Mathematics Subject Classification (2010) MSC 34D23 · MSC 92D30 · MSC 92D25
In the classical compartmental models of epidemiology, the key parameters such as the transmission and vaccination rates are assumed to remain the same during an epidemic. As an example, consider a SIR model with vaccination of the formṠ = − βSI N − vS − µS + µN,
where S, I and R are densities of the populations of susceptible, infected and recovered individuals, respectively; N = S + I + R is the total population density; β is the transmission rate, which can be expressed as the product of the average number of daily contacts a susceptible individual has with other individuals and the probability of transmission during each contact; v is the vaccination rate; γ is the recovery rate; and, µ is the birth/mortality rate (or immigration/emigration rate; for childhood infections, individuals leave the group at a certain age). The recovered individuals are assumed to be immune to the disease. In this particular model setup, the equality of the birth and mortality rates ensures the conservation of the total population density, i.e. N is constant.
The key parameter is the so-called basic reproduction number R 0 . For model (1) it equals R 0 = βµ/((γ+µ)(µ+v)). If R 0 < 1, the infection dies out in the long run; if R 0 > 1, the infection spreads in the population (Korobeinikov et al. 2002; Korobeinikov et al. 2004; Ullah et al. 2013 ) Due to their simplicity, the standard SIR model and its variants, including (1), assume that the hosts are unable to respond in any way to the advent of an epidemic and disregard the ability of the community to adapt its behavior to the danger. Humans, however, are able to intelligently respond to a threat as they receive and perceive information regarding the epidemic from the "outside" world, the government and health authorities, and can adjust their behavior to avoid or to reduce the risk of being infected. Typical aspects of this adaptability may include simple precautionary measures, such as refraining from potentially dangerous contacts, increasing hygiene, using hand sanitizer and disinfectants, wearing face mask in public places, adjusting a general life style, taking an extra portion of vitamin C in a case of a common cold, or using vaccination in a case of influenza. At the threat of epidemics, the government and health authorities can intervene by promoting immunizations, if available, raising awareness in the population about the current severity of the epidemic, providing access to effective and affordable medicines and tests, working with school authorities, using media and/or administrative pressure, etc. During the covid-19 epidemics we have seen such interventions imposed by the authorities on an unprecedented scale including massive quarantine and social distancing measures, business restrictions, gathering limitations, stay-at-home and shelter-in-place orders and closing the state borders.
In order to account for the adaptability of the population, the assumptions of the standard SIR models, which postulate constant transmission and vaccination rates during the whole time period in question, was revisited in several different ways. The incidence rate of the modified form βSI/(1 + aI) or βSI/(1 + aS) was used to account for the saturation effect with saturation rate a. The first form is based on the assumption that an increase in the number of infective individuals leads to a reduction of the incidence rate; the second form is associated with protective measures taken by susceptible individuals against the infection. The two effects were also combined into the incidence rate of the form βSI/(1 + aS + bI) (Dubey et al. 2015 , Kaddar 2010 for an overview of several models. Their extensions of these models include non-pharmaceutical intervention factors such as quarantine and isolation of patients (Hou et Immunization is a proven and probably the most effective tool for controlling and eliminating infectious diseases. It was hypothesized that a vaccination effort can be more efficient when it is pulsed in time rather than uniform. The effect of pulsed vaccination policy has been studied quite extensively (Agur et al. 1993; Lu et al. 2002) where a detailed comparison of models with constant and pulsed vaccination rate is provided. Piecewise smooth epidemiological models of switched vaccination, implemented once the number of people exposed to a virus reaches a critical level, were studied in Wang et al. (2014) . In yet another class of epidemiological models with adaptive switching behavior, a stochastic switching model was combined with an economic optimal stopping problem to determine the optimal timings for public health interventions (Sims et al. 2016) . Models studied in Chladná et al. (2020) assume that intervention measures are implemented when the number of infected individuals exceeds a critical level, and the intervention stops when this number drops below a different (lower) threshold. Implications of such two-threshold intervention strategies for dynamics of an SIR model were considered both in the case of switched vaccination rate (see Section 2.1) and switched transmission rate.
Multiple factors can influence the willingness of an individual to receive immunization depending on the perceived risk of contracting the disease, risk of possible complications, personal beliefs, etc. These risks vary with age, health conditions, lifestyle and profession. Further, interventions of the health authorities and administrative measures at the level of a county or state can vary in scale depending on the availability of resources, the local economic situation and other factors. All these variations lead to the heterogeneity of the response of the population to the advent of an epidemic.
