Using awareness to Z-control a SEIR model with overexposure: insights on Covid-19 pandemic.

In this paper, we use the Z-control approach to get further insight on the role of awareness in the management of epidemics that, just like Covid-19, display a high rate of overexposure because of the large number of asymptomatic people. We focus on a SEIR model including a overexposure mechanism and consider awareness as a time-dependent variable whose dynamics is not assigned a priori. Exploting the potential of awareness to produce social distancing and self-isolation among susceptibles, we use it as an indirect control on the class of infective individuals and apply the Z-control approach to detect what trend must awareness display over time in order to eradicate the disease. To this aim, we generalize the Z-control procedure to appropriately treat an uncontrolled model with more than two governing equations. Analytical and numerical investigations on the resulting Z-controlled system show its capability in controlling some representative dynamics within both the backward and the forward scenarios. The awareness variable is qualitatively compared to Google Trends data on Covid-19 that are discussed in the perspective of the Z-control approach, inferring qualitative indications in view of the disease control. The cases of Italy and New Zealand in the first phase of the pandemic are analyzed in detail. The theoretical framework of the Z-control approach can hence offer the chance to reflect on the use of Google Trends as a possible indicator of good management of the epidemic.

Introduction and motivations

Since the outbreak of the Covid19 epidemic in China, different mathematical models have been formulated to shed light on the severity of this infection as well as to gain predictions on the course of its spread. In many cases, detailed compartmental models with many classes have been introduced to estimate the risk of transmission of the novel coronavirus [1], [2] and to evaluate the effectiveness, even in the long period, of control measures against epidemic [3], [4]. In other cases, simpler qualitative models have been proposed to clarify the different mechanisms underlying its diffusion and to elucidate their reciprocal interplay.

One of the key point that emerged is that undocumented cases - including the asymptomatic ones - play an important role on the spread of the novel Coronavirus epidemic [5], [6], [7], [8] suggesting the importance to monitor, as best as possible, the real spread of the virus in the population. This marked role played by asymptomatic individuals, along with the peculiar disease transmission mechanism, prompted to consider social-distancing and persistent self-isolation as very effective control measures to fight epidemic. On this line, the combined action of the restrictive government measures (lockdown) and of media-driven awareness programs have played a very crucial role in softening the severity of the epidemic. Most countries and the World Health Organization (WHO) have used the great potential of Internet to promote awareness and educational programs on Covid19 and surveying the Relative Internet Search Volumes (RSV) was deemed as a means to give information on the extent of public attention, with Google Trends as one of the most widely used tools for this aim [9], [10]. In addition, a better understanding of the complex potential provided by awareness for the containment of epidemics is a hot spot that continues to attract attention in many different directions, as testified by a multiplicity of studies involving homogeneous spread qualitative models [11], [12], [13], [14], [15], networks [16], [17], [18] and multilayer or multiplex networks [19], [20], [21], [22], [23]. In evaluating the impact of awareness on disease spread, network-based models can certainly offer a deeply realistic perspective since they account for the heterogeneous structure of society and for the individuality of the members of a population. Nevertheless, qualitative spread models benefit of a simpler mathematical treatment and can offer a profound mechanistic understanding. Moreover, they can be considered as a kind of ‘step zero’ when it comes to the spread analysis over networked systems [24]. This study, that moves exactly in this qualitative perspective, has as a starting point the SEIR-like model developed in [14] that incorporates (i) the role of asymptomatic/undocumented cases in the infection process by means of an overexposure mechanism (ii) awareness, carried out through campaigns or targeted actions, and seen as a tool to increase social distancing. In model (1), the positive parameters involved have the following meaning: is the recruitment rate; is the natural death rate; is the transmission rate; is the overexposure coefficient; is the incubation period; is the disease-related death rate; is the cure rate; is the dissemination rate of awareness among susceptibles; is the depletion rate of the awareness programs whereas is their implementation rate. It is assumed that the many asymptomatic/undocumented individuals can cause an although unconscious overexposure of susceptibles to infection and an incidence rate is considered. This choice reflects an increased rate of infection due to double exposures over a short time period: single contacts make infection develop at the rate whereas new infections are caused by double exposures at a rate [25]. Such incidence rate was linked to the occurrence of hysteretic phenomena through a backward bifurcation scenario [25], [26], a phenomenological framework that involves subcritical persistence of the disease and that needs to be carefully monitored and managed [27], [28], [29], [30]. Media awareness campaigns were instead considered by the means of an awareness variable that was supposed to have the effect of isolating a fraction of susceptibles and whose governing equation was given by the balance between an implementation and a depletion process. In [14], the interplay between overexposure and awareness was recognized to be a point that should not be underestimated both in the current and future management of Covid19 emergency since, if overexposure can lead system dynamics towards a backward phenomenology, a responsible role of information - aimed at a correct management of awareness - can allow to avoid such a scenario. Dangerous dynamical transitions could be hence avoided through a suitable management of people awareness and factors as information dissemination, implementation and depletion must be adequately monitored and balanced.

