Forecasting the spread of the COVID-19 pandemic in Kenya using SEIR and ARIMA models.

COVID-19, a coronavirus disease 2019, is an ongoing pandemic caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The first case in Kenya was identified on March 13, 2020, with the pandemic increasing to about 237,000 confirmed cases and 4,746 deaths by August 2021. We developed an SEIR model forecasting the COVID-19 pandemic in Kenya using an Autoregressive Integrated moving averages (ARIMA) model. The average time difference between the peaks of wave 1 to wave 4 was observed to be about 130 days. The 4th wave was observed to have had the least number of daily cases at the peak. According to the forecasts made for the next 60 days, the pandemic is expected to continue for a while. The 4th wave peaked on August 26, 2021 (498th day). By October 26, 2021 (60th day), the average number of daily infections will be 454 new cases and 40 severe cases, which would require hospitalization, and 16 critically ill cases requiring intensive care unit services. The findings of this study are key in developing informed mitigation strategies to ensure that the pandemic is contained and inform the preparedness of policymakers and health care workers.

Introduction

Coronavirus pandemic is an ongoing pandemic of coronavirus disease 2019 (COVID-19) caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The first reported case was in mainland China, City of Wuhan, Hubei on the December 29, 2019 (Li et al., 2020). The disease quickly expanded into an epidemic in Wuhan and elsewhere in China, with subsequent spread to multiple countries resulting in the World Health Organization (WHO) declaring it a Public Health Emergency of International Concern (PHEIC) on January 30, 2020 (WHO, 2020). On March 11, 2020, the novel coronavirus outbreak was declared a global pandemic (World Health Organization, 2020a, 2020b) by the WHO Director-General, Dr. Tedros Adhanom Ghebreyesus. In his remarks, he noted that over the past two weeks, i.e., between February 25, 2020 and March 10, 2020, the number of cases outside China had increased 13-fold and the number of countries had increased 3-fold. As of July 26, 2021, there were 194,080,019 confirmed cases and 4,162,304 deaths globally, with the European region still taking lead in these infections (World Health Organization, 2020a, 2020b). Case reports were not initially seen in Africa, but after the first case reported in Egypt followed by Algeria (Gilbert et al., 2020) cases quickly climbed in South Africa and Nigeria, though not at the pace seen during early epidemics in Europe and Asia. The first case reported in Kenya was on the March 13, 2020 (Brand et al., 2020) in a traveler returning from the US via the UK. Since then, the cases have gradually increased and by July 26, 2021, there were 197,959 cases and 3872 deaths notified.

SARS-CoV-2 is primarily transmitted through direct and indirect physical contact. In direct physical contact, there occurs physical contact between an infected person and a susceptible person through respiratory droplets produced by an infected person who is either sneezing or coughing. Whereas in indirect physical contact, transmission occurs when susceptible persons touch contaminated surfaces. contact routes (Cohen & Kupferschmidt, 2020; Kumar, Poonam, & Rathi, 2020; Mizumoto & Chowell, 2020; World Health Organization, 2020a, 2020b). Based on available data, Kenya has already experienced 3 waves of the pandemic. The rollout of vaccination against the disease began worldwide and by August 31, 2021 about 5.34B persons had received at least 1 dose and 2.13B (27.3%) persons in the population had been fully vaccinated globally. In Kenya, about 2.7M persons had received the first dose with 804,543 persons fully vaccinated. This represents approximately 3% of the adults and 1.5% of the total population.

