Asymptotic Absorption-Time Distributions in Extinction-Prone Markov Processes

We characterize absorption-time distributions for birth-death Markov chains with an absorbing boundary. For “extinction-prone” chains (which drift on average toward the absorbing state) the asymptotic distribution is Gaussian, Gumbel, or belongs to a family of skewed distributions. The latter two cases arise when the dynamics slow down dramatically near the boundary. Several models of evolution, epidemics, and chemical reactions fall into these classes; in each case we establish new results for the absorption-time distribution. Applications to African sleeping sickness are discussed.

Figure 1

Absorption-time distributions for (a) the random transition matrix model (large black circles) and the evolutionary game on a ring (small red circles), (b) SIS model (large black circles), logistic model (small red circles), and autocatalytic chemical reaction model (cyan triangles), (c) the well-mixed evolutionary game, and (d) the process $b_m=r{d}_m=r{m}^p$, for $r=0$ and $p=0.3$ (blue), $p=0.75$ (orange), $p=1$ (green), and $p=1.8$ (red). The $r=0.8$ distributions are indicated by dotted lines (when they differ from the $r=0$ counterparts). See Ref. [19] for models and parameters. We used system sizes (a)–(b) $N=500$ and (c)–(d) $N=1000$ and simulated (a) $5\times {10}^4$, (b)–(c) ${10}^5$, and (d) ${10}^6$ trials to measure the distributions, which have been standardized to have zero mean and unit variance. In (c) the distributions are a convolution of Gumbel distributions with relative weighting $s\approx 0.73$. Deviations from predicted normal and Gumbel distributions in (a)–(c) are due to finite system size.

Figure 2

Absorption-time skew for the process $b_m=r{d}_m=r{m}^p$ with $r=0$ (blue circles) and $r=0.8$ (red squares), plotted as a function of the power-law exponent $p$. Skews were numerically computed for $N={10}^5$ using the recurrence relation approach described in Ref. [9]. The black line shows the asymptotic skew $2\zeta (3p)/\zeta (2p{)}^{3/2}$ for $r=0$. The curves cross at $p=1$ where the distribution is Gumbel, independent of $r$. For $p\le 0.5$ the skew approaches zero and the distribution is Gaussian. The numerical skew is slightly larger than expected for $p\lesssim 0.6$ due to finite size effects.

Figure 3

Generalizations to high-dimensional models and Markov chains with internal sinks. (a) Extinction-time distributions for sleeping sickness predicted using a 17-dimensional compartmental model that was fit to case data from the Mosango (large black circles) and Kwamouth (small red circles) regions of the Democratic Republic of Congo (data from Ref. [3]). Mean extinction times (measured from 2016) are approximately 9.5 and 31 yr for the Mosango and Kwamouth regions, respectively, with standard deviations of 4.8 and 7.9 yr. Disease eradication times approximately follow a Gumbel distribution (fit using the mean and variance). (b) Simulations of the SIS, logistic, reaction, and well-mixed evolutionary game models have exponential absorption-time distributions (standardized to zero mean and unit variance) if parameters are chosen so that the dynamics have an internal sink state. For each case, we used $N=50$ and simulated ${10}^6$ trials. See Ref. [19] for model details and parameters.