  Social units, such as households and schools, can play an important role in
controlling epidemic outbreaks. In this work, we study an epidemic model with a
prompt quarantine measure on networks with cliques (a $clique$ is a fully
connected subgraph representing a social unit). According to this strategy,
newly infected individuals are detected and quarantined (along with their close
contacts) with probability $f$. Numerical simulations reveal that epidemic
outbreaks in networks with cliques are abruptly suppressed at a transition
point $f_c$. However, small outbreaks show features of a second-order phase
transition around $f_c$. Therefore, our model can exhibit properties of both
discontinuous and continuous phase transitions. Next, we show analytically that
the probability of small outbreaks goes continuously to 1 at $f_c$ in the
thermodynamic limit. Finally, we find that our model exhibits a backward
bifurcation phenomenon.
