# A3.8 Dose response relationships

Many dose-response models have been presented in the published literature. These relationships are typically single hit models fitted to human challenge studies (see Box A3.2). While numerous relationships exist, it is recommended to choose the model that is most representative for the conditions under consideration (WHO 2016). While there are several possible choices available, models consistent with the *Australian Guidelines for Water Recycling* were applied (summarised in Table A3.5) (NRMMC, EPHC and AHMC 2006). For the beta-Poisson and the Exact beta-Poisson, the low dose approximation of using the exponential model with r= α/β and r=α/(α+β) respectively, were used for implementation.

### Table A3.5 Summary of dose-response models that could be used for calculation of LRVs <a href="#table-a3-5" id="table-a3-5"></a>

<table data-header-hidden data-full-width="false"><thead><tr><th width="202"></th><th width="179"></th><th width="103"></th><th width="81"></th><th width="78"></th><th></th></tr></thead><tbody><tr><td><strong>Reference pathogen</strong></td><td><strong>Dose-response model</strong></td><td><strong>α</strong></td><td><strong>β</strong></td><td><strong>r</strong></td><td><span class="math">P_{ill/inf}</span></td></tr><tr><td><em>Campylobacter</em> (cfu)</td><td>beta-Poisson Approximation</td><td>0.145</td><td>7.58</td><td></td><td>0.3</td></tr><tr><td>Norovirus (gec)</td><td>Exact beta-Poisson</td><td>0.0044</td><td>0.002</td><td></td><td>0.7</td></tr><tr><td><em>Cryptosporidium</em> (oocysts)</td><td>Exponential</td><td></td><td></td><td>0.2</td><td>0.7 </td></tr></tbody></table>

Abbreviations: cfu colony forming units; gec genomic equivalent copies; α ; β ; r fixed probability that a single microorganism is able to successfully cause infection; $$P\_{ill/inf}$$ probability of illness following infection.

{% tabs %}
{% tab title="Box A3.2" %}

#### Dose-response relationships <a href="#box-a3-2" id="box-a3-2"></a>

Human challenge data refer to controlled studies where volunteers were administered a known dose of pathogens. The responses of the cohort are followed in terms of infection (measured as excretion and or serological response) and illness. Dose-response relationships for many enteric pathogens have been published in the literature. Explanations of the dose response modelling approach are included in a number of papers (Haas 1983; Haas et al. 1993; Teunis et al. 2000). The application of these models is summarised in the WHO guidance (WHO 2016). Briefly, the most commonly applied models are “single hit” models which rely on the assumption that every microorganism acts independently and has a certain probability of passing host defences, colonising and causing an infection. The simplest single hit model is the exponential model:

$$
\text{P}\_{inf} = \text{1 - e}^{-r.D}
$$

Where $$\text{P}\_{inf}$$ is the probability of infection; r is the fixed probability that a single microorganism is able to successfully cause infection, noting that when r = 1 this function is maximised (referred to as the maximum risk curve); and D is the mean dose. When r is assumed to vary according to a beta distribution the single hit model becomes the more complicated:

$$
\text{P}*{inf} = 1 - 1F*{1}(α,α+β,–D)
$$

Where 1F₁ is the Kummer confluent hypergeometric function; referred to as the “hypergeometric dose-response relation”, α and β are the parameters of a beta distribution. This function can be approximated at low doses by the exponential function with <img src="/files/YuK95agbq5DaQ3tIkFSU" alt="" data-size="line">. For certain parameter values (i.e. when β»1 and α « β) the hypergeometric dose-response relation can be simplified to:

$$
\text{P}\_{inf} = 1 - \left( 1 + \dfrac{D}{β} \right)^{-α}
$$

Which is referred to as the Beta Poisson approximation. Similarly, this function may be approximated at low doses with <img src="/files/MDY4gJsCx5Tqfdsiyl00" alt="" data-size="line">.

To develop dose-response relationships that can be applied for QMRA, these models are fitted to human challenge data, finding the best parameter values (r; or α and β) for the dataset.
{% endtab %}
{% endtabs %}


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