In this paper, we propose a variant of the SIR model with a heterogeneous response and analyze how the heterogeneity affects the dynamics. We focus on the scenario when interventions of the health authorities affect the vaccination rate (assuming vaccines are available) while the transmission rate remains constant; the case of variable transmission rate will be considered elsewhere. As a starting point, we adopt the approach of Chladná et al. (2020) to modeling the homogeneous switched response of a subpopulation to the varying number of infected individuals by a two-threshold two-state relay operator (as described in Section 2.1). To reflect the heterogeneity, multiple subpopulations are considered, each characterized by a different pair of switching thresholds. In order to keep the model relatively simple, we apply averaging under further simplifying assumptions. The main simplification is that perfect mixing of the population is assumed. This leads to a differential model with just two variables, S and I, but with a complex operator relationship between the vaccination rate v and the density of the infected individuals I. As such, this operator relationship, known as the Preisach operator, accounts for the heterogeneity of the response. Heterogeneity of intervention policies can be modeled in a similar fashion (see Section 2.2).
We show that the system with the Preisach operator is amenable to analysis when interpreted as a switched system (Bernardo et al. 2008 ) associated with a one-parameter family of nonlinear planar vector fields Φ u = Φ u (I, S) (where u ∈ R is a parameter). Between the switching moments, a trajectory of the system is an integral curve of a particular vector field. The Preisach operator imposes non-trivial rules for switching from one vector field to another. Some intuition can be drawn from dynamics of systems with dry friction such as models of presliding friction behavior (Al-Bender et al. 2004; Ruderman 2011), and population models with theresholds (Meza et al. 2009 ).
Using the switched systems approach, we show that if R 0 ≤ 1, then the infection-free equilibrium is globally stable. In the case of R 0 > 1, the bi-stable nature of an individual response leads to multi-stability in the aggregated model. Namely, the system has a connected set of endemic equilibrium states characterized by different proportion of infected and immune individuals 1 . In this case, we show that every trajectory converges either to one of the endemic equilibrium states or to a periodic orbit corresponding to the recurrence of the disease. Under additional natural assumptions, we prove the global stability of the set of endemic equilibrium states by adapting the method of Lyapunov functions. Each vector field Φ u has a global Lyapunov function V u = V u (I, S). We establish the global stability of the switched system by controlling the increment V u (I 1 , S 1 ) − V u (I 2 , S 2 ) of the Lyapunov function along a trajectory between the switching points, and the difference V u1 (I, S) − V u2 (I, S) of the Lyapunov functions at those points. Numerical analysis of SIR models with the Preisach operator was previously performed in (Pimenov et al. 2010 , Pimenov et al. 2012 The paper is organized as follows. In the next section we present the model and remind the definition of the Preisach operator. In Section 3 some preliminary properties of the Preisach operator and the model are discussed, including hysteresis loops, equilibrium states and the global stability of the infection-free equilibrium in the case R 0 ≤ 1. Sections 4 and 5 present the main results on dynamics in the case R 0 > 1 and their proofs.
We consider the following SIR model
with an additional feedback loop, which relates the variable vaccination rate v = v(t) to the concurrent and past values of the density I = I(t) of the infected population. In the main part of the paper, it is assumed that the function I : R + → R is mapped to the function v : R + → R by the so-called continuous Preisach operator, which is described in the following sections. In order to motivate and explain the nature of the assumed operator relationship between I and v, we first briefly discuss the non-ideal relay operator in the same context. Regardless of the specific form of the feedback, one can see that the sum I + S + R is conserved by system (2) , and the last equation is redundant. Without loss of generality, we can interpret S, I, R as relative densities assuming that I + S + R = 1 at all times. We denote δ = µ + γ and rewrite the system asİ
Note that the domain
is flow-invariant for this system. Indeed, in this region,İ ≤ (β − δ)I anḋ
(where we use δ > µ), which implies the statement. We will consider trajectories from the domain D only.
