# A Phenomenological Epidemic Model Based On the Spatio-Temporal Evolution of a Gaussian Probability Density Function

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. A New Phenomenological Epidemic Model

#### 3.1. The Partial Differential Equation for a Random Epidemic Variable

#### 3.2. A Gaussian Analytical Solution for the PDE

## 4. Numerical Model

## 5. Numerical Experiments

- Obtain real data in the time series of an epidemic outbreak.
- Select the model and provide values for initial parameters, in addition to the lower and upper bounds for final parameters.
- Estimate model parameters and their confidence intervals.
- Quantify the error of the model fit to real data.
- Compare the quality of the fits and the errors yielded by the models across all of the epidemics.

#### 5.1. Epidemic Datasets

#### 5.2. Estimation of the Model Parameters

#### 5.3. Errors of the Model Fits

## 6. Results

#### 6.1. Parameter Estimates with Quantified Uncertainty

#### 6.2. RMSE Errors

#### 6.3. Residuals

#### 6.4. Computational Load

#### 6.5. Forecasts

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EGM | Exponential-Growth Model |

GES | Gaussian Epidemic Solution |

GGM | Generalized-Growth Model |

GMM | Gaussian Mixture Model |

GRM | Generalized Richards Model |

LGM | Logistic Growth Model |

ORM | Original Richards Model |

PDE | partial differential equation |

RMSE | root mean square error |

## Appendix A. Derivation of the PDE for a Random Epidemic Variable

## References

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**Figure 1.**Example of numerical approximation of a simulated epidemic variable, ${v}^{\prime}$, using Equation (15) and ${N}_{v}=3,{C}_{v}^{\prime}=1.17\phantom{\rule{4pt}{0ex}}{10}^{4},{\mu}_{v}\left({x}_{i}\right)=\{40,50,80\}$, ${\sigma}_{v}\left({x}_{i}\right)=\{8,9,10\},\phantom{\rule{0.166667em}{0ex}}i=1,2,3$. Each Gaussian function is identified by the legend $\u2018\u2018i=a",\phantom{\rule{0.166667em}{0ex}}a\in \{1,2,3\}$.

**Figure 6.**Results of the Gaussian model fit to the other three COVID-19 epidemic datasets: deaths (C19DeSp), admitted-to-the-ICU (C19ICUSp), and hospital discharge (C19HDSp).

**Figure 7.**Forecast-1 for the C19InSp COVID-19 dataset using ${N}_{t}=28$ data for calibrating all phenomenological models, from 4 March to 31 March 2020. The forecasting period is ${N}_{h}=50$, from 1 April to 20 May 2020. The solid red line that represents long-term forecasts was extended beyond the last time horizon.

**Figure 8.**Forecast-2 for the C19InSp COVID-19 dataset using ${N}_{t}=38$ data for calibrating all phenomenological models, from 4 March to 10 April 2020. The forecasting period is ${N}_{h}=40$, from 11 April to 20 May 2020. The solid red line that represents long-term forecasts was extended beyond the last time horizon.

**Figure 9.**Forecasts-3 for the C19InSp COVID-19 dataset using ${N}_{t}=58$ data for calibrating all phenomenological models, from 4 March to 30 April 2020. The forecasting period is ${N}_{h}=20$, from 1 May to 20 May 2020. The solid red line that represents long-term forecasts was extended beyond the last time horizon.

**Table 1.**Six real datasets used in numerical experiments. The information for each epidemic time-series data includes the name of the associated disease, the location where the outbreak occurred, temporal resolution (days, weeks), type of individual (infected, dead), date range, number of data points, and data source. ICU is Intensive Care Unit.

