  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.
