# Analysis of the Epidemic Curve of the Waves of COVID-19 Using Integration of Functions and Neural Networks in Peru

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Materials and Methods

#### 3.1. Dataset

_{1}was indexed with consecutive numbers (the first wave with 1, the second with 2, the third with 3, and so on), and the second variable X

_{2}was indexed with powers of 2 (the first wave with 1, the second with 2, the third with 4, …, the nth with 2

^{n−1}).

#### 3.2. Inflection Points

^{2}+ C·x

^{3}(using the R statistical software, version 4.3.3). It was an elementary function within the set of functions that had at least one inflection point. In addition, the second derivative was easy to obtain ${\mathrm{P}}^{\u2033}(\mathrm{x})=4\xb7B+6\xb7C\xb7\mathrm{x}$. There were other functions, for example, $\mathrm{P}\left(\mathrm{x}\right)=\mathrm{x}/(1-{\mathrm{x}}^{2}),$ with one inflection point and $\mathrm{P}(\mathrm{x})=x/(\sqrt[3]{1-{\mathrm{x}}^{2}})$ with three inflection points. Finally, ${P}^{\u2033}({x}_{0})=0$ was solved, and the solution was the inflection point ${x}_{0}=-b/(3c)$ (since it satisfied ${P}^{\u2034}({x}_{0})\ne 0$).

#### 3.3. The Sigmoidal–Boltzmann Function

**e**is Euler’s number.

#### 3.4. The Program

Algorithm 1. Algorithm for modeling COVID-19 | |

1 | Preprocessing: Load data, prepare data structures, scale data, and configure constants and variables. |

2 | Calculation of model parameters: Translate coordinates and estimate parameters (with an auxiliary function). |

3 | Construction of the Boltzmann function. Build the integrated function (with a loop according to the number of waves). |

4 | Results: Report the model parameters and the correlation coefficient. Additionally, it can be exported in LaTeX format. |

5 | Graphics: Set graphics plot style (e.g., line and color) and save them to a JPG file format. |

## 4. Results

#### 4.1. Integration of Functions

#### 4.1.1. The Mathematical Version

_{1}(x), H

_{2}(x), …, H

_{n}(x) be functions for n consecutive waves, which are combined into a single function and activated by the following Dp coefficient: F(x) = D

_{1}× H

_{1}(x) + D

_{2}× H

_{2}(x) + … + D

_{n}× H

_{n}(x), adding feature q to the training data (for example, consecutive numbers) [14], the coefficients are presented in Table 1.

_{w}(x) is the sigmoidal function, and x is the day number since the first case. Specifically, for $n=5$ waves, the coefficients are as follows:

#### 4.1.2. The Computational Version

## 5. Results: The Case Study

#### 5.1. Modeling the Number of Deaths with the Sigmoidal–Boltzmann Model

^{2}= 0.9998, which are acceptable measurements. The correlation was not equal to zero.

#### 5.2. Comparison with Classic Models

^{2}= 0.9958928, respectively. Another drawback is that it required ten segments (or nine inflection points), whereas the Boltzmann model only required five segments (specifically, four horizontal inflection points). Finally, the model had no epidemic parameters.

#### 5.3. Correlation between Isolation Measures and the Mortality Rate

#### 5.3.1. Statistical Results

#### 5.3.2. Discussion

#### 5.4. Modeling the Number of Deaths with the ANN

^{2}).

#### The Procedure and Architecture of the ANN

#### 5.5. Comparison

**the artificial neural network model**had a slightly better Pearson correlation coefficient. It did not require the data to follow an epidemic pattern. In general, it was also useful when the behavior of the data series was unknown.

**the sigmoidal–Boltzmann model**was useful to form an explanatory model. Statistical inference could be used, and it could even be used for prediction (with the confidence interval). It also provided with pandemic parameters. The Pearson’s product–moment correlation was $\delta =0.9999$. The procedure is deterministic, i.e., each execution always reports the same result (e.g., parameters, p-value, t-Student, degrees of freedom, amongst others). The drawback is that if the data do not follow a sigmoidal pattern, then the model might not fit. In this case, it is replaced with another function depending on the pattern of the data, and classical models (for example, exponential, logarithmic, square root, and polynomial), advanced models (for example, logistics, Poisson, Negative Binomial, Dirichlet), time series, and others can be used. However, in practically all countries, COVID-19 registered sigmoidal behavior, which represents a future avenue of study.

