Epidemic extinction in a generalized susceptible-infected-susceptible model

Abstract

We study the extinction of epidemics in a generalized susceptible-infected-susceptible model, where a susceptible individual becomes infected with the rate $\lambda$ when contacting $m$ infective individual(s) simultaneously, and an infected individual spontaneously recovers with the rate $\mu$. By employing the Wentzel-Kramers-Brillouin approximation for the master equation, the problem is reduced to finding the zero-energy trajectories in an effective Hamiltonian system, and the mean extinction time $⟨T⟩$ depends exponentially on the associated action $\mathcal{S}$ and the size of the population $N$, $⟨T⟩\sim \exp (N\mathcal{S})$. Because of qualitatively different bifurcation features for $m=1$ and $m\ge 2$, we derive independently the expressions of $\mathcal{S}$ as a function of the rescaled infection rate $\lambda /\mu$. For the weak infection, $\mathcal{S}$ scales to the distance to the bifurcation with an exponent $2$ for $m=1$ and $3/2$ for $m\ge 2$. Finally, a rare-event simulation method is used to validate the theory.

14 pages, 5 figures

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• Tricritical behavior in epidemic dynamics with vaccination
• Extremal statistics for first-passage trajectories of drifted Brownian motion under stochastic resetting
• Epidemic extinction in a simplicial susceptible-infected-susceptible model
• Expected Extinction Times of Epidemics with State-Dependent Infectiousness