# A Bimodal Lognormal Distribution Model for the Prediction of COVID-19 Deaths
## 1 Introduction
discussed about the choice of asymmetric functions to model the COVID-19 death rate, but eventually decided in favour of the simplest symmetric, bell-shaped Gaussian function. Based on data available as of 2 April 2020, they predicted the final number of deaths for 25 countries. In a note added in proof, the authors compare their theoretical predictions with the data available as of 2 June 2020. Good agreement is obtained for some countries, such as France and Germany. Instead, the model appears to underestimate the real data of some other countries, such as Italy and Spain [13]. Interestingly, less accurate predictions are obtained for countries, where the actual time distribution of deaths presents a higher degree of asymmetry. As a matter of fact, upon inspection of the available data, the number of infections or deaths in a given country (or geographical region) does not follow in general a symmetric distribution in time. Rather, after reaching the peak value, a descending branch less steeper than the ascending one can be often observed. In such cases, a better fit of the real data can be obtained by using asymmetric distribution functions--such as, for instance, the lognormal distribution--instead of the symmetric logistic and Gaussian functions. As an example, Figure 1 shows the daily number of COVID-19 deaths reported in Italy in the first semester of year 2020. Data have been fitted by using both a Gaussian distribution function (Figure 1a) and a lognormal distribution function (Figure 1b). The residual norms computed in the two cases were 202,818,961 and 63,831,005, respectively. Application of the same distribution functions to fit the data of all of the 210 countries in the Our World in Data (OWID) dataset [14] for the same period furnished cumulative residual norms equal to 3,467,666,524 and 1,365,052,591, respectively. Such results prove, beyond the qualitative comparison between the curves, the superiority of the lognormal distribution function to fit the data of the present problem. In a preliminary analysis of the Italian epidemic, the author used a lognormal distribution function to fit the number of deaths per day reported in each Italian administrative region. By processing the data available as of 2 May 2020, the total forecast number of deaths in Italy at the end of the first wave of the epidemic was 35,261 [15]. As of 31 August 2020, the official data reported 35,477 deaths with an error of less than 1% [16]. When looking at the time evolutions of the number of COVID-19 cases and deaths around the world, different trends are observed for each country and geographic region. As of September 2020, China and many European countries have overcome the first wave of the epidemic. Other countries--such as, for instance, Australia, Israel, Japan, Romania, and the United States--are being hit by second waves of the epidemic. In Iran, a third wave of the epidemic is going on. Such new waves appear as second (or third) peaks in the time distributions of infections and deaths. In some cases, e.g., the United States, the second waves could be the result of first waves affecting different internal states or regions. Anyway, a better data fitting for such countries requires the adoption of suitable bimodal (or multimodal) distribution functions. Any theoretical model aimed at describing the pandemic at the global level should account for the variability in both space and time of the monitored trends (cases, hospitalisations, recoveries, fatalities, etc.). The aim of the present work is to develop a phenomenological epidemiological model for the description of the worldwide trends of COVID-19 deaths and their prediction in the short-to-medium (1 and 3 months, respectively) term in a business-as-usual scenario. The model assumes a bimodal lognormal distribution in time of the deaths per country. A nonlinear least squares procedure is used to fit the available data and determine the six model parameters for each country. Then, the obtained curves are extrapolated to forecast the future trends.