While in [14] awareness is considered as a system variable ruled by an assigned evolution equation, in this paper we want to regard the role of awareness from a different perspective. To this aim, we consider awareness as a time-dependent variable whose dynamics is not assigned a priori and, exploiting its potential to produce social-distancing and self-isolation among susceptibles, we use it as an indirect control on the class of infective individuals. In the following, we hence focus on the modeland use the Z-control approach to detect what trend over time must awareness display in order to eradicate the disease. The Z-type control approach, an error-based method proposed to solve dynamic problems [31], [32], [33], has recently extended its range of applications moving from engineering to life sciences problems. In population dynamics, it has been applied to the classical and generalized Lotka-Volterra model [34], [35] in order to promote ecological coexistence; in epidemics, to control disease in simple susceptible-infected models [36], [37] and as a means to control the occurrence of backward scenario [38]; in eco-epidemiology, to prevent system dynamics by chaotic oscillations in a predator-prey model with disease infection in the prey [39], [40].

If applied to the ecological or epidemiological field, the Z-type control can be exerted as (i) direct control for all the species, when exogenous measures are applied to the involved populations for the simultaneous control of their dynamics; (ii) as indirect control of one population, when the control measure is only applied to one species with the ultimate purpose to control the other one. The strength of the approach is its interesting property to stabilize the system dynamics towards a desired population density. This is due to the capability of the Z-control approach to change the dynamics of the uncontrolled system [35] driving it towards a desired controlled equilibrium that becomes the only attractor for the Z-controlled model. In the process it is crucial to carefully calibrate the strength of the control exerted on the system.

The existing literature on the Z-control approach has been mainly focused on these dynamical points, dealing with uncontrolled models that can be essentially reduced to two governing equations. As far as we know, an extension of the method to higher-dimensional systems has not yet been provided. Moreover, little attention has been given to the qualitative features of the Z-controller and to its possible practical applications. With this study, that considers awareness as an indirect Z-control variable for a SEIR-like system, we aim to move a step in both these directions.

On this line, the key contributions of this study, can be summarized as follows:

(i) taking a cue from the structure of the SEIR model, we rigorously extend the Z-control procedure to face the further complication of dealing with a number of governing equations greater than two;

(ii) taking into account the interpretation of the Z-controller in terms of awareness, we focus on its qualitative features and compare it to some representative trends of interest on the topic ‘Coronavirus’, obtained through a Google Trends analysis. In Google Trends, the RSV trend for a certain search query can in fact be indirectly related to the level of people awareness on that specific topic.

(iii) exploiting the theoretical framework of the Z-control, combined with the comparison of the RSV in the cases of Italy and New Zealand, we take the chance to reflect on the use of Google Trends not only as a tool for monitoring the level of awareness but also as a possible indicator of good management of the epidemic. By good management, we mean here the global set of actions ranging from information campaigns to restrictive actions aimed at containing the virus.

The paper is hence structured as follows. In Section 2 we introduce the uncontrolled SEIR model and briefly recall its properties in terms of equilibria and bifurcations. In Section 3, awareness is introduced in the model as an indirect control on the class of infective population and the Z-controlled model is derived. In Section 4, the Z-controlled model is analyzed in terms of its equilibria, stability properties and positiveness of solutions. In Section 5 investigations in the time dependent regimes are provided to show the effects of the Z-control approach on some representative dynamics of the original model. In Section 6, we combine our findings from the Z-control method with the RSV trends related to the search query ‘Coronavirus’ for Italy and New Zealand to get further insight on the role of awareness in the management of the pandemic. In Section 7, some concluding remarks end the paper.

