Epidemic extinction in a generalized susceptible-infected-susceptible model

TL;DR

This paper analyzes the extinction time of epidemics in a generalized SIS model with multi-infection contact, deriving analytical expressions for the mean extinction time and validating them with simulations.

Contribution

It introduces a novel analytical approach for calculating epidemic extinction times in a generalized SIS model with multiple contacts, revealing different bifurcation behaviors.

Findings

Mean extinction time scales exponentially with population size and action.

Different bifurcation behaviors for single and multiple contact infection models.

Analytical expressions for the action as a function of infection rate.

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 $\langle T\rangle$ depends exponentially on the associated action $\mathcal {S}$ and the size of the population $N$, $\langle T\rangle \sim \exp(N\mathcal {S})$. Because of qualitatively different bifurcation features for $m=1$ and $m\geq2$, 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 $m\geq2$. Finally, a rare-event simulation method is used to validate the theory.