Cutoff for the logistic SIS epidemic model with self-infection

TL;DR

This paper analyzes a variant of the logistic SIS epidemic model with self-infection on complete graphs, demonstrating the cutoff phenomenon and providing precise asymptotics for the mixing time as population size grows.

Contribution

It introduces the self-infection variant, derives exact cutoff time constants, and illustrates a formalized methodology for proving cutoff using concentration and coupling techniques.

Findings

The model exhibits the cutoff phenomenon with a sharp transition in mixing time.

Exact leading constant for the cutoff time is derived.

The window size for cutoff is constant and optimal.

Abstract

We study a variant of the classical Markovian logistic SIS epidemic model on a complete graph, which has the additional feature that healthy individuals can become infected without contacting an infected member of the population. This additional ``self-infection'' is used to model situations where there is an unknown source of infection or an external disease reservoir, such as an animal carrier population. In contrast to the classical logistic SIS epidemic model, the version with self-infection has a non-degenerate stationary distribution, and we derive precise asymptotics for the time to converge to stationarity (mixing time) as the population size becomes large. It turns out that the chain exhibits the cutoff phenomenon, which is a sharp transition in time from one to zero of the total variation distance to stationarity. We obtain the exact leading constant for the cutoff time, and show the window size is constant (optimal) order. While this result is interesting in its own right, an additional contribution of our work is that the proof illustrates a recently formalised methodology of Barbour, Brightwell and Luczak, which can be used to show cutoff via a combination of concentration of measure inequalities for the trajectory of the chain, and coupling techniques.