Late December 2019 health facilities reported cluster of patients with pneumonia of unknown origin epidemiologically linked to a seafood and wet animal wholesale market in Wuhan China. A new previous unknown betacoronavirus was discovered and identified as the etiologic agent of this new pneumonia diagnosed in Wuhan [1] . Coronaviruses (CoV) are a large family of viruses that cause illness ranging from the common cold to more severe diseases. The new virus was provisory named 2019-nCoV by World Health Organizzation on 12 January 2020 [2] and sun after the Coronavirus Study Group (CSG) based on phylogeny, taxonomy and established practice, formally recognized this virus as a sister to severe acute respiratory syndrome coronaviruses (SARS-CoVs) of the species Severe acute respiratory syndrome-related coronavirus and designates it as severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) [3] . The outbreak of the SARS-CoV-2 due to the global spread has been defined pandemia on March 11, 2020. In Europe Italy is becoming a particularly alarming and interesting place to study the evolution of the epidemic also thanks to the rapid reaction of the Italian Health organizations and the relevant control measure to prevent transmission adopted. Based on Chinese experience and the estimation of transmission rate published by Tang B et al.. [4] . Here we develop models for the evolution of the SARS-CoV-2 during the early stages of transmission in Italy; models which may be useful for inference, forecasting or scenario analysis. Despite the fact that epidemic is changing rapidly and our results have been considered preliminary, the models we are using are considered strongly predictive and useful for the interpretation of such an unexpected event in a country like Italy.
A. The SEIR model and its limiting approximation
In the SEIR framework [5] , individuals in the population are classified according to their infectious status: Susceptible (S), Exposed (E) (infected but not infectious), Infectious (I) and Recovered (R). In the case of a new infection (as in the present case) the population has no prior immunity, consequently the population starts out at the disease-free equilibrium (S ≈ N, E = 0, I = 0, R = 0), where N is the total population size, and the dynamics are determined by the following equations describing the rates of change of each simplified classes:
(1)
After introduction of an infectious individual (and if the basic reproductive ratio, R 0 = β/γ, is greater than 1), the infection develops through the population by contacts between susceptible and infected individuals to sustain further transmission. Neglecting background birth and death processes (because of the short period analyzed w.r.t. the typical vital periods), the population eventually reaches the following state (S = S ∞ , E = 0, I = 0,
The SIR model is a limiting case of the previous framework and assumes E + I → I at any time, without dis-. CC-BY 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.
The copyright holder for this preprint this version posted March 17, 2020. . https://doi.org/10.1101/2020.03.14.20034884 doi: medRxiv preprint tinguishing the classes E and I. The previous equations reduce to
In Fig. 1 we show the available data [6, 7] for the initial period February 20th -March 11th (21 days) and their comparison with the simplest SIR approximation (5), (6) , and (7) (iteratively solved by means of a MATLAB code) for a first intuitive analysis. The continuous black curve shows the solution assuming a I(0) = 3 number of infectious individuals at day=0, exactly equivalent to the official detected cases at the same day. No attention is paid to a best fit of the data, also because they are rather sparse, however the quality of the simple (SIR) model is evident also at the first stages of the epidemic. The parameter values sound also reasonable: with a recovering average time T r = 7 days (γ = 1/T r ) and R 0 = 3.4 and no further corrections: a brilliant example of the predictive force of the SIR model. However Fig. 1 has a more specific aim: showing the rather delicate rôle played by a possible underestimation of the infectious individual at the early stages of the infection outbreak. In fact one can assumes a different (possible) scenario based on the difficulty of a precise estimation of the initial number of infectious individuals. Such a possible scenario is illustrated by the the full blue curve showing a prediction which results from a misleading estimation of I(0). The effective number of initial infectious is 30 times larger (I(0) = 90), but the curve has been translated to a fictitiousĪ(0) = 3, the reveled number, from the dashed mathematical result. A change in R 0 from R 0 = 3.4 to R 0 = 1.7 would give a reason- able fit of the early stage of the outbreak. The effect is rather well known as discussed in ref. [8] in the case of the epidemic event in China. In that case a proposed exponential fit of the precocious data (January 10th to January 24th) results in a values of R 0 ≈ 3.58 which is reduced to R 0 ≈ 2.24 is the associated initial condition on I(0) is multiplied by an increasing factor from 2 to 8. In Fig. 2 we show the time behavior of the infectious population within the simple SIR model of Eqs. (5)-(7) (T r = 1/γ = 7 days, R 0 = 3.4) both in logarithmic (upper panel) and linear scale (lower panel). The data are the data of Fig. 1 .
The exponential behavior predicted by the model in the growing part of the outbreak description is evident and can be compared with the recent fit to the data by . CC-BY 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.
The copyright holder for this preprint this version posted March 17, 2020. Bucci and Marinari in ref. [9] . While the present simplified SIR model predict a doubling infectious population each (roughly) 2 days, the Roma fit [9] gives an estimation of 2.6 days (see Fig. 3 ).
The model [4] is a generalized SEIR-type epidemiological model which incorporates appropriate compartments relevant to intervention such as quarantine, isolation and treatment. The population is stratified in Susceptible 
The authors of ref. [4] have calibrated the model on the data of the 2019-nCoV as emerged in Wuhan the last two months, therefore in a situations that has the same basic parameters. In table I The results are shown in Figs 4 and 5. The drastic reduction of the infectious population is evident as well as the long tail of the distribution.
The quarantine imposed to the infectious individuals, the effects of an intensive contact tracing and isolation reduce the maximum value of the distribution by a factor ∼ 26 (see Fig. 5 ).
The present calculation strongly supports the interventions and restrictions adopted in Italy to reduce the outbreak of the infectious population of the 2019-nCoV. . CC-BY 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 March 17, 2020. . https://doi.org/10.1101/2020.03.14.20034884 doi: medRxiv preprint