Dataset ID | Disease | Outbreak | Individuals | Resolution | Dates | Total Data | Source |
---|---|---|---|---|---|---|---|

C19InSp | COVID-19 | Spain | infected | day | 4/3/20–20/5/20 | 78 | [22] |

C19DeSp | COVID-19 | Spain | dead | day | 20/2/20–20/5/20 | 91 | [22] |

EboInSL | Ebola | Sierra Leone | infected | week | 65 weeks, 2014–2016 | 65 | [4] |

ZikInCo | Zika | Colombia | infected | day | 27/12/15–8/4/16 | 104 | [14] |

C19ICUSp | COVID-19 | Spain | admitted-to-the-ICU | day | 8/3/20–23/5/20 | 77 | [22] |

C19HDSp | COVID-19 | Spain | hospital-discharge | day | 8/3/20–17/5/20 | 71 | [22] |

Dataset ID | Data Fitting Method | Error Structure | Variance/Mean | Number of Model Realizations |
---|---|---|---|---|

C19InSp | curve_fit | NegativeBinomial | 400 | 200 |

EboInSL | curve_fit | NegativeBinomial | 20 | 200 |

ZikInCo | curve_fit | NegativeBinomial | 5 | 200 |

C19DeSp | curve_fit | NegativeBinomial | 40 | 200 |

C19ICUSp | curve_fit | NegativeBinomial | 10 | 200 |

C19HDSp | curve_fit | NegativeBinomial | 80 | 200 |

Dataset ID | Parameter Median | Parameter 95% Confidence Interval | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Gaussian Model | LGM | Gaussian Model | LGM | |||||||||||

${\mathbf{N}}_{\mathbf{v}}$ | ${\mathbf{C}}_{\mathbf{v}}^{\prime}$ | ${\mathbf{\mu}}_{\mathbf{v}}$ | ${\mathbf{\sigma}}_{\mathbf{v}}$ | r | K | ${\mathbf{C}}_{\mathbf{v}}^{\prime}$ | ${\mathbf{\mu}}_{\mathbf{v}}$ | ${\mathbf{\sigma}}_{\mathbf{v}}$ | r | K | ||||

C19InSp | 16 | $1.9\xb7{10}^{4}$ | 34.4 | 1.9 | $8.8\xb7{10}^{-2}$ | $3.0\xb7{10}^{5}$ | $1.7\xb7{10}^{4}$–$2.4\xb7{10}^{4}$ | $10.8$–$2.1\xb7{10}^{5}$ | $0.2$–$1.7\xb7{10}^{4}$ | $7.8\xb7{10}^{-2}$–$9.9\xb7{10}^{-2}$ | $2.7\xb7{10}^{5}$–$3.3\xb7{10}^{5}$ | |||

EboInSL | 6 | $1.9\xb7{10}^{3}$ | 25.3 | 2.2 | $0.21$ | $1.2\xb7{10}^{4}$ | $1.8\xb7{10}^{3}$–$2.1\xb7{10}^{4}$ | $11.3$–$38.0$ | $1.2$–$12.1$ | $0.19$–$0.22$ | $1.1\xb7{10}^{4}$–$1.3\xb7{10}^{4}$ | |||

ZikInCo | 14 | $2.4\xb7{10}^{2}$ | 72.1 | 11.1 | $7.7\xb7{10}^{-2}$ | $1.9\xb7{10}^{3}$ | $1.9\xb7{10}^{2}$–$3.2\xb7{10}^{2}$ | $3.0$–$1.0\xb7{10}^{3}$ | $3.0$–$1.0\phantom{\rule{4pt}{0ex}}{10}^{3}$ | $6.7\xb7{10}^{-2}$–$8.8\xb7{10}^{-2}$ | $1.7\xb7{10}^{3}$–$2.1\xb7{10}^{3}$ | |||

C19DeSp | 14 | $2.1\xb7{10}^{3}$ | 47.9 | 1.7 | - | - | $1.8\xb7{10}^{3}$–$2.3\xb7{10}^{3}$ | $28.8$–$90.0$ | $0.6$–$14.4$ | - | - | |||

C19ICUSp | 8 | $1.5\xb7{10}^{3}$ | 20.7 | 4.1 | - | - | $1.4\xb7{10}^{3}$–$1.9\xb7{10}^{3}$ | $0.0$–$94.4$ | $1.2$–$37.6$ | - | - | |||

C19HDSp | 23 | $7.0\xb7{10}^{3}$ | 42.4 | 1.2 | - | - | $6.4\xb7{10}^{3}$–$1.1\xb7{10}^{4}$ | $14.8$–$1.4\xb7{10}^{2}$ | $0.4$–$25.1$ | - | - | |||

