key: cord-0838751-3tqxsm6q
authors: Lacitignola, Deborah; Saccomandi, Giuseppe
title: Managing awareness can avoid hysteresis in disease spread: an application to Coronavirus Covid-19
date: 2021-02-01
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2021.110739
sha: 1d5df66428d94638c7e9d80730df6f7bea24a7a8
doc_id: 838751
cord_uid: 3tqxsm6q

A SEIR-type model is investigated to evaluate the effects of awareness campaigns in the presence of factors that can induce overexposure to disease. We find that high levels of overexposure can drive system dynamics towards a backward phenomenology and that increasing people awareness through balanced and aware information can be crucial to avoid dangerous dynamical transitions as hysteresis or transient oscillations before disease eradication. Investigations in the time dependent regimes are provided to support the results. Google Trends data in the context of Covid19 are also used to stress how low levels of awareness, combined with high overexposure, can be related to recent episodes of epidemic resurgence in Europe. Our results suggest that the interplay between overexposure and awareness is a point that should not be underestimated both in the current and future management of the Covid19 emergency.

Despite major medical and health advances, infectious diseases continue to be among the leading causes of mortality in the world as the recent Coronavirus Covid19 pandemic dramatically testifies. It seems now a long story but only few months ago, in December 2019, a first Covid19 coronavirus outbreak occurred in Wuhan, China. The high virulence and the high proportion of asymptomatic cases caused the outbreaks to soon spread all over the world. On 30 January 2020, the World Health Organization declared that the Covid19 outbreak constituted a Public Health Emergency of International Concern and on 11 March the Covid19 outbreak, spanning 112 countries and regions, was declared a Pandemic by World Health Organization that encouraged countries to change the course of this pandemic [1] . The emergency situation that Italy, as well as the rest of the world, is still been experiencing has highlighted a growing need for qualitative and quantitative methods to guide the necessary prevention, treatment and control interventions to counteract the spread of Covid19. In this perspective, the development of epidemic models for the transmission of infectious diseases assumes a key role, both in theory and in practice, becoming an integral part of the policy decision-making of many nations. Mathematical models can in fact be of strategic importance for evaluating evidence-based decisions by policy-makers as well as to provide possible scenarios linked to the adoption of specific measures. To this aim, mathematical models can act in two directions: those based on more sophisticated mathematical tools can give a contribution in terms of quantitative predictions, but simpler qualitative models can more easily shed light on the constitutive mechanisms, highlighting their role and reciprocal interactions. On this line, their simplicity might be an intriguing point of strength -a common lesson to many areas of applied mathematics [2] -that biological insights even related to very complex topics might be obtained, at least qualitatively, through relatively simple mathematical tools. 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 (i.e. susceptibles, exposed, symptomatic infectives, asymptomatic infectives, susceptibles in quarantine, exposed in quarantine, hospitalized, recovered) has been introduced to estimate the risk of transmission of the novel coronavirus [3, 4] and to evaluate the effectiveness, even in the long period, of control measures against epidemic [5, 6] . The role played by asymptomatic individuals was widely stressed, recognizing how a prompt isolation of asymptomatic infectives would change the dynamics of Covid19 spread [7, 8, 9] . In particular, the impact of not officially documented cases on the spread of the novel Coronavirus epidemic was recognized to be enormous since, without the transmission due to the undocumented cases, the number of confirmed infections would have been far less in the whole of China with an estimated decrease of about 80 % [10] . One of the key point is that undocumented cases (including the asymptomatic ones) still play an important role even in an advanced stage of the epidemic, confirming the importance to monitor, as best as possible, the real spread of the virus in the population. Therefore, due to the disease transmission mechanism and to the marked role played by asymptomatic individuals in the spread of the disease, social-distancing as well as persistent and strict self-isolation have been recognized as effective control measures to fight epidemic. For example in Italy, during the outbreak, the combined action of the restrictive government measures and of media-driven awareness programs have played a very crucial role in softening the severity of the epidemic. In fact if compulsory restrictive measures (lockdown) have reduced disease transmission because of the dramatic change in people everyday behaviors [11, 12] , media-driven programs have greatly contributed to the maintenance over time of a responsible and respectful behavior by the means of a constant awareness action according to the basic idea that, if carrying out tens of thousands of tampons for day could not stop the virus alone, our behaviors can strongly help to do it. On this line, most countries and the 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 [13, 14] . On the other hand, the belief that awareness could be used as a weapon to better manage epidemics is