Disease Extinction in the Presence of Random Vaccination

We investigate disease extinction in an epidemic model described by a birth-death process. We show that, in the absence of vaccination, the effective entropic barrier for extinction displays scaling with the distance to the bifurcation point, with an unusual critical exponent. Even a comparatively weak Poisson-distributed random vaccination leads to an exponential increase in the extinction rate, with the exponent that strongly depends on the vaccination parameters.

Figure 1

The scaled barrier for extinction $s_{\mathrm{ext}}^{(0)}$ vs the reproductive rate of infection $R_0$. The dashed line shows expression (10). The data points for different ${\mu }^{\prime }=\mu /(\mu +\mathit{\varkappa })$ are obtained from a numerical solution of Eqs. (6, 7).

Figure 2

The scaled fluctuation-induced change of the barrier for extinction, ${\Delta }_{\mathrm{fl}}=(\eta /\nu )\log A_{\mathrm{fl}}$, vs the scaled intensity of noise pulses $\sigma =\overline{g}{\eta }^{2}N/\mu \beta$ (solid line). The results refer to $\eta \ll 1$. The dashed lines show the asymptotic behavior of ${\Delta }_{\mathrm{fl}}$ for small and large $\sigma$.