  Standard epidemic models exhibit one continuous, second order phase
transition to macroscopic outbreaks. However, interventions to control
outbreaks may fundamentally alter epidemic dynamics. Here we reveal how such
interventions modify the type of phase transition. In particular, we uncover
three distinct types of explosive phase transitions for epidemic dynamics with
capacity-limited interventions. Depending on the capacity limit, interventions
may (i) leave the standard second order phase transition unchanged but
exponentially suppress the probability of large outbreaks, (ii) induce a
first-order discontinuous transition to macroscopic outbreaks, or (iii) cause a
secondary explosive yet continuous third-order transition. These insights
highlight inherent limitations in predicting and containing epidemic outbreaks.
More generally our study offers a cornerstone example of a third order
explosive phase transition in complex systems.