Infections can be categorized as asymptomatic (sub-clinical) or symptomatic. The incubation period for COVID-19, which is the time between exposure to the virus (becoming infected) and symptom onset, is on average 5–6 days, however, it can be up to 14 days (Bai et al., 2020; Hu et al., 2020; Lin et al., 2020; Singhal, 2020; Tian et al., 2020). Various authors gave the incubation period of 7.2 days (Ferguson et al.), 5.1 days (Verity et al.), 5.5 days (Hamzah et al.), 3.1 days (Ojal, John, et al.), 6 days (Li, Lixiang, et al.), 5 days (Kotwal, Atul, et al.) and 5.2 days (Xi He et al.). In our work, we used the median incubation period of 5 days from what had been indicated by various authors over time. It is now known that some infected persons can be contagious before symptom onset, resulting in pre-symptomatic transmission among those who will eventually present symptoms. Asymptomatic transmission refers to transmission of the virus from a person who never develops symptoms (WHO, 2020) and is currently poorly understood both in terms of the proportion of asymptomatic cases and their relative infectiousness compared with symptomatic disease, though there is evidence that symptomatic illness is more common in older cases (Brand et al., 2020) and investigators have used models based on cases in Wuhan to show that the majority of infections are likely asymptomatic, especially in the young (Brand et al., 2020). Time from symptom onset to hospitalization is likely dependent on available medical resources and care-seeking behaviours, studies (Hu et al., 2020; Jia & Lu, 2020; Viner et al., 2020; Wynants et al., 2020) have found that the median time from onset of symptoms to first hospital admission was 7·0 days, to shortness of breath was 8·0 days, to acute respiratory distress syndrome (ARDS) was 9·0 days, to mechanical ventilation was 10·5 days and to ICU admission was 10·5 days. Further, the mortality rate depends on the severity of the disease. The mitigation measures have been found to serve in two folds; one, reduction of the rate of illnesses that could be associated with greater attention to hygiene (handwashing), and two, disrupted utilization of inpatient services in Kenya that would eventually lead to increased avoidable mortality and morbidity due to non-COVID-19 related illnesses (Wambua et al., 2021).

The modeling of infectious diseases has become paramount in studying disease transmission dynamics (Prem et al., 2020). It is can be used to predict the future course of an outbreak and to evaluate strategies that can effectively control an epidemic. Mathematical models have been used to describe the transmission dynamics and the spread of the COVID-19 across the population (Brand et al., 2020; Jia & Lu, 2020; Mizumoto & Chowell, 2020; Tuite, Fisman, & Greer, 2020; Viner et al., 2020; Wynants et al., 2020). These models are either deterministic, where there is only one exact solution or stochastic where there is a range of solutions. The degree of the pandemic is typically presented by the basic reproductive number (R0), which is defined as the average number of secondary infections produced by a typical case of an infection in a population where everyone is susceptible (Anderson & May, 1979; Chowell, Hengartner, Castillo-Chavez, Fenimore, & Hyman, 2004; Delamater, Street, Leslie, Yang, & Jacobsen, 2019). This is affected by the rate of contact in the host population, the probability of infection being transmitted during contact, and the duration of infectiousness. The basic reproductive number of a disease cannot be measured directly and must be estimated from disease transmission models. One previous study for COVID-19 in Kenya estimated that R0 ranges from 1.78 (95% CI 1.44–2.14) to 3.46 (95% CI 2.81–4.17) [56].

With the emergence of the COVID-19 pandemic, several models have been developed to assess the impact of various intervention measures (Brand et al., 2020; Chowell et al., 2004; Delamater et al., 2019; Mizumoto & Chowell, 2020; Tandon, Ranjan, Chakraborty, & Suhag, 2020; Wynants et al., 2020). SEIR models are primarily used in modeling diseases because of their simplicity in providing the dynamics of a disease using compartments. A plausible SEIR mathematical prediction model can assist in determining the progression of the pandemic. Non-pharmaceutical Interventions (NPIs) and vaccination have a strong potential to reduce the magnitude of the epidemic peak of COVID-19 and lead to a smaller number of overall cases by reducing the reproductive number (R0). This is because the attack rate of COVID-19 is influenced by the value of R0.

Social distancing emerged as the most reliable NPI for the mitigation and control of COVID-19 (Mart í n-Calvo et al., 2020; Prem et al., 2020; Tandon et al., 2020; Viner et al., 2020). Reduction of social contacts in schools, workplaces, hospitals, and markets among other public places were the main targets for achieving social distancing. Suppression, which aimed at reversing the epidemic growth by driving the reproductive ratio below one, could help in lowering and flattening the epidemic peak, thereby reducing the acute pressure on the healthcare system. However, the negative economic impact of the pandemic was not sustainable. This led to the relaxation of most of the social distancing measures including reopening of schools, allowing social gatherings, and re-opening of religious gatherings. The sudden lifting of NPIs could lead to an earlier secondary peak, which could be flattened by relaxing the interventions intermittently (Prem et al., 2020). A previous study in Kenya also predicted the risk of epidemic rebound after the social distancing measures are lifted prematurely (Brand et al., 2020). An integrated approach to reversing the pandemic peaks which includes social distancing measures, hand hygiene, and vaccination is inevitable.

