The dose-response relationship attempts to estimate the probability of illness upon exposure to a hazard. The overall probability of illness is dependent upon the occurrence of three conditional probabilities:

- The probability that the organism is ingested. (This is an output from the exposure assessment and all the modelling and estimates prior to the dose-response.)
- The probability that the organism is able to survive and infect the host once it is ingested.
- The probability of the host becoming ill once infected.

The microbial disease process that the dose-response analysis attempts to characterize is dependent on:

- The characteristics of the organism itself, e.g. mechanism of pathogenesis, virulence factors, and resistance to host factors such as gastric acidity and immune response.
- The susceptibility of the host; e.g. immunocompetence or nutritional status.
- The characteristics of the food in which the pathogen is carried, e.g. a food with high fat content will protect the organism from the gastric acidity of the stomach.

The dose-response relationship is a function that provides a link between the dose that is ingested and the response that occurs. This relationship is shown graphically in Figure 4.1.

FIGURE 4.1

**Illustration of link between dose-response function and
dose and response information**

In order to derive the functional relationship between dose and response we can proceed along two pathways. First, we can use data that report on a range of doses and of responses and fit a purely empirical model to this data. Typically, dose-response data tends to have a sigmoidal shape, and as such any one of numerous mathematical functions that exhibit this sort of shape could be used. An example of typical dose-response data is shown in Figure 4.2.

The dose-response data itself can come
from several sources provided they can be adapted to produce the basic
relationship depicted in Figure 4.2. Typically, human feeding trial data has
been used to describe this relationship for several microbial pathogens.
However, more of this type of data is unlikely to be produced due to ethical
concerns, especially for highly virulent pathogens. Some new data could be
generated from vaccine trial information, but for the most part, most of the
human feeding trial data have been analysed and dose-response functions have been
fit to most of these. Teunis *et al.* (1996) provide
a good reference source with a summary of all the available data and
dose-response curves that can be used to describe the dose-response data. The
second source of data can come from animal feeding trial data. This type of
data can provide all the necessary information to fit a dose-response model.
However, since the data is derived from animal feeding trials, a conversion or
translation factor needs to be derived that will allow the estimates to be
applied to human responses. Finally, another source of data that can be used is
epidemiological, or outbreak data. Outbreak data is unique in that it typically
involves a cross segment of the population, or at least, members of the general
population. This is in contrast to feeding trials, which typically involve
healthy male volunteers (perhaps the least susceptible segment of the
population). It is also unique because the data is being generated every day
around the world, although it is just not being collected. When outbreaks
occur, if data is collected on the incriminated dose and the proportion of the
population that ate the contaminated food and became sick, then a database of
dose-response information could be gradually compiled and eventually used to
gain a better understanding of the dose-response curve.

FIGURE 4.2

**Example of typical dose-response data**

Although it is possible to use a purely empirical function to describe the dose-response relationship by simply fitting the curve to the data, it is generally not a recommended approach. The dose-response data is often collected using higher doses than would be observed in the real world, while the dose-response function is often used to extrapolate results in dose ranges beyond the observed data. As a result, using a purely empirical function fit to the observed data does not give us a lot of confidence in our estimates when outside the observed range.

The second approach is to develop a dose-response function that is more mechanistic in nature, or based upon our understanding of how the infection process works, and to translate that into a mathematical function. Figure 4.3 illustrates the concept of how infection and illness are assumed to occur as a result of the ingestion of a pathogen.

FIGURE 4.3

**Schematic of infection/illness process as a result of
ingestion of pathogenic organisms**

There are multiple barriers within the human body that the ingested pathogen needs to successfully pass in order to arrive at an appropriate site in the body at which the infection and subsequent illness can be initiated. There is some finite probability that the pathogen will succeed in passing each of the barriers, and taking this into consideration, the concept can be translated into probability statements and subsequently mathematical functions. In order to get a complete treatment of the derivation of dose-response functions, the reader is referred to Haas et al. (1999). This reference also provides an excellent overview of dose-response modelling in general.

Briefly, we can say that there is some probability of ingesting “j” organisms when we are exposed to a mean dose “d” . This can be represented by P1, which essentially captures the variation in the actual number of organisms ingested, and in probability terms can be written as:

P1 ( j | d)

which can be read as the probability of “j” cells being ingested given that the dose had an average of “d” cells.

In addition, once ingested there is some probability that “k” organisms will survive to initiate an infection or illness. This can be represented by P2, which captures the host and microbe interaction and can be written as:

P2 ( k | j)

and which can be read as the probability of “k” cells surviving given that “j” cells were ingested.

If both processes, P1 and P2, are independent, then the overall probability can be written as:

Response occurs if some critical k_{min} organisms survive.

There are two hypotheses that can be used to describe the way in which infection and illness occur upon ingestion of the pathogens. The first, known as “cooperative interaction” (or a “threshold assumption” ), assumes that organisms act together (cooperate) in order to overcome barriers, and that a minimum dose is required in order for a response to occur. In this assumption, kmin is greater than 1. The second, known as the “independent action” hypothesis assumes that the organisms act independently; there is no threshold, but rather, that the minimum dose for the possibility that a response will occur is 1.In this assumption kmin is equal to 1.

The independent action theory is the theory currently accepted for microbial infection. This is reasonable in terms of biological plausibility and on the grounds of conservatism in the face of no additional information (the threshold approach would predict a lower risk than non-threshold). This theory assumes that one cell is capable of initiating a response because the survival of one cell has the ability to initiate an infectious process since pathogens have the ability to multiply, unlike chemicals which may have a threshold. In the non-threshold assumption, it is recognized that although the probability that a single ingested cell is able to successfully survive all the barriers in the body is small, it is non-zero.

