Modelling information-dependent social behaviors in response to lockdowns: the case of COVID-19 epidemic in Italy

The COVID-19 pandemic started in January 2020 has not only threatened world public health, but severely impacted almost every facet of lives including behavioral and psychological aspects. In this paper we focus on the 'human element' and propose a mathematical model to investigate the effects on the COVID-19 epidemic of social behavioral changes in response to lockdowns. We consider a SEIR-like epidemic model where that contact and quarantine rates are assumed to depend on the available information and rumors about the disease status in the community. The model is applied to the case of COVID-19 epidemic in Italy. We consider the period that stretches between February 24, 2020 when the first bulletin by the Italian Civil Protection was reported and May 18, 2020 when the lockdown restrictions have been mostly removed. The role played by the information-related parameters is determined by evaluating how they affect suitable outbreak-severity indicators. We estimated that citizens compliance with mitigation measures played a decisive role in curbing the epidemic curve by preventing a duplication of deaths and about 46% more contagions.


Introduction
In December 2019, the Municipal Health Commission of Wuhan, China, reported to the World Health Organization a cluster of viral pneumonia of unknown aetiology in Wuhan City, Hubei province. On January 9, 2020, the China CDC reported that the respiratory disease, later named COVID19, was caused by the novel coronavirus SARSCoV2 [52]. The outbreak of COVID19 rapidly expanded from Hubei province to the rest of China and then to other countries. Finally, it developed in a devastating pandemics aecting almost all the countries of the world [10]. As of May 18, 2020 a total of 4,7 million of cases of COVID19 and 315,131 related deaths have been reported worldwide [10]. In absence of treatment and vaccine, the mitigation strategy enforced by many countries during the COVID19 pandemic have been based on social distancing. Each government enacted a series of restriction aecting billions of people including recommendation of restricted movements for some or all of their citizens, and localized or national lockdown with the partial or full closingo of nonessential companies and manufacturing plants [5]. Italy has been the rst European country aected by COVID19. The country has been strongly hit by the epidemic which has triggered progressively stricter restrictions aimed at minimizing the spread of 1 the coronavirus. The actions enacted by the Italian government began with reducing social interactions through quarantine and isolation till to reach the full lockdown [1,30]. On May 4, 2020, the phase two began, marking a gradual reopening of the economy and restriction easing for residents. One week later, shops also reopened and the restrictions on mobility were essentially eliminated, with the only obligation in many regions to use protective masks [32]. During the period that stretches between January 22, 2020 and May 18, 2020, Italy suered 225,886 ocial COVID19 cases and 32,007 deaths [33]. The scientic community has promptly reacted to the COVID19 pandemic. Since the early stage of the pandemic a number of mathematical models and methods has been used. Among the main concerns raised were: predicting the evolution of the COVID19 pandemic wave worldwide or in specic countries [12,25,42]; predicting epidemic peaks and ICU accesses [46]; assessing the eects of containment measures [12,14,24,25,35,42,45] and, more generally, assessing the impact on populations in terms of economics, societal needs, employment, health care, deaths toll, etc [20,36]. Among the mathematical approaches used, many authors relied on deterministic compartmental models. This approach has been revealed successful for reproducing epidemic curves in the past SARSCoV outbreak in 20022003 [26] and has been employed also for COVID19. Specic studies focused on the case of epidemic in Italy: Gatto et al. [24] studied the transmission between a network of 107 Italian provinces by using a SEPIA model as core model. Their SEPIA model discriminates between infectious individuals depending on presence and severity of their symptoms. They examined the eects of the intervention measures in terms of number of averted cases and hospitalizations in the period February 22 March 25, 2020. Giordano et al. [25] proposed an even more detailed model, named SIDARTHE, in which the distinction between diagnosed and nondiagnosed individuals plays an important role. They used the SIDARTHE model to predict the course of the epidemic and to show the need to use testing and contact tracing combined to social distancing measures. The mitigation measures like social distancing, quarantine and selfisolation may be encouraged or mandated [42]. However, although the vast majority of people were following the rules, even in this last case there are many reports of people breaching restrictions [4,47]. Local authorities needed to continuously verifying compliance with mitigation measures through monitoring by health ocials and police actions (checkpoints, use of drones, ne or jail threats, etc). This behavior might be related to costs that individuals aected by epidemic control measures pay in terms of health, including loss of social relationships, psychological pressure, increasing stress and health hazards resulting in a substantial damage to population wellbeing [6,20,53]. As far as we know, the mathematically oriented papers on COVID19 nowadays available in the literature do not explicitly take into account of the fraction of individuals that change their social behaviors solely in response to social alarm. From a mathematical point of view, the change in social behaviors may be described by employing the method of informationdependent models [40,54] which is based on the introduction of a suitable information index M (t) (see [40,54]). This method has been applied to vaccine preventable childhood diseases [16,54] and is currently under development (see [8,37,59] for very recent contributions). In this paper, the main goal is to assess the eects on the COVID19 epidemic of human behavioral changes during the lockdowns. To this aim we build up an informationdependent SEIRlike model which is based on the key assumption that the choice to respect the lockdown restrictions, specically the social distance and the quarantine, is partially determined on fully voluntary basis and depends on the available information and rumors concerning the spread of the COVID19 disease in the community. A second goal of this manuscript is to provide an application of the information index to a specic eldcase, where the model is parametrized and the solutions compared with ocial data.
We focus on the case of COVID19 epidemic in Italy during the period that begins on February 24, 2020, when the rst bulletin by the Italian Civil Protection was reported [33], includes the partial and full lockdown restrictions, and ends on May 18, 2020 when the lockdown restrictions have been mostly removed. We stress the role played by circulating information by evaluating the absolute and relative variations of diseaseseverity indicators as functions of the informationrelated parameters. The rest of the paper is organized as follows: in Section 2 we introduce the model balance equations and informationdependent processes. Two critical epidemiological thresholds are computed in Section 3. Section 4 is devoted to model parametrization for numerical simulations, that are then shown and discussed in Section 5. Conclusions and future perspective are given in Section 6.
2 Model formulation

