# A Bimodal Lognormal Distribution Model for the Prediction of COVID-19 Deaths
## 7 Conclusions  
A phenomenological epidemiological model has been presented for the description and prediction of COVID-19 deaths worldwide. The model assumes a bimodal lognormal distribution for the number of daily deaths in time in each country. Thanks to the asymmetry of the lognormal distribution function, the model is able to fit the available data better than other phenomenological models based on either the logistic or Gaussian distribution functions. Furthermore, thanks to the presence of a second mode in the time distribution, the model is able to describe the trends observed in countries hit by second waves of the epidemic. Besides, if a second wave is at an early stage of development, the model can estimate the time position of its peak. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Location** & **Population 1** & **Actual Fatality Rate 2** & **Predicted Fatality Rate 3** \\ \hline Africa & 1,339,423,921 & 2.7 & 3.4 + (\(-\)0.5, +1.0) \\ Asia & 4,607,388,081 & 4.2 & 6.7 + (\(-\)1.0, +2.1) \\ Europe & 748,506,210 & 29.7 & 35.9 + (\(-\)5.5, +11.0) \\ North America & 591,242,473 & 51.7 & 62.3 + (\(-\)9.5, +19.1) \\ Oceania & 40,958,320 & 2.4 & 2.9 + (\(-\)0.4, +0.9) \\ South America & 430,461,090 & 58.4 & 76.9 + (\(-\)11.8, +23.6) \\ \hline World & 7,757,980,095 & 13.0 & 17.1 + (\(-\)2.6, +5.2) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Actual and predicted fatality rates (deaths per 100,000 people). The model has been validated by using the available data to predict retrospectively the known number of deaths on 30 September 2020. Thus, an estimate has been obtained of the error rate for the extrapolation of future trends. The forecast number of official deaths due to COVID-19 on 31 December 2020 in the World has been estimated at 1,325,614 + (\(-\)203,261, +406,522). The proposed model is purely phenomenological. Thus, it cannot be used to predict the effects of any change of government policies or human behaviour. To this aim, transmission dynamics models should be used instead. Some transmission dynamics models only require a few parameters, while other ones require the specification of a large number of parameters, sometimes difficult to quantify. In the latter case, the numerical implementation of transmission dynamics models may be computationally very expensive. Conversely, the proposed model proved to be computationally very effective. It is able to process a large amount of (publicly available) data from more than two hundred countries and provide extrapolations in a few minutes by using a common personal computer. To sum up, notwithstanding some obvious limitations due to the adopted approach, it is believed that the proposed model can be useful to forecast the trends of COVID-19 mortality in the short-to-medium term in a business-as-usual scenario. The computational effectiveness and the demand for only publicly available data are undoubted strengths of the proposed strategy. Some future developments of the present work may be envisaged:  
1. the predictions of the proposed model shall be checked against the real data that will be available at the end of year 2020. Consequently, corrections will be considered to improve the accuracy of future forecasts;
2. for large federal nations (e.g., Brazil, India, Mexico, and the United States), improved predictions could be obtained by applying the bimodal lognormal distribution model first for each federated state and then summing the results at the national level. A similar approach could be useful also for medium-sized countries (e.g., Italy and the United Kingdom), where national estimates could be improved through the preliminary processing of regional data;
3. for countries facing a third wave of the epidemic (e.g., Iran), the model could be modified by introducing a third mode in the time distribution of daily deaths. To this aim, a suitable linear combination of three lognormal distributions could be used. If further waves should be observed, a model based on a multimodal distribution could be set up.