# Dynamics of Sir Model with Switching Transmission Rate and Vaccination Rate Characterized by a Relay System or Preisach Operator

2021-04-22

## Abstract

We propose three new epidemiological models, which extend the classical SIR model. We study their global dynamics. The first model takes into account a switching prevention strategy by introducing two distinct thresholds set by the density of infected individuals at which the transmission rate switches between two distinct values. The thresholds determine the beginning and end of the intervention. We show that the epidemic trajectory converges to either an endemic equilibrium or a periodic orbit depending on the threshold values. This switched system models an ideally homogeneous hysteretic response of the public to the epidemic. The next model assumes that individuals respond differently to the dynamics of the epidemic. We model the heterogeneous public response by the Preisach hysteresis operator. The degree of heterogeneity of the response is measured by the variance σ 2 of the corresponding distribution (the Preisach density function). The model has a continuum of endemic equilibrium states characterized by different proportions of susceptible, infected and recovered populations. We consider how the limit point of the epidemic trajectory and the infection peak along this trajectory depend on σ. The third model introduces a vaccination process, which is modeled by a switched system in the case of a homogeneous public response and by the Preisach operator in the case of a heterogeneous response. The response exhibits hysteresis in either case. We show that every trajectory converges to either an endemic equilibrium state or a periodic orbit. Under additional natural assumptions, the periodic attractor is excluded, and we guarantee the convergence of each trajectory to an endemic equilibrium state. The global stability analysis uses a family of Lyapunov functions corresponding to the continuum of branches of the hysteresis operator. As such, it is based on a new method of the local and global stability analysis of non-smooth dynamical systems with infinite-dimensional hysteresis operators.

## Keywords

Hysteresis, Lyapunov functions, Epidemiology -- Mathematical models, Communicable diseases -- Transmission