State variables and balance equations
We assume that the total population is divided into seven disjoint compartments, susceptibles S, exposed E, presymptomatic I p , asymptomatic/mildly symptomatic I m , severely symptomatic (hospitalized) I s , quarantined Q and recovered R. Any individual of the population belongs to one (and only one) compartment. The size of each compartment at time t represents a state variable of a mathematical model. The state variables and the processes included in the model are illustrated in the ow chart in Fig. 1. The model is given by the following system of nonlinear ordinary dierential equations, where each (balance equation) rules the rate of change of a state variable.
The model formulation is described in detail in the next subsections.

The role of information
As mentioned in the introduction, the mitigation strategy enforced by many countries during the COVID 19 pandemic has been based on social distancing and quarantine. Motivated by the discussion above, we assume that the nal choice to adhere or not to adhere lockdown restrictions is partially determined on fully voluntary basis and depends on the available information and rumors concerning the spread of the disease in the community. From a mathematical point of view, we describe the change in social behaviors by employing the method of informationdependent models [40,54]. The information is mathematically represented by an information Figure 1: Flow chart for the COVID19 model (1)(3). The population is divided into seven disjoint compartments of individuals: susceptible S(t), exposed E(t), presymptomatic I p (t), asymptomatic/mildly symptomatic I m (t), severely symptomatic/hospitalized I s (t), quarantined Q(t) and recovered R(t). Blue colour indicates the informationdependent processes in model (see (7)(8)(9), with M (t) ruled by (3)).
index M (t) (see Appendix A for the general denition), which summarizes the information about the current and past values of the disease [8,1517] and is given by the following distributed delay This formulation may be interpreted as follows: the rst order Erlang distribution Erl 1,a (x) represents an exponentially fading memory, where the parameter a is the inverse of the average time delay of the collected information on the status of the disease in the community (see Appendix A). On the other hand, we assume that people react in response to information and rumors regarding the daily number of quarantined and hospitalized individuals. The information coverage k is assumed to be positive and k ≤ 1, which mimics the evidence that COVID19 ocial data could be underreported in many cases [38,42]. With this choice, by applying the linear chain trick [39], we obtain the dierential equation ruling the the dynamics of the information index M :Ṁ

Formulation of the balance equations
In this section we derive in details each balance equation of model (1).