In this study, we assess the dynamics of an SEIR mathematical model that forecasts the progression of the disease intending to achieve the following; (i) To determine the COVID-19 peak days, daily and cumulative cases in Kenya for each wave, (ii) To provide a 60-days forecast of the number of daily and cumulative COVID-19 cases. This will be useful in determining the preparedness of the health facilities and health care workers and planning of vaccination rollout. This study is also a working tool for policy formulation that will enable Kenya to delay and eventually flatten the epidemic peak. The epidemiological model used is the usual SEIR model stratified for predicting severe and critical cases.

Materials and methods

The model consisted of two stages with the first one using a Susceptible-Exposed-Infectious-Recovered (SEIR) model and the second stage using Autoregressive Integrated moving average (ARIMA) model to forecast the cases beyond the observed data. The model stages are described as follows:

Firstly, an estimation of the SEIR model parameters was done using the historical data. This would be useful in obtaining a perfect fit for predicting future cases and scenarios.

Secondly estimating the effective reproductive number (Re(t)) using the ARIMA model

Finally using the estimated Re(t) to forecast the pandemic 60 days beyond the observed data.

SEIR model formulation

SEIR model determines the flow of individuals between four phases: susceptible (S), exposed (E), infected (I), and recovered or removed (R). The SEIR model governs how fast individuals move from being susceptible to exposed, from exposure to infected, and from infected to recovered.

In this case, the infectious class is divided into Asymptomatic (A) and Mild (M) symptomatic cases. The mild symptomatic cases can then progress to severe cases (H) that are hospitalized, who in turn may progress to critical (C) who require ventilators and specialized treatment in ICU. Mild, severe, and critical can all progress to the recovery compartment (R), while those in a critical state can also die. This model is represented schematically in Fig. 1.

Fig. 1

A schematic illustration of the underlying model.

The model makes the following assumptions:

The disease is transmitted through human-human transmissions.

There is no reservoir for indirect physical transmission. There is no cross-infection occurring from neither pathogen in the environment (reservoir) nor human-animal transmissions.

Susceptible individuals (S) are exposed to infection through contact with infectious individuals. At the beginning of the epidemic, each infectious individual cause on average R0 secondary infections.

After an average incubation period of 5 days, exposed individuals (E) either become asymptomatic (A) or exhibit mild symptomatic infections (M).

The virus-infected person is not infectious during the incubation period.

Mild infectious individuals (M) either recover (R) or progress to severe disease (H).

Severely sick individuals (H) either recover (R) or deteriorate and turn critical (C).

Critically ill (C) individuals either recover (R) or die (D).

Recovered individuals (R) cannot be infected again.

The total population (N(t)) is described in equation (1) as follows:

And the ordinary differential equations are formulated as follows:

Where the conditions are:

The list of parameters used in the model is shown in Table 1. These parameters are estimated from historical data using the ordinary least-squares method (OLS). The parameter values were first obtained from literature and then using the OLS method, we determined the parameters that best match the observed data.

Table 1

List of the parameters used in the SEIR model; a description of both the literature values, their sources, and the OLS values.

Parameter description	Symbol	Literature Value	Source	OLS Value (calculated)
Rate of transmission from S to E due to contact with IA	β1	0.15	Mwalili et al., Lixiang et al. & Ojal et al.	0.151
Rate of transmission from S to E due to contact with IM	β2	0.08	Ojal et al., Lixiang et. Al, Mwalili et al.	0.436
Proportion of symptomatic infectious people	Δ	0.85	Khalil et al.	0.803
Progression rate from E to either IA or IS	Ω	0.196	Ferguson et al., Mwalili et al.	0.15
The recovery rate of the asymptomatic infected individuals	γA	1	Hamzah et al.	0.9495
Recovery rate of the symptomatic infected individuals	γM	0.9815	Khalil et al., Mwalili et al.	0.4996
Recovery rate of the hospitalized individuals	γH	0.1	Ferguson et al., Kotwal et al., Lixiang et al.	0.5003
The recovery rate of the critically ill individuals	γC	0.5	Hawryluk et al.	0.55
Rate of movement from hospitalization to critical illness condition	ζ	0.3	Ferguson et al.	0.4994
The hospitalization rate of the symptomatic infected individuals	k	0.044	Ferguson et al.	0.02598
The death rate of the critically ill due to the virus	λC	0.25	Ferguson et al.	0.4999

ARIMA model formulation

An autoregressive integrated moving average (ARIMA) statistical model that uses data collected over time (time-series data) was adopted. The model's purpose is either to provide a better understanding of the data or to forecast the future based on past values.

