# A Bimodal Lognormal Distribution Model for the Prediction of COVID-19 Deaths

## Abstract

**:**

## 1. Introduction

## 2. Data

## 3. Methods

#### 3.1. Mathematical Model

#### 3.2. Numerical Implementation

## 4. Validation

#### 4.1. Correlation Study

#### 4.2. Retrospective Predictions

#### 4.3. Extrapolation Error

## 5. Results

#### 5.1. Asia

#### 5.2. Europe

#### 5.3. North America

#### 5.4. South America

#### 5.5. Africa

#### 5.6. Oceania

#### 5.7. World

## 6. Discussion

#### 6.1. Parameter Identifiability

#### 6.2. Observed Trends

#### 6.3. Computational Efficiency

#### 6.4. Representativeness of Predictions

## 7. Conclusions

- 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;
- 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;
- 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. However, if the multiple waves observed for a country are the result of first waves hitting different internal regions, a better strategy would be to process regional data first and then sum up the results at the national level;
- future studies could assess the applicability and accuracy of bimodal (or multimodal) lognormal distribution models to describe the time trends of other epidemiologic parameters (e.g., number of cases, hospitalisations, etc.), as well as for diseases other than COVID-19, such as, for instance, seasonal influenza.

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

COVID-19 | Coronavirus Disease 2019 |

CSV | Comma Separated Values |

ECDC | European Center for Disease Prevention and Control |

OWID | Our World in Data |

PNG | Portable Network Graphics |

SIR | Susceptible-Infected-Recovered |

## Appendix A. Comparison of Predictions with Actual Data

Location | Actual Deaths (No.) | Predicted Deaths (No.) | Error (%) |
---|---|---|---|

Africa | 48,762 | 42,289 | +15.3% |

Asia | 273,170 | 270,181 | +1.1% |

Europe | 343,380 | 246,839 | +39.1% |

North America | 380,946 | 350,253 | +8.8% |

Oceania | 1106 | 1110 | −0.4% |

South America | 314,544 | 300,753 | +4.6% |

World | 1,361,915 | 1,211,432 | +12.4% |

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**Figure 1.**Fitting of daily deaths data from Italy: (

**a**) Gaussian distribution function, (

**b**) lognormal distribution function.

Parameter | Minimum | Guess | Maximum |
---|---|---|---|

A (No.) | $1.10\phantom{\rule{0.166667em}{0ex}}D\left({t}_{\mathrm{fit}}\right)$ | $max\left\{6\phantom{\rule{0.166667em}{0ex}}D\left({t}_{1}\right),\phantom{\rule{0.277778em}{0ex}}2\phantom{\rule{0.166667em}{0ex}}D\left({t}_{\mathrm{fit}}\right)\right\}$ | $max\left\{12\phantom{\rule{0.166667em}{0ex}}D\left({t}_{1}\right),\phantom{\rule{0.277778em}{0ex}}4\phantom{\rule{0.166667em}{0ex}}D\left({t}_{\mathrm{fit}}\right)\right\}$ |

${M}_{1}$ (days) | 1 | ${t}_{1}$ | ${t}_{\mathrm{max}}$ |

${\sigma}_{t1}$ (days) | 0 | 1 | ${t}_{\mathrm{max}}$ |

${M}_{2}$ (days) | 1 | ${t}_{2}$ | ${t}_{\mathrm{max}}$ |

${\sigma}_{t2}$ (days) | 0 | 1 | ${t}_{\mathrm{max}}$ |

$\alpha $ (–) | 0 | $\frac{1}{1+d\left({t}_{2}\right)/d\left({t}_{1}\right)}$ | 1 |

Location | Actual Deaths | Forecast Deaths | ||||||
---|---|---|---|---|---|---|---|---|

End Date of Fitting Range: ${}^{1}$ | 30 June | 15 July | 31 Juyl | 15 August | 31 August | 15 September | ||

