Global Stability of SIR Model with Heterogeneous Transmission Rate Modeled by the Preisach Operator

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1199-1241, June 2024.
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 the hysteresis effects that have been well studied in control theory. To illustrate the importance of history-dependence for epidemic theory, we study the 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. In particular, our results suggest that heterogeneity promotes the convergence of the epidemic trajectory to an equilibrium state. Heterogeneity can be achieved by selective intervention policies targeting specific population groups. On the other hand, heterogeneous responses can lead to a higher peak of infection during the epidemic and a higher prevalence at the endemic equilibrium after the epidemic. These results support the argument that public health responses during the emergence of a new disease have long-term consequences for subsequent management efforts. The main mathematical contribution of this work is a new method of global stability analysis, which uses a family of Lyapunov functions corresponding to different branches of the hysteresis operator. It is well known that instability can result from switching from one flow to another even though each flow is stable (if the flows have different Lyapunov functions). We provide sufficient conditions for the convergence of trajectories to the equilibrium set for switched systems with the Preisach hysteresis operator.