To date, no treatment of vaccine is available for COVID-19 7 . Efforts to contain the virus and to not overwhelm intensive care facilities are based on quarantine measures which have proven very effective in several countries [8] [9] [10] . Despite this, lockdown measures entail enormous economical, social and psychological costs. Recent estimates of the International Monetary Fund recently announced a global recession that will drag global GDP lower by 3% in 2020, although continuously developing and changing as well as significantly depending country-by-country 11 . More than 20 million people have lost their job in United States 12 and a large percentage of Italians have developed psychological disturbances such as insomnia or anxiety due to the strict lockdown measures 13 . Those measures have been taken on the basis of epidemics models, which are fitted on the available data 14 . In Italy, initial lockdown measures started on February 23rd for 11 municipalities in both Lombardia and Veneto which were identified as the two main Italian clusters.
After the initial spread of the epidemics into different regions all Italian territory was placed into a quarantine on March 9th, with total lockdown measures including all commercial activities (apart supermarkets and pharmacies), non-essential businesses and industries, and severe restrictions to transports and movements of people at regional, national, and extra-national levels 15 . People were asked to stay at home or near for sporting activities and dog hygiene (within 200 m from home), to reduce as much as possible their movements (only for food shopping and care reasons), and smart-working was especially encouraged in both public and private administrations and companies. At the early stages of epidemics intensive cares were almost saturated with a peak of 4000 people on April 3rd and a peak of hospitalisations of 30000 on April 4th, significantly reducing after these dates, reaching 1500 and 17000, respectively, at the beginning of phase 2 on May 4th, and 750 and 1000 on May 18th when lockdown measures on commercial activities were relaxed.
These numbers, continuously declining during the next days and weeks, confirmed the benefit of lockdown measures 16 . 18 . Trips of up to 100 kilometres from home are allowed without justification, as will gatherings of up to 10 people. Longer trips will still be allowed only for work or for compelling family reasons, as justified by a signed form. Guiding the government's plans for easing the lockdown is the division of the country into two zones, green and red, based on health indicators. Paris region (Ile de France), with about 12 milions inhabitants is flagged, to date, as an orange zone.
In both countries, the release of lockdown measures has been authorised by authorities after consulting scientific committees which were monitoring the behavior of the curve of infections using COVID-19 data. Those data are provided daily, following a request of the WHO. To date, the WHO guidelines require countries to report, at each day t, the total number of infected patients I(t) as well as the number of deaths D(t). Large uncertainties have been documented in the count of I(t) 19 . Whereas in the early stage of the epidemic several countries tested asymptomatic individuals to track back the infection chain, recent policies to estimate I(t) have changed. Most of the western countries have previously tested only patients displaying severe SARS-CoV-2 symptoms 20 . In an effort of tracking all the chain of infections, Italy and France are now testing all individuals displaying COVID-19 symptoms and those who had strict contacts with infected individuals. The importance of tracking asymptomatic patients has been proven in a recent study 21 .
The authors have estimated that an enormous part of total infections were undocumented (80% to 90%) and that those undetected infections were the source for 79% of documented cases in China.
Tracking strategies have proven effective in supporting actions to reduce the rate of new infections, without the need of lockdown measures, as in South Korea 22 .
The goal of this paper is to explore possible future epidemics scenarios of the long term behavior of the COVID-19 epidemic 23 but taking into account the role of uncertainties in both the parameters value and the infection counts to investigate different outcomes of the epidemics leading or not to a second wave of infections. To this purpose we use a stochastic Susceptible-Exposed-Infected-Recovered (SEIR) model 24 which consist in a set of ordinary differential equations where control parameters are time-dependent modelled via a stochastic process. This allows to mimic the dependence on control parameters on some additional/external factors as super-spreaders 25 and the enforcing/relaxing of confinement measures 24 . As for the classical SEIR models 26 the population is divided into four compartmental groups, i.e., Susceptible, Exposed, Infected, and Recovered individuals. The stochastic SEIR model shows that long-term extrapolation is sensitive to both the initial conditions and the value of control parameters 24 , with asymptotic estimates fluctuating on the order of ten millions units in both countries, leading or not a second wave of infections. This sensitivity arising from both poor data quality and inadequate estimations of model parameters has been also recently investigated by means of a statistical model based on a generalized logistic distribution 27 . The paper is organised as follows: in Section III we discuss the various sources of data for COVID-19 and their shortcomings, and then we discuss in detail the SEIR model and its statistical modelling. In Section IV we discuss the results focusing on the statistical sensitivity of the modelling, and apply it to data from France and Italy. We finish, in Section V, with some remarks and point out some limitiations of our study.