#### 5.6. Limitations

#### 5.7. Future Research Directions

## 6. Conclusions

**the ANN model**had a slightly better Pearson correlation coefficient. Moreover, it did not require the data to follow an epidemic pattern. However, the model required much more time to perform the calculation than the sigmoidal–Boltzmann model.

**the sigmoidal–Boltzmann model**may be useful to form an explanatory model; specifically, the formula provides pandemic parameters. Statistical inference can be used, and it could even be used for predictions. The drawback is that if the data do not follow a sigmoidal pattern, then the model might not fit; however, there are other functions depending on the pattern of the data, and classical models (for example, exponential, logarithmic, square root, and polynomial), advanced models (for example, logistics, Poisson, Negative Binomial, Dirichlet), and time series can be used.

**The proposed method**can be useful for other pandemics and for many general applications that involve certain patterns in the data. The procedure and program only require a change in the type of function.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AI | Artificial intelligence |

ANN | Artificial neural network |

COVID-19 | Coronavirus disease |

CQ | Chloroquine |

HCQ | Hydroxychloroquine |

IMF | International Monetary Fund |

INEI | Instituto Nacional de Estadística e Informática |

JHU | Johns Hopkins University |

LSTM | Long Short-Term Memory |

MINSA | Ministerio de Salud |

ML | Machine learning |

NN | Neural network |

UNA | Universidad Nacional del Altiplano |

WHO | World Health Organization |

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**Figure 1.**The cumulative number of deaths (in thousands) with respect to the day number (day one is the first death). The observed data (or dataset) are in gray (day 740 to 874), the regression function is in solid red, and the third inflection point is the blue diamond.

**Figure 2.**The sigmoidal–Boltzmann model for the cumulative number of confirmed deaths (in thousands) per day (2020–2023). The observed data in grey, the Boltzmann1 function for the initial wave in solid green, the Boltzmann2 function for the second wave (in dashed red), the Boltzmann3 function for the third wave (in solid brown), the Boltzmann4 function for the fourth wave (in dashed black), and the Boltzmann5 function for the fifth wave (in solid orange). The blue diamonds indicate the beginning and end of the waves.

**Figure 3.**The sigmoidal–Boltzmann model for the cumulative number of confirmed deaths (in thousands) according to days. The fourth and fifth waves are on the left and right sides, respectively. The observed data are in grey; the Boltzmann function for the fourth and fifth waves is in solid red.

**Figure 4.**The spline model for the cumulative number of confirmed deaths (in thousands) per day (2020–2023). The observed data are in black, and the spline function is in red. The vertical blue lines are the projected inflection points.

**Figure 5.**Comparison between the death rate function F′(x,p) (in blue) and days with social isolation in shaded bands (yellow and orange).

**Figure 6.**Dataset of the cumulative number of deaths (scaled and in red) according to days and those estimated by the ANN (blue)—Peru (2020–2023). Train data in green and test data in blue.

W | H_{1}(x) | H_{2}(x) | H_{3}(x) | H_{4}(x) | H_{5}(x) |
---|---|---|---|---|---|

1 | 1 − q | 1 − q | 1 − q | 1 − q | |

2 | 2 − q | 2 − q | 2 − q | 2 − q | |

3 | 3 − q | 3 − q | 3 − q | 3 − q | |

4 | 4 − q | 4 − q | 4 − q | 4 − q | |

5 | 5 − q | 5 − q | 5 − q | 5 − q |

Models | Model Adjustment to COVID-19 Data |
---|---|

Exponential, logarithmic, square root model. | The functions are not suitable because they have no inflection points and represent a single curve. Note: the observed data represent a sequence of sigmoidal shapes. |

