Background

Corona virus disease (COVID-2019) is a severe ongoing novel pandemic that is spreading rapidly across the world. Analyzing and predicting COVID-19 prevalence will help governments to take necessary actions in controlling the spread. Time series analysis has proved to be efficient for estimating future impact in such circumstances.

Methods

Auto Regressive Integrated Moving Average (ARIMA) models were developed to predict the epidemiological trend of pandemic in India, The USA and Italy up-to 20 July 2020. A 15 day forecast were made to study the future scenario.

Results

We could identify that all the countries had a linear trend in the increase of daily confirmed COVID-19 cases. Our studies produced a 15 day forecast results for India, US and Italy. the 95% PI for 20 July 2020 are, (25821.72 , 32991.38), (44278.89 , 87095.68) and (-2935.20, 2685.14) respectively for the three countries.

Conclusion

The US and India will have a rise in number of daily counts of COVID-19. Italy will gradually slow the pace. It is supposed that the present prediction models will assist the government and medical personnel to be prepared for the upcoming conditions and have more readiness in healthcare systems.

 Keywords: COVID-19, forecasting, ARIMA models, time series

The COVID-19 pandemic has spread rapidly across the world. It was initially identified in the Hubei province of China. COVID-19 has a dynamic structure and spreads more abruptly than SARS-CoV and MERS-CoV; two zoonotic viruses identified in the past decade_­­­­.¹ COVID-19 has now, as of 06 July 2020, affected worldwide among 213 countries, territories and two international conveyances.² For people residing in or travelling to these countries, the risk of incidence is higher. Older people and those with pre-existing medical conditions are much likely to be affected.³

As on 1 September 2020, a total of about 11,563,000 confirmed cases and 536,800 deaths were reported globally.² At a point, Italy was the epicentre of the disease. The US is now, as of July 4^th 2020, leading with over 3 million confirmed COVID-19 cases and about 1,29,000 deaths. India, having a different population dynamics compared to The US and Italy had a lower number of confirmed COVID-19 cases and deaths. The number of confirmed cases vary due to differences in epidemiological surveillance and healthcare capacities between countries. Up-to-date, no treatments or vaccination were found effective in curing the illness. In this situation, preventing the infection and preparations in health care services is of great importance.³ Modelling and predicting confirmed COVID-19 cases will help the healthcare personnel to be prepared for the worst scenario. Statistical models that could predict the forthcoming situation and model the present situation will be of great help.

In this study, Auto Regressive Integrated Moving Average (ARIMA) models are used in modelling and forecasting confirmed COVID-19 cases in India, USA and Italy. This model has proved to be more efficient than some prediction models such as Wavelet Neural Network (WNN) and the Support Vector Machine (SVM) in prediction of natural disasters.⁴

How different are the pandemic trends in these countries? What will be the future scenario in these countries? So far no serious investigations were taken to analyse and compare the differences in COVID-19 status between these three countries. Therefore, the main aim of this paper is to study and make forecasts of confirmed COVID-19 cases. We also study the relation between COVID-19 mortality and older age group in these countries.

The remainder of the paper is unfolded as follows: In Section 2, materials and methods used for the study is briefly discussed. Section 3 lists out the relevant results of the current study along with tables and figures. In the final Section 4, methods and findings are discussed which ends with the conclusion of the study.

a) Data

The data of cumulative confirmed cases, deaths and recoveries and testing details of India, the USA and Italy was obtained from the Kaggle repository which they derived from Centre for Systems Science and Engineering (CSSE) at Johns Hopkins University, and MS Excel was used to build a time series database. Data was filtered starting from the 50th confirmed case for each of the country till 05 July 2020. We used the ARIMA model with R programming (version 1.2.5042.0) and validated it using Akaike information criterion (AIC. New variables namely CFR (Case Fatality Rate) and CMR (Crude Mortality Rate) were calculated.

b)ARIMA models  

The ARIMA class of models is an important forecasting tool, and is the basis of many fundamental ideas in time-series analysis. The acronym ARIMA stands for auto-regressive integrated moving average. The model includes an auto-regressive (AR) model, moving average (MA) model and an integrated part that will account for the non-stationary behaviour of data.⁵

An ARIMA (p, d, q) model is given by

$\Phi (B){(1-B)}^d{X}_t=\theta (B)$     (1)

where

$\phi (B)=1-{\phi }_{{}_1}B-{\phi }_{{}_2}{B}^2-{\phi }_p{B}^p$      (2)

 is a polynomial in B of order p, B is the backshift operator, such that $B{X}_t =X_{t-1}$ and  is the predicted number of variable,

