# Entropy Ratio and Entropy Concentration Coefficient, with Application to the COVID-19 Pandemic

## Abstract

**:**

## 1. Introduction

## 2. Basic Concepts

**Proposition**

**1.**

- (i)
- $0\le U\le 1$ and $0\le C\le 1$
- (ii)
- $U=1$ and $C=0$ if and only if $P=Q.$
- (iii)
- $U=0$ and $C=1$ if and only if there is an i with ${p}_{i}=1.$
- (iv)
- The values of U and C do not depend on the base of the logarithm.

**Proof.**

## 3. Examples

## 4. Product Measures

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

## 5. Entropy Ratios for Processes and Fractal Measures

**Proposition**

**4.**

**Proposition**

**5.**

## 6. The Global Covid-19 Pandemic

## 7. States and Counties in the USA

## 8. States and Districts in Germany

## 9. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Gini Coefficient

**Proposition**

**A1.**

**Proof.**

## References

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**Figure 1.**Gini coefficient and entropy concentration coefficient as functions of $p={p}_{1}$ for fixed values of $q.$ Left: $q=0.5.$ Right: $q=0.8.$

**Figure 2.**Time series of the pandemic. The upper row describes the intensity of the pandemic, with cases on the left and deaths on the right. The lower row specifies the non-uniformity of the distribution of cases, with Gini coefficients on the left and entropy concentration coefficients on the right. Data from Johns Hopkins University [17], updated 10 November 2020.

**Figure 3.**

**Upper row**: cases and deaths for four representative states of the USA.

**Middle row**: entropy concentration coefficients for the partitions into 51 states and 3143 counties on logarithmic scale.

**Lower row**: Gini coefficients are very similar, but larger and not adapted to sample size. Data from Johns Hopkins University [17], 10 November 2020.

**Figure 4.**

**Upper left**: cases and deaths in Germany.

**Upper right**: entropy concentration coefficients for the 16 federal states and 412 districts.

**Lower row**: Gini coefficient and coefficient of variation are biased by sample size. Data from Robert Koch Institut [22], updated 10 Nov 2020.

**Figure 5.**

**Left**: case incidence with respect to age.

**Right**: entropy concentration coefficients of cases for age groups. These figures were updated in proof on 16 November since the most recent concentration coefficients already indicate improvement in part of the country while case incidences are still rising.

Parameter | Germany | without Berlin | East with Berlin | East without Berlin |
---|---|---|---|---|

D for states | 0.260 | 0.184 | 0.618 | 0.089 |

D for districts | 0.650 | 0.580 | 0.971 | 0.469 |

G for states | 0.336 | 0.309 | 0.390 | 0.227 |

G for districts | 0.528 | 0.509 | 0.539 | 0.414 |

C for states | 0.098 | 0.073 | 0.262 | 0.054 |

C for districts | 0.101 | 0.092 | 0.183 | 0.101 |

Parameter | World | Asia | Africa | Europe | America | without US, Ca |
---|---|---|---|---|---|---|

number of regions/countries | 19/159 | 5/47 | 5/49 | 4/35 | 4/25 | 3/23 |

G for regions | 0.575 | 0.379 | 0.291 | 0.282 | 0.441 | 0.012 |

G for countries | 0.621 | 0.496 | 0.412 | 0.340 | 0.471 | 0.190 |

C for regions | 0.218 | 0.210 | 0.092 | 0.102 | 0.382 | 0.0004 |

C for countries | 0.182 | 0.177 | 0.088 | 0.067 | 0.269 | 0.034 |

k | 1 | 2 | 3 | 4 | 5 | 6 | ∞ |
---|---|---|---|---|---|---|---|

${C}_{k}$ | 0.0406 | 0.2382 | 0.2588 | 0.2671 | 0.2739 | 0.2783 | 0.30576 |

${C}_{k}-{C}_{k-1}$ | 0.1977 | 0.0206 | 0.0083 | 0.0068 | 0.0044 |

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Bandt, C.
Entropy Ratio and Entropy Concentration Coefficient, with Application to the COVID-19 Pandemic. *Entropy* **2020**, *22*, 1315.
https://doi.org/10.3390/e22111315

**AMA Style**

Bandt C.
Entropy Ratio and Entropy Concentration Coefficient, with Application to the COVID-19 Pandemic. *Entropy*. 2020; 22(11):1315.
https://doi.org/10.3390/e22111315

**Chicago/Turabian Style**

Bandt, Christoph.
2020. "Entropy Ratio and Entropy Concentration Coefficient, with Application to the COVID-19 Pandemic" *Entropy* 22, no. 11: 1315.
https://doi.org/10.3390/e22111315