been ingested, a birth-death process then occurs, in which organisms may survive to colonize and proliferate or may be extinguished from the host before proliferation. These two processes can be combined to yield dose-response relationships that are biologically founded (Armitage, Meynell et al., 1965; Williams, 1965a; Williams, 1965b; Haas, 1983; Haas, Rose et al., 1999).

### Mechanistic Models

The mechanistic features of a dose-response model, as depicted above, can be captured in a straightforward relationship. From this general relationship, a number of specific dose-response models can be derived. Define the following:

1. The probability of ingesting precisely j organisms from an exposure in which the mean dose (perhaps the product of volume and density) is d is written as P\(j\d),

2. The probability of k organisms of the j ingested surviving to initiate an infectious process (the second step) is written as P2(k\j).

3. Infection occurs when at least some critical number of organisms survive to initiate infection. If this minimum number is denoted as /cm;n then the probability of infection (i.e., the fraction of subjects who are exposed to an average dose d who become infected) may be written as:

It should be emphasized that the use of km^n in the precise sense of Eq. 4.3 does not correspond to the often-used term "minimal infectious dose" (Duncan and Edberg, 1995; Edberg, 1996). The latter term refers to the average dose administered, and most frequently really relates the average dose required to cause one-half of the subjects to experience a response; the term "median infectious dose" is preferred. If it is understood that /cmin may not be a single number, but may in fact be a probability distribution, then Eq. 4.3, or a generalization of Eq. 4.3, is expected to be sufficiently broad to encompass all plausible dose-response models. By specifying functional forms for P\ and Pj, as well as numerical values of kmin, we can derive a number of specific useful dose-response relationships.

Exponential The simplest dose-response model that can be formulated assumes that the distribution of organisms between doses is random, namely, Poisson, that each organism has an independent and identical survival probability r (strictly, this is the probability that the organism survives to initiate an infectious focus), and that A:min equals one. From the Poisson assumption, we have:

Finally, with the assumption of km\n = 1, this yields

Or this may equivalently be written as:

This is the exponential dose-response relationship. It has one parameter, r (or k), that characterizes the process. The median infectious dose (N50) can be given by:

The exponential dose-response relationship has the property of low-dose linearity. If rd « 1, then exp(-rc/) ^ 1 - rd, and Eq. 4.4 can be approximated as:

Another property of this and other dose-response curves that we will examine is the slope of the curve at the median point (Pi = 0.5). Differentiation of Eq. 4.4 produces:

Because, at the median point, exp(—rd) = 0.5 (see Eq. 4.4), this can also be written as:

By similar analysis, the slope of a log-log plot at the median point for the exponential dose-response equation can be determined to be:

Beta Poisson The exponential model assumes constancy of the pathogen-host survival probability (r). For some agents, and populations of human hosts, there may be variation in this success rate. Such variation may be due to diversity in human responses, diversity of pathogen competence, or both. This variation can be captured by allowing r to be governed by a probability distribution. This phenomenon of host variability was perhaps first invoked by Moran (1954). Armitage and Spicer (1956) appear to have been the first to characterize this variability by a beta distribution; however, computational limitations precluded the use of this model—beta Poisson and other tolerance distributions. Furomoto and Mickey (1967a; 1967b) appear to be the first to have used this model in the context of microbial dose-response relationships.

Under the above assumptions, the dose-response relationship can be expressed as a confluent hypergeometric function as follows:

Properties of this function are given in standard references [Johnson, 1994 #1139], Furomoto and Mickey (1967a; 1967b) derived the following approximation to equation (7-18):

It is convenient to rewrite Eq. 4.10 by redefining the parameters in terms of the median infectious dose. By solving, it can be determined that:

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