INTRODUCTION

The new coronavirus (severe acute respiratory syndrome coronavirus 2-SARS) was reported in December 2019 in Wuhan-China, with the appearance of several cases of pneumonia of unknown etiology which caused severe acute respiratory problems^1-7. This infection is transmitted by inhalation of respiratory droplets, close contact with the infected individual, and contact with contaminated surfaces or objects^1,8.

On March 11, 2020, the World Health Organization (WHO) declared COVID19 a pandemic⁹. Cases are increasing worldwide, including Peru, where the first case was announced on March 6; As of July 12, the total number of confirmed cases was 326,326¹⁰.

There is great concern about the Peruvian health system’s response capacity to effectively meet the needs of people with COVID-19 as the number of cases increases. Despite all the measures taken by the government, it does not stop. The outcome is worst now that the government “opened the doors” for the population to a “new normal.”

Mathematical models are used to understand critical epidemiological transitions^11-14, and in recent months, researchers have been using mathematical methods to forecast the number of COVID-19 cases worldwide. The Autoregressive Integrated Method of Moving Averages (ARIMA) is the one that has been used the most to make forecasts, such is the case of those proposed by Singh RK et al.¹¹, for the United States, Spain, Italy, France, Germany, the United Kingdom, Turkey, Iran, China, Russia, Brazil, Canada, Belgium, the Netherlands, and Switzerland; by Ceylan Z¹⁵, for Italy, Spain, and France; by Moftakhar L and Seif M¹, for Iran; by Benvenuto D et al.¹⁶, for Italy; by Yousaf M et al.¹⁷, for Pakistan; by Hiteshi Tandon¹⁸and Rishabh Tyagi et al.¹⁹, for India and by Perone G²⁰, for Italy

Time series constitute data collections in a period in which trends or patterns are observed to forecast some future values²¹. The ARIMA model has three main parameters; the parameter p: associated with the autoregressive process (AR), the number of differences that must be taken from the series to be stationary. The parameter d: associated with the integrated part (I) y; the parameter q: related to the part of the moving average (MA), which is the current value of a series, which is defined as a linear combination of past errors^11,12,18.

To estimate an ARIMA model, the Box-Jenkins methodology^11,22-23, is used, which consists of 4 stages. Identification: which consists of identifying the model that can tentatively be considered. Estimation: which consists of estimating the parameters of the model tentatively considered; validation: which consists in performing diagnostic tests to check if the model fits the data. Finally, prediction, which consists of obtaining forecasts in probabilistic terms of future values and the model’s predictive capacity, is evaluated⁽²⁴⁾. One of the concerns in Peru and any other country, is to know how many people will be infected with COVID-19 in time; and this could be answered with predictive models²⁵; Therefore, the present study aimed to estimate an Autoregressive Integrated Moving Average model (ARIMA) for the analysis of series of COVID-19 cases, in Peru, t seek an approximation between the results obtained with the model and observed data.

METHODS

RESULTS

Table 1shows the daily count of COVID 19 cases between March 6, 2020, and June 11, 2020. InFigure 1, it is shown that the autocorrelation coefficients (ACF) slowly decrease as linear, so they correspond to a non-stationary time series, this being corroborated by the Augmented Dickey-Fuller unit root contrast (p> 0.01). InFigure 2, the second-order stationary transformation is shown. The time series was stabilized by eliminating trends and obtaining the appropriate model with the orders p, d, q, which was ARIMA (0.2, 9).Table 2shows the parameters (p, d, q) obtained in the modeling process; the Normalized Bayesian Information Criterion (BIC) which indicates that 13,475 was the lowest value obtained indicating the best ARIMA model; the mean absolute percentage error (MAPE) that indicates that the forecast of the model is wrong by 7.775%; and the Ljung-Box test that suggests that the model is capable of reproducing the systematic behavior pattern of the time series. Figure 3andTable 3show the prognosis of the number of cases with 95% confidence intervals and the observed cases, from June 12 to July 11, 2020, with data from March 6, 2020, to June 11, 2020. The observed cases’ values are lower than the predicted cases’ values, with those observed within the minimum and maximum values of the estimates.

Figure 1. Autocorrelation correlogram (ACF) estimated for the time series 

Figure 2. Autocorrelation correlogram (ACF) estimated for the time series, with a second-order transformation 

Figure 3. Forecast of the number of COVID 19 cases, with 95% confidence intervals and observed cases of COVID 19; from June 12 to July 11, 2020 

DISCUSSION

This research was carried out to predict the number of cases with COVID-19 from June 12 to July 11, 2020, in Peru using the Autoregressive Integrated Method of Moving Averages (ARIMA), to later compare them with the cases observed in order to find an approximation between both results.