The ARIMA model has three key parameters:

p which is the order of the autoregressive (AR) term

q the order of the moving average (MA)

d is the integer that controls the level of differencing

The autoregressive (AR) model represents the past values (lags) that are used to forecast the future values such that:

for i = 1, 2 … p. It can therefore be described as the linear combination of the past values.

The moving averages (MA) model makes predictions based on the past forecast errors such that:

for i = 1, 2 … q which is equivalent to a linear combination of q past forecast errors. Using the non-seasonal ARIMA model [ARIMA (p, d, q)] where p is the order of autoregressive (AR) terms, d is the differences required for stationarity, and q is forecasted based on the past data. We estimated the R effective (Re(t)) as follows:Where Re(t) is the effective reproductive number at time t,

The model selection was based on Akaike's Information Criterion (AIC); where, the model with the least AIC was considered. The Augmented Dickey-Fuller (ADF) test (Lopez, 1997) and the partial autocorrelation function (PACF) were used to perform the stationarity checks. Other checks that were performed include the goodness of fit test to estimate the prediction accuracy of the model and normality tests using a qq plot and a histogram.

Results and discussions

In this section, we describe the results of the SEIR and ARIMA models in forecasting the COVID-19 situation in the next 60 days starting September to the end of October 2021. All analyses were performed using R statistical software version 4.1.0. The packages used were ‘deSolve’, ‘tseries’, ‘ggplot2’, ‘forecast’, ‘nortsTest’, ‘reshape2’ and ‘plotly’.

Estimation of the effective reproductive number from the ARIMA model

We used the time-series ARIMA model to predict the effective reproductive number over time (Rt) through simulation of different ARIMA models. The Akaike's Information Criterion (AIC), Mean Error (ME), Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Mean Percentage Error (MPE), Mean Absolute Percentage Error (MAPE), and Autocorrelation Function (ACF) were estimated as shown in Table A.1 (Appendix section). The AIC is very useful in determining the order of the ARIMA model and the least it is then the better the model. The ARIMA model that was identified as the best fit was ARIMA (2,2,0) which is AR (2) with 2 times difference and was used to estimate the Re(t) values up to 60days from the last day of the observed data.

SEIR model using the estimated effective reproductive number

In this subsection, predictions of the new cases, severe, critical, cumulative deaths, and cumulative infections were made. A forecast of the next 60 days was also done from September to the end of October 2021.

A summary of the predictions and forecasts is presented in Table 2 The table is updated based on the results presented in Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6. The peak dates of all the waves were projected as July 13, 2020, October 18, 2020, March 9, 2021, August 26, 2021, and October 26, 2021 for the 1st, 2nd, 3rd, 4th, and 60th-day forecast respectively. The number of daily new infections was 849 and 454 at the peak of wave 4 and the 60th-day forecast respectively. It was forecasted that by October 26, 2021, the country expects to find 40 severe cases per day and 16 cases of critically ill patients daily. The cumulative deaths by October 26, 2021 were forecasted to be 4,764 and the cumulative infections will be about 278,193.

Table 2

COVID-19 peak days, daily and cumulative cases in Kenya.

Pandemic Peak days and number of Daily cases by Wave in Kenya
	1st Wave	2nd Wave	3rd Wave	4th Wave	60th day Forecast
Peak Date	13-07-2020	19-10-2020	9/3/2021	26-08-2021	26-10-2021
Peak Day	122	221	358	498	566
Daily New Cases	1163	1464	1443	849	470
Daily Severe Cases	89	119	104	64	41
Daily Critical Cases	34	46	40	25	16
Cumulative Peak Cases by Wave and 60th day Forecast
Cumulative new cases	30,697	83,194	144,745	204,275	241,145
Cumulative Deaths	398	1,459	2,774	4,094	4,764

Fig. 2

This chart presents a scatter plot (orange dots) of the measured/observed number of COVID-19 cases. The predicted cases are presented in a line chart (blue line) and the forecasted daily cases in Kenya (in red) for the next 60 days.

Fig. 3

This chart presents the forecast of the severe daily cases of COVID-19 for the next 60 days in Kenya. It also contains a scatter plot of the observed cases (orange dots) and the predictions (blue line)., predicted, and forecasted daily severe cases in Kenya (red line).

