Extinction of an infectious disease: a large fluctuation in a nonequilibrium system

Time-resolved extinction rates of stochastic populations

Extinction of a long-lived isolated stochastic population can be described as an exponentially slow decay of quasistationary probability distribution of the population size. We address extinction of a population in a two-population system in the case when the population turnover-renewal and removal--is much slower than all other processes. In this case there is a time-scale separation in the system which enables one to introduce a short-time quasistationary extinction rate W1 and a long-time quasistationary extinction rate W2, and to develop a time-dependent theory of the transition between the two rates. It is shown that W1 and W2 coincide with the extinction rates when the population turnover is absent and present, but very slow, respectively. The exponentially large disparity between the two rates reflects fragility of the extinction rate in the population dynamics without turnover.

Rewiring for adaptation

The idea behind adaptive behavioral epidemiology is that groups and individuals respond to the knowledge of a disease threat by changing their habits to avoid interactions with those who are contagious. Network-based models take this adaptive behavior into account by allowing the network to "rewire" its connections.

WKB theory of epidemic fade-out in stochastic populations

Stochastic effects may cause "fade-out" of an infectious disease in a population immediately after an epidemic outbreak. We evaluate the epidemic fade-out probability by a WKB method and find that the most probable path to extinction of the disease comes from an instantonlike orbit in the phase space of an underlying Hamiltonian flow.

Extinction in the Lotka-Volterra model

Birth-death processes often exhibit an oscillatory behavior. We investigate a particular case where the oscillation cycles are marginally stable on the mean-field level. An iconic example of such a system is the Lotka-Volterra model of predator-prey interaction. Fluctuation effects due to discreteness of the populations destroy the mean-field stability and eventually drive the system toward extinction of one or both species. We show that the corresponding extinction time scales as a certain power-law of the population sizes. This behavior should be contrasted with the extinction of models stable in the mean-field approximation. In the latter case the extinction time scales exponentially with size.

Population extinction risk in the aftermath of a catastrophic event

We investigate how a catastrophic event (modeled as a temporary fall of the reproduction rate) increases the extinction probability of an isolated self-regulated stochastic population. Using a variant of the Verhulst logistic model as an example, we combine the probability generating function technique with an eikonal approximation to evaluate the exponentially large increase in the extinction probability caused by the catastrophe. This quantity is given by the eikonal action computed over "the optimal path" (instanton) of an effective classical Hamiltonian system with a time-dependent Hamiltonian. For a general catastrophe the eikonal equations can be solved numerically. For simple models of catastrophic events analytic solutions can be obtained. One such solution becomes quite simple close to the bifurcation point of the Verhulst model. The eikonal results for the increase in the extinction probability caused by a catastrophe agree well with numerical solutions of the master equation.

Population extinction in a time-modulated environment

The extinction time of an isolated population can be exponentially reduced by a periodic modulation of its environment. We investigate this effect using, as an example, a stochastic branching-annihilation process with a time-dependent branching rate. The population extinction is treated in eikonal approximation, where it is described as an instanton trajectory of a proper reaction Hamiltonian. The modulation of the environment perturbs this trajectory and synchronizes it with the modulation phase. We calculate the corresponding change in the action along the instanton using perturbation techniques supported by numerical calculations. The techniques include a first-order theory with respect to the modulation amplitude, a second-order theory in the spirit of the Kapitsa pendulum effect, and adiabatic theory valid for low modulation frequencies.

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.