Oscillations in epidemic models with spread of awareness

We study ODE models of epidemic spreading with a preventive behavioral response that is triggered by awareness of the infection. Previous studies of such models have mostly focused on the impact of the response on the initial growth of an outbreak and the existence and location of endemic equilibria. Here we study the question whether this type of response is sufficient to prevent future flare-ups from low endemic levels if awareness is assumed to decay over time. In the ODE context, such flare-ups would translate into sustained oscillations with significant amplitudes. Our results show that such oscillations are ruled out in Susceptible-Aware-Infectious-Susceptible models with a single compartment of aware hosts, but can occur if we consider two distinct compartments of aware hosts who differ in their willingness to alert other susceptible hosts.


Introduction
The impact of behavioral responses on the progress of an infectious disease has received growing attention in epidemic modeling in recent years [2,8,10,13]. Responses arising from risk perception of the disease are, for instance, avoidance behavior implying breaking off connections with infectious acquaintances (social distancing), and preventive behaviors, like handwashing or wearing face masks [21,22]. In epidemic network models, which take into account the contact network in a population, social avoidance has been modeled by means of several mechanisms of dynamic rewiring [15,19,23,28,32]. Other approaches cast preventive actions into classic epidemic models by dividing each of the susceptible/infectious/removed classes between responsive and non-responsive individuals [12,20], or by explicitly considering a new class of individuals who are both susceptible and aware, with a lowered susceptibility to infection relative to unaware hosts. This approach leads to the class of SAIS models [18,29,30,31].
Following [11,12], among others, here we will consider awareness as a state of knowledge about the prevalence of the disease that will both induce a behavioral response in the given host and that this host is willing to transmit to other hosts. To distinguish this more stringent notion of awareness from the one made in the SAIS models of the previous paragraph, we call the latter type of models reactive SAIS models. These models incorporate direct transmission of information about the disease, so that the spread of awareness is somewhat similar but not entirely analogous to the spread of an infectious disease. See Section 2 for details.
We are interested in two aspects of the question under what circumstances a behavioral response that is induced by awareness can be an effective control measure: whether such models would predict a lowered epidemic threshold and whether the response would prevent future flare-ups from low endemic levels.
For non-reactive SAIS models without decay of awareness, it was shown in [29] that there is a second epidemic threshold that takes into account the spread of awareness. Even if R 0 > 1 in the underlying model of disease transmission, an endemic equilibrium only appears once this second threshold is surpassed. However, this result requires the assumption that awareness will not decay over time. In [18], it was proved that this second epidemic threshold disappears if one assumes that awareness will decay over time. Under this assumption, when R 0 > 1, awareness may drive the endemic equilibrium to very low levels of disease prevalence, but may not eliminate it or change its stability. Here we show that reactive SAIS models with awareness decay in some cases permit a second epidemic threshold so that the endemic equilibrium will disappear and the disease will be driven towards extinction from almost all initial states, even when R 0 > 1.
Will a behavioral response that is induced by awareness prevent, all by itself, future flare-ups from low endemic levels? When awareness is assumed to be permanent and demography is not considered, as in [29,31], the answer is obviously yes. But it is less clear what to expect when awareness decays over time. In the context of ODE models the question translates into one about the existence of sustained oscillations with a significant amplitude. In Section 2 we show that such oscillations are ruled out in SAIS models. This result applies to both reactive and non-reactive SAIS models and holds even if we allow more general functional responses rather than rate constants in the models.
However, in Section 3 we introduce a more general class of models that we call SAUIS models. They have two distinct compartments of aware hosts who differ in their willingness to alert other susceptible hosts. Thus these models embody the degradation of quality of information as it is transmitted from one individual to another. Our approach for modeling this phenomenon adopts, albeit in greatly simplified form, an idea that was introduced in [1] and adopted in [11] for an epidemic context. We show that sustained oscillations can occur in SAUIS models, even if all rate coefficients are constants. In particular, our results clearly demonstrate that degradation of information during transmission processes can be the driving mechanism for the existence of periodic waves of infection, supporting the claim in [1] that such a degradation can reveal important qualitative and quantitative effects.
The remainder of the paper is organized as follows: In Section 2 we formally define reactive SAIS models and prove the results about these models that were described above. In Section 3 we formally define reactive SAUIS models and examine some of there basic properties. In particular, we present numerical explorations both of the dynamics in SAUIS models and of regions of the parameter space that are bounded by Hopf bifurcation points. These results reveal intricate possibilities for the dynamics in SAUIS models.
In Section 4 we briefly review some related models of behavioral epidemiology that predict oscillations and discuss how they differ from ours. We also discuss some possible implications for public health policy and directions of future work.