The uncontrolled SEIR model

The SEIR-like model (1), without the contribution of awareness, becomes:where the total population N(t) is divided into the susceptible (S), exposed (E), infective (I), recovered (R) individuals and is assumed to be varying and homogeneously mixed so that all people are equally likely to be infected by the infectious individuals if they come into contact.

In the following, we will refer to model (3) as the uncontrolled model. To better highlight the impact of the Z-control on this model, we briefly recall its properties in terms of equilibria and bifurcations. As already highlighted in [14] for model (1), also in system (3) the overexposure mechanism gives the ability to exhibit a multiplicity of endemic equilibria subthreshold and a backward phenomenology.

It is easy to show that solutions of the uncontrolled system (3) with non negative initial data, are positive and bounded and that the setis positively invariant and globally attractive for model (3). We refer to Appendix A for the detailed proof of these preliminary results. Since the first three equations in system (3) are independent on the last one, in the next we will consider the following SEI model:System (4) admits the disease-free equilibrium . Denoting withwe introduce the basic reproduction numberand state the following result:

The disease-free equilibrium is locally asymptotically stable for and unstable for .

The Jacobian matrix of model (4), is given byand, when evaluated at , admits as eigenvalues whereas are solutions of the following second order algebraic equation:

It follows that is locally asymptotically stable if and unstable otherwise. □

We can express the bifurcation threshold in terms of the parameter . In fact: where

An endemic equilibrium can also be found for system (4) whereand where is a real strictly positive solution of the second order algebraic equation:withWe preliminary observe that and where is defined in (7). When , then and is a positive quantity. Applying the Descartes’s rule of signs, it follows that the algebraic Eq. (9) admits a unique real positive root independently of the sign of . Taking into account of Remark 2.1, the following result holds:

If then model (4) admits one endemic equilibrium with and given by (8) .

When then and, according to the sign of the coefficient in Eq. (9), two or zero real positive solutions are possible. To detect a possible multiplicity of endemic equilibria under the threshold , or equivalently , we hence consider the following quantities:where is defined in (5). We observe that the inequality is always verified sinceis a positive quantity. We can hence state the following results:

If and , then Eq. (9) admits no positive solutions so that model (4) admits none endemic equilibrium.

We stress that implies that the sign of the coefficient C in (9) is always positive. Moreover, condition implies that the sign of the coefficient B is positive too. Therefore the thesis follows by the Descartes’ rule of signs. □

Let .

(i) If , then model (4) admits two endemic equilibria and with and , given by (8) .

(ii) If then model (4) admits none endemic equilibrium.

We first observe that being and , we have that the sign of the coefficient C in (9) is always positive and the sign of the coefficient B is negative. Therefore equation (9) admits two positive real solutions if and only if and admits no real solution otherwise. Observing that the thesis easily follows. □

The above results suggest the occurrence of a backward bifurcation scenario for the uncontrolled model, with a saddle-node bifurcation at and a transcritical bifurcation at . More precisely, as stated in the following theorem, the overexposure parameter can make the difference between the forward and the backward scenario:

(i) If , then system (4) exhibits a forward bifurcation at . (ii) If then system (4) exhibits a backward bifurcation at .

The occurrence of backward or forward scenario is investigated by the method proposed in [41]. The coefficients in the equilibrium Eq. (9) can be regarded as functions of the parameter so that, at , and Eq. (9) becomeswith roots and . The root refers to the disease-free equilibrium whereas refers to a positive endemic equilibrium only if and have opposite signs. Therefore, being , then must hold. Implicit differentiation of Eq. (9) with respect to givesthat evaluated at and for , returns:since, by (10), . It follows that the slope of the bifurcation curve at must have the same sign of . Therefore: (i) if then a backward bifurcation occurs at ; (ii) if , then system displays a forward bifurcation at . By (10), it follows that so that if and only if . □

To show that both the forward and backward scenarios can be compatible with the Covid19 spread in Italy, we choose the parameter values in way to be in line with the Covid19 spread in Lombardy, one of the northern regions of Italy most affected by the virus. The demographic parameters and are chosen such that (as the total population of Lombardy in 2019) [42]. The average lifespan is taken to be years (i.e. per day) [42] so that people per day. The incubation period (time from exposure to the development of symptoms) can vary greatly among patients and it is estimated to be between 2 and 14 days [43] with an average of 5.2 days [44]. Therefore days. The WHO reports time from onset of symptoms to death of about 2 weeks [45] and the disease-induced death rate for infectious individuals is assumed to be days [46]. Moreover, the average hospital length of stay of discharged patients is 13 days [47], so that we assume the cure rate to be days. The parameters and are instead assumed to be varying.

