We study implications of the dynamical and spatial contact structure between Brazilian escorts and sex-buyers for the spreading of sexually transmitted infections (STI). Despite a highly skewed degree distribution diseases spreading in this contact structure have rather well-defined epidemic thresholds. Temporal effects create a broad distribution of outbreak sizes even if the transmission probability is taken to the hypothetical value of 100%. Temporal correlations speed up outbreaks, especially in the early phase, compared to randomized contact structures. The time-ordering and the network topology, on the other hand, slow down the epidemics. Studying compartmental models we show that the contact structure can probably not support the spread of HIV, not even if individuals were sexually active during the acute infection. We investigate hypothetical means of containing an outbreak and find that travel restrictions are about as efficient as removal of the vertices of highest degree. In general, the type of commercial sex we study seems not like a major factor in STI epidemics.