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

“…Further, assume that each intervention policy is guided by the two-threshold start/stop rule, such as in ( 10 ), associated with a particular pair of thresholds . Under these assumptions, the vaccination rate in system ( 3 ) is given by This formula defines a mapping from the space of continuous inputs to the space of piecewise constant outputs , which is known as the discrete Preisach operator (Krasnosel’skii and Pokrovskii 1989 ), see Fig. 2 .…”

“…Let us consider a rigorous definition of the Preisach operator ( 12 ) with an absolutely continuous measure F (Krasnosel’skii and Pokrovskii 1989 ). It involves a collection of non-ideal relays , which respond to the same continuous input independently.…”

“…The Preisach operator ( 14 ) is globally Lipschitz continuous (Krasnosel’skii and Pokrovskii 1989 ). More precisely, the relations imply for any with Let us denote by the set of all triplets , where (see ( 4 )) and the initial state of the Preisach operator is compatible with .…”

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

“…As a simplification, let us assume that this measure is independent of time (in particular, the distribution does not change with variations of I). Then, the mapping of the space of continuous inputs I : R + → R to the space of outputs v : R + → R defined by ( 12) is known as the general Preisach operator, which includes the discrete Preisach operator (11) and a continuous Preisach model (corresponding to an absolutely continuous measure F ) as particular cases. In either case, the Preisach operator is referred to as a superposition (or parallel connection) of weighted non-ideal relays.…”

Dynamics of SIR model with heterogeneous response to intervention policy

“…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. As such, this operator relationship, known as the Preisach hysteresis operator (see, for example, Krasnosel'skii et al, 1989;Mayergoyz 1993;Visintin 1994;Brokate et al, 1996;Krejčí, 1996) accounts for the heterogeneity of the response. This approach is similar to the one in Pimenov et al, 2012.…”