currently a hot spot that continues to capture attention as the growing number of scientific studies on the subject testify, e.g. [15, 16, 17, 18] In this paper we use a qualitative SEIR-type model to give a contribution for the understanding of the possible mechanisms that can lead to the spread and control of Covid-19. We focus on the interplay between two key processes: overexposure (that incorporates the role of asymptomatic/undocumented cases in the infection process) and awareness, carried out through campaigns or targeted actions, as a tool aimed at increasing social distancing. Although multigroups models are certainly more realistic, we choose not to divide the infected class in different subclasses and the action of asymptomatic/undocumented individuals is instead accounted by allowing a nonlinear functional form for the contact rate. We assume that the many asymptomatic/undocumented individuals, increasing the ambient viral load [19] , can cause an unconscious overexposure of susceptibles to infection and consider an incidence rate β(I) = β I (1 + a I). 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 β I S whereas new infections are caused by double exposures at a rate β a I 2 S [20] . Such incidence rate was linked to the occurrence of hysteretic phenomena through a backward bifurcation scenario [20, 21, 22] , a phenomenological framework that involves multiplicity of endemic equilibria and subcritical persistence of the disease. Within the backward scenario, a transcritical bifurcation involving an endemic equilibrium and the disease-free equilibrium occurs at R 0 = 1 and a saddle-node bifurcation of endemic equilibria occurs at R 0 = R sn < 1. As a consequence, to eradicate the disease, it is not sufficient to reduce R 0 below 1 since it needs to be lowered below the critical value R sn . In the perspectives of disease control, detecting and managing the occurrence of backward bifurcations has hence an unquestionable importance [23, 24, 25, 26, 27] . Media awareness campaigns are instead considered by the means of an awareness variable m(t) that has the effect of isolating a fraction of susceptibles and whose governing equation is given by the balance between an implementation process (which is supposed to be proportional to the number of infected individuals) and a depletion process (which is related to ineffectiveness, misleading information, fading factors). The increased awareness about the disease has the direct effect to push susceptibles towards social distancing and isolation. A similar dynamics for media-driven awareness programs has been considered for example in [18] where the impact of information has been evaluated in a SIS model with variable population and immigration. Awareness programs were recognized to be helpful in controlling the spread of infectious disease in the sense that, when disease is endemic, increasing awareness makes the number of infective individuals to decrease. In this paper, we show that when the level of overexposure is sufficiently high, Covid-19 spread can display an hysteretic behavior. In this case, dangerous dynamical transitions could be avoided through a suitable management of people awareness. To this aim, information dissemination, implementation and depletion are factors that must be adequately monitored and balanced. In addition, we use Google Trends data on Internet user search in the context of Covid-19 to stress how a low level of awareness, combined with high overexposure, can be related to the emergence of hysteretic behaviors. For all the above reasons -despite the many papers dedicated to the study of Covid-19 -we feel we might have added something new to the existing literature on the subject. The paper is structured as follows. In Section 2 we introduce the model, detect its equilibria and derive conditions for which the system exhibits a multiplicity of endemic equilibria. This circumstance is compatible with the backward scenario that will be investigated in details in Section 3 where conditions for the occurrence of the backward or forward bifurcation are derived in terms of the system parameters. Numerical validations of the obtained results as well as investigations in the time dependent regimes have been performed by choosing parameter values in correspondence of the Coronavirus Covid19 spread in Lombardy, one of the northern regions of Italy most affected by the virus. In Section 4 the general implications of the obtained results are discussed, with the role of awareness adequately stressed. In Section 5, a Google Trends analysis using Coronavirus as search query allowed us to stress the role of awareness in some recent episodes of epidemic resurgence in Europe. To this aim, we specifically consider data ranging from the beginning of the pandemic until August 2020. Conclusions, in Section 6, end the paper.

We consider the total population N(t) as divided into the susceptible (S), exposed (E), infective (I), recovered (R) individuals and assume the total population 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. We suppose that the dynamics of the different classes as well as that for the awareness variable m(t) is governed by the following system of differential equations:Ṡ

All the parameters in model (1) are assumed to be positive constants with the following meaning: Λ is the recruitment rate; µ is the natural death rate; β is the transmission rate; a is the overexposure coefficient; 1/σ is the incubation period; α is the disease-related death rate; k is the cure rate; γ is the dissemination rate of awareness among susceptibles; ρ 0 is the depletion rate of the awareness programs whereas ρ is their implementation rate.