The model
An SAIS model has three compartments: S (susceptible), A (aware) and I (infectious). Susceptible hosts can move to the A-compartment or to the I-compartment, aware hosts can move to the S-compartment due to awareness decay or to the I-compartment due to infection (albeit at a lower rate than susceptible hosts), and infectious hosts will move to the S-compartment upon recovery.
As was mentioned in the introduction, we will consider awareness as a state of knowledge about the prevalence of the disease that will both induce a behavioral response in the given host and that this host is willing to transmit to other hosts. So, it is natural to assume that awareness can be passed on from aware to unaware individuals like a contagious disease [8]. However, while the force of infection in transmission of actual diseases can usually be assumed to be a linear function of the prevalence, the rate at which susceptible individuals become aware usually will show more pronounced nonlinearities. On the one hand, there will be a saturation effect arising from overexposure to information when the proportion of infectious hosts is very high. On the other hand, when the prevalence of the disease is very low, a careless attitude may prevail that renders the channels of transmitting awareness ineffective. Details may significantly vary between different populations and diseases. In order to allow for maximum flexibility in incorporating these effects, we will investigate models where parameters for transmission of awareness are functions of the disease prevalence rather than rate constants. The sources of nonlinearities mentioned above suggest that these parameters may exhibit near switch-like behavior.
The proportions of hosts in the S-, A-, and I-compartments will be denoted by s, a, i respectively. The rates of change of these fractions are governed by the following ODE model: Here we assume that α i (i) ≥ 0 and δ a (i) > 0 are differentiable functions, α a (i) > 0 is Lipschitz-continuous, and β, β a , δ are constants such that 0 ≤ β a < β and 0 < δ.
The term α i (i)i represents the rate at which a susceptible host becomes aware due to direct information about the disease prevalence. As α i (i) may depend on i, the factor i is strictly speaking redundant in this term. But its inclusion simplifies some calculations. Also, inclusion of the factor i makes the similarity with disease transmission more explicit. If α i (i) is not constant, one can think of the process of obtaining direct information as encountering at least one infectious host and then seeking or re-interpreting independent information about the overall disease prevalence. Thus it seems plausible to assume that α i (i) is low when infectious hosts are observed so rarely that they are not considered indicative of an outbreak, steeply increases around a critical level of disease prevalence, and then levels off.
Similarly, the term α a (i)a represents the rate at which susceptible hosts become aware due to a contact with an aware host during which the latter transmits information about the disease.
The assumption that α a (i) > 0 is the one that really distinguishes our models from non-reactive SAIS models. The latter can be obtained by simply removing the term α a (i) s a from (1).
In view of the above discussion we tend to think of α i (i) and α a (i) as increasing in a sigmoidlike fashion from near 0 when i = 0 to near a saturation constant when i = 1, but these properties are not needed for the results of this section.
The term δ a (i) represents the decay of awareness. It could be a constant or any other positive Lipschitz-continuous function, but in realistic models higher disease prevalence should result in slower awareness decay so that δ a (i) will be a decreasing function of i.
The inequality β a < β embodies the assumption that awareness will lead to adoption of a behavioral response that decreases the rate at which hosts contract the infection.

Nullclines and equilibria
By solving di/dt = 0 we find two parts of the i-nullcline. The first one is given by the horizontal axis i = 0 and the second one is the straight line which has a slope between −1 and 0 under the assumption β a < β. It intersects the horizontal axis i = 0 at the point By solving da/dt = 0 we find the a-nullcline. This curve is implicitly defined by the following equation in the variables i and a: Here we use the assumption that α a (i) > 0. The point (0, 0) always satisfies this equation. The other part of the a-nullcline is given by the graph of the following function a(i) on [0, 1]: Note that a(1) = 0 and a(0) = 1 − δ a (0)/α a (0) if δ a (0) ≤ α a (0), while a(0) = 0 if δ a (0) ≥ α a (0). Moreover, a(i) is continuous and takes on positive values for all i ∈ (0, 1).
By sketching these nullclines in the i-a plane, one can see that the system has three possible equilibria in the first quadrant, namely, Here (a * , i * ) denotes the only equilibrium inside Ω. It exists if and only if Note that this condition guarantees the existence of the endemic equilibrium because the function a(i) is continuous and satisfies a(1) = 0, whereas the i-nullcline is a straight line such that i(0) < 1 with a slope larger than −1 and a-intercept β−δ β−βa (see Figure 1). If we define the basic reproduction numbers of the disease and awareness as and R a 0 := α a (0)/δ a (0), we can interpret intuitively the conditions for the existence of these equilibria. The disease-andawareness-free equilibrium P 1 always exists. The equilibrium P 2 is also disease-free, but has a positive fraction of aware hosts. It exists if, and only if, R a 0 > 1, that is, if in a large and otherwise susceptible population with one aware host, awareness will on average increase. The existence of P 3 is guaranteed if the disease spreads faster than awareness in the early stages of an outbreak, i.e., if R 0 > max{1, R a 0 }. In addition, it is also possible to have an endemic equilibrium when R a 0 > R 0 if β a is close enough to β, i.e., when the reduction of the transmission probability due to awareness is not significant enough.