This paper relies on data stored into the Visual Dashboard repository of the Johns Hopkins University Center for Systems Science and Engineering (JHU CSSE) supported by ESRI Living Atlas Team and the Johns Hopkins University Applied Physics Lab (JHU APL). Data can be freely accessed and downloaded at https://systems.jhu.edu/research/public-health/ncov/, and refers to the confirmed cases by means of a laboratory test 3 . Nevertheless there are some inconsistencies between countries due to different protocols in testing patients (suspected symptoms, tracing-back procedures, wide range tests) 28, 29 , as well as, to local management of health infrastructures and institutions. As an example due to the regional-level system of Italian healthcare data are collected at a regional level and then reported to the National level via the Protezione Civile transferring them to WHO. These processes could be affected by some inconsistencies and delays 30 , especially during the most critical phase of the epidemic diffusion that could introduce errors and biases into the daily data. These incongruities mostly affected the period between February 23rd and March 10th, particularly regarding the counts of deaths due to a protocol change from the Italian Ministry of Health 31 . A similar situation occurs in France where the initial testing strategy was based only on detecting those individuals experiencing severe COVID19 symptoms 32 .
In the post lockdown phase, France has extended its testing capacity to asymptomatic individuals who have been in contact with infected patients 33 .
One of the most used epidemiological models is the so-called Susceptible-Exposed-Infected-Recovered (SEIR) model belonging to the class of compartmental models 26 . It assumes that the total population N can be divided into four classes of individuals that are susceptible S, exposed E, infected I, and recovered or dead R (assumed to be not susceptible to reinfection). The model is based on the following assumptions:
1. the total population does not vary in time, e.g., dN/dt = dS/dt + dE/dt + dI/dt + dR/dt = 0, ∀t ≥ 0; 2. susceptible individuals become infected that then can only recover or die, e.g., S → I → R; 3. exposed individuals E encountered an infected person but are not themselves infectious; 4. recovered or died individuals R are forever immune. Thus, the model reads as
where γ > 0 is the recovery/death rate, λ = λ 0 /S(0) > 0 is the infection rate rescaled by the initial number of susceptible individuals S(0), and α is the inverse of the incubation period. Its discrete version can be simply obtained via an Euler Scheme as
in which we fixed dt = 1 day that is the time resolution of COVID-19 counts. By means of γ and λ 0 the model also allows to derived the so-called R 0 parameter, e.g., R 0 = λ 0 /γ, representing the average reproduction number of the virus. It is related to the number of cases that can potentially (on average) caused from an infected individual during its infectious period (τ in f = γ −1 ). Early estimates in Wuhan 34 on January 2020 reported R 0 = 2.68 2.86 2.47 which lead to γ = 0.37 fixing λ 1 as in 35 and a 95% confidence level range for the incubation period between 2 and 11 days 36 .
However, the R 0 parameter as well as models parameters λ , γ, and α can vary in time during the epidemics due to different factors as the possible presence of the so-called super-spreaders 25 , intrinsic changes of the SARS-CoV-2 features, lockdown measures, asymptomatic individuals who are not tracked out, counting procedures and protocols, and so on 37 .
To deal with uncertainties in long-term extrapolations and with the time-dependency of control parameters a stochastic approach could provide new insights in modeling epidemics [38] [39] [40] , especially when epidemics show a wide range of spatial and temporal variability [41] [42] [43] . However, instead of investigating how to get a realistic behavior by stochastically perturbing control parameters, here we investigate how uncertainties into the final counts C(t) are controlled by model parameters 24 . Thus, we use a stochastic version of the SEIR model in which the set of control parameters κ ∈ {α, λ , γ} is modelled via a stochastic process
with being σ the intensity of the perturbation, ξ (t) a random variable from a collection of normally distributed N(0, 1) values at each time t, and since κ(t) ≥ 0 we introduced the absolute value in Eq. (9) . In this way we can introduce instantaneous daily discrete jumps (e.g., take
into account daily uncertainties) in the control parameters to properly model detection errors on infection counts, appropriately described through a discrete process 44 than a continuous one 45 .
We begin this section by validating the SEIR stochastic model on the first wave of infections.
We have therefore to chose the initial conditions, and then introduce the lockdown measures in the parameters.