We preliminary observe that the set Ω = (S, E, I, R, m) ∈ R 5 + :

is positively invariant and absorbing with respect to system (1), as a consequence the orbits of (1) with non negative initial data are bounded. Moreover, since the first three equations as well as the last one are independent on the forth equation, it suffices to consider the following SEIm model:

System (2) admits the disease-free equilibrium P 0 = (S 0 , E 0 , I 0 , m 0 ) = (Λ/µ, 0, 0, 0). Introducing the basic reproduction number R 0 :

we can state the following result:

Theorem 2.1. The disease-free equilibrium P 0 is locally asymptotically stable for R 0 < 1 and unstable for R 0 > 1.

Proof. The Jacobian matrix of model (2) , evaluated at P 0 , is given by:

which allows us to obtain the following eigenvalues: λ 1 = −µ, λ 2 = −ρ 0 , whereas λ 3,4 are solutions of the following second order algebraic equation:

It then follows that P 0 is locally asymptotically stable if R 0 < 1 and unstable otherwise. We also observe that, R 0 = 1 is a bifurcation value.

Remark 2.1. The bifurcation threshold R 0 = 1 can equivalently be expressed in terms of the parameter β. In fact:

System (2) also admits an endemic endemic equilibrium P * = (S * , E * , I * , m * ) with

and where I * is a real positive solution of the following second order algebraic equation:

When R 0 > 1, then C < 0 and hence ∆ = B 2 − 4 A C is a positive quantity. According to the Descartes's rule of signs, the algebraic equation (7) admits a unique real positive root, independently of the sign of B. Therefore, the following result holds:

If β > β c , then equation (7) admits only one positive real solution, so that model (2) admits one endemic equilibrium P * .

When R 0 < 1 then C > 0 and, according to the sign of the coefficient B in equation (7), two or zero real positive solutions are possible. To show this, we preliminary introduce the following quantities:

where ζ has the role of an awareness parameter and Γ 1 is defined in (5).

We can hence state the following results:

If β < β c and a < a * , then equation (7) admits no positive solutions so that model (2) admits none endemic equilibrium.

Proof. Observe that condition β < β c implies that the sign of the coefficient C in (7) is always positive. Moreover, condition a < a * implies that the sign of the coefficient B is positive too. Hence the thesis follows by the Descartes' rule of signs.

We observe that if a > a * thenβ is a positive quantity. Moreover,

Theorem 2.4. Let a > a * and ζ < µ (a − a * ).

(i) If β * 2 < β < β c , then model (2) admits two endemic equilibria P * 1 and P * 2 . (ii) If β < β * 2 then model (2) admits none endemic equilibrium.

Proof. We first observe that, because of Remark 2.2,β < β c . Therefore, being β < β c and a > a * , we have that the sign of the coefficient C in (7) is always positive but the sign of the coefficient B is positive if β <β and it is negative if β >β. In the former case, equation (7) has no positive roots whereas in the latter case it admits two positive solutions that are real if and only if ∆ = B 2 − 4 A C > 0. At this regards we observe that

with

Let us observe that

because of the assumption ζ < µ(a − a * ). Therefore inequality (10) 

By direct computation (see Remark2.3), it is possible to prove that β * 1 <β < β * 2 < β c , so that the thesis follows. Remark 2.3. Let a > a * and ζ < µ (a − a * ). Then:

The above results suggest the occurrence of a backward bifurcation scenario, with a saddlenode bifurcation at β = β * 2 and a transcritical bifurcation at β = β c . In the next section, we investigate such features in more details.

We derive conditions for the occurrence of backward or forward scenario for model (2) by using the method proposed in [28] . We observe that all the coefficients in the equilibrium equation (7) may be regarded as functions of the parameter β. Moreover at β = β c , C(β c ) = 0 and equation (7) becomes

with roots I = 0 and I = − B(βc)

A(βc) . The former is related to the disease-free equilibrium and the latter refers to a positive endemic equilibrium only if B(β c ) and A(β c ) have opposite signs. Since A(β c ) > 0, then B(β c ) < 0 must hold in order to have a positive endemic equilibrium. Implicit differentiation of equation (7) with respect to β gives:

so that looking at the equilibrium I = 0, at β = β c , one has:

since, recalling (8), it holds dC dβ < 0. Therefore the slope of the bifurcation curve at I = 0 must have the same sign with respect to the coefficient B(β c ). As a consequence,

For model (2), B(β c ) < 0 is hence a necessary and sufficient condition for the occurrence of the backward bifurcation at β = β c . From (8), it follows that:

with a > a * . Therefore B(β c ) < 0 if and only if ζ < µ (a − a * ).

