# A Bimodal Lognormal Distribution Model for the Prediction of COVID-19 Deaths
## 1 Introduction  
For over a century, the use of mathematical models has been introduced to describe and predict the course of epidemics [1]. Following the first pioneering studies, epidemiologic models of growing complexity have been developed [2]. Models may be classified as either based on transmission dynamics or phenomenological. Transmission dynamics models aim at describing the physical, biological, and societal mechanisms underlying the outbreak and spread of a disease. On the one hand, they can be very helpful in understanding the behaviour of epidemics and the effects of human interventions. On the other hand, transmission dynamics models rely sometimes on a large number of parameters, which may be difficult to evaluate in practice with the necessary accuracy [3]. As an alternative, phenomenological models can be used. These are based on a limited number of parameters that can be calibrated through fitting of the available data. Phenomenological models do not aim at understanding the mechanisms underlying an epidemic, but can be used for efficient and rapid forecasts [4]. For instance, [PERSON] et al. used phenomenological models to forecast the evolution of the 2015 Ebola epidemic [5]. Both transmission dynamics models and phenomenological models were used to describe the 2020 COVID-19 pandemic caused by the SARS-CoV-2 coronavirus [6]. Among the former, the classic susceptible-infected-recovered (SIR) model and modifications thereof have been widely used. Among the latter, models based on either the logistic or Gaussian distribution functions have been proposed. [PERSON] applied both the SIR and logistic models to predict the final size of the COVID-19 epidemic in China. By processing data available up to 20 February 2020, he estimated a total number of cases at the end of the epidemic equal to approximately 84,500 from the SIR model and 83,700 fromthe logistic model [7]. Retrospectively, such predictions turned out to be very accurate if compared to the official number of cases reported four months later. As of 20 June 2020, the official number of cases in China was 84,524. [PERSON] and [PERSON] used the SIR and logistic models to describe the COVID-19 outbreak India. They processed the official data available from 9 March to 9 May 2020 for three states (Karnataka, Kerala, and Maharashtra), characterised by a significant spread of the epidemic. For the most affected state, Maharashtra, the SIR model estimated the final number of cases at 46,678, whereas the logistic model predicted 43,993 cases [8]. Retrospectively, such predictions turned out to be dramatically underestimated. One month later, as of 9 June 2020, 90,787 cases were reported. As of early August 2020, more than 500,000 cases were registered [9]. [PERSON] used particle swarm optimisation to identify the parameters of a SIR model of the COVID-19 outbreak in Italy. He focused on the four most affected regions (Lombardy, Emilia Romagna, Valle d'Aosta, and Veneto). The machine learning approach was found to be robust for the identification of a large set of model parameters. The model results showed that lockdown measures had a major impact especially in Lombardy [10]. [PERSON] and [PERSON] used the logistic model to analyse the evolution of the COVID-19 epidemic in Italy. They processed data available up to 1 April 2020 and forecast about 138,500 cases and 19,900 deaths by the end of the epidemic [11]. Retrospectively, such figures turned out to be quite underestimated. One month later, as of 1 May 2020, official statistics reported 205,463 cases and 27,967 deaths with the epidemic still spreading in Italy. [PERSON] and [PERSON] used a Gaussian model to predict the development of the COVID-19 epidemic in Germany. Based on the data available as of 30 March 2020, they predicted the peak of the death rate to occur on 18 April 2020 + (-3.4, +5.4) days. Their prediction turned out to be in excellent agreement with the actual peak registered on 16 April 2020 [12]. [PERSON] et al.