With the above parameter values, the transcritical bifurcation threshold is such that whereas . The impact of overexposure for the emergence of a forward or a backward scenario is shown in Fig. 1 . For , the classic forward scenario is obtained where no endemic equilibria can be detected subthreshold, Fig. 1 (Left). Differently for , a backward scenario is obtained where a multiplicity of endemic equilibria can be detected subthreshold, Fig. 1 (Right). We want now to show how such phenomenologies can be Z-controlled with using awareness as an indirect control variable. To this aim, we have first to build the Z-controlled model.

Fig. 1

(Left) Low overexposure: . Bifurcation diagram in the plane () showing a transcritical bifurcation occurring at . For a unique endemic equilibrium exists whereas no endemic equilibrium can be found subthreshold. The numerical value for the other parameters are as fixed in the main text. (Right) High overexposure: . Bifurcation diagram in the plane () showing the coalescence of the two endemic equilibria and at the saddle-node bifurcation threshold . A transcritical bifurcation occurs at .

The Z-controlled model: system derivation

Starting by model (4), we want to apply the Z-control approach to perform an indirect control on the infectives by acting on the susceptible individuals. Therefore, we focus on the model:where the awareness control variable may assume both positive and negative values. Starting from the initial conditions at , we want to drive the infective population toward a desired constant level , namely , with .

To derive the related Z-controlled model, we need to develop for model (13) an analysis similar to that we carried out in [38]. In this case, however, the Z-control procedure needs to be adequately extended in order to face the further complication of dealing with a number of governing equations greater than two. Therefore, we preliminary provide a generalization of the Z-control procedure. To the best of our knowledge, this is an original result within the theoretical literature on the subject.

A generalization of the Z-type control procedure

We start by recalling that the Z-type approach can be used to control systems described by a set of autonomous ordinary differential equationsfor the variable . The idea behind such a error-based dynamic method is to design an analytical expression for the system input , , so that the actual system output is forced to achieve a desired output [33], [34], [48]. Given a vector valued norm , if the error between the actual output of the system and its desired state approaches zero, i.e.then the Z-control turns out to be successful. To achieve this goal we define the vector as first error function and force this quantity to converge to zero exponentially. This is obtained by requiring that the following differential equationis verified. Equation (15) is referred as design formula and the strictly positive parameter , called design parameter, gives a measure of the convergence rate. If the design formula yields equations that explicitly include the system input variables , then we obtain an explicit expression for . Differently, if Eq. (15) does not explicitly include the system input , another error function needs to be defined and the design formula is adopted again. To go into more details, we recursively define the th error function, namely the vectorand state the following result:

Requiring that, for , the th error function verifies the design formula is equivalent to require that the first error function verifies the following equation where we convey that .

The proposition is proved by induction. □

For , and , there exist and , such that for .

For a continuously differentiable and bounded desired state that starts from a positive initial state , the first error function verifying Eq. (17) , converges to zero exponentially.

Solving the th-order differential Eq. (17), for all , we obtainFrom Lemma 3.1 there exists and , such that for . Therefore, the first error function exponentially converges to zero with rate . □

The design procedure for the control law of the Z-type dynamic method is based on the choice that and in such a way that Eq. (17) includes the system input . For example, if we consider the application of the Z-control method to the SIR model in [38], we observe that the value of was set to 1 as the control was done implicitly only on the susceptible population, whereas in (17) was set to 2 in order to obtain the equation for ,where . This allowed to get the analytical expression for the control in order to drive the infective population towards a desired constant level .

Now we explicitly show that, with imposing the third error function to decade exponentially as in (16), we can get the analytical expression for the control in model (13).

Building the Z-controlled SEIR model

We first regard model (13) aswhereandWe hence make use of the indirect Z-type control procedure by following the key steps introduced in the previous subsection. Let the desired state for the infective population. We require that the first error function verifies (17) with , namelyand rewrite the above formula in terms of susceptible-infective populations. First observe that, Then, with replacing in (20) the expressions (21) and (22), we obtainwhere , .

Therefore, for all such that , we can get the expression of in (23) so that the modified system (18) can be regarded aswhereIn the next, we refer to system (24)-(25) as to the Z-controlled model.

