Forecasting COVID-19 cases in Algeria using logistic growth and polynomial regression models

The novel Coronavirus respiratory disease 2019 (COVID-19) is still expanding through the world since it started in Wuhan (China) on December 2019 reporting a number of more than 84.4 millions cases and 1.8 millions deaths on January 3 rd 2021. In this work and to forecast the COVID-19 cases in Algeria, we used two models: the logistic growth model and the polynomial regression model using data of COVID-19 cases reported by the Algerian ministry of health from February 25 th to December 2 nd , 2020. Results showed that the polynomial regression model tted better the data of COVID-19 in Algeria the Logistic model. The rst model estimated the number of cases on January, 19 th 2021 at 387673 cases. This model could help the Algerian authorities in the ghting against this disease.


Introduction
The COVID-19 is the third epidemic reported in the 21 st century caused by a virus from the family of Coronaviridae after the SARS (Severe Acute Respiratory Syndrome) in 2003 and MERS (Middle-East Respiratory Syndrome) in 2012 [1]. However, the propagation of this disease is faster than the two others.
Since its rst report in the city of Wuhan in China on December, 2019, the disease has been widely and rapidly reported in all countries and regions of the planet gaining the statute of a pandemic since March, 2020. The number of cases continues to increase rising a total of more than 84.4 millions and near than 2 millions deaths on January 3 rd , 2021 [2].
In this way, it is important to understand the epidemic evolution and evaluate the protective measures applied by the national authorities. To attempt these objectives, mathematical and statistical models represents an inescapable part not only for prediction but also for planning control and mitigation actions. These models which use various sources of data allow us opportunities to test various strategies in simulations before their application in populations or individuals [3].
Since the rst reports of COVID-19, various models were used to predict the curve evolution and to estimate the nal size of the epidemic. Among these approaches, logistic growth models have been previously used to describe epidemics spread [4,5]. These simpler tools are very popular and have been applied for COVID-19 short-term forecasting [6,7,8,9] and also to predict the nal size of the epidemic [10,11,12]. Roosa et al. [13] used a generalized logistic growth model to assess the impact of containment and predict nal case numbers in China. Vattay [14] used the logistic growth model to analyze the similarity in Hubei, China and Italy in term of COVID-19 death numbers, and predicted the end date in Italy. Wu et al. [15] used four logistic growth models (the classical logistic growth model, the generalized logistic model, the generalized Richards model and the generalized growth model) to analyze the growth of COVID-19 in Chinese provinces using and further applied them to predict the number sizes in other European, American and Asian countries. Roosa et al [16] used a generalized logistic growth model, the Richards growth model, and a sub-epidemic wave model for a 5-and 10-day prediction of cumulative cases in both Guangdong and Zhejiang (China).
They observed that the GLM and Richards model showed comparable predictions, while the sub-epidemic model forecasts showed signi cantly greater uncertainty.
Balaban, 2020 [17]compared the performance of 5 growth models like logistic growth, Von Bertalanffy growth, exponential growth, Gaussian growth and Richards growth models and used them for short term forecasting of COVID-19 in Turkey. The author showed that Von Bertalanffy model has the best performance but the exponential model has predicted the total deaths and cases better than the others. In the study of Zhou et al. [18], the authors used the logistic growth and the SEIR models to forecast the spread of COVID-19. They reported that the pandemic size estimated by the logistic model was considerably smaller than the SEIR models.
In another study, Batista [12] tried to estimate the nal size of the epidemic for the whole World using logistic and SIR models. The same authors used the logistic model to forecast daily predictions and the epidemic size for China, South Korea, and the rest of the World [12].

Malhotra and Kashyap [19] calibrated the Susceptible-Infected-Recovered (SIR) model and the Logistic
Growth model to forecast the endpoint of COVID-19 in India and three states.
Another tool regarded as one of the best tools to analyze and predict the pandemic growth is polynomial regression. This special type of multiple regression method was applied in several studies to analyze the behavior of COVID-19 [20,21,22,23,24]. It has shown 99.85% accuracy in the study of Yadav [25].
Prakash [26] used ANN and regression to model the COVID-19 pandemic in India and other countries like The USA, Italy and Spain. The authors observed that the results of polynomial regression follow the ground truths in India and the USA but not in Italy and Spain. They use also these models to estimate the peak of the epidemic in the cited countries.
Bel n et al, 2002 [27] have used a SEIR and polynomial regression models to estimate the pick of the COVID-19 epidemic and the basic reproduction number in India respectively. Also, Amar et al, 2020 [28] applied seven regression models for the COVID-19 dataset including exponential polynomial, quadratic, third degree, fourth-degree, fth-degree, sixth-degree, and logit growth.
They reported that, the exponential, fourth-degree, fth-degree, and sixth-degree polynomial regression models are excellent models (especially fourth-degree model).
In the same way Chakraborty et al, 2020 [29] compared the performance between Linear Regression model, Granular Box Regression (GBR) and the Polynomial Regression model in predicting the spread of COVID-19 in India. The authors reported that the Polynomial regression model surpassed the two other models.
Algeria repotted its rst case on February 25 th , 2020. Since then, the actual situation shows a total number of 99897 cases and 2762 deaths [30]. Despite the different preventive measures applied since March, 2020 the number of total cases is still increasing. Thus short and long term estimation of the number of cases and prediction of the curve evolution is of great importance to understand the epidemic curve. The current work is conducted to predict the number of case using a logistic growth model.