$\theta (B)=1+{\theta }_{1}B+\mathrm{...}+{\theta }_q{B}^q$      (3)

 is a polynomial in B of order q. p is the order of the auto-regressive (AR) part of the ARIMA model, q is the order of the moving average (MA) part and d refers the degree of trend difference. The error terms are generally assumed to be independent identically distributed random variables (i.i.d.) sampled from a normal distribution with zero mean: follows  N (0,${\sigma }^2$ ) where${\sigma }^2$ is the variance.⁶

 Before analyzing, time series must become stationary based on mean and variance. The Augmented Dickey-Fuller (ADF) is used in recognizing stationary behavior in the mean, the tests generally take the null hypothesis to be that there is a unit root (so that d = 1) and Box Cox test is used for testing whether the time series is stationary based on variance or not.⁷

After achieving stationarity, to determine p, d, q values, ACF and PACF of the differenced data is plotted. Once p, d, q are determined, the data can be fitted using the ARIMA (p, d, q) model. A conditional sum of squares method or method of maximum likelihood is used to fit the data. 5x5 matrices can be created for p, q values and all the models should be fitted. The model with the lowest AIC value whose residuals are white noise is selected as the best model. When the first 25 models were not appropriate, the matrices could be extended and the best model should be selected by AIC criterion.

 $$AIC=-2ln\left(max.likelihood\right)+2m$$     (4)

where m denotes the number of independent parameters estimated in the model.⁶

The most optimized and efficient model could thus be selected. The insignificant coefficients should be excluded. Model diagnostics is the next step. ACF plots are used to find if the residuals of fitted time series model are a white noise. Box-Ljung’s test is used for the same. A significant p value suggests a lack of fit, implying the wrong model selection.⁹

As a first step in ARIMA modelling, the stationarity was checked using ACF and PACF plots. To confirm the findings, ADF test was performed. The test showed non stationarity in the three countries with 95% confidence. All the p values were non-significant, indicating non stationary data. The first order differencing stabilized the mean of daily confirmed COVID-19 cases. The trends in data were thus eliminated. A significant p value by ADF test supported the finding. ARIMA models were determined according to ACF and PACF plots. In addition to the selected ARIMA models, several models were developed and those models with lowest AIC values were chosen. Minimum AIC values of the adopted models were 1838.1_India, 2426.72_USA, 1837.26_Italy. The AIC values of the competing models are given in Table 1. ARIMA (9,1,9) _India, ARIMA (7,1,2) _USA, ARIMA (5,1,7) _Italy­ were the best fit models.

The models fitted the data reasonably well. Box-Ljung’s test was done to test the independence of residuals of the fitted model. All the p values were statistically non-significant indicating that the residuals are white noise.

The fitted models for India, USA and Italy are given below.

$\Delta {\text{x}}_{\text{t}\left(\text{India}\right)}=\text{C}-0.7030\Delta {\text{x}}_{\text{t}-1}-0.1522\Delta {\text{x}}_{\text{t}-2}+0.032\Delta {\text{x}}_{\text{t}-3}-0.1477\Delta {\text{x}}_{\text{t}-4}-0.0637\Delta {\text{x}}_{\text{t}-5}+0.3183\Delta {\text{x}}_{\text{t}-6}+0.6727\Delta {\text{x}}_{\text{t}-7}+0.402\Delta {\text{x}}_{\text{t}-8}+0.4946\Delta {\text{x}}_{\text{t}-9}+0.7642{\text{ε}}_{\text{t}-1}$
$+0.2420{\text{ε}}_{\text{t}-2}-0.1702{\text{ε}}_{\text{t}-3}+0.0093{\text{ε}}_{\text{t}-4}-0.0538{\text{ε}}_{\text{t}-5}+0.3227{\text{ε}}_{\text{t}-6}+0.169{\text{ε}}_{\text{t}-7}-0.3191{\text{ε}}_{\text{t}-8}-0.8093{\text{ε}}_{\text{t}-9}+{\text{ε}}_{\text{t}}$      (5)

