  We study the extinction of epidemics in a simplicial
susceptible-infected-susceptible model, where each susceptible individual
becomes infected either by two-body interactions ($S+I \to 2I$) with a rate
$\beta$ or by three-body interactions ($S+2I \to 3I$) with a rate $\beta
(1+\delta)$, and each infected individual spontaneously recovers ($I \to S$)
with a rate $\mu$. We focus on the case $\delta>0$ that embodies a synergistic
reinforcement effect in the group interactions. By using the theory of large
fluctuations to solve approximately for the master equation, we reveal two
different scenarios for optimal path to extinction, and derive the associated
action $\mathcal{S}$ for $\beta_b<\beta<\beta_c$ and for $\beta>\beta_c$, where
$\beta_b=4 (1+\delta)/(2+\delta)^2$ and $\beta_c=1$ are two different
bifurcation points. The action $\mathcal{S}$ shows different scaling laws with
the distance of the infectious rate to the transition points $\beta_b$ and
$\beta_c$, characterized by two different exponents: 3/2 and 1, respectively.
Interestingly, the second-order derivative of $\mathcal{S}$ with respect to
$\beta$ is discontinuous at $\beta=\beta_c$, while $\mathcal{S}$ and its
first-order derivative are both continuous, reminiscent of the second-order
phase transitions in equilibrium systems. Finally, a rare-event simulation
method is used to compute the mean extinction time, which depends exponentially
on $\mathcal{S}$ and the size $N$ of the population. The simulations results
are in well agreement with the proposed theory.