Equation (1a): Susceptible individuals, S(t)
Susceptibles are the individuals who are healthy but can contract the disease. Demography is incorporated in the model so that a net inow rate bN due to births is considered, where b is the birth rate andN denotes the total population at the beginning of the epidemic. We also consider an inow term due to immigration, Λ 0 . Since global travel restrictions were implemented during the COVID19 epidemic outbreak [50], we assume that Λ 0 accounts only of repatriation of citizens to their countries of origin due to the COVID19 pandemic [21]. In all airports, train stations, ports and land borders travellers' health conditions were tested via thermal scanners. Although the eectiveness of such screening method is largely debated [43], for the sake of simplicity, we assume that the inow enters only into the susceptibles compartment.
In summary, we assume that the total inow rate Λ is given by: The susceptible population decreases by natural death, with death rate µ and following infection. It is believed that COVID19 is primarily transmitted from symptomatic people (mildly or severely symptomatic). In particular, although severely symptomatic individuals are isolated from the general population by hospitalization, they are still able to infect hospitals and medical personnel [2,23] and, in turn, give rise to transmission from hospital to the community. The presymptomatic transmission (i.e. the transmission from infected people before they develop signicant symptoms) is also relevant: specic studies revealed an estimate of 44% of secondary cases during the presymptomatic stage from index cases [27]. On the contrary, the asymptomatic transmission (i.e. the contagion from a person infected with COVID 19 who does not develop symptoms) seems to play a negligible role [55]. We also assume that quarantined individuals are fully isolated and therefore unable to transmit the disease. The routes of transmission from COVID19 patients as described above are included in the Force of Infection (FoI) function, i.e. the per capita rate at which susceptibles contract the infection: The transmission coecients for these three classes of infectious individuals are informationdependent and given by ε p β(M ), ε m β(M ) and ε s β(M ), respectively, with 0 ≤ ε p , ε m , ε s < 1.
The function β(M ), which models how the information aects the transmission rate, is dened as follows: the baseline transmission rate β(·) is a piecewise continuous, dierentiable and decreasing function of the information index M , with β(max(M )) > 0. We assume that where π is the probability of getting infected during a persontoperson contact and c b is the baseline contact rate. In (6) the reduction in social contacts is assumed to be the sum of a constant rate c 0 , which represent the individuals' choice to selfisolate regardless of rumors and information about the status of the disease in the population, and an informationdependent rate c 1 (M ), being c 1 (·) increasing with M and c 1 (0) = 0. In order to guarantee the positiveness, we assume c b > c 0 + max(c 1 (M )). Following [15], we nally set namely

5
. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.  Presymptomatic individuals are infectious people that have not yet developed signicant symptoms. Such individuals lie in a stage between the exposed and the expected symptomatic ones. They remain in this compartment, I p , during the postlatent incubation period and diminish due to natural death or progress to become asymptomatic or symptomatic infectious individuals (at a rate η).

Equation (1d): Asymptomatic/mildly symptomatic individuals, I m (t)
This compartment includes both the asymptomatic individuals, that is infected individuals who do not develop symptoms, and mildly symptomatic individuals [24]. As mentioned above, the asymptomatic transmission seems to play a negligible role in COVID19 transmission. However, asymptomatic individuals are infected people which results positive cases to screening (positive pharyngeal swabs) and therefore enter in the ocial data count of conrmed diagnosis. Members of this class come from presymptomatic stage and get out due to quarantine (at an informationdependent rate γ(M )), worsening symptoms (at rate σ m ) and recovery (at rate ν m ). Equation (1e): Severely symptomatic individuals (Hospitalized), I s (t) Severely symptomatic individuals are isolated from the general population by hospitalization. They arise: (i) as consequence of the development of severe symptoms by mild illness (the infectious of the class I m or the quarantined Q); (ii) directly from the fraction 1 − p of presymptomatic individuals that rapidly develop in severe illness. This class diminishes by recovery (at rate ν s ), natural death and diseaseinduced death (at rate d).

Equation (1f): Quarantined individuals, Q(t)
Quarantined individuals Q are those who are separated from the general population. We assume that quarantined are asymptomatic/mildly asymptomatic individuals. This population is diminished by nat-6 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint this version posted May 25, 2020. . https://doi.org/10.1101/2020.05.20.20107573 doi: medRxiv preprint ural deaths, aggravation of symptoms (at rate σ q , so that they move to I s ) and recovery (at a rate ν q ). For simplicity, we assume that the quarantine is 100% eective, i.e. with no possibility of contagion. Quarantine may arise in two dierent ways. From one hand, individuals may be detected by health authorities and daily checked. Such health active surveillance ensures also that the quarantine is, in some extent, respected. On the other hand, a fraction of quarantined individuals choose selfisolation since they are condent in the government handling of crisis or just believe the public health messaging and act in accordance [3]. As mentioned in subsection 2.2, we assume that the nal choice to respect or not respect the selfisolation depends on the awareness about the status of the disease in the community. Therefore, we dene the informationdependent quarantine rate as follows. We assume that where the rate γ 0 mimics the fraction of the asymptomatic/mildly symptomatic individuals I m that has been detected through screening tests and is`forced' to home isolation. The rate γ 1 (M ) represents the undetected fraction of individuals that adopt quarantine by voluntary choice as result of the inuence of the circulating information M . The function γ 1 (·) is required to be a continuous, dierentiable and increasing function w.r.t. M , with γ 1 (0) = 0. As in [8,16], we set with D > 0, 0 < ζ < 1 − γ 0 , potentially implying a roof of 1 − ζ in quarantine rate under circumstances of high perceived risk. A representative trend of γ(M ) is displayed in Fig. 2, bottom panel.

Equation (1g): Recovered individuals, R(t)
After the infectious period, individuals from the compartments I m , I s and Q recover at rates ν m , ν s and ν q , respectively. Natural death rate is also considered. We assume that individuals who recover from COVID19 acquire long lasting immunity against COVID19 although this is a currently debated question (as of 16 May, 2020) and there is still no evidence that COVID19 antibodies may protect from reinfection [58].