Dataset ID | Parameter Median | Parameter 95% Confidence Interval | ||||||||||||

ORM | GRM | ORM | GRM | |||||||||||

r | K | a | r | K | a | p | r | K | a | r | K | a | p | |

C19InSp | 213.6 | $2.9\xb7{10}^{5}$ | $3.1\xb7{10}^{-4}$ | 51.8 | $29.7\xb7{10}^{5}$ | $4.6\xb7{10}^{-3}$ | $0.89$ | $0.3$–280 | $2.6\xb7{10}^{5}$–$3.0\xb7{10}^{5}$ | $2.4\xb7{10}^{-4}$–$0.29$ | $19.6$–$60.6$ | $2.7\xb7{10}^{5}$–$3.2\xb7{10}^{5}$ | $4.3\xb7{10}^{-3}$–$6\xb7{10}^{-3}$ | $0.82$–$0.91$ |

EboInSL | 0.21 | $1.2\xb7{10}^{4}$ | $0.98$ | 0.95 | $1.2\xb7{10}^{4}$ | $0.98$ | $0.82$ | $0.15$–$0.34$ | $1.1\xb7{10}^{4}$–$1.3\xb7{10}^{4}$ | $0.5$–$1.4$ | $0.61$–$1.22$ | $1.1\xb7{10}^{4}$–$1.3\xb7{10}^{4}$ | $0.25$–$0.99$ | $0.78$–$1.0$ |

ZikInCo | 0.23 | $1.9\xb7{10}^{3}$ | $0.28$ | 0.72 | $1.9\xb7{10}^{3}$ | $0.66$ | $0.74$ | $8.4\xb7{10}^{-2}$–$1.5\xb7{10}^{2}$ | $1.7\xb7{10}^{3}$–$2.1\xb7{10}^{3}$ | $3.3\xb7{10}^{-4}$–$0.93$ | $0.46$–$1.81$ | $1.7\xb7{10}^{3}$–$2.1\xb7{10}^{3}$ | $3.6\xb7{10}^{-3}$–$1.0$ | $0.62$–$1.0$ |

**Table 4.**Median and 95% confidence interval of the RMSE errors (Equation (16)) of the best model fits to the real datasets.

Dataset ID | Median RMSE | RMSE 95% Confidence Interval | ||||||
---|---|---|---|---|---|---|---|---|

Gaussian | LGM | ORM | GRM | Gaussian | LGM | ORM | GRM | |

C19InSp | 783.74 | 1151.6 | 1161.12 | 1777.9 | 610.5–1021.9 | 922.0–1379.2 | 942.4–1443.2 | 1201.6–2691.3 |

EboInSL | 44.67 | 56.11 | 54.32 | 68.32 | 33.41–61.13 | 42.2–76.2 | 42.3–67.4 | 44.0–110.4 |

ZikInCo | 8.37 | 9.06 | 9.13 | 15.61 | 7.0–10.2 | 7.2–11.2 | 7.6–11.1 | 10.2–22.8 |

C19DeSp | 79.05 | - | - | - | 58.1–104.3 | - | - | - |

C19ICUSp | 29.84 | - | - | - | 24.5–35.6 | - | - | - |

C19HDSp | 180.16 | - | - | - | 84.7–311.7 | - | - | - |

**Table 5.**Mean and standard deviation of the residuals (Equation (17)) obtained for the best fits of the epidemic models that are compared in this work. Additionally, the execution times needed by all model fittings to data are included. Time was measured in seconds (s).

Dataset ID | Mean of Residuals | Standard Deviation of Residuals | Execution Time [s] | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Gaussian | LGM | ORM | GRM | Gaussian | LGM | ORM | GRM | Gaussian | LGM | ORM | GRM | |

C19InSp | 66.18 | 57.53 | 68.5 | 32.18 | 372.39 | 1349.44 | 1048.28 | 981.80 | 762 | 5 | 42 | 1472 |

EboInSL | 5.32 | 5.50 | 6.56 | 0.31 | 27.81 | 40.0 | 39.93 | 51.54 | 255 | 3 | 36 | 270 |

ZikInCo | 0.20 | 0.55 | 0.36 | 0.16 | 4.60 | 5.68 | 5.39 | 5.55 | 3945 | 5 | 48 | 858 |

