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VNso, v ' p,^ 0.5 On log-log coordinates, the slope is:

1"

Because a is non-negative, Eqs. 4.13 and 4.14 always yield slopes less than the respective exponential, Eqs. 4.8 and 4.9. In other words, the beta-Poisson model is shallower than the exponential model. This is shown in Figure 4.1, in which it is also shown that as a increases, the beta-Poisson model approaches the exponential model. Figure 4.1 (top) also shows that all models at sufficiently low doses yield a slope of 1—indicating linearity, on a log-log plot.

Empirical Models

A dose-response model is fundamentally of the same mathematical form as a cumulative probability distribution function (cdf) defined over the positive real line. Hence, any cdf can be explored as a dose-response function; however such

d/N50

Figure 4.1. Effect of a on dose-response relationship.

d/N50

Figure 4.1. Effect of a on dose-response relationship.

empirical models may not have concordance with underlying biological bases of infection. Examples of such empirical dose-response functions are shown in Table 4.1.

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