The Z-controlled model: analytical results

In this section, we investigate the Z-controlled model (24)-(25) in terms of positiveness of solutions, equilibria and stability properties. As far as the former is concerned, we start by observing that, solving (20), we obtain the exact solution for the dynamics of infectives:where ,   , andLet us consider with given in (26) wherePositiveness of is guaranteed if and only ifSince the function for all and , the following result holds:

Set . If and , then in (26) assumes positive values for all .

From it follows that the discriminant of assumes negative values. Together with the assumption , this ensures the positivity of for all . Moreover,If and then and the polynomial assumes positive values for all . From inequality (28), the result follows.

Now, let us consider with given in (26) From the dynamics of we can evaluate:Then is positive when  □

As the right-hand term in the above expression is negative, we can prove the following theorem:

If , and , then in (31) assumes positive values for all .

From it follows and . The bounds on imply and . Hence the quantityassumes positive values for all . □

To establish the positivity of , we observe thatand evaluate with given in (30):Then, from (26) and (30), it results thatFrom (31) we have thatUnder the hypothesis the quantity is positive ifWe can hence prove the following theorem:

If , and , then assumes positive values for all .

The hypothesis and the lower bound on guarantee that for . Moreover, the upper bound on provides the positiveness of both coefficients and of the polynomial in (33), this assuring the positiveness of for all . □

In order to obtain the positiveness of solutions for model (24)-(25), we first evaluate and then state the following theorem, expressed in terms of the original parameters:

Set . Suppose and . Let and verify then , and in (24) - (25) assume positive values for all .

It is easy to verify that, under the assumption , the discriminant of the second order polynomial defined in (34) is positive. Hence assumes negative values forand the result follows from Theorem 4.1, Theorem 4.2. 4.3. □

In the sequel, we setand, similarlyThe expression of the unique equilibrium of the Z-controlled model (24)-(25) is given bythat can be easly deduced from the formula of the endemic equilibrium (8) with . Then, the variational matrix of the Z-controlled model (24)-(25), evaluated at is given bywhereAs far as the stability of is concerned, the following result holds:

The equilibrium (35) of the Z-controlled model (24) - (25) is locally asymptotically stable.

Let us evaluate and so that the ratio . Moreover, given it results thatthis concluding the proof. □

Investigations in the time dependent regimes

In this section, we show that the Z-control approach can be used to control some representative dynamics within both the backward and the forward scenarios. For the numerical simulations, as explained in Appendix B, a positive, non-standard, first order accurate numerical integrator was used. In the time-dependent investigations, we consider and as bifurcation parameters whereas the others are fixed as in Section 2. We also recall that, for this choice of the parameters, in the uncontrolled model the transcritical bifurcation threshold is such that whereas the overexposure threshold . According to our theoretical results, low levels of overexposure () make the uncontrolled model exhibit a classic forward scenario where no endemic equilibria can be detected subthreshold. Differently, when overexposure is above a certain threshold (), the uncontrolled model displays a backward scenario, where a multiplicity of endemic equilibria can be detected subthreshold. In the following, we specifically focus on the case where the level of overexposure is high.

High level of overexposure

In this case, we fix and select three values of the parameter that correspond to some representative dynamical behaviors above and below the transcritical threshold .

Case I. The uncontrolled model exhibits a stable endemic equilibrium super-threshold

We fix the parameter such that . For this choice of the parameters, we have and model (4) admits two feasible equilibria: the disease-free equilibrium that is a saddle point and the endemic equilibrium that is a stable focus. Initial conditions and , with , in the neighboring of , drive the trajectories of the uncontrolled model (4) towards a situation of stationary endemic disease. The phase portrait related to this situation is represented in Fig. 2 , first column, first row. Now, we consider the Z-controlled model (24)-(25) and, as before, we start from and with . Since , the initial value is chosen close to the mean value between and . Moreover, we set at the mean value between and . In so doing, the conditions of Theorem 4.4 are all verified and the positiveness of the controlled susceptible population is theoretically guaranteed. The Z-control forces system dynamics towards the stable equilibrium with and its effects can be observed by looking at the phase-space of the Z-controlled model in Fig. 2, first column, second row.

Fig. 2

High level of overexposure, . Each row in the figure shows, for three different values of the parameter : the phase-space trajectories for the uncontrolled model (4), the phase-space trajectories for the Z-controlled model (24) when , and the time dependent behavior of the control variable exerted on the susceptible population when . In the phase-plane, the initial point is plotted as a pink diamond, the uncontrolled equilibrium is plotted as a red star and the Z-controlled equilibrium is plotted as a blu circle. First row: . Second row: . Third row . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Case II.