In 1.1 before reaching the final value of R 0 = 0.7 0.9 0.5 . This sort of "step" into the R 0 time behavior corresponds to the time interval between semi-and full-lockdown measures, whose efficiency significantly increases after March 24th, 2020, also corresponding to the peak value of infections. This is confirmed by looking at daily infections distributions (Figure 2c ) that shows a peak value near March 24th, 2020, also observed in real COVID-19 data 27 . Finally, the cumulative number of infections (Figure 2b) shows that, on average, almost 10 millions people have been infected by SARS-CoV-2 in Italy, ranging between few hundred thousands up to 25 millions due to the the error propagation by the stochastic fluctuations of model parameters (see 24 for explanations). Nevertheless the wide range of uncertainty the average value is close to the value estimated from a team of experts of the Imperial College London according to which the 9.6% of Italian population has been infected, with a 95% confidence level ranging between 3.2% and 26% 52 . These estimates correspond to cumulative infections of ∼6 millions, ranging from ∼2 and ∼16 millions, well in agreement with our model and other statistical estimates 53 .
After lockdown measures are released, for both countries, we model three different scenarios: a first one where all restrictions are lifted (back to normality), a second one where strict measures are taken (semi-lockdown) and a third one where the population remains mostly confined (lockdown).
Results for France are shown in Figure 3 . Lockdown is released at t = 136, corresponding to May 11th, 2020. The back to normality (red) scenario clearly shows a second wave of infections peaking in summer (early July) and forcing group immunity in the French population. The semilockdown (green) scenario, corresponding to a reduction of the mobility of about 50%, leads to a second wave as intense as the first wave, but longer, at the end of August. As in the previous scenario, the semi-lockdown scenario allows to reach a group immunity in France. A third lockdown scenario is modelled (blue). This latter scenario simulates an R 0 1, that can be achieved by imposing strict distancing measures, contact tracking as well as reduction in mobility. It results in a linear modest increase of the total number of infections that does not produce a proper wave of infections. As in the first wave modelling, large uncertainties are also present in future scenarios although the three distinct behaviors clearly appear.
b. Italy Figure 4 shows the results for modeling future epidemic scenarios for Italy. The first relaxation of lockdown measures started at t = 131, corresponding to May 4th, 2020, while strict measures were finally released at t = 146, corresponding to May 18th, 2020. The back to normality (red) scenario moves towards a second wave of infections whose peak occurs at t = 193, corresponding to July 4th, 2020, exactly three months after initial lockdown measures were released (May 4th, 2020). This would lead the so-called herd immunity for the whole Italian population (see Figure 4b) , with a peak of daily infections near 5 millions of people (Figure 4c) , and R 0 re-approaching the initial value (R 0 = 2.68). The semi-lockdown (green) scenario produces a second wave mostly similar, in terms of intensity, as the first wave, but occurring at t = 246, e.g., August 26th, 2020. This scenario will lead to 40 millions infected people, spanning between 25 and 55 millions, thus producing a group immunity in Italy. A third scenario is modelled in which complete lockdown measures are still considered (blue). This latter scenario leads to a more controlled evolution of cumulative infections which still remain practically unchanged with respect to the first wave cumulative number. It has been obtained by simulating an R 0 1, resulting from strict distancing measures and reduced mobility, and does not produce a proper wave of infections. However, all scenarios are clearly characterized by a wide range of uncertainties, although producing three well distinct behaviors in both cumulative and daily infections.
In the previous section we have seen that increasing R 0 above 1 can or not produce a second wave of infections and introduce also a time delay in the appearance of a second wave of infections. We now analyse this effect in a complete phase diagram fashion. disease in both countries 18 . Furthermore, we also assume that fluctuations on the parameters of the SEIR model are Gaussian, although there are good reasons to think that they should be heavy tailed distributions 50, 55 . We would like to remark however that, to overcome these limitations, one would need to fit more complex models and introduce additional parameters which can, at the present stage, barely inferred by the data.
Our choice to stick the stochastic SEIR model is indeed driven by its simplicity and the possi-bility of modeling realistic the uncertainties with the stochastic fluctuations instead of adding new parameters whose inference may affect the results. This study can be applied to other countries, and this is why we publish along the code of our analysis alongside with the paper. To date, Northern Europe, UK, US and other American countries are still facing the first wave of infections, so that future scenarios cannot be devised with the same clarity as those outlined in this study for France and Italy. 
The data that support the findings of this study are openly available in https://systems. jhu.edu/research/public-health/ncov/, maintained by Johns Hopkins University Center for Systems Science. 