Geometric Singular Perturbation Theory Analysis of an Epidemic Model with Spontaneous Human Behavioral Change

TL;DR

This paper applies geometric singular perturbation theory to analyze an epidemic model that incorporates spontaneous human behavioral changes, revealing how behavior delays influence epidemic dynamics.

Contribution

It introduces a novel application of geometric singular perturbation theory to analyze a behavioral epidemic model with fast-changing human behavior.

Findings

Behavior changes are delayed according to the entry-exit function.

Behavior remains 'sticky' until payoff differences justify change.

The model captures the delay in behavioral response during epidemics.

Abstract

We consider a model due to Piero Poletti and collaborators that adds spontaneous human behavioral change to the standard SIR epidemic model. In its simplest form, the Poletti model adds one differential equation, motivated by evolutionary game theory, to the SIR model. The new equation describes the evolution of a variable $x$ that represents the fraction of the population using normal behavior. The remaining fraction $1-x$ uses altered behavior such as staying home, social isolation, mask wearing, etc. Normal behavior offers a higher payoff when the number of infectives is low; altered behavior offers a higher payoff when the number is high. We show that the entry-exit function of geometric singular perturbation theory can be used to analyze the model in the limit in which behavior changes on a much faster time scale than that of the epidemic. In particular, behavior does not change as soon as a different behavior has a higher payoff; current behavior is sticky. The delay until behavior changes in predicted by the entry-exit function.