Fig. 4

The scatter plot of the measured (orange dots), predicted (blue line), and forecasted (red line); daily critical cases (those that require ICU admissions intervention) in Kenya.

Fig. 5

The chart presents the cumulative plot of the observed cases (orange line), predicted (blue line), and the 60 days forecast (red line) of the daily new cases in Kenya.

Fig. 6

Cumulative plot of the measured, predicted and the forecasted deaths in Kenya; the orange line represents the observed, blue line the predicted and the red is the forecasted COVID-19 cases for the next 60 days.

The COVID-19 pandemic has persisted for more than 450 days since it was declared a global epidemic. The trends of its spread and progression in Kenya were evaluated using SEIR and ARIMA models. The best ARIMA model to forecast the reproductive effective number Re(t) was identified as ARIMA (1,0,1). The results of Rt were used to fit an SEIR model to forecast the expected scenarios 60 days from August 29, 2021.

The average time difference between the peaks of wave 1 to wave 4 was observed to be about 130 days (See Fig. 1, Fig. 2, Fig. 3). As shown in Fig. 1, the 4th wave was observed to have the least number of daily cases at the peak. This could be probably associated with the mitigation measures going on in the country including vaccination, social distancing measures, and prioritization of the high-risk groups in the vaccination rollout plan.

According to the forecasts made for the next 60 days, the pandemic is expected to continue. The peak of the 4th wave is currently being experienced with the peak day being August 26, 2021 (498th day). As can be seen in Fig. 2, Fig. 3, by October 26, 2021 (60th day), the average number of daily infections will be 454 new cases and 40 severe cases, that would require hospitalization, and 16 critically ill cases requiring intensive care unit services. As observed from our findings, the estimated number of severe cases is 2 times more than the critically ill cases. This, therefore, implies that there is a need for preparedness; prioritization of supplemental oxygen support in general hospital wards to handle more of the severe cases. The findings of our study agree with those of Kairu et al. (Kairu et al., 2021); that the priority of investments should be on essential care for COVID-19 cases with severe infections. Our study is also comparable to that of Musa et al. (Musa et al., 2021) which recommended forecasting as an important tool to inform future epidemiological investigations and mitigation plans. Further, Kenya has been ranked as one of the Countries in Africa with a moderate risk for COVID-19 importation (Musa et al., 2020), there is a need for taking more serious control measures based on WHO recommendations, to ensure that the pandemic does not progress further.

Time series stationarity assessment

The stationarity was evaluated by PACF plot and ADF test. The decay of the PACF (P-value = 0.062) and ADF (P-value = 0.0529) with 2-order differencing test both confirmed the stationarity of Re(t). The goodness of fit was assessed using the Lobato test (Lobato & Velasco, 2004), which gave a p-value of 0.0734; this implies that Re(t) follows a Gaussian distribution. We have additional diagnostic plots which are shown in Figure A.1 in the appendix, supporting the stationarity and normality assumptions.

Cross-validation and Ljung-Box test of residuals

We performed a 5-fold cross-validation test to estimate the prediction accuracy of the model. The p-value of the Ljung-Box test of residuals was 0.746, which implies that the result of the cross-validation should be used as the model is fitting the data well. The Ljung-Box test p-value (P-value = 0.746) shows that the errors are iid. i.e., without white noise.

Limitations

We acknowledge some limitations in our study. In our analysis, we obtained open access data that was reported daily by the ministry of health Kenya. Some data elements were missing, especially at the beginning of the pandemic, there was no distinction between zeroes and missing values. These limitations are similar to those recorded in some other works (Kucharski et al., 2020; Shinde et al., 2020).

Conclusions

The findings of this study are key in developing informed mitigation strategies to ensure that the pandemic is contained. Since the SEIR parameters were obtained from real data-driven OLS, our predictions and forecasts were more realistic, and consistent. According to the forecasted trends, the disease is expected to progress in the next 60 days. This implies that there is a need for the country to continue implementing the ongoing mitigation measures. The forecasted results will also influence the heightening of the key mitigation strategies that include an emphasis on the accelerated vaccine uptake, and prioritization of investments to fast-track essential care packages in health facilities at both national and county levels. This includes installation of oxygen plants, assessment of ICU capacity in County hospitals, and creation of more ICU capacity, in preparedness for the unprecedented COVID-19 waves. Other key strategies include continuous sensitization of healthcare workers on preparedness to control the pandemic and continuity of the social distancing measures as per WHO recommendations.

