# On Estimating the Number of Deaths Related to Covid-19

## Abstract

**:**

## 1. Introduction

## 2. Model Development on Estimating the Number of Deaths

#### 2.1. Model Considerations

- There are a few people in the population who have already been infected with Covid-19, and are spreading the virus into the community but do not know that they are infected with the virus. The virus is spreading through people who are in close contact with one another. An infected person may, for example, cough or sneeze, spreading the virus through the bacteria eventually coming in contact with the mouths or noses of other people who are nearby or possibly directly inhaled into their lungs. A person can get Covid-19 by touching an infected surface or object and then touching their own mouth, nose, or possibly their eyes [2].
- The virus is spreading throughout the areas based on a time-dependent infection rate per person in which it will spread at a very slow rate from the beginning due to a small number of infected people and will spread at a growth rate much faster due to a higher number of people who have already been infected with the virus and who are in close contact with non-infected individuals as time progresses. The growth rate will then continue to grow slowly until it reaches the maximum total number of Covid-19 deaths.
- The rate of change of the death is the derivative of the number of deaths p’(t) is directly proportional to both the number of deaths p(t) who have infected the virus and the number of people in the susceptible population who have not yet been infected, based on the time-dependent rate infections per person per unit time.
- Deaths are proportional to infections, but with a lag. There can be a significant time lag between when someone is infected and when they die. We assume that death data is more reliable than the reported number of cases and hospitalizations due to the uncertainty of testing mechanisms and the recognized symptoms and treatments. Additionally, it is easier to determine cause of death than cause of hospitalizations and test cases. In fact, we need to know how tests are being conducted; otherwise, there will be a lot of uncertainty about the number of Covid-19 cases, so they will not be very useful indicators.

#### 2.2. Model Development

## 3. Modeling Analysis and Prediction Results

## 4. Conclusions

## Funding

## Conflicts of Interest

## Abbreviations

SSE | sum of squared error |

MSE | mean squared error |

AIC | Akaike’s information criterion |

BIC | Bayesian information criterion |

LSE | least square estimate |

PC | Pham’s criterion |

PP | the predictive power |

PIC | Pham’s information criterion |

PRR | the predictive ratio risk |

## References

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Model | p(t) |
---|---|

Four-parameter logistic fault-detection model [13] (Model 1) | $p(t)=\frac{a}{1+d\left(\frac{1+\beta}{\beta +{e}^{bt}}\right)}$ |

Modified model (Model 2) | $p(t)=c+\frac{a}{1+d\left(\frac{1+\beta}{\beta +{e}^{bt}}\right)}$ |

Five-parameter logistic model (New Model) | $p(t)=\frac{a}{1+d\left(\frac{1+c}{\beta +{e}^{bt}}\right)}$ |

No. | Criteria | Formula | |
---|---|---|---|

1 | SSE [10] | $\mathrm{SSE}={\displaystyle \sum _{i=1}^{n}{\left({y}_{i}-{\widehat{y}}_{i}\right)}^{2}}$ | Measures the total deviations between the predicted values with the actual data observation. |

2 | MSE [10] | $\mathrm{MSE}=\frac{{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{\left({y}_{i}-{\widehat{y}}_{i}\right)}^{2}}{n-k}$ | Measures the difference between the estimated values and the actual observation. |

3 | AIC [11] | $\mathit{AIC}=-2\mathrm{log}\left(L\right)+2k$ | Takes into account the penalty term by adding more parameters. |

4 | BIC [12] | $\mathit{BIC}=-2\mathrm{log}\left(L\right)+k\mathrm{log}\left(n\right)$ | Takes into account the penalty based on the sample size and the number of parameters in the model. |

5 | PIC [10] | $\begin{array}{c}\mathrm{PIC}=SSE+k\left(\frac{n-1}{n-k}\right)\\ \mathrm{where}\text{}SSE={\displaystyle \sum _{i=1}^{n}{\left({y}_{i}-{\widehat{y}}_{i}\right)}^{2}}\end{array}$ | Takes into account more penalty when adding too many parameters in the model where the sample is considerably too small. |

6 | PRR [14] | $PRR={\displaystyle \sum _{i=1}^{n}{\left(\frac{\widehat{m}({t}_{i})-{y}_{i}}{\widehat{m}({t}_{i})}\right)}^{2}}$ | Measures the distance of model estimates from the actual data against the model estimate. |

7 | PP [14] | $PP={\displaystyle \sum _{i=1}^{n}{\left(\frac{\widehat{m}({t}_{i})-{y}_{i}}{{y}_{i}}\right)}^{2}}$ | Measures the distance of model estimates from the actual data against the actual data. |