In Chladná et al. 2020 , the relationship between the density of the infected population, I = I(t), and the vaccination rate was modeled by the simplest hysteretic operator called the non-ideal relay, which is also known as a rectangular hysteresis loop or a lazy switch (Visintin 1994) . The relay operator is characterized by two scalar parameters α 1 and α 2 , the threshold values, with α 1 < α 2 . We will use the notation α = (α 1 , α 2 ). The input of the relay is an arbitrary continuous function of time, I : R + → R. The state ν α (·) equals either 0 or 1 at any moment t ∈ R + . If the input value at some instant is below the lower threshold value α 1 , then the state at this instant is 0 and it remains equal to 0 as long as the input is below the upper threshold value α 2 . When the input reaches the value α 2 , the state switches instantaneously to the value 1. Then, the state remains equal to 1 as long as the input stays above the lower threshold value α 1 . When the input reaches the value α 1 , the state switches back to 0. This dynamics is captured by the input-state diagram shown in Figure 1 . In particular, the input-state pair (I(t), ν α (t)) belongs to the union of the two horizontal rays shown in bold in Figure 1 at all times.
The above description results in the following definition. Given any continuous input I : R + → R and an initial value of the state, ν α (0) = ν 0 α , satisfying the constraints
the state of the relay at the future moments t > 0 is defined by
This time series of the state, which depends both on the input I(t) (t ≥ 0) and the initial state ν α (0) = ν 0 α of the relay, will be denoted by
By definition, the state satisfies the constraints ν α (t) = 1 whenever I(t) ≥ α 2 ; ν α (t) = 0 whenever I(t) ≤ α 1 (9) at all times. Further, the function (7) has at most a finite number of jumps between the values 0 and 1 on any finite time interval t 0 ≤ t ≤ t 1 . If the input oscillates between two values I 1 , I 2 , such that I 1 < α 1 < α 2 < I 2 , then the point (I(t), ν α (t)) moves counterclockwise along the rectangular hysteresis loop shown in Figure 1 .
In Chladná et al. (2020) , it was assumed that interventions of the health authority change the vaccination rate according to the following rules. The vaccination rate is switched from a lower rate v nat to a higher rate v int := v nat + q 0 , Fig. 1 The non-ideal relay operator defined by (7) q 0 > 0, when the density of the infected population reaches a threshold value α 2 . The intervention stops when the number of infected individuals drops below a lower threshold value α 1 , at which point the vaccination rate returns to its lower value v nat . Using the definition (7) of the non-ideal relay operator (8) , this leads to the formula
Coupling of this operator equation with dynamic equations (3) results in a switched system. As shown in Chladná et al. (2020), switched system (3), (10) exhibits different dynamic scenarios depending on its parameters. In particular, it can have a globally stable endemic equilibrium. Alternatively, a locally stable endemic equilibrium coexists with a stable periodic orbit. Along this orbit, the vaccination rate (10) switches between the values v nat and v nat + q 0 twice per period.
Now, we consider a model, in which several vaccination laws of the form (10), with different thresholds α, are combined either because the health authority employs multiple intervention strategies or because different individuals respond differently to the advent of an epidemic.
Assume that the health authority has multiple intervention policies (numbered n = 1, . . . , N ) in place, each increasing the vaccination rate by a certain amount q n while the intervention is implemented, in order to provide a response, which is adequate to the severity of the epidemic. 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 α n = (α n 1 , α n 2 ). Under these assumptions, the vaccination rate in system (3) is given by
This formula defines a mapping from the space of continuous inputs I : R + → R to the space of piecewise constant outputs v : R + → R, which is known as the discrete Preisach operator ( Krasnosel'skii et al. 1983), see Figure 2 .
In particular, the vaccination rate (11) is set to change at multiple thresholds α n 1 , α n 2 . On the other hand, individuals can respond differently to dynamics of the epidemic and interventions of the health authority. In particular, the willingness to receive vaccination can vary significantly from individual to individual for the same level of threat of contracting the disease. In order to account for the heterogeneity of the individual response, let us divide the susceptible population into non-intersecting subpopulations parametrized by points α of a subset Π ⊂ {α = (α 1 , α 2 ) : α 1 < α 2 } of the α-plane. Assuming that the vaccination rate for a subpopulation labeled α is given by (10) with q = q(α), the total vaccination rate equals
where the probability measure F describes the distribution of the susceptible population over the index set Π (the set of threshold pairs). 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.