We can hence state the following theorem:

Theorem 3.1. Let a > a * (i) If ζ > µ (a−a * ) then system (2) exhibits a forward bifurcation at β = β c . (ii) If ζ < µ (a − a * ) then system (2) exhibits a backward bifurcation at β = β c .

Proof. It follows from (8) by direct computations.

Therefore if the overexposure parameter a is above the threshold a * , the awareness parameter ζ can make the difference between the forward and the backward scenario. In fact, in the case of efficient awareness campaigns, ζ > µ (a − a * ) and a classical forward scenario is obtained. On the contrary, for ζ < µ (a − a * ), inadequate awareness campaigns induce a backward scenario. To validate numerically the above results we use the parameter values summarized in Table  1 and chosen 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 Λ and µ are chosen such that Λ/µ = 10036258 (as the total population of Lombardy in 2019) [29] . The average lifespan is taken to be 1/µ = 83 years (i.e. µ = 0.000033 per day) [29] so that Λ = 331 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 [30] with an average of 5.2 days [31] . Therefore σ = 1/(5.2) days −1 . The WHO report time from onset of symptoms to death of about 2 weeks [32] and the disease-induced death rate for infectious individuals is assumed to be α = 1/8 days −1 [33] . Moreover, the average hospital length of stay of discharged patients is 13 days [34] , so that we assume the cure rate to be k = 1/13 days −1 . The parameters a, β and the awareness parameter ζ are instead assumed to be varying. To give a more 'quantitative' measure of the impact of the above parameters on the basic reproduction number R 0 , we test its robustness by the means of a sensitivity analysis [17, 35] . The sensitivity of a certain variable with respect to system parameters can be in fact measured through a sensitivity index that provides a quantitative measure of the relative change in a variable when a parameter changes. Therefore, introducing the normalized forward sensitivity of R 0 with respect a given parameter p [35] ,

we easily obtain the sensitivity index of R 0 with respect to system parameters. We found that the most sensitive parameters are the transmission rate β and the recruitment rate Λ with φ R 0 β = 1 and φ R 0 Λ = 1: when these parameters are increased by 10%, also R 0 increases by 10%. The incubation period has instead a very low impact on R 0 , being φ R 0 σ = 0.00017 so that increasing the parameter σ by 10%, R 0 increases by 0.0017%. We also observe that the natural death rate µ and the disease-related death rate α negatively impact R 0 , being φ R 0 µ = −1.0003 and φ R 0 α = −0.61. The cure rate k also negatively impacts R 0 since φ R 0 k = −0.38 so that, increasing k by 10%, has the effect of reducing R 0 by 3.8%. As far as the threshold values are concerned, with the parameter values as in Table1, the Table 1. transcritical bifurcation threshold β c is such that β c = 2.0143 · 10 −8 whereas a * = 0.0006. We choose a = 10 > a * so that µ (a − a * ) = 0.00033 and show the impact of information in determining a forward or a backward scenario. In the presence of efficient awareness campaigns, ζ is such that ζ > µ (a − a * ). In this case, we consider ζ = 0.0004 (by setting γ = 4 · 10 −2 , ρ = 10 −1 , ρ 0 = 10) obtaining the classic forward scenario where no endemic equilibria can be detected subtreshold, Fig.1 . For inadequate awareness campaigns, ζ is instead such that ζ < µ (a − a * ). In this case, we consider ζ = 0.0002 (by setting γ = 2 · 10 −2 , ρ = 10 −1 , ρ 0 = 10) and obtain a backward scenario, where a multiplicity of endemic equilibria can be detected subtreshold, Fig.2 . Moreover, increasing awareness has the effect to mitigate the related phenomenology by decreasing the length of the range (β * 2 , β c ) as shown in Fig.3 .