8 | PC (Pham’s criterion) | $\begin{array}{c}PC=\left(\frac{n-k}{2}\right)\mathrm{log}\left(\frac{SSE}{n}\right)+k\left(\frac{n-1}{n-k}\right)\\ \mathrm{where}\text{}SSE={\displaystyle \sum _{i=1}^{n}{\left({y}_{i}-{\widehat{y}}_{i}\right)}^{2}}\end{array}$ | Takes into account the tradeoff between the uncertainty in the model and the number of parameters in the model by slightly increasing the penalty when each time adding parameters in the model where the sample is considerably too small. |

**Table 3.**US deaths data [20] during 2/29/20—4/22/20.

Date | Cumulative Number of Deaths | Date | Cumulative Number of Deaths |
---|---|---|---|

2/29 | 1 | 3/27 | 2110 |

3/1 | 1 | 3/28 | 2754 |

3/2 | 6 | 3/29 | 3251 |

3/3 | 9 | 3/30 | 3948 |

3/4 | 11 | 3/31 | 5027 |

3/5 | 12 | 4/1 | 6263 |

3/6 | 15 | 4/2 | 7438 |

3/7 | 19 | 4/3 | 8694 |

3/8 | 22 | 4/4 | 10,231 |

3/9 | 26 | 4/5 | 11,632 |

3/10 | 30 | 4/6 | 13,128 |

3/11 | 38 | 4/7 | 15,347 |

3/12 | 41 | 4/8 | 17,503 |

3/13 | 48 | 4/9 | 19,604 |

3/14 | 58 | 4/10 | 21,830 |

3/15 | 73 | 4/11 | 23,843 |

3/16 | 95 | 4/12 | 25,558 |

3/17 | 121 | 4/13 | 27,272 |

3/18 | 171 | 4/14 | 29,825 |

3/19 | 239 | 4/15 | 32,443 |

3/20 | 309 | 4/16 | 34,619 |

3/21 | 374 | 4/17 | 37,147 |

3/22 | 509 | 4/18 | 39,014 |

3/23 | 689 | 4/19 | 40,575 |

3/24 | 957 | 4/20 | 42,514 |

3/25 | 1260 | 4/21 | 45,179 |

3/26 | 1614 | 4/22 | 47,520 |

Model | p(t) | Parameter Estimates |
---|---|---|

Model 1 | $p(t)=\frac{a}{1+d\left(\frac{1+\beta}{\beta +{e}^{bt}}\right)}$ | $\begin{array}{l}a=54900,b=0.1774159\\ d=400.013,\beta =5.977112\end{array}$ |

Model 2 | $p(t)=c+\frac{a}{1+d\left(\frac{1+\beta}{\beta +{e}^{bt}}\right)}$ | $\begin{array}{l}a=54800,\text{}b=0.17794\\ c=0.49804,d=342.0186\\ \beta =7.32222\end{array}$ |

New model | $p(t)=\frac{a}{1+d\left(\frac{1+c}{\beta +{e}^{bt}}\right)}$ | $\begin{array}{l}a=62100,b=0.1535604\\ c=2.6586221,d=338.99688\\ \beta =-11.9747477\end{array}$ |

Criteria | Model 1 (Rank) | Model 2 (Rank) | New Model (Rank) |
---|---|---|---|

SSE | 16,888,788 (2) | 17,383,120 (3) | 16,165,633 (1) |

MSE | 337,775.8 (2) | 354,757.5 (3) | 329,910.9 (1) |

AIC | 691.2715 (2) | 694.8294 (3) | 690.9084 (1) |

BIC | 699.2275 (1) | 704.7743 (3) | 700.8533 (2) |

PIC | 16,888,792 (2) | 17,383,125 (3) | 16,165,638 (1) |

PRR | 17.66833 (1) | 17.94312 (2) | 54.3795 (3) |

PP | 42,211.26 (1) | 57,031.17 (2) | 605,026.3 (3) |

PC | 320.5694 (3) | 316.1178 (2) | 314.3388 (1) |

Estimation | Real Observation | Model 1 | Model 2 | New Model |
---|---|---|---|---|

Fitted value | ||||

#54 (4/22/20) | 47,520 | 46,030.9 | 46,011.8 | 47,348.4 |

Predicted value | ||||

#55 (4/23/20) | 49,845 | 47,272.0 | 47,246.5 | 49,009.9 |

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**MDPI and ACS Style**

Pham, H.
On Estimating the Number of Deaths Related to Covid-19. *Mathematics* **2020**, *8*, 655.
https://doi.org/10.3390/math8050655

**AMA Style**

Pham H.
On Estimating the Number of Deaths Related to Covid-19. *Mathematics*. 2020; 8(5):655.
https://doi.org/10.3390/math8050655

**Chicago/Turabian Style**

Pham, Hoang.
2020. "On Estimating the Number of Deaths Related to Covid-19" *Mathematics* 8, no. 5: 655.
https://doi.org/10.3390/math8050655