The reproduction numbers
A frequently used indicator for measuring the potential spread of an infectious disease in a community is the basic reproduction number, R 0 , namely the average number of secondary cases produced by one primary infection over the course of infectious period in a fully susceptible population. If the system incorporates control strategies, then the corresponding quantity is named control reproduction number and usually denoted by R C (obviously, R C < R 0 ). The reproduction number can be calculated as the spectral radius of the next generation matrix F V −1 , where F and V are dened as Jacobian matrices of the new infections appearance and the other rates of transfer, respectively, calculated for infected compartments at the diseasefree equilibrium [13,51]. In the specic case, if β(M ) = β b and γ(M ) = 0 in (1)(3), namely when containment interventions are not enacted, we obtain the expression of R 0 ; otherwise, the corresponding R C can be computed. Simple algebra yields 7 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint this version posted May 25, 2020.   (1)(3). and with (for more details see Appendix B). The rst two terms in the r.h.s of (11) describe the contributions of pre symptomatic infectious and asymptomatic/mildly symptomatic infectious, respectively, to the production of new infections close to the diseasefree equilibrium. The last three terms represent the contribution of infectious with severe symptoms, which could onset soon after the incubation phase or more gradually after a moderate symptomatic phase or even during the quarantine period. Note that the latter term is missing in the basic reproduction number (10), where the possibility for people to be quarantined is excluded. Note also that R C = R 0 when β 0 = γ 0 = 0.

Parametrization
Numerical simulations are performed in MATLAB environment [41] with the use of platform integrated functions. A detailed model parametrization is given in the next subsections.

Epidemiological parameters
The epidemiological parameters of the model as well as their baseline values are reported in Table 1. The most recent data by the Italian National Institute of Statistics [29] refer to January 1, 2019 and 8 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint this version posted May 25, 2020. . provide a countrylevel birth rate b = 7.2/1000 years −1 and a death rate µ = 10.7/1000 years −1 , as well as a resident population of aboutN ≈ 60.360 · 10 6 (12) inhabitants. Fluctuations in a time window of just over a year are considered negligible. The immigration inow term Λ 0 accounts for the repatriation of Italians abroad. On the basis of data communicated by the Italian Ministry of Foreign Aairs and International Cooperation [31], a reasonable value for Λ 0 seems to be Λ 0 = 4000/7 days −1 , namely the average number of repatriated citizens is 4000 per week. From (4), we nally obtain Λ ≈ 1.762 · 10 3 days −1 .
Epidemiological data are based on the current estimates disseminated by national and international health organizations [19,28,34,42,56,57] or inferred by modelling studies [12,24,35]. More precisely, the median incubation period is estimated to be from 56 days, with a range from 114 days, and identication of the virus in respiratory tract specimens occurred 12 days before the onset of symptoms [19,56]. Hence, we set the latency (ρ) and prelatency (η) rates to 1/5.25 days −1 and 1/1.25 days −1 , respectively. From [24], the specic informationindependent transmission rates for the presymptomatic (ε p β b ), asymptomatic/mildly symptomatic (ε m β b ) and severely symptomatic (ε s β b ) cases are such that ε m /ε p = 0.033 and ε s /ε m = 1.03. They are in accordance with the observation of high viral load close to symptoms onset (suggesting that SARSCoV2 can be easily transmissible at an early stage of infection), and with the absence of reported signicant dierence in viral load in presymptomatic and symptomatic patients [19]. We set β b = 2.25 days −1 , which, together with the other parameters, leads to the basic reproduction number R 0 ≈ 3.49, a value falling within the ranges estimated in [19,24,42,56]. As in [35], we consider that just 8% of infectious individuals show serious symptoms immediately after the incubation phase, yielding p = 0.92. Nonetheless, people with initial mild symptoms may become seriously ill and develop breathing diculties, requiring hospitalization. It is estimated that about 1 in 5 people with COVID19 show a worsening of symptoms [34] within 45 days from onset [28], giving σ m = 0.2/4.5 ≈ 0.044 days −1 . Instead, the possibility that the aggravation happens during the quarantine period is assumed to be more rare: σ q = 0.001 days −1 . Governmental eorts in identifying and quarantining positive cases were implemented since the early stage of epidemics (at February 24, 94 quarantined people were already registered [33]), hence we consider the daily mandatory quarantine rate of asymptomatic/mildly symptomatic individuals (γ 0 ) for the whole time horizon. From current available data, it seems hard to catch an uniform value for γ 0 because it largely depends on the sampling eort, namely the number of specimen collections (swabs) from persons under investigation, that varies considerably across Italian regions and in the dierent phases of the outbreak [28,33]. Since our model does not account for such territorial peculiarities and in order to reduce the number of parameters to be estimated, we assume that γ 0 = 1.3σ q , namely it is 30% higher than the daily rate at which members of the I m class hospitalize, yielding γ 0 ≈ 0.057 days −1 . Simulations with such a value provide a good approximation of the timeevolution of registered quarantined individuals at national level, as displayed in Fig. 4, second panel.
Following the approach adopted in [26] for a SARS-CoV epidemic model, we estimate the diseaseinduced death rate as where X is the case fatality and T is the expected time from hospitalization until death. From [28], we approximate X = 13% and T = 6 days (it is 9 days for patients that were transferred to intensive care and 5 days for those were not), yielding d ≈ 0.022 days −1 . Similarly, the recovery rates ν j with j ∈ {m, q, s} 9 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.   (11) versus mandatory quarantine and transmission reduction rates. Intersection between dotted black and red (resp. blue) lines indicates the value after the rst (resp. second) step reduction. Other parameters values are given in Table 1. are estimated as where T j is the expected time until recovery or expected time in quarantine/hospitalization. Preliminary data indicate that the virus can persist for up to eight days from the rst detection in moderate cases and for longer periods in more severe cases [19], suggesting T m = 6 days is an appropriate value. As far as the time spent in hospitalization or quarantine, in the lack of exact data, we assume T s < T q , because hospitalized individuals are likely to receive a partly eective, experimental treatment: mainly antibiotics, antivirals and corticosteroids [28]. Moreover, shortages in hospital beds and intensive care units (ICUs) lead to as prompt as possible discharge [22]. In particular, we set T s = 18 and T q = 25 days, by accounting also for prolonged quarantine time due to delays in test response (if any) and for WHO recommendations of an additional two weeks in home isolation even after symptoms resolve [57]. Crucially, we also estimate the initial exponential rate of case increase (say, g 0 ), by computing the dominant eigenvalue of the system's Jacobian matrix, evaluated at the diseasefree equilibrium. It provides g 0 ≈ 0.247 days −1 , in accordance to that given in [24].