The uncontrolled model exhibits a stable endemic equilibrium sub-threshold We now progressively decrease the value of the parameter and fix it such that . For this choice of the parameters, we have and model (4) admits three feasible equilibria: the disease-free equilibrium that is a stable node and the two endemic equilibria: that is a stable focus and that is a saddle point. The model hence exhibits a bistability situation sub-threshold. Initial conditions in the neighboring of drive the trajectories of the uncontrolled model (4) towards a situation of stationary endemic disease through a certain number of damped oscillations. The phase portrait related to this situation is represented in Fig. 2, first column, second row. Changing affects the value of and consequently the values of . However, this has a little effect on and , so that we can consider the Z-controlled model (24)-(25) and start again from . Similarly is set inside the interval and that is unchanged. All the conditions of Theorem 4.4 are verified and the positiveness of susceptible population is theoretically guaranteed. The Z-control forces system dynamics towards the stable equilibrium with and its effects can be observed by looking at the phase-space of the Z-controlled model in Fig. 2, second column, second row.

Case III. The uncontrolled model exhibits a oscillatory behavior sub-threshold

Still decreasing the value of the parameter , we fix it such that . For this choice of the parameters, we have and model (4) admits three feasible equilibria: the disease-free equilibrium that is a stable node and the two endemic equilibria: that is a saddle point and that is now an unstable focus because a couple of complex-conjugate eigenvalues has crossed the imaginary axis. Initial conditions in the neighboring of drive the trajectories of the uncontrolled model (4) towards a situation of oscillatory endemic disease. In fact a stable periodic orbit can be found in the neighboring of because of a Hopf bifurcation that occurs at . The phase portrait related to this situation is represented in Fig. 2, first column, third row. As before, changing has a little effect on the values and so that, for the the controlled system (24)-(25), we can set . Similarly, the value of still lies inside the interval and . All the conditions of Theorem 4.4 are verified and the positiveness of controlled susceptible population is theoretically guaranteed. The Z-control forces system dynamics towards the stable equilibrium with and its effects can be observed by looking at the phase-space of the Z-controlled model in Fig. 2, second column, third row.

Could we associate to awareness a ‘face’?

The temporal evolution of the awareness variable in the third column of Fig. 2, indicates that, to convert an uncontrolled equilibrium super-threshold into a new Z-controlled equilibrium, a stronger effort is required with respect to the one required to change a stationary or an oscillatory uncontrolled behavior sub-threshold. In this sense, dynamical behaviors emerging below the transcritical threshold appear to be more easily modifiable regimes and, among them, the oscillatory trend is slightly less expensive to control. Fig. 2 also shows that, to eradicate the disease, the required trend of awareness is qualitatively similar in the three cases. In fact a phase of low awareness is observed, that grows rapidly until it reaches a maximum and then decreases up to settle - with some oscillations - on a low positive level of constant awareness.

Fig. 3 summarizes the key features of a successful Z-controller when it is used to reduce the infection by a certain percentage (i.e. relative cases reduction) in different endemic situations. This is given by a contour plot (i) of the maximum value and (ii) of the final value of the awareness variable required by the Z-method to obtain relative cases reduction for different values of .

Fig. 3

(a) The color bar represents the maximum value of the control variable during the whole time interval in order to achieve a specific case reduction for different values of . (b) The color bar represents the stabilized value of after achieving a specific case reduction for different values of . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 3(a) shows that - for a fixed value of - if the required percentage of infection reduction increases, then also the maximum value of the removal rate increases. The higher is the value of , the higher is the increasing in the maximum of . A similar behavior for the maximum value of is observed when the percentage of case reduction is fixed and is varied.

Fig. 3(b) shows instead how the final value of varies to achieve different targets and in different endemicity situations. For fixed values of , the final value of varies slightly as the percentage of infected to be reduced varies whereas for higher values of , the variation in the final value of is more pronounced. Moreover, for a fixed percentage of case reduction, the final value of the awareness variable increases with increasing the value of .