$\begin{array}{l}\Delta {\text{x}}_{\text{t}\left(\text{USA}\right)}=\text{C+}0.6956\Delta {\text{x}}_{\text{t}-1}- 0.2044 \Delta {\text{x}}_{\text{t}-2} -0.2995 \Delta {\text{x}}_{\text{t}-3}\\ + 0.2667 \Delta {\text{x}}_{\text{t}-4}0.0153 \Delta {\text{x}}_{\text{t}-5}0.1060 \Delta {\text{x}}_{\text{t}-6}+0.2447\Delta {\text{x}}_{\text{t}-7}1.3843{\text{ε}}_{\text{t}-1}0.7011{\text{ε}}_{\text{t}-2}+{\text{ε}}_{\text{t}}\end{array}$      (6)

- 0.4079 + 0.2441 -+

0.3522  + 0.5738  -  

+ - + +

$\begin{array}{l}\Delta {\text{x}}_{\text{t}\left(\text{Italy}\right)}=\text{C}+0.1659\Delta {\text{x}}_{\text{t}-1}0.2931\Delta {\text{x}}_{\text{t}-2}+0.1289\Delta {\text{x}}_{\text{t}-3}-0.4079\Delta {\text{x}}_{\text{t}-4}+0.2441\Delta {\text{x}}_{\text{t}-5}-0.4766{\text{ε}}_{\text{t}-1}+\\ 0.3522{\text{ε}}_{\text{t}-2}-0.4250{\text{ε}}_{\text{t}-3}+0.5738{\text{ε}}_{\text{t}-4}-0.2148{\text{ε}}_{\text{t}-5}+0.2025{\text{ε}}_{\text{t}-6}+0.5026{\text{ε}}_{\text{t}-7}+{\text{ε}}_{\text{t}}-1.4809{\text{ε}}_{\text{t}-3}+0.04058{\text{ε}}_{\text{t}-4}+{\text{ε}}_{\text{t}}\end{array}$      (7)

Point as well as interval forecasts for the next 15 days were made. Forecasts are presented in Table2.

The first positive case was identified in the USA followed by India and the Italy. The three countries took different paths in the pandemic progress. India had a hike in the number of cases at mid-March. But this hike is not significant compared with the USA and Italy which had enormous increases in the pandemic incidence during the first weeks of March. The USA and Italy has just started flattening the curve during the last weeks of May 2020. As per forecasts on the three countries, Italy will have lower numbers of cases reported daily. But in India and The US, the situation will be just the opposite. Predictions made in India show a slow and steady increase in the number of daily cases during the 2nd and 3rd week of July 2020. A 95% PI and 80% PI also suggests the same results.

The high differences, through another point of view, may be because of the false number of total cases. A case is reported only when a test is done. The total number of cases hence depends directly on the number of tests done. The USA and Italy have the respective first and second places in the number of daily tests performed (Figure 1).

Figure 1 Forecast of daily confirmed cases in India.

Figure 2 Forecast of daily confirmed cases in The USA.

Figure 3 Forecast of daily confirmed cases in Italy.

Figure 4 Total COVID-19 deaths in India, The USA and Italy.

Figure 5 Daily COVID-19 tests per 1000 people.

The USA, Italy and India have done 27.07, 42.44 and 1.17 tests per thousand people. India has a very lower number of daily tests and thus implicating the lower confirmed cases. The study suggests that the true cases in India are concealed. The undetected cases thus are larger in India and can cause a serious and dangerous outbreak than the USA and Italy.

Governments have adopted their own measures for prevention. This could result in changes in the future situation. For e.g. India has started taking NRI’s. This could increase the number of cases. One of the main limitation of our study is that these couldn’t be incorporated in ARIMA model.

The scenario will be in favour of Italy, whereas against USA and India. India should increase the number of tests and reveal the true number of affected patients. Italy has a total of 3.2 hospital beds per 1000 inhabitants. The USA has 2.8 and India has only 0.5 beds per 1000.¹¹ No. of hospital beds, ventilators and other facilities must be dealt with immediately. Italy even though is gaining control, should increase the pace or continue as of now in implementing measures to contain the disease, but never lessen the same. The continuation of the restrictive measures and the strict compliance with the rules, such as traffic and travel restriction, ban on gatherings, and closure of commercial activities, may mitigate the size of the pandemic.

The authors are grateful to Mr. Noel George, p-Value Solutions Ltd Pala for the valuable suggestions during various stages of this research.

This research did not receive any specific grant from any of the funding agencies in public, commercial or any other sector. It is part of PG Project Work.

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