Polynomial model
$$F(x)={\beta}_{0}+{\beta}_{1}\xb7x+{\beta}_{2}\xb7{x}^{2}+\dots +{\beta}_{\mathrm{n}}\xb7{x}^{\mathrm{n}}$$
| The functions are continuous and can have several inflection points; however, they cannot represent a step sequence of sigmoidal functions. Moreover, they do not flatten at the end and do not have horizontal asymptotes; when “x” tends to infinity, “y” tends to ±∞. |

Spline model (segmental polynomial fit). | Because in the spline model, the data on the abscissa axis are divided into segments, the problem of the step sequence of sigmoidal functions is solved. However, this model is limited to the polynomial functions that the spline uses in each segment. |

Group | Supreme Decree (Law) | From | To |
---|---|---|---|

Group 1. Supreme Decree No. 044-2020 and its extensions. It is characterized by strict social isolation measures. | N° 044-2020-PCM | 16 March 2020 | 30 March 2020 |

N° 051-2020-PCM | 31 March 2020 | 12 April 2020 | |

N° 064-2020-PCM | 13 April 2020 | 26 April 2020 | |

N° 075-2020-PCM | 27 April 2020 | 10 May 2020 | |

N° 083-2020-PCM | 11 May 2020 | 24 May 2020 | |

N° 094-2020-PCM | 25 May 2020 | 30 June 2020 | |

N° 116-2020-PCM | 1 July 2020 | 31 July 2020 | |

N° 135-2020-PCM | 1 August 2020 | 31 August 2020 | |

N° 146-2020-PCM | 1 September 2020 | 30 September 2020 | |

N° 156-2020-PCM | 1 October 2020 | 31 October 2020 | |

N° 174-2020-PCM | 1 November 2020 | 30 November 2020 | |

Group 2. Supreme Decree No. 174-2020 and its extensions allow some activities outside the home. | N° 184-2020-PCM | 1 December 2020 | 31 December 2020 |

N° 201-2020-PCM | 1 January 2021 | 31 January 2021 | |

N° 008-2021-PCM | 1 February 2021 | 28 February 2021 | |

N° 036-2021-PCM | 1 March 2021 | 31 March 2021 | |

N° 058-2021-PCM | 1 April 2021 | 30 April 2021 | |

N° 076-2021-PCM | 1 May 2021 | 31 May 2021 |

Comparison Criterion | Sigmoidal Model | ANN Model |
---|---|---|

Pearson correlation coefficient | Fixed value | Can be improved (e.g., with more training) |

Execution time (according to reference computer) | Less than a third of a second | More than 2 min depending on parameters |

Requires data to follow a sigmoidal pattern? | Yes | No |

Statistical report | From a regression (e.g., p-value, t-Student, degrees of freedom, etc.) | It has its own metrics (e.g., accuracy) |

Pandemic parameters | Yes | No |

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## Share and Cite

**MDPI and ACS Style**

Vilca Huayta, O.A.; Jimenez Chura, A.C.; Sosa Maydana, C.B.; Martínez García, A.J.
Analysis of the Epidemic Curve of the Waves of COVID-19 Using Integration of Functions and Neural Networks in Peru. *Informatics* **2024**, *11*, 40.
https://doi.org/10.3390/informatics11020040

**AMA Style**

Vilca Huayta OA, Jimenez Chura AC, Sosa Maydana CB, Martínez García AJ.
Analysis of the Epidemic Curve of the Waves of COVID-19 Using Integration of Functions and Neural Networks in Peru. *Informatics*. 2024; 11(2):40.
https://doi.org/10.3390/informatics11020040

**Chicago/Turabian Style**

Vilca Huayta, Oliver Amadeo, Adolfo Carlos Jimenez Chura, Carlos Boris Sosa Maydana, and Alioska Jessica Martínez García.
2024. "Analysis of the Epidemic Curve of the Waves of COVID-19 Using Integration of Functions and Neural Networks in Peru" *Informatics* 11, no. 2: 40.
https://doi.org/10.3390/informatics11020040