Global stability of SIR model with heterogeneous transmission rate modeled by the Preisach operator

TL;DR

This paper models epidemic dynamics with history-dependent human behavior using the Preisach hysteresis operator, revealing multiple endemic states and analyzing their stability to inform long-term public health strategies.

Contribution

It introduces a novel global stability analysis method for SIR models incorporating hysteresis effects via the Preisach operator, highlighting the impact of adaptive responses.

Findings

Multiple endemic equilibrium states identified

Stability depends on heterogeneity of responses

Long-term public health implications discussed

Abstract

In recent years, classical epidemic models, which assume stationary behavior of individuals, have been extended to include an adaptive heterogeneous response of the population to the current state of the epidemic. However, it is widely accepted that human behavior can exhibit history-dependence as a consequence of learned experiences. This history-dependence is similar to hysteresis effects that have been well-studied in control theory. To illustrate the importance of history-dependence for epidemic theory, we study dynamics of a variant of the SIRS model where individuals exhibit lazy-switch responses to prevalence dynamics. The resulting model, which includes the Preisach hysteresis operator, possesses a continuum of endemic equilibrium states characterized by different proportions of susceptible, infected and recovered populations. We discuss stability properties of the endemic equilibrium set and relate them to the degree of heterogeneity of the adaptive response. Our results support the argument that public health responses during the emergence of a new disease can have long-term consequences for subsequent management efforts. The main mathematical contribution of this work is a method of global stability analysis, which uses a family of Lyapunov functions corresponding to different branches of the hysteresis operator.