Let us consider a rigorous definition of the Preisach operator (12) with an absolutely continuous measure F (Krasnosel'skii et al., 1989). It involves a collection of non-ideal relays R α , which respond to the same continuous input I = I(t) independently. The relays contributing to the system have different pairs of thresholds α = (α 1 , α 2 ) ∈ Π, where we assume that Π is measurable and bounded; the α-plane is called the Preisach plane. The output of the continuous Preisach model is the scalar-valued function
where q : Π → R is a positive bounded measurable function (measure density) representing the weights of the relays; and, ν 0 α is the initial state of the relay R α for any given α ∈ Π. The function ν 0 = ν 0 α : Π → {0, 1} of the variable α = (α 1 , α 2 ) is referred to as the initial state of the Preisach operator. It is assumed to be measurable and satisfy the constraints (5), (6) , in which case the initial state-input pair is called compatible. These requirements ensure that the integral in (13) is well-defined for each t ≥ 0 and, furthermore, the output v(·) of the Preisach model is continuous. The input-output operator of the Preisach model defined by (13) will be denoted by
where both the input I : R + → R and the initial state ν 0 = ν 0 α (which is compatible with the input) are the arguments; the value of this operator is the output v :
In what follows we consider system (3) with the vaccination rate defined by equation (13).
We begin by discussing some of the properties of the Preisach operator and system (3), (13).
The Preisach operator (14) 
for any τ ≥ 0 with
Let us denote by U the set of all triplets (I 0 , S 0 , ν 0 ), where (I 0 , S 0 ) ∈ D (see (4)) and the initial state ν 0 of the Preisach operator is compatible with I 0 . The global Lipschitz estimate (15) ensures (for example, using the Picard-Lindelöf type of argument) that for given (I 0 , S 0 , ν 0 ) ∈ U, system (3) with the Preisach operator (14) has a unique local solution with the initial data I(0) = I 0 , S(0) = S 0 and the initial state ν 0 (see, for example, the survey in Leonov et al. 2017 ). Further, the invariance of D implies that each solution is extendable to the whole semi-axis t ≥ 0. These solutions induce a semi-flow in the set U, which is considered to be the phase space of system (3) and is endowed with a metric by the natural embedding into the space R 2 ×L 1 (Π; R). This leads to the standard definition of local and global stability including stability of equilibrium states and periodic solutions. In particular, an equilibrium is a triplet (I 0 , S 0 , ν 0 ) ∈ U and a periodic solution is a periodic function (I(·), S(·), ν(·)) : R + → U where the last component viewed as a function ν : R + × Π → {0, 1} of two variables t ∈ R + and α ∈ Π is given by (8) .
The vaccination rate (13) at an equilibrium is constant, while for a periodic solution the vaccination rate is also periodic with the period of I and S.
We call a periodic input I = I(t) simple if each local minimum of this input is its global minimum and each local maximum is its global maximum. That is, the input increases from its global minimum value to its global maximum value and then decreases back to its global minimum value over one period. Let us consider inputs I : R + → [0, 1].
The following property of the Preisach operator is called monocyclicity: for any T -periodic input I(t), t ≥ 0, and any admissible initial state of the Preisach operator, the corresponding output v(t) = (P[ν 0 ]I)(t) satisfies v(t + T ) = v(t) for all t ≥ T . Further, for a simple periodic input, the output is defined by
for t ≥ T , where the Lipschitz continuous functionsv(·),v(·) increase and satisfyv (I) <v(I) for I 1 < I < I 2 ;
with I 1 := min t≥0 I(t),
In other words, a simple periodic input produces a closed (hysteresis) loop formed by the graphs of the functionsv(·),v(·) on the input-output diagram (after the moment T ), see Figure 3 (a). 
