# Exploring COVID-19 Daily Records of Diagnosed Cases and Fatalities Based on Simple Nonparametric Methods

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Observational Data

#### 2.2. Mathematical and Statistical Modelling

#### 2.2.1. Asymptotic and Instantaneous Fatality–Case Ratios

- Delay$-\Delta t$ asymptotic fatality–case ratio:$$AFC{R}_{\Delta t}\left(t\right)=\frac{cumDeaths\left(t\right)}{cumCases(t-\Delta t)}\phantom{\rule{2.em}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}t\ge \Delta t$$
- Delay$-\Delta t$ instantaneous fatality–case ratio:$$IFC{R}_{\Delta t}\left(t\right)=\frac{deaths\left(t\right)}{cases(t-\Delta t)}\phantom{\rule{2.em}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}t\ge \Delta t.$$

#### 2.2.2. Diagnosis-to-Death Duration via Maximum Correlation between Deaths and Time-Delayed Cases

#### 2.2.3. Generation Time via Delay-Time Autocorrelation of Cases and Deaths

#### 2.2.4. Piecewise Exponential Growth and the Basic Reproduction Number

## 3. Results

#### 3.1. Fatality–Case Ratios Worldwide and for Eight Selected Countries

#### 3.2. Diagnosis-to-Death Duration for Germany

#### 3.3. Diagnosis-to-Death Duration for the Eight Selected Countries

#### 3.4. Negative Correlation of the Fatality-to-Case Ratio with the Number of Cases

#### 3.5. Estimating Generation Time

#### 3.6. Time-Dependent Infection Rate and the Effective Reproduction Number

#### 3.7. Per Capita Growth Rate as an Alternative for the Reproduction Number

#### 3.8. Spectral Analysis to Confirm Periods

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AFCR | Asymptotic Fatality–Case Ratio |

Generation time | Average time between two consecutive infections |

IFCR | Instantaneous Fatality–Case Ratio |

${R}_{0}$ | Basic Reproduction Number, number of secondary infections emerging |

from an index case in a fully susceptible population | |

Serial interval | Average time between the onset of symptoms of two consecutive |

infections, often used as an approximation to the generation time | |

SIR/SEIR | Susceptible-(exposed)-infected-removed epidemiological |

compartment models | |

Time-to-death duration | Average time between the registrations of new cases and the |

corresponding registrations of deaths, if applicable. |

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**Figure 1.**Time courses of cumulative cases, cumulative deaths, and delay$-0$ asymptotic fatality–case ratios for the entire world and eight selected countries: (

**A**) worldwide cumulative cases and cumulative deaths normalised to the world population (per capita values), (

**B**) worldwide ratio of cumulative deaths to cumulative cases (delay$-0$ asymptotic fatality–case ratio, (

**C**) normalised (per capita) cumulative cases and cumulative deaths of eight selected countries, and (

**D**) melay$-0$ asymptotic fatality–case ratio for the eight selected countries.

**Figure 2.**Asymptotic fatality–case ratio for the German COVID-19 data: (

**A**) logarithmised cumulative deaths, $ln\left(cumDeaths\left(t\right)\right)$, versus time delayed logarithmised cumulative cases, $ln\left(cumCases\right(t-delay\left)\right)$, for different delays as indicated in the panel headers along with linear correlation (regression line plus Pearson’s correlation coefficient printed in the upper left corner of each panel); (

**B**) correlation coefficient as a function of the lengths of the time series (i.e., final observation time) for delays ranging from 10 to 16; (

**C**) delay$-0$ asymptotic fatality–case ratio (black) with time average (blue) and median (red); and (

**D**) delay$-13$ asymptotic fatality–case ratio (black curve) with time average ($0.037$, blue line) and median ($0.04$, red line).

