Considering that β = cp, after fitting contact parameter c(N/A) to contact–density data for the four functions, a transmission rate p had to be determined before being able to compare the effect of the different transmission–density functions. This is exactly equivalent to the situation in which real data are known, but the underlying transmission–density function is not. In such a situation, different transmission–density functions are implemented in a model and fitted to the data. For each contact function c, a different transmission rate p must be fit in order for the transmission–density function to match the real data as closely as possible. If instead we were to use the same transmission rate p for each function, we would, in fact, be comparing entirely different transmission dynamics. Fitting p is typically done using a certain important characteristic of the real data, such as average outbreak size or annual cumulative incidence. For this purpose, we implemented a function-specific constant (qi) that was fitted for each function i to a common characteristic. Out of numerous possible transmission dynamic characteristics to choose for fitting, we opted for one that ensured that β, summed across the probability distribution of population densities occurring during 1 year in a simulated, deterministic model of demography, was the same for each contact function. Formally, this meant that: β=qi×Σj=1300fc(N/A)j×h(N/A)j, where fc is the contact–density function, j is the population density and h is the frequency distribution of densities in a year. We chose this method because it has the advantage of not selecting for certain outbreak characteristics such as prevalence or outbreak size. We nevertheless also examined the effect of using two alternative fitting methods. The first alternative method fits qi so that a deterministic transmission model resulted in a maximum annual prevalence of 40%. The second alternative method fits qi so that the final annual number of infections was 2N0, i.e. twice (an arbitrarily chosen number) the initial number of individuals. While these alternative fitting methods resulted in highly different values for the constant (density-independent) function, the three other functions were always very similar (electronic supplementary material, table S3–1 and figure S3–1). As we are mainly interested in the differences between the three non-constant functions, we did not further investigate the effects of these alternative β-fitting methods.