Abstract

Background: Coronavirus disease has become a worldwide threat affecting almost every country in the world. The spread of the virus is likely to continue unabated. The aim of this study is to compare between Autoregressive Integrated Moving Average (ARIMA) model, Fuzzy time series and Holt-Winters – No seasonal for forecasting the COVID-19 new cases in Algeria. Methods: Three different models to predict the number of Covid-19 new cases in Algeria were used. The number of new cases of COVID-19 in Algeria during the period from 24^th February 2020 to 31th July 2021 was modeled according to ARIMA(4,1,2) model, Five based Fuzzy time series models including the Chen model, Heuristic Huareng model, Singh model, Abbasov-Manedova model and NFTS model, and Holt-Winters – No seasonal. Results: The predictive values were obtained from the 1^st August 2021 to 31th December 2021. According to a set of criteria (ME, MAE, MSE, RMSE, U), we found that the FTNS model is the most accurate and best generating model for the values of the number of new cases of Covid-19. Conclusion: To the best of our knowledge, this is the first comparative study of three models of forecasting of Covid-19 new cases in Algeria. This study shows that ARIMA models with optimally selected covariates are useful tools for monitoring and predicting trends of COVID-19 cases in Algeria. Moreover, this forecast will help the Health authorities to be better prepared to fight the epidemic by engaging their healthcare facilities.

1. Introduction

The world is facing a global health pandemic that health authorities worldwide have been unable to stop its spread. The 2019 novel coronavirus (2019-nCoV) was originally discovered on December in Wuhan, a city in the Hubei province in China ¹. The virus which is linked to the same family of viruses as Severe Acute Respiratory Syndrome (SARS) causes respiratory diseases which may lead to serious complications including death, has become a worldwide threat affecting almost every country in the world. According to the latest data provided by the world health organization (WHO), the total number of confirmed cases with Covid-19 virus has surpasses 219 million, and the number of deaths have reached 4.5 million. Moreover, these numbers continue to grow at a significant speed, indicating that the pandemic is far away from being over. Furthermore, mutant viral strains continue to emerge, and no public policies or treatment strategies seem to be sufficiently effective in blocking Covid-19 ².

On February 25, 2020, Algeria laboratory-reported its first case of Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2), an Italian man who arrived on 17 February ³. Over time, the number of confirmed cases of Covid-19 has increased, whether it is spread locally or imported cases from other countries. China is considered to be the most important economic trader in Africans countries including Algeria, which increases the probability of the virus spreading in Africa. A recent study have shown that Algeria and Egypt have moderate to high capacity to respond to outbreak, yet these two countries have the highest importation risk ³. According to the Algerian government, on 02 August 2021, the number of infections was 170,000 and the death total was 4,189. Moreover, the number are at increase.

The mathematical and statistical models have been used in epidemiology in order to understand the dynamics of infectious diseases. Autoregressive Integral Moving Average model (ARIMA), namely the Box–Jenkins model, is the most common time series prediction model in the statistic model. In fact, several studies have attempted to track the trend of spread of Covid-19 virus and predict future path using statistical and mathematical methods. For instance, a previous research article have analyzed the time series data for the top five countries affected by Covid-19 namely US, Brazil, India, Russia and Spain, for forecasting the spread of the epidemic, using ARIMA models. The forecast accuracy were found within acceptable agreement predictions measured by MAD and MAPE ⁴.

Moreover, based on of the COVID-19 in Hubei the Chinese province, researcher have established the ARIMA model, then used these models to predict the development of the Italian epidemic in the next 10 days, even though the model fits well, yet is more suitable for short-term forecasting ⁵.

Several studies have used the ARIMA models to analyze the spread of Covid-19, using Japanese and South Korean data ⁶, also in Nigeria, using NCDC available data. In order to establish a suitable forecasting model, the daily data from February 27 to April 26, 2020 was collected, and an ARIMA model was constructed using R software. The model also carried out a 10-day forecast. The model showed that the spread of COVID-19 in Nigeria was on the rise during the selected time frame ⁷.

Relevant governments and health authorities can use these predictive models to have a better insight of the pandemic situation, and manage and strengthen their healthcare facilities for a better control and contain the outbreak in an efficient way.

Given the fact that the pandemic is ravaging life in Algeria, it becomes necessary to propose a predictive model for the spread of the pandemic.

A previous study have investigated the trend of spread of the virus in the most affected African countries: Algeria, Egypt, and South Africa. Two statistical approaches were used: a class of ARIMA model: Box-Jenkins and fuzzy time series to predict the incidence of Covid-19 cases. The results showed that trends in all three countries are positive, however, the fuzzy time series is better than the ARIMA model in forecasting. This study was the only one conducted in Algeria, however, the time period used is relatively short, only 32 days, and it was restrained on just two models ⁸.

The aim of this study, is to propose a predictive model for the spread of the SARS-CoV-19 virus resulting new Covid-19 cases in Algeria, using three different statistical and mathematical models, including: ARIMA, Holt-Winter models and fuzzy time series.

To the best of our knowledge, this is the first comparative study based on three different statistical models conducted in Algeria, and over a long time period of 17 months.