Therefore, for a simple periodic input, the difference ∆v =v −v is a nonnegative function of three scalar variables, ∆v = ∆v(I, I 1 , I 2 ), defined on the domain 0 ≤ I 1 ≤ I ≤ I 2 ≤ 1. Since q = q(α 1 , α 2 ) is bounded, the following quantity is well-defined and finite:
In particular, L ≤ K (cf. (16)). The quantity (22) measures the maximal width of the hysteresis loops of the Preisach operator relative to their length. It plays an important role in the following. We will assume that
More generally, let 0 ≤ t 0 < t < t and let us consider an input I = I(t), which increases on the interval [t 0 , t ] and then decreases on the interval [t , t ]. Such inputs will be also called simple on the interval [t 0 , t ]. Suppose that the input values I 0 = I(t 0 ), I 2 = I(t ) and I 1 = I(t ) satisfy I 0 ≤ I 1 < I 2 . Then it follows from the definition of the Preisach operator that formula (17) is valid, where the increasing functionsv(·),v(·) satisfy (18), but relations (19) and (21) do not necessarily hold. Formulas (17) and (18) are also true if the input I = I(t) first decreases on the interval [t 0 , t ] and then increases on the interval [t , t ], and the input values I 0 = I(t 0 ), I 1 = I(t ) and I 2 = I(t ) satisfy I 1 < I 2 ≤ I 0 , see see Figure 3 (b). Functionsv =v(I),v =v(I) in (17) will be referred to as an ascending branch and a descending branch of the Preisach operator. It is important to notice that these functions depend on the states ν α (t 0 ), α ∈ Π, of the relays at the moment t 0 > 0, which in turn depend on the initial states ν 0 α of the relays at the moment t = 0 and the value of the input I on the interval [0, t 0 ] (cf. (8)). As such, on any interval of monotonicity of the input, the input-output pair (I, v) follows one of infinitely many possible branches of the Preisach operator, and a particular branch followed by the input-output pair is uniquely defined by the prior history of the input variations and the initial states of the relays.
We will assume that
Due to this assumption and the compatibility constraint (9), the inclusion α ∈ Π implies that all the relays are in state ν α = 0 when I = 0. Therefore, system (3), (13) has a unique infection-free equilibrium
in which the state ν 0 = ν 0 α : Π → {0, 1} of the Preisach operator is the identical zero. According to (13) , the vaccination rate at this equilibrium is minimal and equals v nat .
In addition, if
then system (3), (13) also has a family of endemic equilibrium states
with (I * , S * ) ∈ D, where the vaccination rate v 0 is related to the state ν 0 = ν 0 α : Π → {0, 1} of the Preisach operator by
These equilibria form a connected set in U. Further, the set (27) includes equilibrium states with different vaccination rates v 0 and different proportions of the infected and recovered populations, while the fraction of the susceptible individuals is the same at all these states.
If Π is different from (24) and includes points (α 1 , α 2 ) with α 1 < 0, 0 ≤ α 2 ≤ 1, then the set of infection-free equilibrium states is also infinite and connected. Depending on the parameters, it can be either an attractor or repeller or include equilibrium states with different stability properties (this is not unlike the classical SIR model with zero mortality rate). Further, in this case, a solution which starts from small infected population and converges to an infection-free equilibrium is characterized by higher vaccination rate at the end than at the beginning. This is because the relays with α 1 < 0 switch from state 0 to state 1 but never switch back due to the positivity of the input I. (3), (13) lie in the infinite-dimensional phase space U of triplets (I, S, ν 0 ). Slightly abusing the notation, we will sometimes also refer to the two-dimensional curve (I(t), S(t)) as to a trajectory, omitting the component (8) in the state space of the Preisach operator.
then the infection-free equilibrium (25) is the global attractor of system (3), (13) . If the opposite inequality (26) holds, then any trajectory of system (3), (13) , which has at most finite number of intersections with the nullclineİ = 0 (the line S = S * = δ/β), converges to an endemic equilibrium.
Proof If δ/β ≥ 1, then the first equation of (3) implies thatİ < 0 in D. Therefore, on any given trajectory of (3), (13), the vaccination rate is defined by v(t) =v(I(t)) (cf. (17)), where the functionv(I) is a continuous descending branch of the Preisach operator; this branch depends on the initial state ν 0 = ν 0 α . Thus, a trajectory of (3), (13) is simultaneously a trajectory of the ordinary differential systemİ = βIS − δI, S = −βIS −ṽ(I)S − µS + µ (30) withṽ(·) =v(·) depending on ν 0 = ν 0 α . Sinceİ < 0, each trajectory of (30) converges to the infection-free equilibrium (I * , S * ) for any branchṽ(·) =v(·), and the result follows.
If 1 > δ/β and (29) holds, then the second equation of (3) impliesṠ < 0 for S ≥ δ/β = S * in D. Hence, all trajectories of (3), (13) enter the domain {(I, S) ∈ D : S < S * } and remain there for all sufficiently large t. In this domain,İ < 0 as follows from the first of equations (3) . Therefore, the same argument as we used in the case δ/β ≥ 1 above shows that all the trajectories of system (3), (13) converge to the infection-free equilibrium (25).