From a practical point of view, it is quite difficult to quantify ‘awareness’. We think however that Google Trends can be used - at least qualitatively - as a mean to associate to the awareness variable ‘a face’. Google Trends expresses the absolute number of searches relative to the total number of searches over the defined period of interest, allowing to study trends and patterns of Google search queries. For the specified search query of interest, Google Trends provides an index that ranges from 0 to 100, with 100 being the highest relative search term activity [10]. Monitoring how this index varies over time gives hence a measure of how the extent of public attention on a specific topic varies over time. The RSV trend over time can therefore be associated with the public level of awareness regarding a given topic in a certain time interval.

Interestingly, the -trend observed in Fig. 2 qualitatively resembles the RSV trend for the search query ‘Coronavirus’ during the first phase of the Coronavirus pandemic in Italy (Fig. 4 , top) and obtained through a Google Trends analysis.Fig. 4 (top) shows that a first sign of interest, detectable in a small peak in the RSV, is found in the week of 26Jan-1Feb (point A in the figure) with an interest level of . This is followed by a decrease until the week of Feb 9-15 (point B in the figure) where the interest level is at and by a rapid growth in the week of Feb 23–29 (point C in the figure) when the interest level reaches , decreases in the week Mar 1–7 up to an interest level of 68% (point D in the figure) and, in the week of Mar 8-14 (point E in the figure), the interest level quickly reaches its highest peak with a relative popularity of . From here, a slow progressive decreasing begins that ends in the week June 14–20 with an interest level of (point F in the figure).

Fig. 4

The case of Italy. (Top) Trend of the Relative Search Volume (RSV) for the search query ‘Coronavirus’ from the beginning up to August 2020. (Inset) Time-dependent behavior of the awareness required to Zcontrol an uncontrolled model with high overexposure and a stable endemic equilibrium super-threshold. (Bottom) Trend over time of the newly confirmed Covid19 cases from the beginning up to August 2020.

The high level of interest, corresponding to an increased awareness, recorded in the period March-April could be thought on the one hand as the product of an extensive information campaign and on the other as an indirect effect of the various restrictive measures (lockdown) imposed by the Government in that period. Such an increased awareness - favoring behaviors such as social distancing or intensive use of masks - contributed to the softening of the pandemic curve, inducing a progressive decrease in infections. This can be observed in Fig. 4(bottom), where we report the trend of the new cases per day up to the beginning of August, when it was essentially stable with fluctuations around the value 200.

What happened next is now history and, in Fig. 5 , we report the RSV trend for the search query ‘Coronavirus’ and that of the newly confirmed Covid19 cases from the beginning up to November 2020.Fig. 5(top) indicates that, since August, the RSV trend remains substantially stable, with an interest level below testifying a prolonged period of low interest in the subject. This despite of the dramatic surge in cases that took place since the end of September. Fig. 5(bottom) in fact indicates that from 23 August to 30 September the number of new cases increased with fluctuations around the value 1000 and, from 30 September onwards, they dramatically grew with marked fluctuations, to the point that 39809 new cases were recorded on November 7th: the highest number since the beginning of the pandemic in Italy. The lower interest in this second phase could be partly explained by the fact that, also due to the prolonged emergency, people have likely learned to live with the idea of Covid by deceptively lowering their defenses and causing a situation of decreased awareness. This latter could also have been favored by the often contradictory information on the topic and by the absence of enough drastic and timely containment measures to the virus.

Fig. 5

The case of Italy. (Top) Trend of the Relative Search Volume (RSV) for the search query ‘Coronavirus’ over the last year, i.e. November2019-November2020. The dashed black vertical line is a watershed between what happens before and after August2020. (Bottom) Newly confirmed Covid19 cases from the beginning up to November2020.

Fig. 5 highlights in fact that a marked awareness can be associated with a softening of the contagion curve (green circles on the left of the vertical bar) whereas a prolonged low level of awareness can instead be related to a dramatic and persistent surge in the new cases (red circles on the right of the vertical bar).

