Table 1. Common dose–response functions and selected properties.

Equations give a probability of infection f(x) (response) for a number of pathogens x (dose). Parameter π = f′(0) for the functions with finite, non-zero slope at the origin. Some functions have been given non-standard parameterizations in order facilitate comparison between functions. In particular, the beta–Poisson functions are often parameterized in terms of α = πβ and N₅₀ = β(2^1/α − 1). Some functions are known by different names with different parameterizations. Hill-n is called log-logistic when written in terms of β = −n and α = 1/π, and log-normal is called log-probit when written in terms of β₀ = −μ/σ and β₁ = 1/σ.

Model	Equation f(x)	Single-hit	Low-dose lineara	Scalable	Concave
Linear	$\{\begin{array}{cc}\pi x\hfill & x\le \frac{1}{\pi }\hfill \\ 1\hfill & x>\pi \hfill \end{array}$	No	Yes	Yes	Yes
Exponential	1 − e−πx	Yes	Yes	Yes	Yes
Exact beta–Poissonb	1 − 1F1(πβ, β(π + 1), − x)	Yes	Yes	No	Yes
Approximate beta–Poisson	${\displaystyle 1-{(1+\frac{x}{\beta })}^{-\pi \beta }}$	Yesc	Yes	Yes	Yes
Hill-1	${\displaystyle \frac{x}{\frac{1}{\pi }+x}}$	Yesc	Yes	Yes	Yes
Hill-n	${\displaystyle \frac{x^n}{{\left(\frac{1}{\pi }\right)}^n+x^n}}$	No	No	Yes	Dependsd
Log-normale	${\displaystyle \Phi \left(\frac{lnx-\mu }{\sigma }\right)}$	No	No	Yes	No
Weibull	1 − e−(πx)n	No	No	Yes	Dependsd

^a: Finite, non-zero low-dose linear.

^b: ₁F₁(a, b, x) is the confluent hypergeometric function of the first kind.

^c: With negative-binomial rather than Poisson distribution of organisms; see text.

^d: Depends on the value of the parameters. Both Hill-n and Weibull are concave for n ≤ 1.

^e: Writing the log-normal function in integral form demonstrates that it is well-defined at x = 0.