Finally, assume that (26) holds, and suppose that a trajectory of system (3), (13) has at most finite number of intersections with the line S = S * whereİ = 0. Then, after the last intersection, the I-component of the trajectory either strictly decreases or strictly increases with t. In either case, the monotonicity of I(t) implies that the trajectory of the (3), (13) (after its last intersection with the line S = S * ) is simultaneously a trajectory of the ordinary differential system (30) whereṽ(·) is either a descending or an ascending branch of the Preisach operator. Due to the fact that any branchṽ(I) increases, relation (26) implies that system (30) has a unique endemic equilibrium (I * , S * ) given by
Furthermore, system (30) has a global Lyapunov function (Korobeinikov et al. 2002 , Korobeinikov et al. 2004 )
This implies convergence to the endemic equilibrium point.
Proposition 1 does not cover those trajectories that have infinitely many intersections with the nullcline S = S * in the case when relation (26) holds.
Theorem 1 Let (26) hold. Any trajectory of system (3), (13) (starting from any initial values from the region I > 0, S > 0, S + I ≤ 1 and any admissible initial state of the Preisach operator) converges either to an endemic equilibrium or to a simple periodic orbit.
Proof Consider a trajectory (I(t), S(t)) which does not converge to an equilibrium point. Due to Proposition 1, it has infinitely many intersections with the line S = S * at points I(t k ) = I k , k = 1, 2, ... with t k < t k+1 (where we can assume without loss of generality that I 1 > I 2 ). If for some k we have I k < I k +2 < I k +1 , then let us compare the arc Γ k +2 of the trajectory (I(t), S(t)) connecting the points (I k +2 , S * ) and (I k +3 , S * ) with its arc Γ k connecting the points (I k , S * ) and (I k +1 , S * ). Note that on each arc Γ k the vaccination rate follows a particular branch of the Preisach operator, which we will denote asv k (cf. (17)).
Both arcs Γ k +2 and Γ k lie above the nullcline S = S * , both go from left to right (hence I k +3 > I k +2 ), and Γ k starts to the left of Γ k +2 . Since for the internal points of these arcs,
andv k +2 (I) >v k (I), we see that Γ k and Γ k +2 cannot intersect except at the end point, hence I k < I k +2 < I k +3 ≤ I k +1 as required. Further, if I k +3 = I k +1 , then due to forward uniqueness the trajectory becomes periodic starting from the moment t = t k +1 , i.e. I k +2j+1 = I k +1 , I k +2j+2 = I k +2 for all j = 1, 2, . . . Similarly, relations I k +1 > I k +3 > I k +2 imply I k +1 > I k +3 > I k +4 ≥ I k +2 , and if I k +4 = I k +2 , then the trajectory becomes periodic after the moment t = t k +2 .
Combining the above two results, we see that if either I k < I k +2 < I k +1 or I k +1 > I k +3 > I k +2 for some k , and the trajectory does not become periodic, then
Therefore, the trajectory converges to a periodic orbit oscillating between the points (I , S * ) and (I , S * ) with I = lim j→∞ I k +2j and I = lim j→∞ I k +2j+1 (or an equilibrium if the two limits coincide).
The only remaining alternative to this scenario is to have either · · · > I 5 > I 3 > I 1 > I 2 > I 4 > I 6 > · · · or · · · < I 5 < I 3 < I 1 < I 2 < I 4 < I 6 < · · · for all I k . In this case, again, the limit is a periodic trajectory oscillating between the points (I , S * ) and (I , S * ) unless inf I k = 0. However, it is easy to see that actually inf I k ≥ ε > 0. Indeed, on each arc Γ k which lies below the line S = S * , the component I(t) of the solution decreases, and the vaccination rate is given by v(t) =v k (I(t)) for some descending branchv k (·) of the Preisach operator. Therefore, the Lyapunov function V k (I, S) given by (31), (32) with v(·) =v k (·) decreases along the segment Γ k of the trajectory, hence the value of V k (·, ·) at the left end of Γ k is less than at the right end. But the functions V k (·, S * ) : (0, 1] → R are uniformly bounded for I ∈ [δ * , 1], δ * > 0, and satisfy V k (I, S * ) ≥ −I * k ln(I/I * k ), where I * k is the solution of (31) forṽ(·) =v k (·). Relation (26) ensures that inf I * k ≥ ε 0 > 0, hence inf k V k (I, S * ) → ∞ as I → 0+, and consequently the fact that the set of values of V k (·, ·) at the right ends of the arcs Γ k is bounded implies that the left ends satisfy inf I k ≥ ε > 0. 4.2 Sufficient conditions for global stability of the set of endemic equilibrium states.
