Analysis of the prediction of the 2021 time-evolution of the Covid-19 pandemic in Italy using a Planck's distribution

In a previous paper, we studied the time-evolution of the Covid-19 pandemic in Italy during the first wave of 2020 using a number of distribution laws. We concluded that the best distribution law to predict the evolution of the pandemic is a distribution of the type of Planck's law with three parameters, provided that the basic conditions of the pandemic (such as social distancing, vaccination campaigns and new Covid variants) do not appreciably change the spread of the pandemic. In our 2020 study, we did not use the number of daily positive cases in Italy but the ratio of daily positive cases per number of daily tests, ratio today sometimes referred to as: "positivity rate." We showed that if basic conditions do not change, the Planck's distribution with three parameters provides very good predictions of the positivity rate about 1 month in advance. In a second paper, using the Planck's distribution with three parameters, we predicted, about 1 month in advance, the spread of the pandemic in Italy during the Christmas 2020 holidays. Here we show that indeed in our second paper the spread of the pandemic in Italy, after 1 month, was well predicted using the Planck's distribution with an error of a few percent only. We then studied the present (October 2021) evolution of the pandemic in Italy, and we showed that the Planck's distribution, based on the data of July and August, predicted well the evolution of the pandemic. We then show that the peak of the positivity rate was approximately reached during the middle of August. However, the end of the Italian holidays and the start of all the activities including schools, intensive use of public transportation and further distancing measures may change again the trend of the positivity rate of the pandemic.