Investigations in the time dependent regimes allow us to get further inside in the consequences of such two phenomenologies. The scenario induced by efficient awareness campaigns is a classical forward scenario: for β < β c the disease-free equilibria P 0 is the only attractor for the system whereas for β > β c , system trajectories tend towards the endemic equilibrium P * , Fig.4 . In this case, the eradication of the disease can be classically obtained by simply reducing the parameter β below the transcritical bifurcation threshold β c . The backward scenario induced by inadequate awareness campaigns presents instead more complex features. With regard to stability properties, the endemic equilibrium E * showing the coalescence of the two endemic equilibria E 1 and E 2 at the saddle-node bifurcation threshold β = β * 2 = 1.23589 · 10 −8 . A transcritical bifurcation occurs at β = β c . (Bottom) Detail of the bifurcation diagram below the transcritical bifurcation threshold β c : for β * 2 < β < β c two endemic endemic equilibria, P * 1 and P * 2 , that coalesce and disappear at the saddle-node bifurcation threshold β * 2 . The numerical value for the awareness parameter is ζ = 0.0002; the other parameters are as in Table 1 . -characterized by a small number of infectives -is always unstable whereas the diseasefree equilibrium P 0 is locally asymptotically stable for β < β c and unstable otherwise. The endemic equilibrium E * 2 -characterized by a large number of infectives -may instead change its stability properties by varying the bifurcation parameter. It is in fact stable for β > β h = 1.609215 · 10 −8 and loses its stability at β = β h because of a subcritical Hopf bifurcation. The interval (β h , β c ) is hence a range of the parameter β where bistability can be found sub-threshold. In this range, initial conditions near the endemic equilibrium P * 2 drive the system towards this same equilibrium. However, when the value of the parameter β is decreased from the transcritical bifurcation threshold β c towards the subcritical Hopf bifurcation threshold β h , we observe that system trajectories approach the endemic equilibrium P * 2 after an ever longer transient and with an increasing number of damped oscillations, Fig.5 (top) . Differently, when the value of β is slightly decreased below the threshold β h , initial conditions near the endemic equilibrium P * 2 drive the system towards the disease-free equilibrium P 0 . This transition occurs after a long transient of large amplitude oscillations that suddenly collapse on the disease-free equilibrium. This is an effect of the subcritical Hopf bifurcation since system trajectories starting near the endemic equilibrium P * 2 remains in the neighboring of the unstable cycle before entering in the basin of attraction of the disease-free equilibrium P 0 . Further lowering β, the transient shortens and the number of oscillations progressively decreases, Fig.5(bottom) . When the value of β is lowered below the saddle-node bifurcation threshold β * 2 , because of a transition focus-node, the disease-free equilibrium P 0 is approached monotonically. Our analysis in the time-dependent regimes hence reveals that, although the bistability range ends with the subcritical Hopf bifurcation β h , to eradicate the disease it would be preferable to reduce β under the saddle-node bifurcation threshold β * 2 in order to avoid the occurrence of dangerous and unwieldy transient oscillations subthreshold before the collapse of system trajectories towards the disease-free equilibrium.

Our analytical results indicate the peculiar role of the overexposure parameter a and of the awareness parameter ζ in discriminating between the forward and the backward scenario. If the overexposure parameter a is sufficiently low, i.e. lower than a threshold value a * , then no endemic equilibrium can be found subthreshold and a forward scenario is obtained that is classically manageable by reducing R 0 below 1. For higher values of the overexposure parameter, the expected phenomenology becomes more complex since either forward and backward scenario can be obtained. In this case, the right and responsible use of mediadriven campaigns aimed to increase collective awareness can make the difference between a more and a less manageable situation. Keeping high the values of the awareness parameter ζ ensures a forward bifurcation scenario; low values of awareness are instead not able to avoid the backward phenomenology which is associated to an endemic persistence below the transcritical threshold R 0 = 1. Our definition (9) of the awareness parameter ζ, i.e. ζ = γ ρ/ρ 0 , elucidates that awareness is based on the interplay between different processes, being increased by increasing both the dissemination and the implementation rate and decreased by increasing the depletion rate because of misleading information campaigns or fading mechanisms. Therefore a serious, coherent and adequately disseminated information, ensuring a high level of awareness in the population, is a powerful weapon to fight possibly dangerous scenarios as the backward bifurcation one. But the situation is even more problematic than what the backward scenario seems to show. Indeed, a detailed analysis in the time dependent regimes has shown that an high overexposure along with little awareness can be doubly harmful because of the combined occurrence of the backward bifurcation and of a subcritical Hopf bifurcation that involves the stable endemic equilibrium subthreshold. Both these bifurcations are in fact associated to catastrophic transitions in dynamical systems. This means that, if the system exhibits a certain asymptotic dynamical regime, it may happen that due to a microscopic variation in some parameters, a transient can occur after which the system is in a macroscopically different regime. We show that this is the case by fixing β < β c , i.e. β = 1.608 · 10 −8 ≈ β h , and varying the awareness parameter ζ by slightly increasing the dissemination rate γ. We want to stress that, interestingly, the same kind of results can be obtained if one introduces the awareness variable m by the means of a distributed delay, i.e.

