The recent diffusion of the COVID-19 Corona virus has renewed the interest of the scientific and political community in the mathematical models for epidemic. Many researchers are making efforts for proposing new refined models to analyse the present situation and predict possible future scenarios.
With this paper we hope to contribute to the ongoing research on this topic and to give a practical instrument for a deeper comprehension of the virus spreading features and behaviour.
We consider here deterministic models based on a system of initial values problems of Ordinary Differential Equations (ODEs). This theory has been studied since about one century by W.O. Kermack and A. G. MacKendrick [1] that proposed the basic Susceptible-Infected-Removed (SIR) model. The SIR model and its later modifications, such as Susceptible-Exposed-Infected-Removed (SEIR) [2] are commonly used by the epidemic medical community in the study of outbreaks diffusion.In these models, the population is divided into groups. For example, the SIR model groups are: Susceptible who can catch the disease, Infected who have the disease and can spread it, and Removed those who have either had the disease, or are recovered, immune or isolated until recovery. The SEIR model proposed by Chowell et al. [3] considers also the Exposed group:
containing individuals who are in the incubation period.
The evolution of the Infected group depends on a key parameter, usually denoted as R0, representing the basic reproductive rate. The value of R0 can be inferred, for example, by epidemic studies or by statistical data from literature or it can be calibrated from the available data. In this paper we use the available data for determining the value of R0 best fitting the data.
Compared to previous outbreaks, such as SARS-CoV or MERS-CoV [4] , when the disease had been stopped after a relatively small number of infected people, we are now experimenting a different situation. Indeed the number of infected people grows exponentially, and apparently, it can be stopped only by a complete lockdown of the affected areas, as evidenced by the COVID-19 outbreak in the Chinese city of Wuhan in December 2019.
Analogously, in the Italian case, in order to limit the virus diffusion all over the Italian area, the government has started to impose more and more severe restrictions since March 6th 2020. Hopefully, these measures will affect the spread of the COVID-19 virus reducing the number of infected people and the value of the parameter R0.
The introduction of different levels of lockdown require an adaptation of the standard epidemic models to this new situation. Some examples about the Chinese outbreak can be found in [5, 4, 6] . Concerning the Italian situation, which is currently evolving, it is possible to model the introduction of restricting measures by introducing a non constant infection rate [7] . In this paper we propose to represent the infection rate as a piecewise function, which reflects the changes of external conditions. The parameter R0, which is proportional to the infection rate, becomes a time dependent parameter R t which follows a different trend each time the external conditions change, depending on the particular situation occurring in that period. For example, if new restrictions are applied to the population movements at time t 1 , we can hopefully argue that R t starts to decrease when t > t 1 .
Finally, we believe that relevant information is not only Infected but also Recovered and Dead numbers we modified SEIR model by splitting the Removed population into Recovered and Dead.
In section 2 we describe the details of SEIRD model with constant and with time dependent infection rate SEIRD(rm). Finally in section 3 we test the model on some regional aggregated data published by the Protezione Civile Italiana [8] . The equations relating the groups are the followings:
where N is the total population, β is the infection rate, a coefficient accounting for the susceptible people get infected by infectious people and γ is the parameter of infectious people which become resistant per unit time. A more refined model is the SEIR model where a new compartment E representing the exposed individuals that are in the incubation period is added.
The resulting equations in the SEIR model are the following:
where α represents the incubation rate. The difference between the exposed (E) and infected (I) is that the former have contracted the disease but are not infectious, and the latter can spread the disease. SEIR has been used to model breakouts, such as Ebola in Congo and Uganda [3] . In [2] the equations are modified by adding the quarantine and vaccination coefficients. In our case, unfortunately, vaccination is not available. A further model, the SIRD, considering the group of Dead (D) in place of the Exposed is analysed in [7] for the forecast of COVID 19 spreading.
In this paper we propose a SEIRD model accounting for five different groups, Susceptible, Exposed, Infected, Recovery and Dead. The system of equations is given by:
In order to consider the restrictions imposed by the Italian government since the infection rate of SEIR equations can be found in [7] , where the function is assumed to have a decreasing exponential form. However, observing the data trend, we believe that β t has a smoother decreasing behavior and we choose to model it as a decreasing rational function:
In the present work we use a constant value ρ = 0.75 but we might calibrate it in future. By substituting β(t) (2) in the S and E equations in (1) we obtain SEIRD rational model SEIRD(rm).
We calibrate the parameters of SEIRD and SEIRD(rm) by solving non-linear least squares problems with positive constraints. For example, in the SEIRD
on the vector of parameters q = (β, α, γ R , γ D ), and the vector y of the acquired data at given times t i , i = 1, . . . n. Let F (u, q) be the function computing the numerical solution u of the differential system (1), the estimation of the parameter q is obtained solving the following non linear least squares problem:
q ≥ 0
where we introduce positivity constraints on q. The constrained optimization problem is solved with a trust-region based method implemented in the lsqnonlin Matlab function. For further details about the optimization problem for identification parameters in differential problems see for example [9] .
In this section we report the results obtained by using the SEIRD model to monitor the Covid-19 outbreak in Italy during the period 24/02/2020-20/03/2020. 
In this section we consider the SEIRD model (1) and use the following different measurement subsets from [8] relative to Lombardia region:
• S1 10 days measurements: 24/02/2020-04/03/2020
• S2 18 days measurements: 24/02/2020-12/03/2020
• S3 23 days measurements: 24/02/2020-17/03/2020 for identification of the parameters and than apply the identified parameters to model the Infected-Recovered-Dead populations up to June 22nd (120 days).
Besides population plots in figures 1 ,2 we collect some meaningful quantitative information about the model parameters ( • The maximum Infected population is reached on April 9th 2020 for S1, April 23rd 2020 for S2 and April 28th 2020 for S3.
• Concerning the values of the model parameters we have that the transmitting rate β can be estimated as β e = 0.3, the incubation rate α is approximately α e = 3 while the recovery rate is approximately (γ R ) e = 0.06 and the dead rate can be approximated as is approximately (γ D ) e = 0.04.
The following conclusions can be drawn:
• The SEIRD parameters, computed from the first 10 days measurements, do not model properly the measurements. Indeed the plots reported in figure 1 show that data of IRD populations have a slower increase rate, compared to the model predictions.
• in figure 5 reproduce the data quite accurately. The peak values reported in table 4 represent the largest percentage of Infected and the longest time required to reach the peak.
In order to improve the data fit we split the parameter identification step into the following two phase process:
• Phase 1 Identification of the parameters of the standard SEIRD model using a the data subset S1, (i.e. up to time t 0 = 10). change of the Reproduction parameter (R0) into a time dependent Reproduction function defined as follows:
The plots of R t for the three regions are reported in figure 6 . In table 6 we Lombardia Emilia Romagna Veneto The improved modeling properties can be appreciated in population plots reported in figures 7 , 8 and 9. We observe that SEIRD(rm) reproduces the data trends more precisely compared to the SEIRD model. 
In this paper we proposed a SEIRD model for (about June 20th), while Veneto has its infection peak around the end of July 2020, probably due to different testing modalities.
We highlight that it is only 12 days since restrictions started in Italy and, maybe in the next few days, the effects of such measures will become more evident, hopefully causing a further decrease in the infection trend. In this case, the previsions shown in this paper should be updated by introducing a new time t 1 at which the the decreasing slope of β t should change, for example by estimating the parameter ρ in (2) with new data.
The proposed model is flexible and we believe it could be easily adapted to monitor various infected areas with different restriction policies.