The Jacobian matrix of system (1) is The eigenvalues of J at P 1 are which shows that conditions for the existence of P 2 and P 3 imply instability of P 1 . At P 2 , the eigenvalues are Note that λ 1 (P 2 ) = −λ 1 (P 1 ) and that λ 2 (P 2 ) > 0 when there exists an interior equilibrium. The former implies that if R a 0 > 1, then P 1 is unstable and P 2 attracts any trajectory on the a-axis; exactly as one would expect from an SIS-like behavior of awareness.
Proof. Let f 1 (a, i) and f 2 (a, i) denote the functions on the right-hand side of the system. The in the interior of Ω, and its divergence is given by for all (a, i) in the interior of Ω. So, the divergence does not change sign and does not take the value 0. Therefore, Dulac's criterion of nonexistence of periodic orbits [26] precludes the existence of a closed orbit lying entirely in Ω.
The next theorem sums up the dynamics of system (1) for the most interesting situation when 0 ≤ β a < δ where the infection could not invade a population that consists entirely of aware hosts.
, β satisfy the conditions that were spelled out below (1). Then the global behavior of the solutions of the system (1) depends as follows on the remaining parameters: (i) If R 0 ≤ 1 and R a 0 ≤ 1, then P 1 is the only equilibrium point and is globally asymptotically stable.
(ii) If R 0 ≤ 1 < R a 0 , then P 1 and P 2 are the only equilibrium points. P 2 is globally asymptotically stable on Ω\{P 1 }. When R 0 < 1, then P 1 is a saddle point. (iii) If R a 0 ≤ 1 < R 0 , then P 1 and P 3 are the only equilibrium points. P 3 is globally asymptotically stable on Ω\{P 1 }. When R a 0 < 1, then P 1 is a saddle point. (iv) If R 0 > 1 and R a 0 > 1, then P 1 is an unstable point and, if λ 2 (P 2 ) < 0, then system (1) has also the equilibrium P 2 which is globally asymptotically stable on Ω\{P 1 }.
For part (ii), if α a (0) > δ a (0), the a-nullcline intersects the a-axis at P 2 , and P 3 is ruled out as in case (i). So P 1 and P 2 are the only equilibria of the system. As we have λ 1 (P 1 ) > 0 and λ 2 (P 1 ) = β − δ ≤ 0, it follows that P 1 is a saddle point when the inequality β < δ is strict. Moreover, as (α i (0)i) ′ ≥ 0, the first row of the Jacobian at P 1 has two nonnegative entries. Thus the eigenvector with eigenvalue λ 1 (P 1 ) cannot intersect the interior of Ω. The eigenvalues of the linearization of the system at P 2 satisfy λ 1 (P 2 ) < 0 and λ 2 (P 2 ) < 0. From Lemma 2.1 and the Poincaré-Bendixson theorem, it follows that P 2 is globally asymptotically stable on Ω\{P 1 }.
Under the assumptions of (iv) and (v), the a-nullcline intersects the horizontal axis at a(0) > 0 and P 2 is always an equilibrium. Since the i-nullcline intersects the a-axis at (β − δ)/(β − β a ), the sign of λ 2 (P 2 ) determines whether this intersection occurs to the left of P 2 , with no interior equilibrium, or to the right of P 2 , with an interior equilibrium P 3 . The former occurs in case (iv) where P 2 is locally asymptotically stable, whereas P 1 is unstable. From Lemma 2.1, Lemma 2.2, and Poincaré-Bendixson theorem, it follows that indeed P 2 is globally asymptotically stable on Ω\{P 1 }. The latter occurs in case (v), where P 2 is a saddle point while P 1 is unstable. Again, from Lemma 2.1, Lemma 2.2, and the Poincaré-Bendixson theorem, it follows that P 3 must be globally asymptotically stable on Ω\{P 1 , P 2 }.
As we already mentioned in the introduction, for non-reactive SAIS models (α a = 0) it was shown in [29] that if there is no awareness decay (δ a = 0), then there is a second epidemic threshold so that all outbreaks will be minor as long as β a < δ < β and α i is sufficiently large. In [18], it was proved that this second epidemic threshold disappears if one assumes that awareness will decay over time. The reason is that in order to suppress epidemic spreading, we need a permanent fraction of aware hosts. If awareness is eventually lost, then the presence of aware individuals created at the early stages of an epidemic is not enough to contain an outbreak, and the classic disease-invasion epidemic threshold β/δ = 1 is recovered.