Methods
Data sources: In the presentstudy, the used datasets of COVID-19 con rmed cases were taken from the daily reports of the Algerian Ministry of Health from February 25 th (when the rst case was reported) to December 2 nd , 2020 [30]. The number of con rmed cases was based on RT-PCR positive tests. The gureof the COVID-19 in Algeria on December 2 nd , 2020 showed a number of 85084 cases. In parallel, the number of recovered persons was 54979 and the number of deaths was 2464 persons.
To predict the cumulative number of cases of COVID 19 in Algeria, we used both Modi ed Logistic Growth model and Polynomial regression.

Modi ed Logistic Growth Model:
Logistic Growth is characterized by increasing growth in the beginning period, but achieve stability at a later stage, as you get closer to a maximum. For example in the Coronavirus case, this maximum limit would be the total number of people in the world, because when everybody is sick, the growth will necessarily diminish. The reason to use Logistic Growth for modeling the Coronavirus outbreak is that epidemiologists have studied those types of outbreaks and it is well known that the rst period of an epidemic follows Exponential Growth and that the total period can be modelled with a Logistic Growth.
The modi ed logistic growth model is presented as follows [31,32]: Where y(t) is the number of cases at any given time t C is the limiting value, the maximum capacity for y a = (C / y 0 ) -1 b rate of change.
the number of cases at the beginning, also called initial value is: C / (1 + a) the maximum growth rate is at t =ln(a) / b When y is equal to c (that is, the population is at maximum size), y /C will be 1. Therefore, the (1-(y/c)) will be 0 and hence the growth will be 0.

Polynomial Regression
Polynomials are widely used in situations where the response is curvilinear, as even complex nonlinear relationships can be adequately modeled by polynomials over reasonably small ranges of the x's. In general, the k th -order polynomial model in one variable is In this data we used Polynomial regression with 6 degree.

Results
Logistic Growth Model: Figure 1 and 2 show that Logistic Growth Model is not working well for COVID-19 cases in Algeria. Model shows attening but in real scenario it's still increasing. To get better t using Logistic Growth Model, We have to wait for some more days to know the trend of the data in upcoming days.

Polynomial Regression:
As shown in Figure 3, the data tted well in polynomial regression model. Predictions of table 1 showed that the number of cases will be estimated at 387673 (372809-402538) cases on January 19 th 2021.

Discussion
In this work, we used two types of models to predict COVD-19 cases in Algeria: Logistic Growth model and polynomial growth model.
Results showed that logistic growth model despite its wide use in forecasting COVID-19 curve has shown discordance between the real and the tted data from March 25 th to December 2 nd , 2020. In the same way,Abusam et al, 2020 showed that both Velhust and Richards models are not well adapted and needs more parameterization for COVID-19 predictions in Kuwait [33].
Regarding polynomial regression model, Results showed that this model tted well COVID-19 data in Algeria. These results are in accordance of those of Amar et al, 2020 who showed that the exponential, fourth-degree, fth-degree, and sixth-degreepolynomial regression models are excellent models and especially fourth-degree model.
However, the limitation of our model is that we cannot predict for more than 60 days. Usually polynomial regression is used for the short term prediction. This type of model was widely used in the case of COVID-19 [20,21,22,23,24] and has shown an excellent accuracy in certain cases [25].

Conclusion
A forecast of COVID-19 cases in Algeria was conducted in this study using logistic growth and polynomial regression models based on data from February, 25 th , and December 2 nd , 2020. Results showed that polynomial regression model is more adapted in COVID-19 forecasting in Algeria. We used then this model to predict the short term future cases until January 19 th , 2020. The total number of cases in this date is estimated at 387,673 cases. This model could help the Algerian Government in adapting the best strategies against the COVID-19 epidemic. Declarations