C19DeSp | 11.36 | - | - | - | 36.95 | - | - | - | 376 | - | - | - |

C19ICUSp | 0.82 | - | - | - | 42.35 | - | - | - | 395 | - | - | - |

C19HDSp | 33.43 | - | - | - | 203.45 | - | - | - | 5777 | - | - | - |

**Table 6.**Model performance measures for the three forecasting experiments shown in Figure 7, Figure 8 and Figure 9. These experiments are called Forecasts-1,2,3, respectively. The first set of three lines of the table show the root mean squared errors provided by four phenomenological models (Gaussian, LGM, ORM, GRM) for the calibration and forecasting periods. The second set of three lines show the 95% confidence intervals during the calibration and forecasting periods. The third set of three lines shows the mean and standard deviation of residuals for the forecasting intervals.

Forecast ID | Median RMSE of Calibration Interval | RMSE 95% Confidence Interval of Calibration Interval | ||||||
---|---|---|---|---|---|---|---|---|

Gaussian | LGM | ORM | GRM | Gaussian | LGM | ORM | GRM | |

Forecast-1 | 1072.9 | 1047.6 | 1062.1 | 1094.9 | 704.0–1634.6 | 698.5–1570.4 | 688.0–1550.2 | 627.0–1766.4 |

Forecast-2 | 1214.4 | 1240.2 | 1221.9 | 1368.2 | 914.7–1639.8 | 946.1–1655.6 | 884.9–1639.3 | 907.6–2331.6 |

Forecast-3 | 1195.3 | 1234.2 | 1257.4 | 1672.2 | 958.4–1492.1 | 963.3–1509.9 | 977.0–1548.5 | 1088.5–2590.0 |

Forecast ID | Median RMSE of Forecasting Interval | RMSE 95% Confidence Interval of Forecasting Interval | ||||||

Gaussian | LGM | ORM | GRM | Gaussian | LGM | ORM | GRM | |

Forecast-1 | 1039.9 | 2977.7 | 1838.1 | 1576.5 | 796.8–1986.5 | 2391.0–3372.7 | 1259.6–3396.7 | 1152.3–5073.0 |

Forecast-2 | 983.1 | 2533.8 | 2013.1 | 1475.8 | 740.7–1239.1 | 2268.1–2768.7 | 1403.5–2668.2 | 955.4–1943.4 |

Forecast-3 | 793.1 | 1478.8 | 1143.4 | 1067.7 | 717.2–1173.0 | 1257.1–1660.3 | 920.0–1448.5 | 795.4–1399.8 |

Forecast ID | Mean of Residuals of Forecasting Interval | Standard Deviation of Residuals of Forecasting Interval | ||||||

Gaussian | LGM | ORM | GRM | Gaussian | LGM | ORM | GRM | |

Forecast-1 | 22.1 | 2977.7 | 1803.7 | -310.1 | 783.1 | 1651.1 | 1165.5 | 1270.7 |

Forecast-2 | 96.2 | 1196.5 | 977.8 | 597.7 | 783.7 | 1530.5 | 1222.1 | 1088.0 |

Forecast-3 | 68.6 | 334.4 | 231.8 | 273.6 | 760.6 | 1530.5 | 1048.5 | 987.4 |

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**MDPI and ACS Style**

Benítez, D.; Montero, G.; Rodríguez, E.; Greiner, D.; Oliver, A.; González, L.; Montenegro, R.
A Phenomenological Epidemic Model Based On the Spatio-Temporal Evolution of a Gaussian Probability Density Function. *Mathematics* **2020**, *8*, 2000.
https://doi.org/10.3390/math8112000

**AMA Style**

Benítez D, Montero G, Rodríguez E, Greiner D, Oliver A, González L, Montenegro R.
A Phenomenological Epidemic Model Based On the Spatio-Temporal Evolution of a Gaussian Probability Density Function. *Mathematics*. 2020; 8(11):2000.
https://doi.org/10.3390/math8112000

**Chicago/Turabian Style**

Benítez, Domingo, Gustavo Montero, Eduardo Rodríguez, David Greiner, Albert Oliver, Luis González, and Rafael Montenegro.
2020. "A Phenomenological Epidemic Model Based On the Spatio-Temporal Evolution of a Gaussian Probability Density Function" *Mathematics* 8, no. 11: 2000.
https://doi.org/10.3390/math8112000