Extinction of an infectious disease: a large fluctuation in a non-equilibrium system

TL;DR

This paper develops a theoretical framework for understanding the stochastic extinction of infectious diseases in non-equilibrium systems, using epidemiological models and advanced mathematical techniques to analyze extinction pathways and times.

Contribution

It introduces a novel method combining probability generating functions and eikonal approximation to identify optimal extinction paths in complex stochastic models.

Findings

Computed the optimal extinction trajectory numerically.

Discovered a non-monotone, spiral path to disease extinction.

Derived analytical results near bifurcation points.

Abstract

We develop a theory of first passage processes in stochastic non-equilibrium systems of birth-death type using two closely related epidemiological models as examples. Our method employs the probability generating function technique in conjunction with the eikonal approximation. In this way the problem is reduced to finding the optimal path to extinction: a heteroclinic trajectory of an effective multi-dimensional classical Hamiltonian system. We compute this trajectory and mean extinction time of the disease numerically and uncover a non-monotone, spiral path to extinction of a disease. We also obtain analytical results close to a bifurcation point, where the problem is described by a Hamiltonian previously identified in one-species population models.