Effects of information-induced behavioural changes during the COVID-19 lockdowns: the case of Italy

The COVID-19 pandemic that started in China in December 2019 has not only threatened world public health, but severely impacted almost every facet of life, including behavioural 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 behavioural changes in response to lockdowns. We consider an SEIR-like epidemic model where the contact and quarantine rates depend on the available information and rumours about the disease status in the community. The model is applied to the case of the COVID-19 epidemic in Italy. We consider the period that stretches between 24 February 2020, when the first bulletin by the Italian Civil Protection was reported and 18 May 2020, when the lockdown restrictions were mostly removed. The role played by the information-related parameters is determined by evaluating how they affect suitable outbreak-severity indicators. We estimate that citizen compliance with mitigation measures played a decisive role in curbing the epidemic curve by preventing a duplication of deaths and about 46% more infections.

(x 1 (τ ), x 2 (τ ), . . . , x n (τ )) K(t − τ )dτ. (S.1) 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 [58], incidence [9] or other relevant quantities like the vaccine side eects [10]. One may assume that: The delay kernel K(·) in (S.1) 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' [11] 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 [3,4].

S.2 The next generation matrix method
Following the procedure and the notations adopted by Diekmann et al. [12] and Van The Jacobian matrices of F and V evaluated at model (1)(2) diseasefree equilibrium DF E = Λ µ , 0, 0, 0, 0, 0, 0, 0 read, respectively, The control reproduction number is given by the spectral radius of the next generation matrix F V −1 [12,13]. 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 , yielding Similarly one can prove that the basic reproduction number is given by The rst two terms in the r.h.s of (S.2) describe the contributions of postlatent infectious and asymptomatic/mildly symptomatic infectious, respectively, to the production of new infections close to the diseasefree equilibrium. The third, the fourth and the fth term in (S.2) represent the contribution of infectious with severe symptoms. The severe symptoms can onset soon after the postlatent phase (third term) or after a mildly symptomatic phase (fourth term). In case of quarantine, severe symptoms can onset also during such a stage (fth term). Note that the last term is missing in the basic reproduction number (S.3), where the possibility for people to be quarantined is excluded. Note also that R C = R 0 when β 0 = γ 0 = 0.

S.3 Epidemiological parameters
As done by Gumel et al. [14], we adopt an SEIRlike model with demography and constant net inow of susceptibles Λ. Including a net inow of susceptible individuals into the model allows to consider not only new births (which can be assumed to be approximately constant due to the short time span of our analysis), but also immigration, which played a role during the lockdown in Italy. Therefore, the net inow of susceptibles is given by In (S.4), the parameter Λ 0 is the inow term due to immigration and bN is the inow term due to births, where b is the birth rate andN denotes the total population at the beginning of the epidemic. The most recent data by the Italian National Institute of Statistics [15] refer to January 1, 2019 and provide a countrylevel birth rate b = 7.2/1, 000 years −1 and a natural death rate µ = 10.7/1, 000 years −1 , as well as a resident population of aboutN ≈ 60.360 · 10 6 (S. 5) inhabitants. Fluctuations in a time window of just over a year are considered negligible. Since global travel restrictions were implemented during the COVID19 epidemic outbreak [16], we assume that the immigration inow term Λ 0 accounts only of repatriation of citizens to their countries of origin (Italy in that case) due to the COVID19 pandemic [17]. In all airports, train stations, ports and land borders travellers' health conditions have been tested via thermal scanners. Although the eectiveness of such screening method is largely debated [18], for the sake of simplicity, we assume that the inow enters only the susceptible compartment. On the basis of data communicated by the Italian Ministry of Foreign Aairs and International Cooperation [19], a reasonable value for Λ 0 seems to be Λ 0 = 4, 000/7 days −1 , namely the average number of repatriated citizens was 4,000 per week. Being Λ = bN + Λ 0 , 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 [2025] or inferred by modelling studies [2628]. 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 occurs 12 days before the onset of symptoms [21,22]. Hence, we set the latency (ρ) and postlatency (η) rates to 1/5.25 days −1 and 1/1.25 days −1 , respectively. From [26], the specic baseline transmission rates for the postlatent (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 asymptomatic and symptomatic patients [22]. 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 several sources [21,22,25,26].
As made by Kantner & Koprucki [27], we consider that just 8% of infectious individuals shows 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 shows a worsening of symptoms [23] within 45 days from onset [20], 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 have been implemented since the early stage of epidemics (at February 24, 94 quarantined people were already registered [29]), 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 varied considerably across Italian regions and in the dierent phases of the outbreak [20,29]. 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 .
Following the approach adopted by Gumel et al. [14] for a SARSCoV epidemic model, based on the formula given by Day [30], we estimate the diseaseinduced death rate as where X is the case fatality and T is the expected time from hospitalization until death. From [20], 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 δ ≈ 0.022 days −1 . Similarly, the recovery rates ν j with j ∈ {m, q, s} 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 [22], suggesting T m = 6 days is an appropriate value. As far as the time spent in hospitalization or quarantine is concerned, 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 [20]. Moreover, shortages in hospital beds and intensive care units (ICUs) led to as prompt as possible discharge [31]. 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 [24]. 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 by Gatto et al. [26].

S.4 Initial conditions
In order to provide appropriate initial conditions, we consider the ocial national data at February 24, 2020 archived on [29]. 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 use system (1)(2) of the main text to 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 (S.5). We assume β 0 = 0 and γ 0 as in Table 1 of the main text (no social distance restrictions were initially implemented, but quarantine eorts were active since then) and disregard the eect of information on the human social behaviors in this phase (α = D = 0 in (1)(2) of the main text). 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 1 of the main text. We obtain ∆t 0 = 31.9, indicating that the virus circulated since the end of January, as predicted also by Gatto et al. [26] and Giordano et al. [32].

S.5 Mitigation measures enacted by the Italian government
Italy was the rst European country aected by COVID19. After the rst ocially conrmed case (the socalled`patient one') on February 21, 2020 in the Lodi province, several suspected cases emerged in the south and southwest of the Lombardy region. A`red zone' encompassing 11 municipalities was instituted on February 22 and put on lockdown to contain the emerging threat. On March 8, the red zone was extended to the entire Lombardy region and 14 more northern Italian provinces, while the rest of Italy implemented social distancing measures. A leak of a draft of this decree prompted a panic reaction with massive movement of people towards Italian regions, especially from the north to the south [33]. 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 enforced with all commercial and retail businesses except those providing essential services closed down [34]. 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 [35]. 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 [36].  Tables 1 and 2 of the main text.