Epidemic extinction in a simplicial susceptible-infected-susceptible model

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.