Here we have seen that such a permanent fraction of aware individuals can be obtained with awareness decay if one assumes the natural hypothesis that awareness is also transmitted from an aware host to a susceptible one (α a > 0). For β a < δ < β and R a 0 > 1, Theorem 2.3 shows that the reactive SAIS-model predicts a second epidemic threshold which is given by λ 2 (P 2 ) = 0. This threshold can be written as When the left-hand side of (2) exceeds the right-hand side, the model predicts a globally stable endemic equilibrium, but if the right-hand side of (2) exceeds the left-hand side, the model predicts that each outbreak will be driven to a disease-free equilibrium with some positive proportion of aware hosts.
This gives a range of parameter choices for which this second epidemic threshold drives the dynamics. While we have not formally proved a result about the case when δ < β a < β, it is fairly straightforward to see that in this case there will be no second epidemic threshold.
It may be of interest to point out that in the limiting case β a = 0 the behavioral response that is triggered by awareness confers perfect immunity to becoming infectious. This is quite similar to the assumption that is often made about vaccinations. In this case (2) can be written as R 0 = R a 0 . A population that initially contains some aware but no infectious hosts will approach the equilibrium P 2 with a proportion of 1 − 1/R a 0 aware hosts, which happens to be exactly the herd immunity threshold for vaccinations that confer perfect protection if R 0 = R a 0 .

SAUIS models
3.1. The model SAIS models ignore the degradation of information quality as it is transmitted from one individual to another. According to these models, aware individuals would always create aware hosts with the same degree of responsiveness, which seems unrealistic. The following approach to modeling degradation of information was introduced [1] and adopted in [11] for an epidemic context: We assume that direct information about the disease prevalence induces awareness of the risk of infection and maximum observance of protective measures, while subsequent awareness transmission, decreases its impact on an individual's reaction by a constant decay factor. Following this idea, but simplifying it in order to get a manageable model, we will introduce a new class of individuals whose behavioral response has been induced indirectly through contacts with aware individuals. In contrast to aware individuals, the latter are assumed to be unwilling to convince other people about the risk. They may also show a weaker behavioral response. We will call them unwilling individuals and let u denote their proportion in the population. Transmission of awareness can create both aware and unwilling hosts.
We will investigate here the simplest equations describing such dynamics that are constructed in direct analogy to the reactive SAIS models of Section 2: Here α u a is the rate at which susceptible hosts become unwilling after having a contact with an aware host, and δ u is the rate of awareness decay of the unwilling hosts. This model implicitly assumes that aware hosts first turn into unwilling hosts before possibly entering the susceptible compartment. All other terms play the same role as the corresponding terms in the reactive SAIS model.
We assume that all rate constants with the possible exception of β a , β u are positive, and 0 ≤ β a , β u < β. Similarly to the SAIS model, one could allow some of the rate coefficients to depend on i; see Section 4 for a brief discussion of these extensions of the model. However, we specifically restrict our attention here to the case of constant rate coefficients to emphasize the point that their dependence on i is not a necessary condition for observing periodic oscillations.
Lemma 3.1. The region Ω = {(a, u, i) ∈ R 3 + | 0 ≤ a + u + i ≤ 1} is positively invariant. Proof. Direct inspection of the system (3) shows that (da/dt)| a=0 ≥ 0 and the inequality is strict when si > 0. Similarly, (du/dt)| u=0 ≥ 0, and the inequality is strict when a > 0. On the other hand, (di/dt)| i=0 = 0. It follows that the (a, i)-and (u, i)-coordinate planes repel the trajectories and that the (a, u)-plane is invariant. So, we only need to see that trajectories cannot cross the boundary a + u + i = 1. If v denotes the vector field defined by the right-hand side of the system, we can see this by computing the scalar product of v on this boundary with the outward-pointing normal vector n = (1, 1, 1). Since s = 0 in this region of the boundary, we get v · n = −δ u u − δ i ≤ 0. The inequality is strict except at the point (1, 0, 0), where da/dt = −du/dt < 0 = di/dt. Therefore the vector field on the boundary a + u + i = 1 never points towards the exterior of Ω.

Possible equilibria
The system (3) can have up to three types of equilibria in Ω. The first one is the disease-free, awareness-free equilibrium P 1 = (0, 0, 0). The second one is the disease-free equilibrium P 2 = (a * 0 , u * 0 , 0). From the first line of (3) we have s * 0 = δ a /α a . In view of the equality u * 0 = 1 − a * 0 − δ a /α a , the second line of (3) implies: The third kind of possible equilibrium is an endemic equilibrium, i.e., an interior point As the upper panel of Figure 2 shows, there may be more than one interior equilibrium.

