Which curve provides the best explanation of the growth in confirmed COVID-19 cases in Chile? *

Objective to explore the best type of curve or trend model that could explain the epidemiological behavior of the infection by COVID-19 and derive the possible causes that contribute to explain the corresponding model and the health implications that can be inferred. Method data were collected from the COVID-19 reports of the Department of Epidemiology, Ministry of Health, Chile. Curve adjustment studies were developed with the data in four different models: quadratic, exponential, simple exponential smoothing, and double exponential smoothing. The significance level used was α≤0.05. Results the curve that best fits the evolution of the accumulated confirmed cases of COVID-19 in Chile is the doubly-smoothed exponential curve. Conclusion the number of infected patients will continue to increase. Chile needs to remain vigilant and adjust the strategies around the prevention and control measures. The behavior of the population plays a fundamental role. We suggest not relaxing restrictions and further improving epidemiological surveillance. Emergency preparations are needed and more resource elements need to be added to the current health support. This prediction is provisional and depends on keeping all intervening variables constant. Any alteration will modify the prediction.


Introduction
The rising pneumonia called COVID-19, caused by SARS-CoV-2, exhibits strong infectivity but less virulence when compared to SARS-CoV-1 and MERS-CoV in terms of morbidity and mortality. It is not only a virus that spreads from one person to another, but probably spreads because many people become infected in various places through different mechanisms. Restricting the movement of people, reducing contact, disseminating key high-frequency prevention information through multiple channels, mobilizing state and local authorities to respond quickly to the contingency, can help contain the pandemic (1)(2)(3)(4)(5) .
The actual number of infected cases is much larger than that reported worldwide. The observed mortality rate of COVID-19 is estimated at around 4.8% worldwide. Although this rate is low in Chile, this estimate may be incorrect due to underestimation, because the likelihood that the health authorities will collect severe cases is higher and, as active cases increase, the health resources do not support the demand and overestimation, considering that the vast majority of cases without symptoms or with mild symptoms are not investigated (6)(7)(8) .
The COVID-19 pandemic is an important international test for the medical and scientific community, as it reveals weaknesses in the management of emerging viral diseases and reminds us that contagious diseases should never be underestimated. In addition, it has strained health systems due to the virulence and excess demand on hospitals (9)(10) .
It is essential to understand the transmission dynamics of the infection, as it could determine whether outbreak control measures are exerting a significant effect.
The numbers of newly infected cases largely depend on the effectiveness of the control measures. Several governments have rapidly incorporated recent scientific findings into public policies at community, regional and national levels to slow down and/or prevent the further spread of COVID-19.
Control measures such as quarantine, travel restrictions and airport inspections for travellers have been widely implemented to contain the spread of infections. The effectiveness of these containment measures to control the outbreak is inconclusive though (1,3,(5)(6)11) .
It is advisable for government entities to report on the current state of the pandemic in daily reports, specifying hospitalized and critical patients with COVID-19. Statistics need to be read carefully though, as it is leading to massive panic without organized solutions related to the redistribution of resources (12) .
Understanding the epidemiological characteristics of COVID-19 transmission in Chile is essential to formulate effective control strategies. The objective is to know what type of curve or model can best explain the epidemiological behavior of the accumulated confirmed cases of COVID-19 and identify the possible causes that contribute to explain the corresponding model and the health implications that can be inferred.

Method
Data were collected from the COVID-19 reports from the Department of Epidemiology of the Ministry of Health in Chile (MINSAL). The data are publicly available (7) .
To adjust the level of smoothing of the data (elimination of irregular fluctuations), the optimal ARIMA model was used for weighting, minimizing the sum of the square residues (18)(19) . The absolute error of each measure was the difference (∆) between the actual observed value and the predicted value of confirmed cases for the same day. The median and interquartile range were estimated after checking for the normality of the absolute errors using the Kolmogorov-Smirnov test. Minitab 18.0 ® software was used. The significance level was α ≤ 0.05. Figure 1 presents the estimated results of the regression equations of the observed data curves of confirmed, adjusted and predicted cases in the quadratic, exponential, simple exponential smoothing and double exponential smoothing models. The MAD, MAPE, and MSD coefficients are lower in the double exponential smoothing curve, which shows that the curve that best fits the evolution of the accumulated confirmed cases of COVID-19 in Chile is the one described above.

Discussion
The progress of the COVID-19 pandemic in Chile fits well into a model and we study its predictive capacity.
By analyzing the epidemiological characteristics and transmission dynamics of an emerging infectious disease, the key to successful outbreak control is obtained through mitigation strategies (6) . The mathematical model of any process tries to describe its basic components and tries to predict some general trends, but it will never be able to provide an exact description and prediction (22) . That happens on May 13 th , 2020 (May 10 th and July 19 th ). It should be mentioned that active cases in Chile represent 47.04% (SD=1.96) of confirmed cases, as of May 5 th , 2020. The propensity of this percentage should tend to decrease gradually and slowly according to international experience, but it has stagnated for 13 days between 45-50%. This stagnation is probably due to the fact that the public health strategies and the population's behavior are not equivalent to that of the countries mentioned (26)(27)(28)(29)(30)(31)(32)(33)(34)(35)(36)(37)(38) .
Chile needs to remain vigilant and adjust strategies around prevention and control measures, which means to improve health actions and to strengthen surveillance systems and public health infrastructure to provide the early detection and rapid response. We hope, for the sake of Chilean health, that the support of infrastructure and health assets achieve the results in terms of effectiveness and do not pose a problem of needs and resources. At the same time, emergency preparations are necessary in response to a more serious outbreak that may occur (1,6) . propose it as an epidemiological method for monitoring and predicting the progression of infectious diseases in the local situation, e.g. regions, provinces and communes (43) .
Our approach has limitations. This study was based on the cases reported by MINSAL and contains the biases of case confirmation and information, which may be delayed with respect to the occurrence of cases, the delay in the appearance of symptoms due to the incubation period and the high proportion of unreported cases as a result of limited detection and testing capacity. strategies at all levels of health care (7) .

Conclusion
The type of curve that best explains the behavior of COVID-19 in Chile is a doubly smoothed exponential curve. From this model, it follows that the number of infected cases will continue to increase. The number of confirmed cases will grow stronger and within a short time if the population behavior and public health measures are not in line with the size of this international public health emergency. We hope that the results will enable the medical staff and leaders to make decisions. In no case should restrictions be relaxed and epidemiological surveillance needs improvements, as this is not the end of the pandemic.