as to summarize information about the current and recent past values of the disease. The importance of considering such kind of models is provided by the fact that the role of delays in biological models is widely recognized [36, 37, 38, 39, 40] , being often appropriate for these kind of problems to allow the rate of change of the system variables to depend in some sense on the previous history. In this case, the distributed lag (14) in the governing equations means that system dynamics at time t is affected by the state variables S, E, I at possibly all previous times τ ≤ t in a way prescribed by the function f (S(τ ), E(τ ), I(τ )) and distributed in the past by the delay kernel K ν (t − τ ), also called 'memory function'. In other words, the delay kernel is a weighting factor that indicates how much emphasis should be given to the size of the population at earlier times to determine the present effect on the awareness variable. The function f (S(τ ), E(τ ), I(τ )) is often assumed to depends only on prevalence [37] so that f (I) = h I where the parameter h ∈ [0, 1] represents the information coverage and can be interpreted as the balance between two opposite phenomena, the disease under-reporting and the level of media coverage of the status of the disease, which tends to amplify the social alarm [37] . To provide a reasonable effect of short term memory, one can choose the Gamma distribution delay kernel

with n = 1, obtaining the so called weak exponential delay kernel

Such a choice qualitatively represents a weak delay in the sense that the maximum (weighted) response of the growth rate is to current population density whereas past densities have exponentially decreasing influence. Moreover, the positive constant ν is related to the average delay T : in fact T = n/ν and, for the weak exponential delay kernel, T = 1/ν. With (16) as delay kernel and by applying the linear chain trick [38, 40] , one explicitly obtains the differential equation prescribing the awareness dynamics,

that is exactly the forth equation in (2) when ρ = ν h and ρ 0 = ν namely when the implementation and depletion rate are assumed to be proportional, with ν as a constant of proportionality. In this case the awareness parameter ζ reduces to ζ = γ h and does not directly involve the parameter ν. This would suggest that in the case of high overexposure, differently from the dissemination rate γ and the information coverage h, the delay parameter ν would have no role in discriminating between the forward or the backward scenario. However, although not impacting the number and the numerical values of the system endemic equilibria, it can have a role in changing their stability properties. In Fig 7, we have considered the same parameter values as in Fig.6 Table  1 . Initial conditions are chosen in the neighboring of the endemic equilibrium P * 2 .

i.e. β = 1.608 · 10 −8 , and the dissemination rate γ = 1.98 · 10 −2 . We further consider ρ = ν h, ρ 0 = ν and choose h = ρ/ρ 0 such that h = 0.1/10. The numerical values for the other parameters are as in Table 1 . In this way, compatibly with Fig. 6 , the awareness parameter ζ = γ h = γ ρ/ρ, is such that ζ < ζ * so that a backward scenario is expected. Nevertheless, reducing the value of the delay parameter ν has a destabilizing effect on the endemic equilibrium P * 2 and for ν less than a critical threshold ν * , system trajectories near P * 2 leads to the disease-free equilibrium P 0 . This makes sense since decreasing ν means increasing the average length T of the historical memory concerning the disease in study, acting in favor of disease eradication.