The lockdowns eect on the transmission
We explicitly reproduce in our simulations the eects of the progressive restrictions posed to human mobility and humantohuman contacts in Italy. Their detailed sequence may be summarized as follows.
After the rst ocially conrmed case (the socalled`patient one') on February 21, 2020 in Lodi province, several suspected cases emerged in the south and southwest of Lombardy region. A`red zone' encompassing 11 municipalities was instituted on February 22 and put on lockdown to contain the emerging 10 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint this version posted May 25, 2020. Reactivity factor of voluntary quarantine 9 · 10 −6 ζ 1 − ζ is the roof of overall quarantine rate 0.01 days −1 a 1/a is the average information delay 1/3 days −1 k Degree of information coverage 0.8 . The next day, a decree evocatively entitled`I'm staying at home' was signed: the lockdown was declared for the whole country with severe limitations to mobility and other progressively stricter restrictions. Soon after, on March 11, 2020, the lockdown was extended to the entire country [30], with all commercial and retail businesses except those providing essential services closed down [30]. Finally, on March 22, 2020, the phase one of restrictions was completed when a full lockdown was imposed by closing all non essential companies and industrial plants [1]. On May 4, Italy entered the phase two, representing the starting point of a gradual relaxation of the restriction measures. One week later, shops also reopened and the restrictions on mobility were essentially eliminated, with the only obligation in many regions to use protective masks [32]. Because data early in an epidemic are inevitably incomplete and inaccurate, our approach has been to try to focus on what we believe to be the essentials in formulating a simple model. Keeping this in mind, we assume that the disease transmission rate incurs in just two step reductions (modelled by the reduction rate β 0 in (7)), corresponding to • March 12 (day 17), when the lockdown decree came into force along with the preceding restrictions, cumulatively resulting in a sharp decrease of SARSCoV2 transmission; • March 23 (day 28), the starting date of the full lockdown that denitely impacted the disease incidence.
In the wake of [25,42], we account for a rst step reduction by 64% (that is β b − β 0 | 17≤t<28 = 0.36β b ), which drops the control reproduction number (3) close to 1 (see Fig. 3, dotted black and red lines). It is then strengthened by an additional 28% about, resulting in a global reduction by 74% ( β b − β 0 | t≥28 = 0.26β b ) that denitely brings R C below 1 (see Fig. 3, dotted black and blue lines).

