The Covid-19 Pandemia, which is worldwide raging, does not allow the development of multiple strategic defensive scenarios. Viruses are entering the human body, cause an infection and there is no available anti-virus drugs to contrast Covid-19.
The restriction of social contacts is therefore the only (hopefully) efficient mean to slow down the diffusion of the infectious disease, thus allowing the sanitary structures to operate without the pressure of an emergency stress.
We have therefore asked ourselves whether tools, borrowed from other disciplines, may be exploited as a predictive mean to understand how the number of Covid-19 infections increases with time, what is the maximum number of expected infections and when the rate of infections is expected to decrease.
In recent times the point of view, we are going to propose has been exploited for the understanding of bacterial growth [1] or of other mechanisms mediated by bacterial growth, such as global biogeochemical cycles [2] , human gut health [3] , wastewater treatment [4] or in totally different fields like cancer growth and metastsys dissemination [5] and in the physics of Free Elecron Laser [6] as well.
In the following we will make use of the logistic function [7] and of its variants [8] , to get a reasonable fit of the data provided by the Italian Department of health 1 and develop some previsions. Before getting into the specific details of the discussion, we give a short resume of the mathematical tools we will employ.
The logistic function (Fig. 1) describing the evolution of a given population N (τ ) is
where τ is the time, measured in some units to be specified. Furthermore r and K accounts for the growth rate and the carrying capacity respectively. Regarding bacteria or viruses, K represents the maximum number of individuals which can be infected. In absence of an ad hoc developed infectious model it cannot be easily specified (see the concluding section for further comments), it (and r as well) will be inferred from the "experimental data".
The logistic equation provides a good model for the exponential growth and it does include the so called saturation mechanism leading to the equilibrium, where the population saturates. It is the solution of the first order non-linear equation [9] 
which states that the growth rate is not simply r but r 1 − N K . The logistic curve describes the total number of infections and its derivative yields the number of infections per unit time, namely
It is a bell shaped curve and it is known as the Hubbert curve, initially exploited as an approximation of the production rate of a resource over time [10] . The location of the peak depends on growth rate and carrying capacity according to the identity
At this time we find for the infected rate
corresponding to a total number of infected
The figures reported below yields a better idea of what has been worded so far.
In Fig. 1 we have shown the growth of infected individual N (τ ) vs. the number of days (τ = 0 has been arbitrarily chosen) for different values of the growth rate r and for the same value of the carrying capacity K.
In Fig. 2 we have reported the Hubbert curve vs. τ , for different values of r and the same K. It is worth noting that, with decreasing r, the peak shifts towards larger τ values. We have fixed a threshold value the maximum rate that a health structure may care (N (τ * )), the number of patients which cannot be treated is given by the total number of infected above this threshold (N ), namely
where τ ± are the times in which the Hubbert and threshold curves intercept.
In Fig. 3 we have reported the number ∆ of patients potentially untreated vs r. This is the minimum Mathematica we need to start our discussion. Further information will be added once needed. 
In this section we apply the formalism of the previous section to the analysis of the behavior of the viral infection in Italy.
The starting point is the number of infected individuals per day, provided by the Italian Ministry of Health 2 . The data are reported in Fig. 4 , with the inclusion of the forecasting of the future trend, which foresees the saturation by the end of April.
It is evident that the plot in Fig. 4 is essentially the Hubbert curve and can be derived by a corresponding Logistic curve, as it shown in Figs. 5 where we have reported the number of infected/day. In 5a we make the comparison between the present model and that in Fig. 4 . To make a correct comparison we have fitted the logistic function parameter using the data of the first 12 days. In Fig. 5b we have extended the fit to 23 days (red istogram) and included the prevision from the our model (blue istogram). The zero day, coincides with that in which the number of positive cases has been started to be officially communicated.
The green dotted curve represents the evolution of the infected according to the official data, while he red continuous is the logistic fit. The agreement looks satisfactory.
The best fit procedure yields N 0 = 294, r = 0.2264, K = 50346.
The Hubbert curve, inferred from the data in Figs. 6 is shown in Fig. 7 in which the continuous and dashed curves are just the analytical derivatives of The conclusions we may draw are that, if we trust this model, the peak of the number of infected per day should be reached in the next few days and that a substantive decline could be experienced at the end of April.
It should be noted that being the baricenter of the data shifted towards those of "Regione Lombardia", the extension of the previous conclusions to the rest of the Nation is not fully justified. In the next days we will presumably observe the impact of the lockdown restriction and have a better picture of its role on parameters like the growth rate r.
For further comments see the concluding section.
The number of positive cases around the peak is close to 3, 000, if 10% needs intensive cares, about 300 patient/per day are supposed to be allocated in health structures with limited resources. Further comments on these points will be given in the forthcoming section.
The model we have exploited is a simplification in many sense. It does not include all the elements of the SIR epidemic models of infections [11] . We considered infected only and we did not include the evolution of healed and mortality.
The logistic model seems to work in a reasonable way and, if further refined, may become a predictive tool. We must however underline that a sigmoid evolution curve and a bell shaped Hubbert form can be derived within different analytical frameworks. The most attractive alternative is the so called Gompertz model [8] , based on the evolution curve
The analytical dependence on time is interesting for our purposes because it displays an exponential term, which simultaneously includes the growth rate and the saturation. Regarding the meaning of the parameters we note that
We recognize α as the carrying capacity so that
and eventually, with N 0 = G 0 , we can write Eq. (9) as
If we keep N 0 and K the same as those predicted by the Logistic model and derive γ from the data, we can evaluate the corresponding Hubbert curve, as reported in Figs. 6 in which we have made a comparison with the logistic case.
The most remarkable difference is that the Gompertz-Hubbert curve is more asymmetric and wider than the Logistic counterpart, this behavior seems more similar to the case reported in Fig. 4 (which is also the result of a computational model). The drawback is that the two models predict different values of the peaks which, by using parametrs from Eq. (8), in the case of the logistic is max(N (τ )) = 2850 reached at τ * = 23,
and of the Gompertz
is max(G (τ )) = 1648 reached at τ * G = 18,
The conclusions we can draw confirm essentially what we have underscored in the previous section, with the caveat that the data are referring to the whole national territory, in which the infection is proceeding in a non homogeneous way.
Our analysis is strongly influenced by the growth of the disease in Lombardia (and more in general in the north of Italy). The evolution in the national territory might qualitatively reproduce that reported in Fig. 9 , which refers to the world-wide landscape. The growth of the disease in other regions, owing to the lockdown, may follow a different pattern. We can foresee a kind of saturation and then a resurgence at lower increasing rate, if the government prescription are duly observed. In qualitative terms we may observe the same bi-logistic behavior [12] , with China replacing the north of Italy and the superimposed plot yielding the diffusion in the rest of Italy. In a forthcoming note we will analyze the evolution per individual region.
We must finally stress that the spatial diffusion of the disease, not contained in the present analysis, requires the use of models including non-linear Fisher type evolution equations [13, 14] , which will be touched in forthcoming studies. 