# A Parsimonious Description and Cross-Country Analysis of COVID-19 Epidemic Curves

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data

#### 2.2. The Gompertz Model

#### 2.3. The Evolution of the Reproduction Number

#### 2.4. Fitting Procedure

## 3. Results

#### 3.1. Epidemic Curves of Sweden and Norway

#### 3.2. Analysis of Data from 73 Countries

## 4. Discussion

#### 4.1. How to Interpret the Gompertz Model

#### 4.2. How to Interpret Figure 6 and Figure 7

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. From the SIR Model to the Gompertz Model

#### Appendix A.1. Nonlinear Evolution of an “Old” Disease

#### Appendix A.2. Evolution of an Epidemic Due to a “New” Pathogen

#### Appendix A.3. The Reproductive Number

#### Appendix A.4. Reduction to a Generalized Logistic Growth Model

#### Appendix A.5. Richards’ Growth Model

**Figure A1.**(

**a**): Normalized growth rate $\gamma /{\gamma}^{\left(0\right)}$ as a function of $J/{J}_{\infty}$ for varying $\nu $. Light blue; $\nu =1/8$. Brown; $\nu =1/4$. Magenta; $\nu =1/2$. Red; $\nu =1$. Green; $\nu =2$. Orange; $n=4$. Dark blue; $\nu =8$. For $\nu =1$ we have the symmetric logistic growth curve [11]. Increasing $\nu >1$ shifts the inflection point ${J}_{s}$ closer to the limiting value ${J}_{\infty}$. Decreasing $\nu <1$ shifts it towards ${J}_{0}$. (

**b**): Normalized Richards growth curve $J/{J}_{\infty}$ as a function of time t for varying $\nu $ with same color coding as in (

**a**).

#### Appendix A.6. The Gompertz Limit

## Appendix B. Gompertz fits to 73 Countries

**Figure A2.**Shows cumulative COVID19 deaths (black) and the estimated Gompertz curves for different countries (red).

**Figure A3.**Shows cumulative COVID19 deaths (black) and the estimated Gompertz curves for different countries (red).

**Figure A4.**Shows cumulative COVID19 deaths (black) and the estimated Gompertz curves for different countries (red).

**Figure A5.**Shows cumulative COVID19 deaths (black) and the estimated Gompertz curves for different countries (red).

**Figure A6.**Shows cumulative COVID19 deaths (black) and the estimated Gompertz curves for different countries (red).

**Figure A7.**Shows cumulative COVID19 deaths (black) and the estimated Gompertz curves for different countries (red).

## References

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**Figure 1.**(

**a**): Evolution of cumulative deaths per million $J\left(t\right)$ in a log-plot expressed as a Gompertz function. (

**b**): The evolution of the daily death number ${d}_{t}J\left(t\right)$ in a log-plot with the tangents at $t=0$ and $t=\infty $ marked. The slopes of these tangents are ${\Gamma}_{0}$ and ${\gamma}_{\infty}$, respectively.

**Figure 2.**(

**a**): Red bullets represent cumulative COVID-19 related deaths per million inhabitants in Norway from March 23 and onwards, and the full curve is the Gompertz curve $J\left(t\right)$ fitted to these data. The blue bullets and curve are the corresponding for Sweden (shifted 3 days forwards). Note that the plot is logarithmic, and that cumulated death toll per million in Sweden in early July is 12 times that in Norway. (

**b**): The relative growth rate $\gamma \left(t\right)={d}_{t}(lnJ)$ as given by Equation (6) for Norway (red) and Sweden (blue). These growth rates are the slopes of the curves in (

**a**).

**Figure 3.**(

**a**): Evolution of daily deaths per million, $dJ/dt$, computed numerically from Equation (12) for Norway (red) and for Sweden (blue). (

**b**): The relative growth rate $\Gamma \left(t\right)={d}_{t}(ln{d}_{t}J)$ for Norway (red) and for Sweden (blue).

**Figure 4.**(

**a**): Evolution of $\mathcal{R}\left(t\right)$ computed numerically from Equation (12) for Norway (red) and for Sweden (blue). (

**b**): Evolution of ratio between accumulated deaths in Sweden and Norway.

**Figure 5.**Density plot of ${J}_{\infty}({\Gamma}_{1},{\gamma}_{\infty})$ with some isolines marked, along with the points for Sweden and Norway. The crosses mark the end states for paths where the inital and final phase of the two countries are mixed.

**Figure 6.**Density plot of ${J}_{\infty}({\Gamma}_{1},{\gamma}_{\infty})$ with some isolines marked, along with the points for 73 countries. The legend shows the positions for some selected countries.

**Figure 7.**(

**a**): Estimated ${\Gamma}_{1}$ for the 73 countries ranked from lower to higher values. (

**b**): Estimated ${\gamma}_{\infty}$ ranked from high to low values. (

**c**): Estimated ${\Gamma}_{1}/{\Gamma}_{\infty}=ln{J}_{\infty}$ ranked from lower to higher values.

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**MDPI and ACS Style**

Rypdal, K.; Rypdal, M.
A Parsimonious Description and Cross-Country Analysis of COVID-19 Epidemic Curves. *Int. J. Environ. Res. Public Health* **2020**, *17*, 6487.
https://doi.org/10.3390/ijerph17186487

**AMA Style**

Rypdal K, Rypdal M.
A Parsimonious Description and Cross-Country Analysis of COVID-19 Epidemic Curves. *International Journal of Environmental Research and Public Health*. 2020; 17(18):6487.
https://doi.org/10.3390/ijerph17186487

**Chicago/Turabian Style**

Rypdal, Kristoffer, and Martin Rypdal.
2020. "A Parsimonious Description and Cross-Country Analysis of COVID-19 Epidemic Curves" *International Journal of Environmental Research and Public Health* 17, no. 18: 6487.
https://doi.org/10.3390/ijerph17186487