It must be taken into account that the forecasts obtained from a univariate time series model are extrapolations of the observed data until the time the series ends, being in many cases very effective providing a reference point to predict future values of confirmed cases. Most series are stochastic, in that their past values can partially determine the future, so accurate forecasts are impossible²⁴.

Various authors^1,11,15-20, worldwide, have developed models for the prognosis of COVID-19 cases based on the ARIMA method. They have used certain statistics to evaluate the goodness of fit of the model showing good results in short-term forecasts.

The modeling carried out in this study has followed all the stages of the Box-Jenkins methodology^11,22-23, to obtain the best predictive model. Although, some researchers have applied these stages, they have only mentioned some statistical parameters used in their construction.

Moftakhar L and Seif M¹, mention the use of the analysis of the autocorrelation coefficients (ACF), partial autocorrelation (PACF) and the Box-Ljung test; Singh RK et al¹¹, mention the use of the Akaike information criterion (AIC); Ceylan Z¹⁵, mentions the use of the mean square error (RMSE), mean absolute error (MAE) and mean absolute percentage error (MAPE); Benvenuto D¹⁶, mentions the use of the Augmented Dickey-Fuller unit root test (ADF) and the analysis of the autocorrelation coefficients (ACF) and partial autocorrelation (PACF); Yousaf M et al¹⁷, mention the use of the analysis of the autocorrelation coefficients (ACF) and partial autocorrelation (ACF and PACF) and the Akaike information criterion (AIC); Hiteshi Tandon¹⁸, mentions the use of the analysis of the autocorrelation coefficients (ACF) and partial autocorrelation (PACF) in addition to the variance, normality test, the mean absolute percentage error (MAPE), the Absolute Deviation of the Mean ( DAM) and mean squared deviation (MSD); Rishabh Tyagi et al¹⁹, mention the use of the analysis of the autocorrelation coefficients (ACF) and partial autocorrelation (PACF), the Augmented Dickey-Fuller unit root test (ADF), and provide the confidence intervals (CI ) 95% for point estimates; Finally, Perone G²⁰⁾ mentions the use of the Akaike information criterion (AIC), the analysis of the autocorrelation coefficients (ACF) and partial autocorrelation (PACF), the Augmented Dickey-Fuller unit root test (ADF ), the modified Elliott-Rothenberg-Stock (DF-GLS) unit root test, the mean absolute error (MAE), the normality test, the homoscedasticity, the Engle Lagrange multiplier test, and the Box test -Ljun, many of which coincide with those used in this investigation.

Although it is true that within the mathematical forecasting models, there are a variety of models, the ARIMA model has a higher fitting precision than the others²⁶. It captures seasonal and non-seasonal forecast trends; This study focuses on a non-seasonal model to describe the growth pattern over time. On the other hand, if a model uses statistical approaches for its selection and evaluation, it must be recognized that the number of cases is not true. Since ARIMA is the only official source of information available in the country, it can lead to an inaccurate forecast. Even with this limitation, the results suggest that the method can help estimate the outbreak dynamics and provide guidelines to stop or better control the increase in infections in the country. It is also evident that any attempt to control depends on the public policies dictated by the authorities and above all. The awareness of each citizen about the spread of COVID-19 and the lethal effects that an irresponsible behavior or practice can have, as a consequence, the health of an individual, a family, and community.

Likewise, this model allows to analyze the possible evolution of the contagion curve and to be able to compare the results with those of other countries to evaluate the actions that are being taken.

It should be clarified that this estimate is strongly related to the trend of the original series data, and the result of the forecasts can help to understand any sudden change in the prognosis of COVID-19 cases.

Within the limitation of the study, it can be mentioned that the precision of the forecast depends on the precision of the applied data, this means that there is no other additional evidence that can estimate the exact number of patients with COVID-19 since only the number of official cases reported by the National Institute of Health and the National Center for Epidemiology, Prevention and Control of Diseases of the Ministry of Health was considered.

CONCLUSION

Compared with the observed data, the results obtained with the ARIMA model show an adequate adjustment of the values. This model is easy to apply and interpret, but does not simulate the exact behavior over time. It can be considered a simple and immediate tool to approximate the number of cases.