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We study the extinction of epidemics in a simplicialsusceptible-infected-susceptible model, where each susceptible individualbecomes 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 synergisticreinforcement effect in the group interactions. By using the theory of largefluctuations to solve approximately for the master equation, we reveal twodifferent scenarios for optimal path to extinction, and derive the associatedaction $\mathcal{S}$ 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 differentbifurcation points. The action $\mathcal{S}$ shows different scaling laws withthe 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 itsfirst-order derivative are both continuous, reminiscent of the second-orderphase transitions in equilibrium systems. Finally, a rare-event simulationmethod is used to compute the mean extinction time, which depends exponentiallyon $\mathcal{S}$ and the size $N$ of the population. The simulations resultsare in well agreement with the proposed theory.

Epidemic extinction in a simplicial susceptible-infected-susceptible model