Africa | 35,673 | 28,172 | 32,456 | 41,066 | 37,198 | 36,288 | 35,604 | |

(26.6%) | (9.9%) | (−13.1%) | (−4.1%) | (−1.7%) | (0.2%) | |||

Asia | 193,257 | 156,591 | 171,482 | 174,028 | 173,256 | 178,301 | 188,030 | |

(23.4%) | (12.7%) | (11.0%) | (11.5%) | (8.4%) | (2.8%) | |||

Europe | 222,139 | 246,838 | 215,591 | 213,573 | 218,079 | 218,907 | 219,429 | |

(−10.0%) | (3.0%) | (4.0%) | (1.9%) | (1.5%) | (1.2%) | |||

North America | 305,746 | 259,154 | 236,142 | 270,083 | 318,303 | 322,088 | 309,580 | |

(18.0%) | (29.5%) | (13.2%) | (−3.9%) | (−5.1%) | (−1.2%) | |||

Oceania | 972 | 140 | 144 | 656 | 897 | 823 | 1032 | |

(594.3%) | (575.0%) | (48.2%) | (8.4%) | (18.1%) | (−5.8%) | |||

South America | 251,324 | 175,060 | 196,979 | 217,521 | 271,055 | 254,636 | 247,064 | |

(43.6%) | (27.6%) | (15.5%) | (−7.3%) | (−1.3%) | (1.7%) | |||

World ^{2} | 1,009,118 | 865,962 | 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%) |

Continent | Country | A | ${\mathit{M}}_{1}$ | ${\mathit{\sigma}}_{\mathit{t}1}$ | ${\mathit{M}}_{2}$ | ${\mathit{\sigma}}_{\mathit{t}2}$ | $\mathit{\alpha}$ | Actual Deaths ^{1} | Predicted Deaths ^{2} |
---|---|---|---|---|---|---|---|---|---|

(No.) | (Days) | (Days) | (Days) | (Days) | (–) | ||||

Africa | South Africa | 19,585.9 | 214.7 | 55.8 | 217.5 | 13.9 | 0.730 | 16,667 | 19,289 |

Egypt | 7105.3 | 181.3 | 15.9 | 184.0 | 97.0 | 0.368 | 5914 | 6694 | |

Asia | India | 187,322.1 | 235.3 | 86.7 | 276.9 | 48.3 | 0.658 | 97,497 | 167,051 |

Iran | 28,584.6 | 93.3 | 19.8 | 211.6 | 48.7 | 0.238 | 25,986 | 28,394 | |

Europe | United Kingdom | 46,279.2 | 111.3 | 19.0 | 205.3 | 166.9 | 0.823 | 42,072 | 44,095 |

Italy | 39,462.5 | 96.9 | 20.3 | 199.1 | 224.4 | 0.836 | 35,875 | 37,280 | |

North America | United States | 253,254.2 | 115.0 | 18.5 | 221.7 | 69.2 | 0.396 | 205,998 | 244,290 |

Mexico | 98,272.7 | 166.2 | 10.9 | 203.7 | 66.9 | 0.031 | 77,163 | 94,706 | |

Oceania | Australia | 970.2 | 110.4 | 187.5 | 237.0 | 17.6 | 0.232 | 882 | 934 |

Guam | 228.0 | 278.1 | 0.2 | 305.2 | 48.7 | 0.000 | 47 | 195 | |

South America | Brazil | 166,675.9 | 150.2 | 29.1 | 223.4 | 50.5 | 0.333 | 142,921 | 164,969 |

Peru | 35,635.6 | 185.0 | 61.5 | 218.9 | 15.2 | 0.701 | 32,396 | 35,184 |

Location | Population ${}^{1}$ | Actual Fatality Rate ${}^{2}$ | Predicted Fatality Rate ${}^{3}$ |
---|---|---|---|

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) |

World | 7,757,980,095 | 13.0 | 17.1 + (−2.6, +5.2) |

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Valvo, P.S.
A Bimodal Lognormal Distribution Model for the Prediction of COVID-19 Deaths. *Appl. Sci.* **2020**, *10*, 8500.
https://doi.org/10.3390/app10238500

**AMA Style**

Valvo PS.
A Bimodal Lognormal Distribution Model for the Prediction of COVID-19 Deaths. *Applied Sciences*. 2020; 10(23):8500.
https://doi.org/10.3390/app10238500

**Chicago/Turabian Style**

Valvo, Paolo S.
2020. "A Bimodal Lognormal Distribution Model for the Prediction of COVID-19 Deaths" *Applied Sciences* 10, no. 23: 8500.
https://doi.org/10.3390/app10238500