Informationdependent parameters
The informationrelated parameter values are reported in Table 2 together with their baseline values. Following [8,16], we set ζ = 0.01 days −1 potentially implying an asymptotic quarantine rate of 0.99 days −1 if we could let M go to +∞. The positive constants α e D tune the informationdependent reactivity, respectively, of susceptible and infectious people in reducing mutual social contacts and of individuals with no/mild symptoms in selfisolating. In virtue of the order of magnitude of the information index M (ranging between 10 and 10 5 ), we set α = 6 · 10 −7 and D = 9 · 10 −6 , resulting in a receptive propensity to selfisolation for asymptomatics/mildly symptomatics and less evident degree of variability in contact 11 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint this version posted May 25, 2020.  rate, being the latter already impacted and constrained by government laws (as shown later in Fig. 7). Values range for the information coverage k and the average time delay of information 1/a are mainly guessed or taken from papers where the information index M is used [7,8,15,16]. The former may be seen as a`summary' of two opposite phenomena: the disease underreporting and the level of media coverage of the status of the disease, which tends to amplify the social alarm. It is assumed to range from a minimum of 0.2 (i.e. the public awareness is of 20%) to 1. The latter ranges from the hypothetical case of prompt communication (a = 1 days −1 ) to a delay of 60 days. We tune these two parameters within their values range in order to reproduce the curves that best t with the number of hospitalized individuals (I s ) and the cumulative deaths as released every day at 6 p.m. (UTC+1 h) since February 24, 2020 by the Italian Civil Protection Department and archived on GitHub [33]. We nd a good approximation by setting k = 0.8 and a = 1/3 days −1 , meaning a level of awareness about the daily number of quarantined and hospitalized of 80%, resulting from the balance between underestimates and media amplication and inevitably aected by rumors and misinformation spreading on the web (the socalled`infodemic' [44]). Such awareness is not immediate, but information takes on average 3 days to be publicly disseminated, being the communication slowed by a series of articulated procedures: timing for swab tests results, notication of cases, reporting delays between surveillance and public health authorities, and so on. Of course, parameters setting is inuenced by the choice of curves to t. Available data seem to provide an idea about the number of identied infectious people who have developed mild/moderate symptoms (the fraction that mandatorily stays in Q) or more serious symptoms (the hospitalized, I s ) and the number of deaths, but much less about those asymptomatics or with very mild symptoms who are not always subjected to a screening test.

Initial conditions
In order to provide appropriate initial conditions, we consider the ocial national data at February 24, 2020 archived on [33]. In particular, we take the number of mandatorily quarantined individuals (at that time, they coincide with Q being the voluntary component negligible) and the hospitalized people (I s ). Then, we simulate the temporal evolution of the epidemics prior to February 24, by imposing an initial condition of one exposed case ∆t 0 days before in a population ofN individuals, withN given in (12). We assume β 0 = 0 and γ 0 as in Table 1 (no social distance restrictions were initially implemented, but quarantine eorts were active since then) and disregard the eect of information on the human social 12 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint this version posted May 25, 2020.  Tables 1, 2 and 3. behaviors in this phase (α = D = 0 in (1)(3)). The length of temporal interval ∆t 0 is tuned in order to reproduce the ocial values released for Q and I s at February 24 and provide estimations for the other state variables, as reported in Table 3. We obtain ∆t 0 = 31.9, indicating that the virus circulated since the end of January, as predicted also in [11,24].

Numerical results
Let us consider the time frame [t 0 , t], where t 0 ≤ t ≤ t f . We consider two relevant quantities, the cumulative incidence CI(t), i.e. the total number of new cases in [t 0 , t], and the cumulative deaths CD(t), i.e. the diseaseinduced deaths in [t 0 , t].
For model (1)(3) we have, respectively: where β(M ) is given in (7), and In Fig. 4 the time evolution in [t 0 , t f ] of CI(t) and CD(t) is shown (rst and fourth panel from the left), along with that of quarantined individuals Q(t) (second panel) and hospitalized I s (t) (third panel). The role played by information on the public compliance with mitigation measures is stressed by the comparison with the absolute unresponsive case (α = D = 0 in (1)(3)). Corresponding dynamics are 13 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Other parameter values are given in Tables 1, 2 and 3. labelled by black solid and red dashed lines, respectively. In absence of reactivity to information, the cumulative incidence would have been much less impacted by lockdowns restrictions (11.4 · 10 5 vs 7.8 · 10 5 on May 18) and the number of quarantined would have been reduced to those forced by surveillance authorities. As a consequence, the peak of hospitalized patients would have been about 77% higher and 10 days timedelayed, with a corresponding increase in cumulative death of more than 100%. For all reported dynamics, the detachment between the responsive and unresponsive case starts to be clearly distinguishable after the rst step reduction of 64% in transmission rate (on March 12). Trends are also compared with ocially disseminated data [33] (Fig. 4, blue dots), which seem to conform accordingly for most of the time horizon, except for CI, that suers from an inevitable and probably high underestimation [24,25,38,42]. As of May 18, 2020, we estimate about 780,000 contagions, whereas the ocial count of conrmed infections is 225,886 [33].
We now investigate how the information parameters k and a may aect the epidemic course. More precisely, we assess how changing these parameters aects some relevant quantities: the peak of quarantined individuals, max(Q) (i.e., the maximum value reached by the quarantined curve in [t 0 , t f ]), the peak of hospitalized individuals, max(I s ), the cumulative incidence CI(t f ) evaluated at the last day of the considered time frame, i.e. t f = 84 (corresponding to , and the cumulative deaths CD(t f ).
The results are shown in the contour plots in Fig. 5. As expected, CI(t f ), max(I s ) and CD(t f ) decrease proportionally to the information coverage k and inversely to the information delay a −1 : they reach the minimum for k = 1 and a −1 = 1 days. Dierently, the quantity max(Q) may not monotonically depend 14 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.  Tables 1, 2 and 3. on k and a −1 as it happens for k ≥ 0.6 and a −1 ≤ 15 days (see Fig. 5, second panel, lower right corner). In such parameter region, for a given value of k (resp. a) there are two dierent values of a (resp. k) which correspond to the same value of max(Q). The absolute maximum (max [k,a −1 ] (max(Q))) is obtained for k = 1 and a −1 ≈ 7 days. Note that the couple of values k = 1, a −1 = 1 days corresponds to the less severe outbreak, but not with the highest peak of quarantined individuals.
In the next, we compare the relative changes for these quantities w.r.t the case when circulating information does not aect disease dynamics. In other words, we introduce the index  Fig. 6. However, we report in Table 4 three exemplary cases, the baseline and two extremal ones: (i) the baseline scenario k = 0.8, a −1 = 3 days, representing a rather accurate and shortdelayed communication; (ii) the case of highest information coverage and lowest information delay, k = 1, a −1 = 1 days; (iii) the case of lowest information coverage and highest information delay, k = 0.2, a −1 = 60 days.