In the rest of the paper, we derive sufficient conditions which ensure the global convergence to endemic equilibrium states. We make the following assumption:
(A) Each input-output loop of the Preisach operator corresponding to a simple periodic input is convex. In other words, the functionv(·) in (17) is convex and the functionv(·) in (17) is concave. 
We need to show that κ < 0, i.e. V SS (V I ) 2 + V II (V S ) 2 > 0, which for the function (32) is equivalent to Due to Theorem 1, it suffices to show that system (3), (13) has no simple periodic solutions if the quantity L defined by (22) is sufficiently small. An explicit estimate for L will be established in the proof.
To explore how the heterogeneity of the response of the susceptible population to the advent of an epidemic can affect dynamics of system (3), we consider the aggregate vaccination rate (13) with the Gaussian density
where the normalizing parameter A = A(α m1 , α m2 , σ) ensures that the integral of q over the domain (24) equals 1. The limit σ → 0 corresponds to the perfectly homogeneous response (10), i.e. the vaccination rate switches from the value v nat = 0 to the value v int = 1 at the switching threshold I = α m2 and switches backwards at the threshold I = α m1 . Increasing the variance σ 2 of the (truncated) Gaussian distribution corresponds to increasing the heterogeneity of the response within the susceptible population. Figure 4 presents an example of the convergence to a periodic cyclic behavior for small σ > 0. This scenario for system (3), (10) with one non-ideal relay was studied in Chladná et al. (2020) . Further, Figure 4 shows that as σ increases, the periodic behavior is replaced with the convergence to an endemic equilibrium. This is in agreement with Theorem 2 because the quantity (22) decreases with increasing σ. Indeed, Theorem 2 ensures the global stability of the set of endemic equilibrium states when the quantity (22) becomes sufficiently small.
The trajectory corresponding to the more heterogeneous response (σ = 0.01) in Figure 4 converges to an endemic equilibrium state with the densities I * = 0.00063, S * = 0.056 of the infected and susceptible populations, respectively. The vaccination rate at this equilibrium is v * = 0.0041 (week −1 ). The trajectory corresponding to the more homogeneous response (σ = 0.0009), which converges to the periodic orbit, exhibits a lower initial peak of the infected population during the transient than the trajectory of the heterogeneous system. The average density of the susceptible population for the periodic trajectory,S = 0.057, is close to S * . The density of the infected population along the periodic trajectory is much higher than I * at its peaks, but the average densityĪ = 0.00016 is significantly lower than I * . This agrees with the fact that the average vaccination ratev = 0.023 (week −1 ) is significantly higher than v * .
The following inequalities will be systematically used: 
We prove the theorem by contradiction. Let us assume that there exists a simple periodic solution (I(t), S(t)), for which I(t) increases from I 1 to I 2 and then decreases from I 2 to I 1 on a period with I 1 < I 2 . Let us denote byv(I) andv(I) the two branches of the Preisach operator corresponding to the increasing I and decreasing I of this solution, respectively, hence relations (17) -(20) hold. Then system (3), (13) has an endemic equilibrium (Ī * , S * ) defined by equation (31) withṽ(·) =v(·) and an endemic equilibrium (Î * , S * ) defined by equation (31) withṽ(·) =v(·). Further, since (I 1 , S * ) and (I 2 , S * ) are the turning points of the periodic solution,
On the other hand, according to (31),
Combining these four relations with (18) and recalling thatv andv are increasing functions, we see that
Recall the definition of the functionsV (I, S) andV (I, S), see Lemma 1. We now consider the following trivial identity
and estimate the differences participating in it to establish a lower bound for L. 