Existence and linear stability of equilibria
The disease-free, awareness-free equilibrium P 1 = (0, 0, 0) always exists. Evaluating the Jacobian matrix of system (3) at P 1 we have whose eigenvalues are So, as expected, when β > δ, i.e., when R 0 > 1, this equilibrium is unstable. Moreover, as in the reactive SAIS model, when R a 0 := α a /δ a > 1, this equilibrium is unstable independently of the sign of β − δ and P 2 becomes biologically meaningful. By (4), the equilibrium P 2 exists in Ω\{P 1 } if, and only if, R a 0 = α a /δ a > 1. In this case it must be in the interior of the intersection of the a-u-plane with Ω. The Jacobian matrix of (3) at this equilibrium is with s * 0 = δ a /α a . Here we have used the observation that the expression α a (s * 0 −a * 0 )−δ a obtained from direct computation of the partial derivative in the upper left corner of J(P 2 ) simplifies to −α a a * 0 . It follows that the determinant of the submatrix formed by the intersection of the first two rows of J(P 2 ) with its first two columns is positive and its trace is negative, so J(P 2 ) has two eigenvalues that are either negative or have negative real parts.
The third eigenvalue λ 3 ( then λ 3 (P 2 ) will be positive. By (4), the values of a * 0 , u * 0 do not depend on the disease transmission parameters, and it can be seen from (8) that there are large regions of the parameter space where λ 3 (P 2 ) is positive and large regions where it is negative while β > δ.

Transcritical bifurcations
3.4.1. Transcritical bifurcation at R a 0 = 1 Assume β < δ so that λ 3 (P 1 ) < 0. As λ 1 (P 1 ) changes from negative to positive when the bifurcation parameter R a 0 = αa δa increases past 1, the equilibrium P 1 loses its stability at the bifurcation value 1. Simultaneously, P 2 enters the biologically feasible region Ω and becomes locally asymptotically stable as explained in the previous subsection. Moreover, at the bifurcation value P 1 and P 2 coincide. Note also that the plane i = 0 is invariant for the system (3), and when a * 0 is negative, then the determinant of the submatrix formed by the intersection of the first two rows of J(P 2 ) with its first two columns is negative. This implies that at least right before it crosses into the biologically feasible region, P 2 is unstable.
For the analysis of the bifurcations of the endemic equilibrium P 3 from P 1 and P 2 , we will use standard results for the existence of a transcritical bifurcation (see the criterion that is given right after Sotomayor's Theorem in [26]). In order to use them, we introduce some notation. Let f denote the vector defined by the right-hand side of system (3) and let f µ be the vector of partial derivatives of its components f i with respect to a bifurcation parameter µ. Moreover, let Df µ be the Jacobian matrix of f µ and let D 2 f (y, y) be the column vector with components D 2 f (y, y) k := j,l ∂ 2 f k ∂x j ∂x l y j y l , where y is a vector in R 3 , x 1 = a, x 2 = u, and x 3 = i. We will use f BP , f BP µ to indicate that the relevant objects are computed at the bifurcation point.

Transcritical bifurcation at
The endemic equilibrium P 3 can bifurcate from P 1 when α a < δ a (i.e., R a 0 < 1). In particular, since λ 3 (P 1 ) = β − δ is a simple eigenvalue, a bifurcation occurs for β = δ (i.e., at R 0 = 1). Taking β as the bifurcation parameter and evaluating the Jacobian matrix J(P 1 ) at the bifurcation point, it follows that the row vector u = (0, 0, 1) and the column vector v = (1, (δ a + α u )/δ u , (δ a − α a )/α i ) T are the left and right eigenvectors for λ 3 = 0, respectively. Moreover, f β = (0, 0, (1 − a − u − i)i) T . A straightforward computation at the bifurcation point leads to Note that the inequality α a < δ a and the assumption 0 ≤ β a , β u < β of the SAUIS model imply that all coordinates of v and coefficients in 3) are positive, so that the whole expression becomes negative and, in particular, nonzero. So, when R a 0 < 1 system (3) experiences a transcritical bifurcation as β crosses the bifurcation value β = δ [26]. Moreover, Theorem 4.1 in [5], together with the inequality > 0 in 2) and the inequality < 0 in 3), implies that the direction of the bifurcation is always the same, namely, system (3) experiences a forward bifurcation at R 0 = 1.

Transcritical bifurcation at λ 3 (P 2 ) = 0
Let us now assume α a > δ a to guarantee the existence of a positive P 2 , and β > δ to allow λ 3 (P 2 ) to be positive for some parameters values. From the discussion surrounding (8) with a * 0 and u * 0 given by (4). That is, at this parameter combination, the endemic equilibrium P 3 bifurcates from P 2 , the disease-free equilibrium with aware individuals.