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(which was not certified by peer review)
The copyright holder for this preprint this version posted May 25, 2020. . https://doi.org/10.1101/2020.05.20.20107573 doi: medRxiv preprint 11.42 · 10 5 0 0.82 · 10 5 0 6.09 · 10 4 0 6.99 · 10 4 0 k = 0.8 7.82 · 10 5 -0.32 2.08 · 10 5 1.53 3.45 · 10 4 -0.43 3.47 · 10 4 -0.50 a −1 = 3 days k = 1 7.24 · 10 5 -0.37 2.04 · 10 5 1.49 3.09 · 10 4 -0.49 3.11 · 10 4 -0.56 a −1 = 1 days k = 0.2 10.79 · 10 5 -0.06 1.29 · 10 5 0.57 5.70 · 10 4 -0.06 6.19 · 10 4 -0.11 a −1 = 60 days Table 4: Exact and relative values of nal cumulative incidence, CI(t f ), quarantined peak, max(Q), Under circumstances of very quick and fully accurate communication (case (ii)), CI(t f ), max(I s ) and CD(t f ) may reduce till 37%, 49% and 56%, respectively (see Table 4, third line). On the other hand, even in case of low coverage and high delay (case (iii)), the information still has a not negligible impact on disease dynamics: nal cumulative incidence and hospitalized peak reduce till 6%, nal cumulative deaths till 11% and quarantined peak increases of 57% about (Table 4, fourth line). As mentioned above, information and rumors regarding the status of the disease in the community aect the transmission rate β(M ) (as given in (7)) and the quarantine rate γ(M ) (as given in (8)). In our last simulation we want to emphasize the role of the information coverage on the quarantine and transmission rates. In Fig. 7 a comparison with the case of low information coverage, k = 0.2, is given assuming a xed information delay a −1 = 3 days (blue dotted lines). It can be seen that more informed people react and quarantine: an increasing of the maximum quarantine rate from 0.32 to 0.69 days −1 (which is also reached a week earlier) can be observed when by increasing the value of k till k = 1 (Fig.  7, second panel). The eect of social behavioral changes is less evident in the transmission rate where increasing the information coverage produces a slight reduction of the transmission rate mainly during the full lockdown phase (Fig. 7, rst panel). This reects the circumstance that the citizens compliance with social distancing is not enhanced by the informationinduced behavioral changes during the rst stages of the epidemic. On the other hand, a widespread panic reaction may lead people to`do it as long as you can' (see, for example, the case of stormed supermarkets at early stage of the epidemic [48]).