To concretely show that, in some conditions, a decreased awareness can be associated with a drastic resumption of the epidemic, we consider the case of Spain that has experienced a drastic re-surgence of epidemic since July 2020. To this aim, we perform a Google Trends analysis using Coronavirus as search query and combine it with the trend of the newly confirmed Covid19 cases. We specifically use data ranging from the beginning of the pandemic until August 2020. Google Trends data allows to study trends and patterns of Google search queries [14] . It makes use of million of user searches and has been recently employed within the health issue framework to monitor disease control, to detect the success rates of awareness programs and to predict outbreaks worldwide [41, 42, 43] . Google Trends expresses the absolute number of searches relative to the total number of searches over the defined period of interest. For the specified search query in the time of interest, Google Trends provides an index that ranges from 0 to 100, with 100 being the highest relative search term activity [14] . 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. Looking at the trend of the Relative Search Volume (RSV) for the search query 'Coronavirus' over the last year in Spain (Fig. 8, top) , we see that a first sign of popularity, 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 8%. This is followed by a decrease until the week of figure) , the interest level quickly reaches its highest peak with a relative popularity of 100%. From here a slow progressive decreasing begins that will end in the week June 7-13 with an interest level of 9% (point F in the figure) . This trend will remain substantially stable until 30 August (point G in the figure), resulting in a prolonged period of low interest in the subject (ζ < ζ * ). This lower interest, at first sight rather strange, can be partly explained by the fact that, also due to the prolonged emergency, people have likely learned to live with the idea of the Covid by deceptively lowering their defenses and causing a situation of decreased awareness. On the other hand the trend of the new cases per day (Fig. 8, bottom) , shows that the first cases appeared in the period 31Jan-25Feb and community transmission exponentially continued in the period 26Feb-12Mar (1 Mar=54; 6 Mar=225; 12 Mar=1472) until it peaked on Mar 26 with 9159 new cases. From here the trend is decreasing with marked fluctuations until June 1st when the number of new cases is 209. From 1st June to 29th June the trend of new cases is essentially stable with small fluctuations around the value 200. However from 3 July onwards, the trend is growing rapidly with marked fluctuations, to the point that 9779 new cases were recorded on 30 August, the highest number since the beginning of the pandemic in Spain. Contextualizing these numbers, we can observe that the low number of new cases per day recorded throughout the month of June -a first sign of success for containing the epidemic -is certainly a direct effect of the restrictive measures implemented by the Spanish government since mid-March. However, the relaxation of the restrictive measures in April, the official end of the emergency decreed on 21 June together with the start of the tourist season, characterized by a greater mobility and a more casual tendency to gather (discos, pubs, summer nightlife etc.), favored a situation of overexposure (a > a * ), also thanks to the large number of asymptomatic people. In line with our findings this condition, combined with the decreased level of awareness (ζ < ζ * ) recorded in this same period, can be responsible of a backward scenario and can be associated to the drastic resumption of epidemic in Spain we are observing still now. However, a decreased awareness does not necessarily produce catastrophic effects and the case of New Zealand is emblematic of this. The trend of the RSV for the search query 'Coronavirus' over the last year in New Zeland (Fig.9, top) shows that a first sign of popularity, detectable in a peak in the RSV, is found in the week of 26Jan-1Feb (point A in the figure) with an interest level of 23%. This is followed by a decrease until the week of Feb 16-22 (point B in the figure) where the interest level is at 9% and by a growth up to the week of Mar 15-21 (point C in the figure), when it reaches its highest peak with a relative popularity of 100%. From here a slow progressive decreasing begins that will end in the week of 7-13 June with an interest level of 9% (point D in the figure). This trend will remain substantially stable until 30 August, resulting in a prolonged period of low interest in the subject (ζ < ζ * ). In New Zeland, the number of new cases per day has remained relatively low until now. Fig.9 (bottom) shows that the period of maximum expansion was from March to the beginning of April: the maximum value of new daily cases was 89, and then it gradually began to decline. From 20 April to early August, the trend remains very low (with less than three new cases per day), while in August there was a very slight increase with the number of new cases per day oscillating around 10. These persistent low numbers can be explained by the drastic planning of restrictive measures in New Zeland that have favored social distancing by strongly reducing the risk of overexposure (i.e. hence keeping low the value of the parameter a). In fact, from the outset, drastic measures were taken well in advance, when the number of cases was still low and the lockdown was started with less than 100 cases in the country. Even more recently, in August, the city of Auckland has returned to a short lockdown after the discovery of 4 new infected. The case of New Zeland therefore supports the idea that if overexposure is kept low (a < a * ), a low level of awareness (ζ < ζ * ) is not capable, alone, to cause a backward scenario. This is exactly online with our findings and seems hence to suggest that, if these conditions continue to be maintained, there would be no risk of a drastic resurgence of the disease for New Zeland.

In this paper we considered a SEIR-type model that (i) incorporates the role of asymptomatic/undocumented cases in the infection process by means of an overexposure mechanism (ii) considers awareness, carried out through campaigns or targeted actions, as a tool aimed at increasing social distancing. We found that high overexposure can lead system dynamics towards a backward phenomenology and that a responsible role of information aimed at a correct management of awareness can allow to avoid such a scenario. Numerical investigations in the time dependent regimes have shown that the emerging phenomenology can be doubly dangerous because of catastrophic dynamic transitions associated with both the backward bifurcation and a subcritical Hopf bifurcation that the system experiences subthreshold. In fact, if the former indicates that lowering R 0 below the classical threshold R 0 = 1 is not enough to eradicate the disease, the latter can be responsible for the emergence of transient large-amplitude oscillations subthreshold before disease eradication.

