Time Series Models for COVID-19 New Cases in Top Seven Infected African Countries

Introduction: In Africa region on the date of December 09, 2021 at 14:46 GMT, the total cumulative cases of COVID-19 was 8,889,437 with total deaths and total recoveries of 224,731 (2.5% of death rate) and 8,185,382 (92% of recovery rate) respectively. Thus, this study aimed modelling and forecasting of COVID-19 new cases in top seven infected African countries using time series models. Methods: The top seven infected African countries COVID-19 new cases dataset was taken from our World COVID-19 dataset. The study period was from February 14 to September 06, 2020. Different time series models were used for modelling and forecasting of COVID-19 new cases data. Models comparisons were done by normalized BIC, root mean square error (RMSE), mean absolute percentage error (MAPE), and R-squared values. Results: The COVID-19 new cases data of Algeria, Egypt, Ethiopia, Morocco, and South Africa were fitted by ARIMA (0,1,0), ARIMA (0,1,0), Damped trend, Brown, and ARIMA (0,1,14) models in the study period, respectively. Whereas Ghana, and Nigeria COVID-19 new cases data were followed by simple exponential smoothing models. The 95% confidence levels for lowest to highest forecasted COVID-19 new cases were 258 to 197 with decreasing trend in Algeria, 63 to 933 with increasing trend in Egypt, 636 to 2,141 with increasing trend in Ethiopia, 0 to 1,022 with constant trend in Ghana, 1,900 to 2,807 with increasing trend in Morocco, 0 to 543 with constant trend in Nigeria, and 2,056 to 2,444 with increasing trend in South Africa for the next one month (from September 7 to October 6, 2020). Conclusion: The findings of the study used for preparedness planning against further spread of the COVID-19 epidemic in African countries. The author recommends that as many countries continue to relax restrictions on movement and mass gatherings, and more are opening their air Original Research Article Argawu; JPRI, 33(60B): 983-992, 2021; Article no.JPRI.78043 984 spaces, and the countries’ other public and private sectors are reopening and then strong appropriate public health and social measures must be instituted on the ground again and again before the virus is distributed and attacked more and more peoples in the region. And, the researcher recommended that risk factors of COVID-19 new cases should be conducted for next time in Africa countries.


Background of the Study
The COVID-19 was first identified on 31 December 2019 in the city of Wuhan, which is the capital of Hubei Province in China [1]. The World Health Organization (WHO) on March 11, 2020, has declared the novel coronavirus (COVID-19) outbreak a global pandemic [2]. It caused the severe acute respiratory syndrome coronavirus 2 (SARSCoV-2) has hit the world severely on early of 2020. Many countries are facing a rapid increasing trend of confirmed cases. The case-fatality-rate varies wildly from country to country [3,4].
Globally on the date of December 09, 2021 at 14:46 GMT, the total cumulative cases of COVID-19 was 268,326,045 with total deaths and total recoveries of 5,298,933 (2% of death rate) and 241,501,470 (90% of recovery rate) respectively as Worldometer report shown [5].
In Africa region on this date, the total cumulative cases of COVID-19 was 8,889,437 with total deaths and total recoveries of 224,731 (2.5% of death rate) and 8,185,382 (92% of recovery rate) respectively as this report shown. In the region, there was some rates increment on both sides. On this date, the top seven infected African countries were South Africa On this date, the COVID-19 cases and rates were highly increased as compared with the study period from February 14 to September 06, 2020. The increments may be due to the increasing of daily laboratory tests in the country [6,7]. And, it's nature of rapid distribution and less protectiveness of individual in the countries.
The African region was described as one of the vulnerable with the COVID-19 infection in the initial phase, due to the fact that Africa is important commercial partner of China and as a result, large volumes of business people travel to the region. Since the epicentre is now in Europe and America, due to the close tie between Africa and countries, African countries face even bigger threat [4].  [18]. In the region, it was 1,297,434 total cases. And, South Africa covered almost 50% of regional COVID-19 total cases. The report was charted on Fig. 2.