Conclusions
In this work we propose a mathematical approach to investigate the eects on the COVID19 epidemic of social behavioral changes in response to lockdowns. Starting from a SEIRlike model, we assumed that the transmission and quarantine rates are partially determined on voluntary basis and depend on the circulating information and rumors about the disease, modeled by a suitable timedependent information index. We focused on the case of COVID19 epidemic in Italy and explicitly incorporated the progressively stricter restrictions enacted by Italian government, by considering two step reductions in contact rate (the partial and full lockdowns). The main results are the followings: • we estimated two fundamental informationrelated parameters: the information coverage regarding 16 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.  Tables 1, 2, 3. the daily number of quarantined and hospitalized individuals (i.e. the parameter k) and the information delay (the quantity a −1 ). The estimate is performed by tting the model's solution with ocial data. We found k = 0.8, which means that the public was aware of 80% of real data and a −1 = 3 days, the time lag necessary to information to reach the public; • social behavioral changes in response to lockdowns played a decisive role in curbing the epidemic curve: the combined action of voluntary compliance with social distance and quarantine resulted in preventing a duplication of deaths and about 46% more contagions (i.e. approximately 360,000 more infections and 35,000 more deaths compared with the total unresponsive case, as of May 18, 2020); • even under circumstances of low information coverage and high information delay (k = 0.2, a = 1/60 days −1 ), there would have been a benecial impact of social behavioral response on disease containment: as of May 18, cumulative incidence would be reduced of 6% and deaths of 11% about.
Shaping the complex interaction between circulating information, human behavior and epidemic disease is challenging. In this manuscript we give a contribution in this direction. We provide an application of the information index to a specic eldcase, the COVID19 epidemic in Italy, where the information dependent model is parametrized and the solutions compared with ocial data.
Our study presents limitations that leave the possibility of future developments. In particular: (i) the model captures the epidemics at a country level but it does not account for regional or local dierences and for internal human mobility (the latter having been crucial in Italy at early stage of COVID19 epidemic); (ii) the model does not explicitly account of ICU admissions. The limited number of ICU beds constituted a main issue during the COVID19 pandemics [22]. This study did not focus on this aspect 17 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

(which was not certified by peer review)
The copyright holder for this preprint this version posted May 25, 2020. . but ICU admissions could be certainly included in the model; (iii) the model could be extended to include age structure. Age has been particularly relevant for COVID19 lethality rate (in Italy the lethality rate for people aged 80 or over is more than double the average value for the whole population [28]). Further developments may also concern the investigation of optimal intervention strategies during the COVID19 epidemics and, to this regard, the assessment of the impact of vaccine arrival. In this case, the approach of informationdependent vaccination could be employed [7,8,18,54].

A The information index
Consider the scenario of an epidemic outbreak that can be addressed by the public health system through campaigns aimed at raising public awareness regarding the use of protective tools (for example, vaccination, social distancing, bednet in case of mosquitoborne diseases, etc). Assume also that the protective actions are not mandatory for the individuals (or else, they are mandatory but local authorities are unable to ensure a fully respect of the rules). Then, the nal choice to use or not use the protective tools is therefore partially or fully determined by the available information on the state of the disease in the community.
The idea is that such information takes time to reach the population (due to timeconsuming procedures such as clinical tests, notication of cases, the collecting and propagation of information and/or rumors, etc) and the population keeps the memory of the past values of the infection (like prevalence or incidence). Therefore, according to the idea of informationdependent epidemic models [40,54], an information index M should be considered, which is dened in terms of a delay τ , a memory kernel K and a functiong which describes the information that is relevant to the public in determining its nal choice to adopt or not to adopt the protective measure (the message function). Therefore, the information index is given by the following distributed delay: M (t) = t −∞g (x 1 (τ ), x 2 (τ ), . . . , x n (τ )) K(t − τ )dτ Here, the message functiong depends generically on the state variables, say x 1 , x 2 , . . . , x n but it may specically depend only on prevalence [8,1517], incidence [9] or other relevant quantities like the vaccine side eects [18]. One may assume that: The delay kernel K(·) in (2) is a positive function such that +∞ 0 K(t)dt = 1. It represents the weight given to past history of the disease. The Erlangian family Erl n,a (t) is a good candidate for delay kernel since it may represent both an exponentially fading memory (when n = 1) and a memory more focused in the past (when n > 1). Moreover, when an Erlangian memory kernel is used, one can apply the so called linear chain trick [39] to obtain a system ruled by ordinary dierential equations. For example, in the case of exponentially fading memory (or weak kernel Erl 1,a (t)), the dynamics of the information index is ruled byṀ = a (g (x 1 , x 2 , . . . , x n ) − M ) .
For further details regarding the information index, see [40,54].

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. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 25, 2020. . https://doi.org/10.1101/2020.05.20.20107573 doi: medRxiv preprint B The next generation matrix method Following the procedure and the notations in [13,51], we prove that the control reproduction number of model (1)(3), R C , is given by (11). Similarly one can prove that the basic reproduction number is given by (10). Let us consider the r.h.s. of equations (1b)(1c)(1d)(1e)(1f), and distinguish the new infections appearance from the other rates of transfer, by dening the vectors As proved in [13,51], the control reproduction number is given by the spectral radius of the next generation matrix F V −1 . It is easy to check that F V −1 has positive elements on the rst row, being the other ones null. Thus, R C = (F V −1 ) 11 , as given in (11).