Dynamics of SIR model with vaccination and heterogeneous behavioral response of individuals modeled by the Preisach operator

Abstract

We study global dynamics of an SIR model with vaccination, where we assume that individuals respond differently to dynamics of the epidemic. Their heterogeneous response is modeled by the Preisach hysteresis operator. We present a condition for the global stability of the infection-free equilibrium state. If this condition does not hold true, the model has a connected set of endemic equilibrium states characterized by different proportion of infected and immune individuals. In this case, we show that every trajectory converges either to an endemic equilibrium or to 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 family of branches of the hysteresis operator.

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

“…the transmission coefficient behaves as a lazy-switch, some threshold pairs lead to the convergence of the epidemic trajectory to a periodic orbit predicting recurrent outbreaks of the epidemic (Chladná et al 2020). Emergence of sustained periodic oscillations of the prevalence was also shown numerically in a similar SIR model with vaccination when the heterogeneity of the response (measured by the variance σ 2 of the probability density function of the Preisach operator) decreases (Kopfová et al 2021). This scenario is in line with earlier findings by Reluga, Bauch and Galvani (2006) who identified stable oscillatory vaccination dynamics in heterogeneous populations and concluded that the more homogeneous the response of a population, the less likely vaccine uptake is to be stable.…”

“…Next, to reflect the heterogeneity of the response, the population is divided into multiple subpopulations, each characterized by a different pair of switching thresholds. In order to keep the model relatively simple, we apply averaging under additional simplifying assumptions as in Kopfová et al (2021); Rouf and Rachinskii (2021). This leads to a differential model with just two variables, S and I, but with a complex operator relationship between the transmission rate and the density of the infected population I.…”

“…This convergence can be shown using the energy potential as a Lyapunov function (for the construction of hysteretic thermodynamic potentials and examples, see Brokate and Sprekels (1996); Krejčí et al (2013); Krejčí et al (2019); related methods of the construction of the Lyapunov function for the problem of absolute stability of Lurie systems with hysteresis in the feedback loop can be found in Gelig and Yakubovich (1978) and the review Leonov et al(2017). However, for SIR systems, hysteresis is a source of instability, which can lead to periodic oscillations corresponding to recurrent waves of infection (Chladná et al 2020;Kopfová et al 2021). The objective of this work is to show the convergence to the equilibrium set if hysteresis has a limited magnitude (as quantified by an appropriate measure) and provide estimates for the magnitude of the hysteresis effect, which guarantee this stable scenario globally.…”

Dynamics of SIR model with heterogeneous response to intervention policy

“…Krasnosel'skii et al 1983). Under the above assumptions, the dynamics of the epidemic is modeled by system (2) where R 0 (t) is related to the I(t) by the Preisach operator (15), which accounts for the heterogeneity of the transmission coefficient.…”

“…Below we consider absolutely continuous measures F . The corresponding operator (15), which is called the continuous Preisach model, can be written in the equivalent form…”