This result is not in contrast with the indications provided - in terms of awareness - by the Z-control. In fact, the Z-method suggested that, starting from a situation of endemicity super-threshold, a sufficiently low level of awareness can persist only after the disease has been eradicated (), something that, for example, did not happen in Italy. Contextualizing the Italian situation within the theoretical framework of the Z-control and imagining that a first effect of the method was to produce a daily level of new infections just under 200, a reapplication of the method would have been desirable, to progressively drive the disease towards the eradication. In this way, the control variable would have settled on a low level of awareness after a small number of damped oscillations. This kind of behavior can be observed for example in the RSV trend concerning the search query ‘Coronavirus’ and ‘Covid’ in New Zealand, a country associated with a successful management of the Covid epidemic [49] since the number of new cases per day remained very low until now, Fig. 6 (bottom). Fig. 6(top) shows in fact that the RSV trend concerning the search query ‘Coronavirus’ and, even more markedly, that for the search query ‘Covid’ displays a damped oscillatory pattern. This peculiar shape of the RSV trend could be the indirect effect of the specific set of non-farmaceutical interventions adopted in New Zealand, from the very beginning, with the aim to bring Covid-19 incidence to zero [49], [50]. It is in fact reasonable to think that factors as early decisive reactions from health authorities, performant surveillance systems, targeted testing strategies as well as stringency have fed, when necessary, the level of people awareness, preventing it from settling on a fading value that, as in Italy, soon proved to be inadequate with respect to the dangerousness of the situation.

Fig. 6

The case of New Zealand. (Top) Trend of the Relative Search Volume (RSV) for the search query ‘Coronavirus’ and ‘Covid’ over the last year, i.e. November 2019 - November. (Bottom) Newly confirmed Covid19 cases from the beginning up to November 2020.

Concluding Remarks

Many times, throughout the course of this Covid-19 pandemic, we were told that its course would depend mainly on our behaviors. This study is, in a certain a sense, a further validation of this idea. In this paper, we used the Z-control approach to stress the peculiar role that awareness can play for the management of epidemics that, just like Covid-19, display a high rate of overexposure also because of the large number of asymptomatic people.

We considered a SEIR model including a overexposure mechanism and used the potential of awareness to produce social distancing and self-isolation among susceptibles as an indirect control on the class of infective individuals. We generalized the Z-control procedure to appropriately treat an uncontrolled model with more than two governing equations and analytically investigated the resulting Z-controlled system in terms of equilibria and stability properties. Numerical simulations allowed us to show that the Z-control approach can be used to control some representative dynamics within both the backward and the forward scenarios when the level of overexposure is sufficiently high. Time dependent investigations confirmed in fact that the application of the Z-control method can change the uncontrolled model - potentially capable to exhibit a multiplicity of equilibria and both stationary and oscillatory behaviors - into a Z-controlled model that admits the Z-controlled equilibrium as a unique attractor. In so doing, it effectively neutralizes the occurrence of any backward scenario. The trend of the awareness variable was shown to display some qualitative features common to all the representative dynamical situations we investigated: it before grows and then decreases until settling, after some oscillations, on a low positive level of constant awareness. We hence investigated the relation between the infection reduction and awareness with respect to the value of , and how awareness changes in the different endemicity situations.

From a more practical point of view, we qualitatively related the awareness variable to RSV Google Trends data on Covid-19 and discussed them in the perspective of the Z-control approach, inferring some qualitative indications in view of the disease control.

Our analysis would seem to suggest that an RSV characterized, over time, by damped oscillations can be considered an indicator of good management and could hence be linked to the containment of the epidemic. Differently, an RSV index that quickly settles on low constant levels and remains in this state despite the disease has not been eradicated at all, would indicate that something in the management is not working well. This situation would therefore be connected to possible critical re-emergences of the epidemic.

Indeed, the analysis developed in the present paper and the obtained results might be refined by pushing the simple interplay model here considered in the more involved framework of the network-based models. Multilayer or multiplex networks may in fact lead to some hidden patterns of influences and could provide different outcomes, hence improving the qualitative suggestions provided through the presented approach. An interesting future development of our research goes exactly in this direction.

With this study, we feel to have added further evidence to the idea that suitably strengthening awareness can be crucial for a good management of the Covid-19 pandemic. These months of pandemic have in fact taught us that Covid19 is a complex system and it must be treated such as, acting on several fronts through a multidisciplinary approach. Among the various weapons at our disposal, awareness is perhaps the most powerful. In a recent interview [51], Ilaria Capua, Director of the University of Florida’s One Health Center of Excellence stated: ‘We are Covid-19, its strength is in our behavior, its ability to spread depends on the many variables that characterize our cities’. This is the reason why hundreds of thousands of tampons for day, alone, cannot stop the virus but our behaviors can strongly help to do it.

Credit Author Statements

D.L. and F.D. contributed to the conceptualization of research, to the formal analysis of the results and to the writing, review & editing of the manuscript.

Declaration of Competing Interest

The authors of the manuscript declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.