Motivated by the current Coronavirus Disease (COVID-19) pandemic, which is due to the SARS-CoV-2 virus, and the important problem of forecasting daily deaths and cumulative deaths, this paper examines the construction of prediction regions or intervals under the Poisson regression model and for an over-dispersed Poisson regression model. For the Poisson regression model, several prediction regions are developed and their performance are compared through simulation studies. The methods are applied to the problem of forecasting daily and cumulative deaths in the United States (US) due to COVID-19. To examine their performance relative to what actually happened, daily deaths data until May 15th were used to forecast cumulative deaths by June 1st. It was observed that there is over-dispersion in the observed data relative to the Poisson regression model. An over-dispersed Poisson regression model is therefore proposed. This new model builds on frailty ideas in Survival Analysis and over-dispersion is quantified through an additional parameter. The Poisson regression model is a hidden model in this over-dispersed Poisson regression model and obtains as a limiting case when the over-dispersion parameter increases to infinity. A prediction region for the cumulative number of US deaths due to COVID-19 by July 16th, given the data until July 2nd, is presented. Finally, the paper discusses limitations of proposed procedures and mentions open research problems, as well as the dangers and pitfalls when forecasting on a long horizon, with focus on this pandemic where events, both foreseen and unforeseen, could have huge impacts on point predictions and prediction regions.
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