**Figure 3.**Instantaneous fatality–case ratio for the German COVID-19 data: (

**A**) new deaths ($deaths\left(t\right)$) versus time delayed new cases ($cases(t-delay)$) along with linear correlation (regression line plus Pearson’s correlation coefficient printed in the upper right corner of each panel), (

**B**) correlation coefficient as a function of the lengths of the time series (i.e., final observation time) for delays ranging from 10 to 16, (

**C**) delay$-0$ instantaneous fatality–case ratio (black) with time average (blue) and median (red), and (

**D**) delay$-13$ instantaneous fatality–case ratio (black) with time average ($0.044$, blue) and median ($0.02$, red).

**Figure 4.**Diagnosis-to-death duration for eight selected countries analysed using delay-time correlation: the plot shows the magnitudes of delay-specific correlations between $deaths\left(t\right)$ and $cases(t-\Delta t)$ for the eight selected countries (column labels) in the form of a heatmap. The delays $\Delta t$ (row labels) run from $\Delta t=0d$ through $\Delta t=17d$. Strong correlations are shown in dark red, and declining correlation coefficients gradually fade to blue. Also shown for each country and each delay are the time courses of Delay$-\Delta t$ instantaneous fatality case ratios along with time average (blue line) and median (green).

**Figure 5.**Instantaneous fatality–case ratios stratified for the analysed 8 exemplary epidemics: the corresponding country code is assigned to the top of each panel.

**Figure 6.**Delay-time autocorrelation for German incidence data: (

**left**panel) autocorrelation, $C(\Delta t)$, of cases (blue curve) and deaths (red) as a function of delay $\Delta t$ and (

**right**panel) ratio $\frac{cases\left(t\right)}{cases(t-\Delta t)}$ (primitive approach to estimate the reproduction ratio) for nine different delays $\Delta t$, as indicated in the panel headers. The red curves result from a moving average with a window width of seven days.

**Figure 7.**Delay-time autocorrelation for French incidence data: (

**left**panel) autocorrelation, $C(\Delta t)$, of cases (blue curve) and deaths (red) as a function of delay $\Delta t$ and (

**right**panel) ratio $\frac{cases\left(t\right)}{cases(t-\Delta t)}$ (primitive approach to estimate the reproduction ratio) for nine different delays $\Delta t$, as indicated in the panel headers. The red curves result from a moving average with a window width of seven days.

**Figure 8.**Time-dependent infection rate and approximate effective reproduction number for Germany: (

**A**) time course of the infection rate ${\lambda}_{\Delta t}\left(t\right)$ according to Equation (7) for nine different intervals (delays) $\Delta t$, as indicated in the panel headers. Also shown are lines that correspond to doubling times of either $1d$ or $2d$, respectively. (

**B**) Approximate reproduction numbers calculated according to Equation (8). The inlets show details where R is close to 1, i.e., from May onwards. Of note, computed this way, R has a lower limit of 1.

**Figure 9.**Per capita growth rates by time for the German COVID-19 data: (

**A**) growth rate for cumulative cases, where the inlet shows the tail of the time course for $t>220d$ with adjusted y-axis for better visibility, and (

**B**) growth rate for the daily new cases, where the inlet shows the same time course with a narrow y-axis range around zero. Red curve: moving average with a 7 day window size.

**Figure 10.**Spectral analysis for the German COVID-19 data: (

**A**) spectral density of confirmed cases time series; (

**B**) spectral density of confirmed deaths time series; and (

**C**) cases–deaths coherency, showing the correlation at different frequencies (cross-sprectrum).

**Table 1.**Comparison of correlation coefficients for the cumulative incidence data: column 2 contains the estimated correlation coefficients of the two time series $ln\left(cumCases\right(t-\Delta t\left)\right)$ and $ln\left(cumDeaths\right(t\left)\right)$ with the corresponding delays $\Delta t$ in days listed in the first column. The p-values in the third column refer to a test for difference of any given correlation coefficient with the maximum correlation coefficient, in this case, the one estimated for delay $\Delta t=13d$. The last column contains the corresponding Benjamini–Hochberg adjusted p-values. Some p-values assume $0.000$ after rounding; thus, $p<0.0005$ in such cases.

