# A Bimodal Lognormal Distribution Model for the Prediction of COVID-19 Deaths
## 4 Validation
### Extrapolation Error  
Figure 3 shows the extrapolation errors obtained in the retrospective predictions as functions of the extrapolation interval length, i.e., the number of days included between the end of the fitting range and the forecast date. For the main continents and the World, the extrapolation errors can be enveloped approximately by assuming a lower bound error rate of \(-2.5\%/15\) days and an upper bound error rate of \(+5.0\%/15\) days.  
As concerns future predictions, similar error trends are expected for the extrapolation of the fitted curves. Extrapolation errors should increase with the length of the extrapolation intervals. Conversely, errors should decrease with an increase in the amount of fitted data. Hence, extrapolations from data known as of 30 September 2020 should be more accurate than the extrapolations carried out in the retrospective study. Anyway, to be on the safe side, the same error rates (\(-2.5\%\), \(+5.0\%\))/15 days will be assumed also to define confidence intervals for future predictions.  
\begin{table}
\begin{tabular}{l r r r r r r r r} \hline \hline \multirow{2}{*}{**Location**} & \multirow{2}{*}{\begin{tabular}{c} **Actual** \\ **Deaths** \\ \end{tabular} } & \multicolumn{4}{c}{**Smd Date of**} & \multicolumn{4}{c}{**Smd July**} & \multicolumn{1}{c}{**30 June**} & \multicolumn{1}{c}{**15 July**} & \multicolumn{1}{c}{**31 July**} & \multicolumn{1}{c}{**15 August**} & \multicolumn{1}{c}{**31 August**} & \multicolumn{1}{c}{**15 September**} \\ \cline{3-8}  & & & 28,172 & 32,456 & 41,066 & 37,198 & 36,288 & 35,604 \\ Africa & 35,673 & & (26.6\%) & (9.9\%) & (\(-13.1\%\)) & (\(-4.1\%\)) & (\(-1.7\%\)) & (0.2\%) \\ \hline \multirow{2}{*}{Asia} & \multirow{2}{*}{193,257} & \multirow{2}{*}{\begin{tabular}{c} 156,591 \\ (23.4\%) \\ \end{tabular} } & 156,591 & 171,482 & 174,028 & 173,256 & 178,301 & 188,030 \\  & & (23.4\%) & (12.7\%) & (11.0\%) & (11.5\%) & (8.4\%) & (2.8\%) \\ \hline \multirow{2}{*}{Europe} & \multirow{2}{*}{222,139} & \multirow{2}{*}{\begin{tabular}{c} 246,838 \\ (\(-10.0\%\)) \\ \end{tabular} } & 246,838 & 215,591 & 213,573 & 218,079 & 218,907 & 219,429 \\  & & (\(-10.0\%\)) & (3.0\%) & (4.0\%) & (1.9\%) & (1.5\%) & (1.2\%) \\ \hline \multirow{2}{*}{\begin{tabular}{c} North \\ America \\ \end{tabular} } & \multirow{2}{*}{305,746} & \multirow{2}{*}{\begin{tabular}{c} 259,154 \\ (18.0\%) \\ \end{tabular} } & 236,142 & 270,083 & 318,303 & 322,088 & 309,580 \\  & & (18.0\%) & (29.5\%) & (13.2\%) & (\(-3.9\%\)) & (\(-5.1\%\)) & (\(-1.2\%\)) \\ \hline \multirow{2}{*}{\begin{tabular}{c} Oceania \\ America \\ \end{tabular} } & \multirow{2}{*}{972} & \multirow{2}{*}{\begin{tabular}{c} 140 \\ (594.3\%) \\ \end{tabular} } & 144 & 656 & 897 & 823 & 1032 \\  & & (594.3\%) & (575.0\%) & (48.2\%) & (8.4\%) & (18.1\%) & (\(-5.8\%\)) \\ \hline \multirow{2}{*}{\begin{tabular}{c} South \\ America \\ \end{tabular} } & \multirow{2}{*}{251,324} & \multirow{2}{*}{\begin{tabular}{c} 175,060 \\ (43.6\%) \\ \end{tabular} } & 196,979 & 217,521 & 271,055 & 254,636 & 247,064 \\  & & (43.6\%) & (27.6\%) & (15.5\%) & (\(-7.3\%\)) & (\(-1.3\%\)) & (1.7\%) \\ \hline \multirow{2}{*}{\
\begin{tabular}{c} World 2 \\ \end{tabular} } & \multirow{2}{*}{1,009,118} & \multirow{2}{*}{
\begin{tabular}{c} 865,962 \\ (16.5\%) \\ \end{tabular} } & 852,801 & 916,934 & 1,018,795 & 1,011,050 & 1,000,746 \\  & & (16.5\%) & (18.3\%) & (10.1\%) & (\(-0.9\%\)) & (\(-0.2\%\)) & (0.8\%) \\ \hline \hline \end{tabular}  
* The start date of the fitting range is 31 December 2019. 2 The World summary includes also seven international deaths [14].  
\end{table}
Table 2: Actual and forecast deaths on 30 September 2020.