CRediT authorship contribution statement

Joyce Kiarie: Conceptualization of this study, Methodology, Results, and Discussion. Samuel Mwalili: Conceptualization of this study, Data curation, writing – review, and editing. Rachel Mbogo: Conceptualization of this study, Writing - review, and editing.

Declaration of competing interest

Declare no conflict of interest in publication of the manuscript titled:

Table A.1

Accuracy measures for selecting the best order of the ARIMA model. ∗

	ME	RMSE	MAE	MPE	MAPE	MASE	ACF1	loglikelihood	AIC	p	q	d
Training set	0.00000	0.228	0.194	−4.971	19.574	12.675	0.994	29.503	−55.005	0	0	0
Training set1	0.00040	0.020	0.015	0.026	1.438	0.998	0.782	1250.254	−2498.509	0	0	1
Training set2	−0.00004	0.013	0.010	0.024	0.933	0.643	−0.165	1457.594	−2913.187	0	0	2
Training set3	0.00010	0.117	0.099	−2.499	9.955	6.452	0.940	367.451	−728.902	0	1	0
Training set4	0.00024	0.016	0.012	0.022	1.151	0.795	0.295	1384.887	−2765.774	0	1	1
Training set5	−0.00004	0.013	0.010	0.027	0.939	0.649	−0.003	1464.242	−2924.485	0	1	2
Training set6	0.00020	0.067	0.056	−1.363	5.639	3.664	0.794	646.417	−1284.833	0	2	0
Training set7	0.00018	0.014	0.011	0.022	1.031	0.712	0.191	1446.981	−2887.962	0	2	1
Training set8	−0.00004	0.013	0.010	0.027	0.938	0.648	−0.003	1464.440	−2922.879	0	2	2
Training set9	0.00065	0.020	0.016	0.027	1.464	1.014	0.780	1251.076	−2496.153	1	0	0
Training set10	0.00007	0.013	0.010	0.025	0.934	0.644	−0.049	1489.515	−2975.030	1	0	1
Training set11	−0.00004	0.013	0.010	0.027	0.936	0.648	0.005	1464.607	−2925.213	1	0	2
Training set12	0.00044	0.016	0.012	0.021	1.151	0.794	0.294	1385.906	−2763.812	1	1	0
Training set13	0.00006	0.013	0.010	0.026	0.929	0.642	−0.006	1490.299	−2974.598	1	1	1
Training set14	−0.00004	0.013	0.010	0.027	0.940	0.650	0.001	1464.871	−2923.742	1	1	2
Training set15	0.00037	0.014	0.011	0.020	1.031	0.712	0.190	1448.245	−2886.491	1	2	0
Training set16	0.00007	0.013	0.010	0.024	0.925	0.638	0.001	1494.855	−2981.709	1	2	1
Training set17	−0.00004	0.013	0.010	0.027	0.940	0.650	0.001	1464.871	−2921.742	1	2	2
Training set18	0.00006	0.013	0.010	0.021	0.935	0.645	−0.051	1489.578	−2971.156	2	0	0
Training set19	0.00006	0.013	0.010	0.027	0.927	0.640	0.009	1490.558	−2975.115	2	0	1
Training set20	−0.00004	0.013	0.010	0.027	0.939	0.650	0.002	1464.788	−2923.575	2	0	2
Training set21	−0.00004	0.013	0.010	−0.022	0.951	0.652	−0.115	1489.952	−2969.903	2	1	0
Training set22	0.00006	0.013	0.010	0.027	0.923	0.638	0.022	1491.764	−2975.528	2	1	1
Training set23	−0.00004	0.013	0.010	0.027	0.940	0.650	0.001	1464.871	−2921.742	2	1	2
Training set24	0.00017	0.013	0.010	0.000	0.922	0.635	−0.002	1498.813	−2985.626	2	2	0
Training set25	0.00007	0.013	0.010	0.024	0.925	0.638	0.001	1494.859	−2979.717	2	2	1
Training set26	−0.00051	0.013	0.010	−0.020	0.922	0.638	0.019	1486.797	−2963.593	2	2	2

∗ME = Mean Error, RMSE = Root Mean Square Error, MAE = Mean Absolute Error, MPE = Mean Percentage Error, MAPE = Mean Absolute Percentage Error, ACF = Autocorrelation Function, AIC = Akaike Information Criterion, and (p, d, q) represent the ARIMA order.