When the Jacobian matrix (7) is evaluated at the bifurcation point, the value in the lower right corner becomes zero. We get which has the row vector u = (0, 0, 1) as left eigenvector with eigenvalue λ 3 = 0. Since the determinant det J BP 2 = α a a * 0 (δ a + δ u + α u δ a /α a ) > 0 of the submatrix J BP 2 of J BP formed by the first two rows and the first two columns is strictly positive, the first two columns of J BP 2 are linearly independent. Therefore, the third component v 3 of the right eigenvector v with eigenvalue λ 3 = 0 must be different from 0.
If we take µ = β a as the bifurcation parameter, then f βa = (−a i, 0, a i) T . As before, a straightforward computation at the bifurcation point leads to: Recall that v is by definition orthogonal to the first two rows of J BP and these are linearly independent. Thus v cannot also be orthogonal to any vector that is linearly independent of the latter two vectors. Therefore, since v 3 = 0, the inequality u · D 2 f BP (v, v) = 0 will follow whenever the vector given by b := (β − β a , β − β u , β) is linearly independent of the two first rows of J BP . This may not always be the case, but it will be generically true. For instance, since the first two columns of J BP are linearly independent and the parameter α i appears in only one position of J BP , while a * 0 and u * 0 that give the location of P 2 do not depend on it. Since it does not appear in b, one can take α i as a free parameter in order to see whether the previous inequality fails for a particular positive choice of α i given any choice of all other parameters.
Thus the criterion given in [26] after Sotomayor's Theorem guarantees that for a nonempty open set of parameter settings at which P 1 = P 2 ∈ Ω the system (3) experiences a transcritical bifurcation as β a passes through the bifurcation value However, in contrast to what happens at R 0 = 1, the direction of the bifurcation is not always the same. To illustrate this fact, Figure 2 shows an example of forward and backward bifurcations occurring at λ 3 (P 2 ) = 0 for β a < β. The same conclusion holds if we use β or β u as a bifurcation parameter. However, the use of β a seems more appropriate because its value is related to the effectiveness of the response adopted by aware hosts.

Hopf bifurcations
We will see that in contrast to the SAIS model, sustained oscillations are possible in the SAUIS model thanks to the occurrence of Hopf bifurcations. A pair (σ, τ ) of parameters of (3) will be called a Hopf pair if there exists an equilibrium point at which the Jacobian matrix has a pair of pure imaginary eigenvalues. This definition of course depends on chosen values of the other parameters. For simplicity we will always implicitly assume that these other parameters are fixed and suppress them in our notation.
An explicit criterion that specifies whether an n × n matrix M , with coefficients that may depend upon parameters, has a pair of pure imaginary eigenvalues is given in [17]. For our purposes the case when n = 3 and M = J is the Jacobian matrix at endemic equilibrium P 3 is relevant. Let p(λ) = λ 3 + c 2 λ 2 + c 1 λ + c 0 be the characteristic polynomial of J. Its coefficients are c 0 = − det(J), the sum c 1 of the principal minors of J, and c 2 = −trace(J). According to Theorem 2.1 and Table 1 in [17], the matrix J has precisely one pair of pure imaginary eigenvalues if and only if c 0 − c 1 c 2 = 0 and c 1 > 0.
Therefore, to locate the Hopf pairs in a (σ, τ ) parameter space, we will use σ as free parameter and solve the system given by the equilibrium equations combined with (5) and (10), for a * , u * , i * , and τ , while all other parameters are set to fixed values.  The component τ of the solution, if any, and the corresponding value of σ define a Hopf pair (σ, τ ) for which a Hopf bifurcation occurs at the endemic equilibrium P 3 = (a * , u * , i * ). The set of Hopf pairs defines the so-called Hopf-bifurcation curve H in (σ, τ ) parameter space, (5), (10), and (11) hold}.
Note that H can be parametrized by i * (or any of the components of P 3 ) [35]. This would be equivalent to parametrizing the set of endemic equilibria where a Hopf bifurcation occurs by one of their coordinates, say x, which satisfies a nonlinear equation F (x, σ, τ ) = 0 derived from equations (5) and (11). We will focus here on the case σ = β a and τ = α i . This choice appears to be of particular interest, as α i may be most amenable to alteration by increased epidemiological monitoring (see Section 4), while β a can be thought of as inversely proportional to the effectiveness of the behavioral response in aware hosts (see Section 4). An example of a curve H in the (β a , α i ) parameter space is shown in Figure 3. For each value of β a within the range [0, 0.4379), there exist two solutions of equations (10) and (11).
