The present study is motivated by the paper [1] that appeared in The Lancet on 13 March 2020, where the authors are calling for trend analysis. At the beginning of the outbreak, Italian data followed closely the exponential growth trend observed in Hubei Province, China. The question is if Italian measures can break the exponential trend? Can they turn the process to a logistic growth with a final saturation value like in China?
Exponential growth turns into logistic growth when the number of susceptible subjects is limited. If the epidemic is uncontrolled, this limit is defined by the herd immunity, and the vast majority of the population gets infected. Another possibility is when physical contacts in the social network are so severely restricted that the system remains below the epidemic threshold [2] so that only finite-sized infected pockets can develop from infected seeds. This second scenario is in China, where it led to the termination of the epidemic in Hubei Province and only small infected pockets in other parts of China. Exponential growth is described by the differential equation
where λ is the rate of growth. If we apply this for the cumulative number of cases N n on a given day n, then the rate of growth can be calculated from the relative growth of the cummulative number of cases When the exponential growth has some limit N n < N ∞ , the growth rate depends on the number of cases λ(N ) and decreases as we approach the limit. Growth should stop when we reach the limit λ(N ∞ ) = 0. We can expand the growth rate near this point and get
where the first derivative is negatíve λ (N ∞ ) < 0.
λ 0 , the second and higher order terms are negligable and we can determine λ 0 and N ∞ from a linear fit
which leads to the logistic growth equation
The solution of the logistic equation is the logistic curve
We can define the end time of the epidemic when the number of cases reaches the fraction 1 − p of total cases. The solution of the equation gives and
where p 1 is a very small fraction. The solution for very small p values is then given by
Next, we show that this is an excellent approximation to the actual data, which we obtained from the JHU site[3].
2 Deaths as seen on March 17, 2020
First, we can do a post-fact analysis of the data from Hubei Province. We have the confirmed cases and deaths data, which both show logistic growth. Both of them have some subjective elements, and never the less the deaths data seems to be more objective. In Fig.1 , we show the daily growth rate of death cases. Since the reporting of death cases is uneven, deaths are sometimes reported a day later. Therefore we calculate the daily rate from the average of two consecutive days λ n = 0.5 · (N n − N n−2 )/N n . The first data point is January 24, 2020, three days after the lockdown of Wuhan. The daily growth rate is fluctuating around 25-30%, which seems to be the "natural" rate of growth in the absence of public actions. Then 14 days after the lockdown at about 500 deaths, the daily growth rate drops to about 10%. Then a linear trend sets in with some fluctuations due to some errors in the reporting. The slope of the linear fit is λ 0 /N ∞ = 4.6 ± 0.4 · 10 −5 , λ 0 ≈ 0.146, and N ∞ ≈ 3158.
We repeat the same analysis for the data from Italy and show the result in Fig.2 . The Italian data starts on February 25, 2020. The beginning of the data fluctuates around 20-30% daily growth. The lockdown of Lombardy starts on March 9, at about 463 deaths, and the linear trend can be observed starting from this date. The slope of the linear trend is λ 0 /N ∞ = 4.05 ± 1.13 · 10 −5 , λ 0 ≈ 0.243, and N ∞ ≈ 5996.
Remarkably, the slope of the curve is the same in Hubei and Italy within the error of fitting. It seems that the lockdown has a similar effect in both cases. The daily rate of growth of deaths appears to be falling 4% for each 1000 new deaths in Italy and the daily growth rate will be reduced from roughly 24% to zero after 6000 deaths. In Hubei the initial daily growth rate of 12% fell to zero after about 3000 deaths. It seems that this is a universal consequence of lockdown, which needs further explanation. 
After releasing the first version of this manuscript, the situation in Italy started to change. In Fig.3 , we show the daily growth rate of fatalities again as a function of the cumulative number of deaths. We keep the original linear fit too, which would lead to about 6000 deaths. Instead, the slope changed sharply on March 17, and the new, clearly visible trend predicts about N ∞ = 19557 ± 1670 deaths. The new slope is λ 0 /N ∞ = 8.1 · 10 −6 , and the rate of deaths falls only 0.8% for every 1000 new deaths. The reason for this radical change in the slope is probably not related to the progress of the epidemic. The number of deaths is simply a proxy for the size of the epidemic. It depends on the number of people infected and the probability of death. We assume that the latter changed drastically when the Italian health Fig.2 , while the black line is fitted to the data from March 17 to 30, 2020.
system reached its capacity limit on March 17. Without this limit, probably 3500 people would have died. Instead, due to the limitations, 17.000 people are expected to die now, which is a five-fold increase.
4 Length of the outbreak as seen on March 17, 2020
Using the fitted parameters, we attempt to calculate the expected end of the outbreak in Italy. First, we validate the model on the Hubei data. In Fig.1 , the linear regime starts when the number of deaths is N 0 = 549 on February 5. Today, 40 days later, on March 17, the number of deaths is 3111 in Hubei, which is 98.5% of the expected deaths N ∞ = 3158 or p = 0.015. Using these numbers (8) yields t = 39.44 days, which is a very precise estimate for the length of the process. In Italy, the linear regime starts on March 9, and N 0 = 463. Using the same p = 0.015 and the number of ultimate deaths N ∞ = 5996 (8) yields t = 27.49 days that is April 6. The p = 0.001 milestone will be reached around April 15, and the lockdown can probably be partially lifted. 