Forecasting COVID-19 per Regions Using ARIMA Models and Polynomial Functions

Introduction

The COVID-19 pandemic, caused by the novel coronavirus, has had a worldwide impact. Predicting its spread is critical for health resource planning, policy-making, and public safety measures. This paper introduces a two-stage approach for COVID-19 forecasting, combining Auto-Regressive Integrated Moving Average (ARIMA) models and polynomial functions to handle the complexity of different geographic regions.

The core motivation is that countries within the same geographic region (e.g., North America, South America, Africa, etc.) might share certain quantifiable variables, such as climate or population distributions, and non-quantifiable factors such as cultural behavior and social patterns. By grouping countries by region, the proposed system more accurately captures underlying trends.

Databases

The primary source for COVID-19 infection data is ‘‘Our World in Data’’, which aggregates statistics from institutions such as the ECDC, WHO, and Johns Hopkins University. Data used in this paper include:

Daily confirmed COVID-19 cases per country.

Population figures per country (e.g., population per million people).

The time series for each country begins on the day its first COVID-19 case was detected and runs up to a cutoff date (April 25, in the paper’s analysis). A separate dataset from a later date (May 11 or May 28) is used to evaluate forecasting performance.

Methodology

1. Building the Model

This stage begins by applying ARIMA to each country within a given region:

Time Series Split: Each country’s daily confirmed cases are partitioned into a training set (90% of data) and a test set (10% of data). The ARIMA model is fitted to the training set, and its forecasting accuracy is verified on the test set via the Root Mean Square Error (RMSE).
Storing ARIMA Parameters (p, D, q): For every country, the algorithm identifies the best combination of ARIMA hyperparameters p (auto-regression), D (integration), and q (moving average) that minimize the forecast RMSE.
Polynomial Regression for (p, D, q): Next, polynomial functions are learned to represent p, D, and q as a function of the country’s “cases per 1M population × total population” metric. This enables using a region-level model (a polynomial) to look up ARIMA parameters based on that combined variable.

2. Evaluating (Forecasting) with the Regional Polynomial Model

After building the polynomial functions for ARIMA parameters in each region, the model can quickly forecast future cases for any new or existing country data in that region:

Lookup ARIMA Parameters: Given updated total cases and population, the new “combined metric” is fed into the polynomial functions, returning an estimate for p, D, and q.
Forecast Cases: Once p, D, q are obtained, the ARIMA model is instantiated and used to predict future daily cases. The paper typically looks 15+ days ahead.
Compare to Observed Values: The predictions are compared against real data for verification, and the RMSE is reported for performance assessment.

Results

The method was tested on 145 countries grouped into six continental regions (North America, South America, Africa, Oceania, Asia, and Europe). Each country's model parameters (p, D, q) and its error were recorded. The following highlights emerged:

Accuracy: Mean RMSE values vary by region, typically reflecting population size and the intrinsic difficulty of forecasting in areas with rapidly changing cases.

Population as a Key Variable: Larger populations were generally associated with higher RMSEs, although exceptions exist (e.g., some African nations with large populations showed low forecasting errors, suggesting that other factors like climate or intervention policies can affect spread).

Overall RMSE: For the 145 countries under study, the average training-stage RMSE was around 144.81, indicating acceptable short-term predictive performance.

Polynomial Modeling of ARIMA Parameters: By consolidating each region's ARIMA parameters into polynomial functions, the approach offers a generalized model capable of quickly adapting for any country in the same region.

Discussion

The paper demonstrates that countries sharing a continent may have overlapping social, environmental, or cultural characteristics that affect COVID-19 transmission. Modeling these nations jointly through a single polynomial function for ARIMA parameters simplifies forecasting without sacrificing accuracy.

One interesting discovery is the partial correlation between a country’s population size (per million) and the overall forecast error, but climate, testing rates, government response (lockdowns, social distancing), and other interventions also heavily influence the spread.

As the pandemic evolves, re-training the polynomial models with updated data is key to maintaining accuracy in longer-term forecasts. This method can be extended by incorporating additional variables (e.g., humidity, culture, or more sophisticated epidemiological indicators) and by refining the selection of ARIMA hyperparameters.

Conclusion

This study presents a unified, two-stage framework for short-term COVID-19 forecasting. First, country-level ARIMA models are estimated, and then their parameters (p, D, q) are captured as polynomial functions indexed by a “cases × population” metric. By grouping countries into six geographic regions, the system can leverage shared attributes to improve forecasting.

The results confirm that ARIMA models, with the help of polynomial interpolation for parameter selection, can achieve low RMSE in predicting daily case counts across diverse countries. Future research can broaden these polynomial-based parameter mappings and adapt them to a wider set of influencing factors for improved generalization and robustness.