The (sub-or supercritical) character of the Hopf bifurcation has been found by numerical integration of the system (3) for values of α i inside and outside of the region R limited by H and the straight line β a = 0. For each pair (β a , α i ) inside R, the endemic equilibrium P 3 is unstable (J(P 3 ) has two complex eigenvalues with positive real part and a third eigenvalue that is real and negative), while stable periodic orbits appear around P 3 . Outside this region, P 3 is asymptotically stable. Therefore, the points on the curve H define supercritical Hopf bifurcations of the system (3).
The prevalence of the disease at the bifurcation points along the curve H is presented in  Numerical simulations confirm our predictions regarding the Hopf pairs shown in Figure 3. In particular, for β a = 0.2 and the parameter settings as specified in the caption of Figure 3, Hopf bifurcations in system (3) are predicted for α * i = 0.2251 and α * * i = 0.8916. Figure 5 illustrates what happens if we increase α i from below α * i to above α * * i for trajectories with initial condition a(0) = u(0) = 0 and i(0) = 0.1, i.e., far from the endemic equilibrium P 3 .
The top left panel of Figure 5 shows a simulation with α i = 0.2, right below the first Hopf bifurcation. The solution quickly approaches an endemic equilibrium with disease prevalence i * = 0.1249. When we increase α i slightly above α * i , to 0.24, then we observe significant oscillations in the top right panel. Moreover, these oscillations correspond to a stable limit cycle in the phase portrait of the system that attracts other trajectories whose initial conditions are not necessarily close to the endemic equilibrium. For the (unstable) endemic equilibrium we get i * = 0.0651 in this case. Due to the oscillations, the long-term mean prevalence will be even slightly lower, around 0.0592.
A similar picture of persistent oscillations is shown in the bottom left panel of Figure 5 where we chose α i = 0.5, about half-way between α * i and α * * i . The existence of a stable limit cycle becomes clearly noticeable in simulations over a longer time horizon. Here the prevalence at endemic equilibrium is i * = 0.0148, very close to the mean prevalence i = 0.0150 in the long run. Note that for these settings long periods of very low disease prevalence alternate with short spikes that indicate periodic small flare-ups. These flare-ups are followed by a rapid increases in awareness, which indicate a panic-like spread of information.
Finally, when we further increase α i beyond α * * i to 0.94, we initially observe a similar pattern of alternating periods of extremely low disease prevalence, interrupted by small flare-ups with panic-like spread of awareness (see the bottom right panel in Figure 5). The prevalence at endemic equilibrium further decreases to i * = 0.0065. However, as this panel indicates, the amplitude of these oscillations now decreases, albeit very slowly because the pair (β a , α i ) = (0.2, 0.94) is very close to the curve H.

Discussion
As was mentioned in the introduction, previous results on various types of SAIS models had suggested that a behavioral response of hosts who are aware of an ongoing outbreak of a disease can be effective in eliminating the endemic equilibrium or driving it to very low levels. However, the question of when the response will be sufficient to prevent future flare-ups from low endemic levels had not previously been addressed in the literature. This question is of interest for models where awareness will decay over time. Lemma 2.2 of Section 2 indicates that when there is not much differentiation between hosts in their propensity to share information about the disease with other hosts, future flare-ups are ruled out. This result applies both to SAIS models with constant rates and to SAIS models in which the rate coefficients depend on the prevalence of the disease.
However, in most real human populations hosts will significantly differ in how effectively they contribute to growing awareness of other hosts. We constructed a new class of models, SAUIS models, that incorporate this phenomenon. We think of them as the simplest possible straightforward generalization of SAIS models in this direction. SAUIS models permit persistent cycles of flare-ups that induce panic-like spread of information, which will temporarily drive the prevalence to low levels without leading to elimination of the disease or providing permanent protection against future major outbreaks. This phenomenon does not depend on variable rate coefficients, as all of these are assumed constant in the version of SAUIS models presented here. This, together with the contrasting result for the closely related SAIS models clearly demonstrates that the observed oscillations are in fact driven by the unequal propensity of hosts to share information when differences in the willingness to share information arise from the degradation of the information during subsequent transmission events as modeled in [11].
These findings seem obviously relevant from the point of view of designing effective policies for controlling infections. As the two top panels of Figure 5 suggest and our calculations confirmed, the average disease load, as interpreted as the mean prevalence calculated from the area under the solid curve, may in the long run be similar and sometimes even lower in the presence of periodic flare-ups than for similar parameter settings where an endemic equilibrium is approached. Thus if the primary goal of control is a reduction of average load, then elimination of periodic flare-ups may sometimes even be counterproductive. However, peak load may be more important than average load in terms of the danger of overwhelming the health care system [9], and it will be higher under oscillatory dynamics. Thus if the primary goal of control is to reduce peak load of the disease as much as possible, then control measures should be targeted at elimination of possible flare-ups.