Delay | Corr | p | p_adj |
---|---|---|---|

0 | 0.965 | 0.000 | 0.000 |

1 | 0.969 | 0.000 | 0.000 |

2 | 0.972 | 0.000 | 0.000 |

3 | 0.976 | 0.000 | 0.000 |

4 | 0.979 | 0.000 | 0.000 |

5 | 0.982 | 0.000 | 0.000 |

6 | 0.984 | 0.000 | 0.000 |

7 | 0.986 | 0.000 | 0.000 |

8 | 0.989 | 0.000 | 0.000 |

9 | 0.990 | 0.007 | 0.112 |

10 | 0.992 | 0.085 | 1.000 |

11 | 0.993 | 0.416 | 1.000 |

12 | 0.993 | 0.839 | 1.000 |

13 | 0.993 | 1.000 | 1.000 |

14 | 0.993 | 0.855 | 1.000 |

15 | 0.993 | 0.506 | 1.000 |

**Table 2.**Comparison of correlation coefficients for the incidence data: column 2 contains the estimated correlation coefficients of the two time series $cases(t-\Delta t)$ and $deaths\left(t\right)$ with the corresponding delays $\Delta t$ in days listed in the first column. The p-values in the third column refer to a test for difference of any given correlation coefficient with the maximum correlation coefficient, in this case, the one estimated for delay $\Delta t=13d$. The last column contains the corresponding Benjamini–Hochberg adjusted p-values. Some p-values assume $0.000$ after rounding; thus, $p<0.0005$ in such cases.

Delay | Corr | p | p_adj |
---|---|---|---|

0 | 0.711 | 0.065 | 0.087 |

1 | 0.588 | 0.000 | 0.000 |

2 | 0.526 | 0.000 | 0.000 |

3 | 0.478 | 0.000 | 0.000 |

4 | 0.525 | 0.000 | 0.000 |

5 | 0.659 | 0.002 | 0.004 |

6 | 0.735 | 0.234 | 0.288 |

7 | 0.743 | 0.335 | 0.383 |

8 | 0.666 | 0.003 | 0.005 |

9 | 0.571 | 0.000 | 0.000 |

10 | 0.545 | 0.000 | 0.000 |

11 | 0.571 | 0.000 | 0.000 |

12 | 0.704 | 0.045 | 0.065 |

13 | 0.775 | 1.000 | 1.000 |

14 | 0.768 | 0.826 | 0.881 |

15 | 0.693 | 0.023 | 0.037 |

**Table 3.**Linear regression Instantaneous Fatality–Case Ratio (IFCR) by cases × time. The standard errors of the estimates are in parentheses.

IT | DE | |
---|---|---|

(Intercept) | 0.158 ${}^{***}$ | 0.054 ${}^{***}$ |

$\left(0.011\right)$ | $\left(0.014\right)$ | |

1000 cases | $-0.009{}^{\phantom{\rule{0.166667em}{0ex}}***}$ | $-0.005$ |

$\left(0.003\right)$ | $\left(0.004\right)$ | |

time (months) | $-0.020{}^{\phantom{\rule{0.166667em}{0ex}}***}$ | $-0.002$ |

$\left(0.002\right)$ | $\left(0.002\right)$ | |

cases:time | $0.001{}^{\phantom{\rule{0.166667em}{0ex}}***}$ | $0.001$ |

$\left(0.000\right)$ | $\left(0.000\right)$ | |

R${}^{2}$ | $0.329$ | $0.011$ |

Adj. R${}^{2}$ | $0.322$ | $0.001$ |

Num. obs. | 303 | 307 |

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**MDPI and ACS Style**

Diebner, H.H.; Timmesfeld, N. Exploring COVID-19 Daily Records of Diagnosed Cases and Fatalities Based on Simple Nonparametric Methods. *Infect. Dis. Rep.* **2021**, *13*, 302-328.
https://doi.org/10.3390/idr13020031

**AMA Style**

Diebner HH, Timmesfeld N. Exploring COVID-19 Daily Records of Diagnosed Cases and Fatalities Based on Simple Nonparametric Methods. *Infectious Disease Reports*. 2021; 13(2):302-328.
https://doi.org/10.3390/idr13020031

**Chicago/Turabian Style**

Diebner, Hans H., and Nina Timmesfeld. 2021. "Exploring COVID-19 Daily Records of Diagnosed Cases and Fatalities Based on Simple Nonparametric Methods" *Infectious Disease Reports* 13, no. 2: 302-328.
https://doi.org/10.3390/idr13020031