2. Data and Methodology

Data was obtained from the reliable Our World in Data, an official website, which published a daily updated statistics of the coronavirus pandemic. This open access and open-source platform provide official daily and updated statistics for all countries of the world on a wide range of variables related to Coronavirus pandemic, such as: the number of accumulated cases, the number of new cases, the number of deaths, globally and by country ⁹. For our study, we have extracted the number of confirmed new cases of COVID-19, the data obtained extends over a period of 17 months, from the period 24 February 2020 to 30 July 2021.

In our study, we used three different predictive methods to forecast the number of covid-19 new cases in Algeria. Box-Jenkins methodology that relies on ARIMA models, fuzzy time-series method including 5 different approaches: the Chen model, Heuristic-Huareng model, Singh model, Abbasov-Manedova model and NFTS model, and finally, the exponential smoothing models (Holt-Winters – No seasonal). The two most widely used approaches to time series forecasting are exponential smoothing and ARIMA models. ARIMA models try to characterize the autocorrelations in the data, whereas exponential smoothing methods were based on a description of trend and seasonality in the data ¹⁰, in fuzzy time series, forecasting several methods have been developed for constructing fuzzy relationships on linguistic-valued time-series data to obtain predictive values.

In time series analysis, the Box Jenkins (ARIMA) model introduced by Box and Jenkins is one of the most widely used approaches ¹¹. Due to its generality, it can be applied to any series: stationary or not, with or without seasonal elements, Moreover, it can be implemented in a wide range of different programs ¹².

ARMA model is a combination of autoregressive models which regresses on its own lagged terms and can be expressed as follows:

(1)

Where is stationary, and are model parametres.

Introducing the lag operator the equation (1) can be written as

Where:

The second part of combination is Moving-average model builds a function of error terms of the past and can be expressed as follows:

(2)

Where and

The equation (2) can be written as:

The model is defined as:

Or even more concisely as:

If the series is non stationary, we generalize the ARMA model to become the ARIMA model by including integrated components.

According to Box and Jenkins (ARIMA) the steps of modeling a time series consists of three main components: Identification model, estimation model and diagnostic model ¹³.

Based on these main steps, we proposed a predictive model of the number of new cases of COVID-19 in Algeria which follows the steps below:

1. Check for the stationary of the time series through the graphic curve and the autocorrelation function correlogram, as well as carrying out the Dickey-Fuller test, Philips Peron test, KPSS test.

2. Identification of a tentative model by examining the autocorrelation function (ACF) and partial autocorrelation function (PACF)

3. Estimating the various possible models using the method of maximum likelihood and choosing the best model based on Akaike criterion.

4. Estimating the appropriate model using the maximum likelihood method of estimation.

5. Diagnostic checking model.

The second approach we will use in our comparative study is Fuzzy time series, described below:

Let be the universe of discourse, A fuzzy set A of is:

is the grade of membership of in A, with: 0 ≤ μA(ui) ≤ 1 and

For the sake of accuracy, we will give some of the most used definitions ^{14, 15, 16, 17}:

Definition 1. The first order relation defined by:

If is caused by or , or... with m>0.

Definition 2. The m^th order relation defined by:

If is caused by and and… with m>0.

Definition 3. Time variant fuzzy time series:

F(t) is called time variant fuzzy time series If the first and m^th order relation or of F(t) is independent of t.

The third approach is the Holt-Winter method introduced by Holt in 1957 and Winter in 1960 ^{18, 19}, All exponential smoothing methods share the property of giving recent values relatively more weight in prediction than old observations ²⁰. Our study will be limited to the application of the Holt-Winter non seasonal method, which is similar to the double smoothing method that generates predictive values with a linear trend and without the seasonal component. This method use two smoothing constants and and three equation:

Where and denote an estimate of level and slope of series at time t, Predictive value on the horizon

3. Results and Discussion

The first step is visual examination of the data plot and the correlogram of the autocorrelation function (ACF), as it is shown in Figure (1,a) the series does not fluctuate around a fixed average and that the variance of the series does not appear to be constant.

The Figure (1,c,d) represents the correlogam of the covid-19 series, we can deduct that half of the autocorrelations coefficients are significantly different from zero and decrease exponentially. This is indicative of a non-stationary series. However, it is necessary to verify this assumption by applying statistical stationary tests such as: ADF test, Phillips – Perron test and KPSS test.

Table 1 represents the results of the stationary tests (ADF test, PP test, KPSS test) of the new cases series at the level and after the first differentiation, at the level, we notice that the p-value of ADF test and P-P test equal to 0.173 and 0.793 respectively, greater than 0.05 which means that the new cases series is not stationary, and also the results of KPSS test show that the calculated value is equal to 0.174 greater than the tabulated value (0.146) which confirms the results ADF and PP test. The new cases series is not stationary (has a unit root) and the best way to make it stationary is to differentiate it. Table 1 also shows that the p-value of ADF test and PP test of the differentiate series are < 0.001, <0.001 respectively, which indicates that the new cases series is stationary after the first differentiation, same result for the KPSS test.