One possible control strategy is to commit resources to continuous monitoring the prevalence i of the disease and dissemination of this information be health care professionals and the media. In terms of SAUIS models, this can be interpreted as increasing the parameter α i . It seems plausible that this strategy will decrease the value of i * at the endemic equilibrium and the long-term average disease loadī. Our numerical explorations suggest that this intuition is correct. When α i crosses the threshold represented by the lower branch of the curve in Figure 3 the endemic equilibrium will become unstable, with oscillatory dynamics for values above the threshold. As Figure 5 shows, right above the threshold the resulting oscillations may result in an increased peak load relative to values of α i right below it. A further increase of α i beyond a second threshold represented by the upper branch of the curve in Figure 3 will lead to renewed local asymptotic stability of the endemic equilibrium. But as the lower-right panel Figure 5 shows, even above this threshold damped oscillations may remain observable for a considerable time horizon when trajectories start far away from the equilibrium. These and other results presented in Section 3 reveal a surprisingly rich and intricate dynamics of SAUIS models. It becomes clear that optimizing the dynamics by controlling one or more parameters that govern the spread of awareness will usually be quite delicate even if one could treat the model at face value.
Our findings open several avenues of future research. It would be interesting to analytically derive necessary and/or sufficient conditions on the information flow about the disease between different types of hosts that would preclude the existence of sustained oscillations. However, in order to base actual public health policy on such results, they would need to be the sufficiently robust and carry over to wider classes of models of transmission of awareness and degradation of information. The SAUIS models that we introduced and explored here are useful for clearly demonstrating effects that are precluded by the simplifying assumptions of previously studied SAIS models. However, in biologically more realistic models one would presumably want to consider more fine-grained scenarios of degradation of information, perhaps by incorporating more compartments or more directly adapting the model of [11].
Moreover, the assumption of constant rate coefficients is clearly an oversimplification. We already discussed in the context of SAIS models how and why α(i), α a (i), δ a (i) might depend on the actual prevalence i. While at low levels of prevalence the information received by susceptible hosts is unlikely to be confirmed by first-hand knowledge of actual cases, at moderate or high prevalence such confirmation by direct observation is more likely. Thus if such confirmation determines whether receiving information moves a given host into the A-compartment or the U-compartment, the ratio of newly created aware to newly created unwilling hosts may increase with the prevalence of the disease. We intend to explore some of these directions in a follow-up paper.
Predictions of oscillatory dynamics have been reported for a number of previously studied models of behavioral epidemiology, both for models that incorporate demographics, and for models that, similarly to the ones studied here, ignore demographics. However, to the best of our knowledge, variability in the propensity of hosts to further disseminate awareness had not yet been identified as a possible driving mechanism for this phenomenon. We conclude this paper with a brief discussion of some related previous results. This review is not intended to be exhaustive; our goal is only to illustrate the variety of other mechanisms that appear to be capable of generating oscillations in disease prevalence.
Damped oscillations can be observed in reactive SAIS models when the asymptotically stable interior equilibrium has complex eigenvalues [18]. They are also possible when the underlying disease dynamics is of type SIR without demographics. For example [8], considers spatially structured models where flight from endemic regions is identified as a key factor that will drive the overall dynamics.
Sustained oscillations have been found in SIR-based models with demographics where the behavioral response represents compliance with vaccination and the driving mechanism involves real or perceived benefits of failing to adopt the behavioral response [3,6,7,27]. Such benefits are absent in SAUIS models. Sustained oscillations in a fixed population were also reported for models of vaccination compliance for influenza described in [4,36]. In these models, vaccination decisions are made from year to year for different strains, so that the underlying disease dynamics is closer to type SIS than SIR. There is no direct information transfer in these models, and decisions are based on experience with past outbreaks rather than on current incidence levels. The latter features are also present in the models of [6,7].
Sustained oscillations also have been reported for a model that in some aspects resemble a reactive SAIS model, but makes an assumption about a fixed number of susceptibles. The authors of that paper conjectured that this assumption is needed for sustained oscillations in their model (see page 1353 of [25]). Such oscillations can also occur if the underlying disease dynamics is of type SIRS and demographics are considered. In [14] it is argued, based on longitudinal studies of STI data, that this mechanism may be the driving factor even if clear patterns of behavioral changes over time are present.
Finally, there exists a substantial literature on oscillations in models with a behavioral response that involve rewiring the contact network (see, for example, [15,28,33,37]). While these are individual-based models, analytical confirmation can sometimes be obtained by coarsegraining [16] or using pair-approximations to build corresponding ODE-models with nonuniform mixing [34,35]. Thus the mechanism that drives oscillations in these models is very different from the one in our SAUIS models, where uniform mixing is implicitly assumed.