Aim of the Study
This study aimed modelling and forecasting of COVID-19 new cases in top seven infected African countries using time series models. Modelling and forecasting COVID-19 new cases would be a useful guidance for timely prevention and control measure to be effectively planned in advance, and it is also useful for sustaining strict measures in order to curtail spread of the virus.

Study Period
The study period was from February 14 to September 06, 2020.

Source of the Data
The top seven infected African countries COVID-19 new cases dataset was downloaded from our World COVID-19 dataset from February 14 to September 06, 2020. It is available in https://github.com/owid/covid-19-data.

Time Serious Models
A time series is a set of observations x t , each one being recorded at a specific time t. Discretetime time series are recorded when observations are made at fixed time intervals. Continuous-time time series are obtained when observations are recorded continuously over some time interval. Then, different time serious models were used to forecast COVID-19 new cases for the next times [19][20][21][22].

ARIMA models
ARIMA model becomes AR (p), MA (q), or ARMA (p, q) if the time series is stationary. The expression of ARIMA (p, d, q) model can be defined as follows: Where are the parameter values for an autoregressive operator, are the error term coefficients, are the parameter values for moving average operator, and is the time series of the original series difference at the degree.

Exponential smoothing models
There are four types of non-seasonal exponential smoothing models. These are Simple, Holt's linear trend, Brown's linear trend, and Damped trend models.
Simple Model: It is used for forecasting a time series when there is no trend or seasonal pattern. The simple exponential smoothing model is given by the model equation: Where: Holt's Linear Trend Model: This model is appropriate for a series with a linear trend and no seasonality. Its relevant smoothing parameters are level and trend, and, in this model, they are not constrained by each other's values. The estimates are made using the equations below.
Where α and γ are the smoothing constants in the range of [0, 1].

Brown's linear trend model:
In this model, the parameters are assumed that the level and trend are equal. In this method, estimates are made using the equations below.
Damped Trend Model: It is well established for an accurate forecasting method, and it's new stated damped trend model is written as follow: Where Y t is the observed series, l t is its level and b t is the gradient of its linear trend. This model has a single source of error, ε t .
And, models comparisons were done by normalized BIC, root mean square error (RMSE), mean absolute percentage error (MAPE), and Rsquared values.

Fitted Time Series Models
The COVID-19 new cases data of Algeria, Egypt, Ethiopia, Morocco, and South Africa were fitted by ARIMA (0,1,0), ARIMA (0,1,0), Damped trend, Brown, and ARIMA (0,1,14) models in the study period, respectively. Whereas Ghana, and Nigeria COVID-19 new cases data were followed by simple exponential smoothing models. All the fitted models had relatively the smallest normalized BIC, root mean square error (RMSE), mean absolute percentage error (MAPE) values, and with the highest R-squared values. The result was presented in Table 1.

Forecasting and Trends of COVID-19 New Cases
The

Residuals Stationary Tests
The residuals stationary tests were examined and checked by auto-correlation function and partial auto-correlation function graphs. Both were presented in Fig. 4.