Since the partΓ of the periodic trajectory lies outside the level setV (I, S) ≤ V (I 2 , S * ) and the line segment S = S * + h,Ī * ≤ I ≤ I b belongs to this set, we have S ≥ S * + h for (I, S) ∈Γ ,Ī * ≤ I ≤ I b . Therefore, on the interval [Ī * , I b ], relation (40) implies
We integrate this inequality along the trajectory over the interval I ∈ [Ī * , I b ] and using the monotonicity ofV which follows from (41), we obtain
Using inequality (36) and relation (43), we estimate the right hand side of (44) as follows:
Therefore,V
In order to estimate S M − S * , we evaluate the functionV at the point (I 2 , S * ) using (32) and substitute the result into (42) to obtain
We estimate the left hand side of (46) from above using
Further, we find a lower bound of the right hand side of (46) using estimate (37):
Combining the last two inequalities with (46), we obtain
Substituting this relation into (45), we finally arrive at
where the last inequality holds due to I b ≤ I 2 ≤ 1, S M ≤ 1 and S * = δ/β.
Estimation ofV (I 1 , S * ) −V (I 2 , S * ). Now we consider the lower partΓ of the trajectory with decreasing I and S < S * . We slightly modify the above argument. The relation
which is similar to (40), implies dV dI > 0,
Hence,Γ lies outside the level setV (I, S) ≤V (I 1 , S * ). The bottom point (with the smallest S) of the level lineV (I, S) =V (I 1 , S * ) is defined by ∂V /∂I = 0, therefore I =Î * at this point. Denoting its S-component by S m , we obtain V (I 1 , S * ) =V (Î * , S m ) = S m − S * ln S m S * +Î * ,
i.e. I 2 −Ī * ≥ 0,Î * − I 1 ≥ 0, I 2 −Ī * +Î * − I 1 ≥ 1 − L β (I 2 − I 1 ), (60)
where L/β < 1 according to (23) . Let us find the minimum value F min of the function F (x, y) = Ax 2 + By 2 under the constraints x, y ≥ 0 and x + y ≥ (1 − L/β)(I 2 − I 1 ). Clearly, the minimum value is achieved for x + y = (1 − L/β)(I 2 − I 1 ) and equals
Hence, due to (60), the left hand side of (59) satisfies
Recalling the definition of A and B, this is equivalent to
5.4 Estimates of I 1 , S m .
Denote v max := 0≤α1<α2≤1 q(α 1 , α 2 ) dα 1 dα 2 .
hence
From relations (58) andv
(where (65) follows from (15)) it follows that µ(1 − S * ) S * = βÎ * +v(Î * ) ≤ βÎ * + KÎ * + v nat ,
where, due to assumption (26),
If v max S * ≤ µ(1 − S * ), then relations (64), (66) imply
On the other hand, if v max S * > µ(1 − S * ), then
Combining the two cases,
where a + = a for a > 0 and a + = 0 for a ≤ 0.
Finally, we obtain a lower bound for S m . From (54) and (58) it follows that −S * ln S m S * ≤ S * − S m + I 1 −Î * −Î * ln
Therefore,
β(S * ) 2 ln I 1 I * and using (67) we arrive at S m ≥ S * exp −1 + µ(1 − S * ) β(S * ) 2 ln ρ 1 .
Thus, we have shown that the existence of a simple periodic orbit implies estimates (61), (67) and (68), which establish a lower bound L ≥ L 0 > 0 on the quantity (22) . This completes the proof.
We considered an SIR model with vaccination, where we assumed that the vaccination rate changes in response to dynamics of the epidemic. We modeled the adaptive response of an individual to the varying number of active cases by a two-state two-threshold switch, and the aggregate response of the susceptible population by the Preisach operator. This operator relationship between the vaccination rate and the number of active cases accounts for the heterogeneity of the response among the susceptible individuals.
If the basic reproduction number satisfies R 0 > 1, then the infection-free equilibrium is globally stable. On the other hand, if R 0 > 1, then the system has a connected infinite set of endemic equilibrium states. In this case, we showed that each trajectory converges either to an endemic equilibrium or to a periodic orbit. This is in agreement with Chladná et al. (2020) where a simpler system with the homogeneous response modeled by a single twostate two-threshold switch was considered. Further, we showed that the set of endemic equilibrium states is the global attractor if a certain parameter of the Preisach operator, which is associated with the width of the hysteresis loops relative to their length, is sufficiently small. This parameter decreases with increasing heterogeneity of the adaptive response among the susceptible population. Based on these results, one can conclude that the heterogeneity of the response promotes the convergence to an endemic equilibrium state, while the homogeneous response may result in recurrent periodic outbreaks of the epidemic.