It is now common-place that pathogen transmission during an outbreak can be more heterogeneous than what is commonly assumed, and that it can have major consequences on their dynamics. However, previous studies did not explore the impact of the different biological sources of heterogeneity while controlling for the resulting heterogeneity in the number of secondary cases. In this study, we explore the role of individual variation in infection duration and transmission rate on parasite emergence and spread in a population. We simulate outbreaks using a custom stochastic SIR model, with and without evolution of the parasite. We show that for a given mean, the variance in the number of secondary cases is the main driver of the outbreak probability, with or without evolution, while it does not play a role on the outbreak dynamic once it emerged. On the opposite, a smaller and more realistic variance in the infection duration causes a faster outbreak. It is therefore useful to take into consideration more realistic distributions when modelling infectious diseases outbreaks.