If we take the non-threshold assumption, which is the recommended assumption for pathogen infection as stated earlier, then depending upon the assumptions related to P1 and P2, specifically the types of probability distributions used to characterize them, and following some mathematical manipulations, we can arrive at various mathematical functional forms. Two commonly used dose-response functions are described below.

*Exponential dose-response function*

The exponential dose-response function has the following assumptions:

- One cell can initiate infection (no threshold);
- Organisms are randomly distributed in the serving, P1 (The probability of ingesting an organism in a serving is described by a Poisson process.); There is some finite probability of passing each barrier to successfully initiate infection (independent action) - Binomial;
Host-pathogen interaction is a constant, P2. (The probability associated with this parameter is a constant value; there is no variation in the probability.)

Using these assumptions, the exponential dose-response function shown below is derived (Haas, 1999):

*P _{response}* = 1 - exp(-

where D is the dose and r is a parameter of the dose-response function and which is interpreted as the probability for one cell to successfully initiate a response (infection/illness, etc).

Figure 4.4 shows the exponential dose-response curve with various “r” values to indicate how the curve changes with different parameters values.

From Figure 4.4, it can be seen that changing the value of “r” in the exponential dose-response curve tends to create a shift in the curve on a log dose vs. response scale. As the “r” value decreases, the curve shifts to the right on the log dose scale, which as expected translates to a lower probability of response at a given dose. For instance, at 3.0 log dose, the probability of illness with an “r” value of 1e-2 is estimated to be approximately 100 percent, while at an “r” value of 1e-4, the probability at this dose is estimated to be approximately 10 percent.

FIGURE 4.4

**Exponential dose-response function, using three different
parameter values**

*Beta-Poisson dose-response function*

The beta-Poisson dose-response function expands upon the assumptions made in the exponential dose-response by making the following assumptions:

- One cell can initiate infection (no threshold);
- Organisms are randomly distributed in the serving, P1, (the probability of ingesting an organism in a serving is described by a Poisson process);
- There is some finite probability of passing each barrier to successfully initiate infection (independent action) - Binomial;
Host-pathogen interaction is beta distributed, P2 (the probability associated with this parameter is described with a Beta distribution; the variation in host-pathogen interaction is assumed to be beta distributed);

where D is the dose, and β and α are parameters of the beta distribution that describes the host pathogen interaction.

Figure 4.5 shows the beta-Poisson dose-response function with different parameter values and the effect of changes in the parameter values on the shape of the distribution function.

The beta-Poisson function has two parameters, and a change in each translates to a different effect on the dose-response curve. Figure 4.5 shows the effect of changes in the beta parameter. Keeping alpha fixed, changes in the beta parameter produce a shift in the curve similar to that observed in the exponential dose-response function. The effect of changing the value of the alpha parameter is shown in Figure 4.6.

Altering the alpha parameter of the beta-Poisson dose-response function produces a change in the slope of the curve. So, the curve can be both shifted (altering the beta parameter) and its slope increased or decreased (changing the alpha parameter).

Overall, the beta-Poisson model tends to be more flexible in its ability to describe data, primarily as a result of the additional parameters. It should be recognized that as alpha becomes large, the shape of the beta-Poisson model tends towards the exponential model.

FIGURE 4.5

**Beta-Poisson dose-response function**

FIGURE 4.6

**Beta-Poisson dose-response function**

This section illustrates a simple example of a dose-response
model for *Vibrio cholerae*, including the experimental data, and the model fit
to the data that can be used to predict the probability of illness associated
with exposure to a dose of *V. cholerae*. Hornick *et al.* (1971) conducted experiments
with human subjects exposed to various doses of *V. cholerae*, using two
different strains (classical Inaba 569B and classical Ogawa 395). These
researchers performed their experiments with and without the simultaneous
ingestion of sodium bicarbonate, which was used as a stomach pH-buffering
agent. The data from the Inaba 569B strain taken simultaneously with sodium
bicarbonate are shown in Table 4.1.

Table 4.1

**Response to V. cholerae Inaba 569B
ingested with sodium bicarbonate. (Data from Hornick et al. [1971], summarized
from Haas, Rose and Gerba. [1999])**

Dose | Total | Negative | Positive | Prop Responding |

10 | 2 | 2 | 0 | 0.00 |

1 000 | 4 | 1 | 3 | 0.75 |

10 000 | 13 | 2 | 11 | 0.85 |

100 000 | 8 | 1 | 7 | 0.88 |

1 000 000 | 23 | 2 | 21 | 0.91 |

100 000 000 | 2 | 0 | 2 | 1.00 |

Positive response in the data above was defined as either the presence of the organism in the stool or a positive antibody response, or diarrhea with organisms present and of a severity that either did or did not require re-hydration.

Analysis of this data (Haas, Rose and Gerba , 1999) has shown that the
beta-Poisson dose response model provides the best fit. The fitting to the data
can be done using a *maximum likelihood approach*. The beta-Poisson dose-response
function with alpha = 0.25 and beta = 16.2 is found to provide the best fit to the data, as shown in Figure
4.7.

FIGURE 4.7

**Beta-Poisson dose-response model for V. cholerae**

The dose-response function once generated allows us to estimate the risk of a response occurring upon exposure to a specific dose. In this particular example we can now estimate that if an individual ingests 100 cells (2 log), there is an approximately 40 percent chance of a positive response. Interpreted in another way, if we had 10 people ingesting an average of 100 cells, we would expect 4 of them to become sick.

In summary, the dose-response analysis provides a powerful tool that can actually be used independently of the exposure assessment component to determine the level of contamination that corresponds to specific risk levels.