Expected Extinction Times of Epidemics with State-Dependent Infectiousness

TL;DR

This paper models how adaptive behavioral precautions affect epidemic extinction times, revealing a threshold phenomenon where the epidemic either dies out quickly or persists indefinitely depending on the curing rate.

Contribution

It introduces a model with state-dependent infectiousness, demonstrating a sharp threshold for epidemic extinction times and analyzing the impact of precautionary behavior on epidemic dynamics.

Findings

Existence of a sharp threshold for epidemic extinction based on curing rate.

Logarithmic extinction time when curing rate exceeds threshold.

Potential for infinite extinction times when curing rate is below threshold.

Abstract

We model an epidemic where the per-person infectiousness in a network of geographic localities changes with the total number of active cases. This would happen as people adopt more stringent non-pharmaceutical precautions when the population has a larger number of active cases. We show that there exists a sharp threshold such that when the curing rate for the infection is above this threshold, the mean time for the epidemic to die out is logarithmic in the initial infection size, whereas when the curing rate is below this threshold, the mean time for epidemic extinction is infinite. We also show that when the per-person infectiousness goes to zero asymptotically as a function of the number of active cases, the mean extinction times all have the same asymptote independent of network structure. Simulations bear out these results, while also demonstrating that if the per-person infectiousness is large when the epidemic size is small (i.e., the precautions are lax when the epidemic is small and only get stringent after the epidemic has become large), it might take a very long time for the epidemic to die out. We also provide some analytical insight into these observations.