Figure (1, b) shows the graph of the stationary series new cases after the first differentiation, through these stationary results we can conclude that the appropriate model is ARIMA (I = 1).

The second step of the Box-Jenkins methodology is to choose the appropriate model among the possible models by studying the plots of the autocorrelation function (ACF) and partial autocorrelation function (PACF). Then choose the optimal model based on the Akaike criterion which is calculated as follows:

The optimal model to fit the new cases series is the model which has the lowest Akaike information criterion. Relying on the auto-selection of model order based on Python library (pmdarima), we estimated 28 possible models, where we set max_p=7, max_q=7, Based on the minimum value of the Akaike criterion, we found that the appropriate model is ARIMA (4, 1, 2).

Table 2 represents the estimation results of the ARIMA model (4, 1, 2), the results shows that all the estimated parameters are significant at 1%, the p-value of all parameters are completely less than 0.01. The estimated model is of good quality if the calculated series follows the evolutions of the observed series as it is shown in Figure 2, the residuals between these values must therefore behave like white noise. The critical probability of the Ljung-Box statistic is always greater than 0.05 and all the terms of the correlogram are in the confidence interval (Figure 2). After estimating the model and verifying its validity, we predicted the number of cases infected with COVID-19 during the period: 08/01/2021 to 12/31/2021 with the confidence interval for the predicted values as shown in the Figure 3.

The following steps have been performed to estimate new COVID cases in Algeria:

Step 1. Define universe of discourse as

Where and are values of real time new case data respectively.

Step 2. Partitioning U into equal length of sub intervals based on formula statistic

with T: number of observations T=523.

We find Table 3 shows the sub intervals of

Step 3. Defining each fuzzy set whith 1< i <10.

Step 4. Fuzzy logical relationships of 10 groups, obtain a total of 250 logical relations

Step 5: Using the R Package “AnalyseTS”, we have estimated the predictive values of a new cases series using five models in fuzzy time series analysis, the actual and forecast values of new cases covid-19 are showing in Figure 4.

• Chen (1996) model ¹⁹ (Figure 4(a)).

• Singh (2008) model ²⁰ (Figure 4(b)).

• Heuristic Huareng (2001) model ²¹ (Figure 4(c)).

• Abbasov-Manedova (2010) model ²² (Figure 4(d)).

• New fuzzy time series (NFTS) model ²³ (Figure 4(e)).

We used the Eviews 12 program to perform the prediction using the Holt –Winters-No seasonal approach, and obtain the predictive values of the number of cases of Covid-19 during the time period from 08/01/2021 until 12/31/2021. We found that the estimated value of Figure 5 shows actual and forecast values for the number of new COVID-19 cases.

A comparison of different forecasting methods

According to our results, and in order to choose the best model that gives predictive values for COVID-19 cases in Algeria, we calculated five selection criteria: ME, MAE, MSE, RMSE and U.

Table 4 shows the values of the statistical criteria for the model, the Holt-Winter model, and the five fuzzy time-series models. First, comparing with both and Holt winter models the fuzzy time series models show the minimum values of the criteria: As Table 4 shows: the for the NFTS model, while for the model the is Moreover, we can deducted based on the results of these selection criteria that the fuzzy time-series models outperform the ARIMA and Holt-Winter models in predicting the number of new cases of COVID-19 in Algeria.

However, we used five different models of the fuzzy time-series approach. We relied on the same selection criteria to choose the most accurate fuzzy time series model.

According to the criteria MAE, MSE, RMSE, U, it is clear that the NFTS model has the lowest values compared to the Chen model, Heuristic-Huareng model, Singh model, Abbasov-Manedova model. Based on these results, we conclude that the NFTAS model is the most accurate model that provides predictive values for the number of Covid-19 cases in Algeria. Table 5 shows the predictive values of new cases of COVID-19 during the time period from 1^st August 2021 to 31^th December 2021.

4. Conclusion

Finding the pattern or model that generates values for the number of new cases of Covid-19 is very crucial. In our study, we tested and compared the predictive ability of different models using data of Covid-19 infection cases from the period 24 February 2020 to 30 July 2021. The most important results were as follows:

• Following the Box Jenkins methodology, which relies on ARIMA models, our study concluded that the best model that gives predictive values is the model.

• Using the exponential smoothing models and estimating the Holt-Winters – No seasonal model, through which we compare the current and forecast values and generate predictive values during the period 01 August 2021 to 31 December 2021.

• Relying on a set of fuzzy time series models (the Chen model, Heuristic Huareng model, Singh model, Abbasov-Manedova model, and NFTS model) we generated the predictive values and compared them with the actual values.

• Relying on a set of statistical criteria we made a comparison between all these models, and the results revealed that the best model is the NFTS model, then we presented the predictive values during the period 01 Augest 2021 until 31 December 2021.

The main objective of this study is to provide predictive values for the expected number of COVID-19 cases to take the necessary precautions measures. According to our results, the most accurate forecast model shows that the number of cases will increase in the upcoming days, therefore the health authorities should maintain restrictions on gatherings, as well as social distancing and wearing a mask.

References