This intriguing interplay between overexposure and awareness highlights a point that should not be underestimated both in the current and future management of Covid19 emergency. In fact if lowering the basic reproduction number R 0 below the threshold R 0 = 1 has indeed been pursued to ensure its eradication, some factors -such as overexposure -can cause the emergence of less classic scenarios, as the backward one, that is associated with the persistence of the disease subthreshold. Our results demonstrate that in this case dangerous dynamical transitions could be avoided through a suitable management of people awareness and that, to this aim, information dissemination, implementation and depletion are factors that must be adequately monitored and balanced. Moreover, when information is supposed to depend on the past history of the disease, increasing the length of the historical memory concerning the disease can be crucial to suppress dangerous transient oscillations subthreshold before disease eradication. The obtained results shed light on some features that goes beyond the management of the Covid19 emergency. In fact, the importance of managing individuals' behaviors by increasing their awareness through balanced and aware information becomes a key point, to be taken seriously into consideration, for the control of all those diseases with the same transmission characteristics and in which asymptomatic/undocumented cases or other environmental factors can induce high levels of overexposure and increase the probability of disease transmission. Furthermore, the awareness that many clinical aspects of COVID19 are not yet fully understood, suggests that for a correct management of the emergency, those mechanisms believed to be responsible for an hysteretic behavior should be adequately detected and investigated. In fact, in this case, aiming to bring R 0 below 1 can be an insufficient goal for the eradication of the disease. The interplay between awareness campaigns, behaviors and disease transmission is a complex topic and this work merely scratches the surface of the multifarious processes that are part of it. Our results represent a qualitative step to better understand how awareness can impact disease transmission in the presence of factors that can induce overexposure to disease. In order to provide further insights into the topic, the simple SEIR-like model employed in this study could be extended at least in two directions. First, different levels of awareness (i.e. high, moderate and low) could be included so that a more general model with more susceptible classes that practice social distancing at different levels could be investigated. On this line it is also of interest, for awareness, to consider a mechanism somehow similar to 'reinfection', whereby a certain portion of aware people return to being entirely susceptible because their level of commitment to avoid spreading decreases. A further interesting direction of study is to suppose that the awareness variable is not forced to follow a prescribed dynamics but it is only required to verify general conditions such as positiveness, boundedness or even a convergence towards a fixed limit. On the other hand awareness could be considered, to all intents and purposes, as a control variable with the aim to investigate its impact in defusing the mechanism of repeated epidemic waves and to mitigate the spread of the epidemic. An extension of the SEIR-like model according to these points of interest is a course for future work.

(iii) β * 1 <β We observe that β * 1 −β = 1 2 1 ρ 2 0 (Λ a σ + Γ 1 ) 2 (Γ 1 − Λ a σ)

where ∆ * is defined in (11) and Q 2 is given by (20) .

Being a > a * , one has that Γ 1 − Λ a σ < 0 and by (21) , it follows that

We observe that Λ a σ − Γ 1 > 0 and that

that is verified since ζ < µ(a − a * ). Therefore the thesis is verified.

Uses and abuses of mathematics in biology

An updated estimation of the risk of transmission of the novel coronavirus (2019-nCov)

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On the dynamics of an SEIR epidemic model with a convex incidence rate

Epidemic solutions and endemic catastrophes

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Causes of backward bifurcations in some epidemiological models

Van den Driessche, Backward bifurcation in epidemic control

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World Health Organization (WHO) , World Health Organisation, Coronavirus disease 2019 (COVID-19) Situation Report-56

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The Validity of Google Trends Search Volumes for Behavioral Forecasting of National Suicide Rates in Ireland

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The present work has been performed under the auspices of the Italian National Group for Mathematical Physics (GNFM-Indam). We also thank the anonymous referees and the handling Editor for their valuable comments and remarks.

Let a > a * and ζ < µ (a − a * ). Let β c defined in (4) ,β defined in (9) , β * 1 and β * 2 defined as in (12) . We prove that the following inequalities are verified:We observe thatwhere ∆ * is defined in (11) and Q 1 is given by:and is a positive quantity since ζ < µ a. By (18) , it follows thathence the thesis is verified.where ∆ * is defined in (11) and Q 2 is given by:Being a > a * , one has that Γ 1 − Λ a σ < 0 and by (19) , it follows thatso that the thesis is verified.

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.