DISCUSSION
The Earlier study in the African region indicated that the epidemic was controlled in late April with strict control of scenario one, manifested by the circumstance in South Africa and Senegal. Under moderate control of scenario two, the number of infected peoples was increase by 1.43-1.55 times of that in scenario one, the date of the epidemic being controlled was delayed by about 10 days, and Algeria, Nigeria, and Kenya were following this situation. In the third scenario of weak control, the epidemic was controlled by late May, and the total number of infected cases was double that in scenario two, and Egypt was in line with this prediction [14].
Study in the selected G8 European countries (Germany, United Kingdom, France, Italy, Russian, Canada, Japan, and Turkey) for the number of COVID 19 epidemic cases data was fitted the cubic regression models with the curve estimations. The number of COVID 19 epidemic cases data were modelled and forecasted that Japan (Holt Model), Germany (ARIMA (1,4,0) and France (ARIMA (0, 1, 3) were provided statistically significant. UK (Holt Model), Canada (Holt Model), Italy (Holt Model), and Turkey (ARIMA (1, 4, 0) were not statistically significant [15].
Ceylan [17] found that the ARIMA (0, 2, 1), ARIMA (1, 2, 0), and ARIMA (0, 2, 1) models with the lowest MAPE values (4.7520, 5.8486, and 5.6335) were selected as the best models for Italy, Spain, and France, respectively. The ARIMA models are suitable for predicting the prevalence of COVID-19 in future. The results of the analysis can shed light on understanding the trends of the outbreak and give an idea of the epidemiological stage of these regions [17].
Parallel study shown that the ARIMA (0, 2, 3), ARIMA (0, 1, 1), ARIMA (3, 1, 0) and ARIMA (0, 1, 2) models were chosen as the best models for South Africa, Nigeria, and Ghana and Egypt, respectively. Forecasting was made based on the best models. It is noteworthy to claim that the ARIMA models are appropriate for predicting the prevalence of COVID-19. The researchers noticed a form of exponential growth in the trend of this virus in Africa in the days to come [23].
Likassa et al. [24] also revealed that the spatial and temporal pattern of this novel virus was varying, spreading and covering the entire world within a brief time. In the study, the fitting effect of the cubic model (R 2 =99.6%) was the best outperforming compared to the other six families of exponentials [24].
Similar study by Achoki et al. [25] found that spatial pattern of cumulative COVID-19 cases in Morocco was the leading contributor to the burden of COVID-19 in Northern African on June 30, 2020. Morocco had forecasted 4,459,877 cumulative cases of COVID-19 and this was almost double the estimated number for Algeria, a country with the next highest burden, 2,804,674 by the end of June 2020. In Southern Africa, South Africa and Swaziland are the leading contributors to the pandemic. By the end of June 2020, the countries were expected to have 2,581366 and 254,403 cumulative cases, respectively. In the Western Africa sub-region, cumulative cases of infection were dominated by Libya and Ghana, despite Nigeria having a larger population than both countries combined. And the numbers of new COVID-19 infections were expected to increase from 2,453,700 cases in April to 5778830 cases in May to 8,044,927 cases by the end of July [25].
Another similar study using African COVID-19 cases showed that estimated exponential growth rate was 0.22 per day, and the basic reproduction number (R 0 ) was 2.37 based on the assumption that the exponential growth starting from March 1, 2020. With an R 0 at 2.37, the researchers quantified the instantaneous transmissibility of the outbreak by the timevarying effective reproductive number to show the potential of COVID-19 to spread across African region [26].

CONCLUSION
The aim of this investigation was modelling and forecasting of COVID-19 new cases in top seven infected African countries using time series models. The study was based on secondary data obtained from our World COVID-19 dataset. The trends of COVID-19 new cases were increased in Egypt, Ethiopia, Morocco, and South Africa from September 5 to October 6, 2020. But, it was constant in Tunisia, Nigeria, and Libya. And, the measures taken by countries such as the individual attitudes of the societies towards the specified measures and the number of virus tests to be performed are factors that may affect the number of cases. Since this study was conducted with the current measures, the forecasts obtained may differ from the number of cases that occur in the future. Thus, the study findings should be useful in preparedness planning against the further spread of the COVID-19 epidemic in Africa.

RECOMMENDATIONS
The author recommend that as many countries continue to relax restrictions on movement and mass gatherings, and more are opening up their airspaces to international travellers with easing of quarantine measures for returning residents and visitors, and the countries' different public and private sectors (like Schools, Universities, Stadiums, and others) are reopening, then strong appropriate public health and social measures must be instituted on the ground again and again before the virus is distributed and attacked more and more peoples in the region. And, the researcher recommended that risk factors of COVID-19 new cases should be conducted for next time in Africa countries.