June 22, 2024

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Time series analysis and short-term forecasting of monkeypox outbreak trends in the 10 major affected countries | BMC Infectious Diseases

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Exploratory analysis

The dynamic characteristics of the transmission of Mpox in eight countries are investigated by the launch of the well-celebrated general scheme of the ARIMA model. The data regarding the daily reported new cases and cumulative frequencies ranging from May 18, 2022, to December 31, 2022, were assembled from the official website of Our World in Data. The exploratory analysis reveals that the greatest number of confirmed cases during the above-mentioned period was reported in United States which is 29,603 (Fig. 2).

It was then followed by the Brazil (10,508), Spain (7496), France (4114) and Colombia (4021). Where, United Kingdom, Germany, Peru and Mexico all showing more than 3000 confirmed cases of the outbreak. Lastly, Canada reveals confirmed cases of Mpox virus infection as 1460. The display of daily reported cases data convincingly exhibits the time-inflicted characters which are then aggregated in cumulative cases. The time series display daily cases reported in the selected countries showed that the reported numbers gained momentum after the second week of June, however, intensities vary (see Figs. 3 and 4).

These findings remain verifiable from the graph of cumulative frequencies indicating a rise in the degree of flow with varying degrees (Fig. 4). The numerical summaries of the data also pass various behavioural aspects of the time-dependent flow of the viral transmission (Supplementary Table S1).

On average United States seals its most affected situation concerning the outbreak with the highest mean value of 199 cases per day over eight months period. This is then followed by the Brazil, Spain, France, Germany, and the United Kingdom. Whereas Canada showed a minimal average value of almost 9 cases per day. Moreover, the highest number of newly reported cases in a single is also associated with United States as 1500 cases. This number is overwhelmingly distinctive from the following country of France and Spain with a maximum number of cases of 526 and 520 in one day. The least affected Canada projected the maximum number of reported cases associated with a single day as 106.

Dynamic modelling

The dynamic parametric estimation and modelling of the time-dependent viral flow of the Mpox data related to eight selected countries forwarded different degrees of complexities. Both new confirmed cases and cumulative cases of Mpox exhibit time series properties. For the following 20 days, daily new and cumulative cases were predicted using best-fitted ARIMA models. RStudio (version 4.1.2; RStudio, Boston, MA, USA) with forecast package was used for forecasting.

The generation of ARIMA model parameters generally involves the use of statistical methods like performing a difference to remove non-stationarity and plotting ACF and PACF graphs (Figs. 5 and 6). Based on lower AIC, AICc, and BIC, this package intuitively selects the optimal set of parameters (p,d,q) for better forecasting 38. This function searches for a range of \(p, q\) values, after fixing d by Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test. It chooses the model having the lowest AIC score.

Fig. 5
figure 5

ACF and PACF of the residuals of the best fitted ARIMA models on the confirmed monkeypox cases

Fig. 6
figure 6

ACF and PACF of the residuals of the best-fitted ARIMA models on the cumulative monkeypox cases

The daily reported data of Spain is best modelled with ARIMA (5,1,3) indicating the prevalence of auto-regressive factors along with moving average part and non-seasonal effects. Similarly, ARIMA (2,1,2), ARIMA (2,1,1) and ARIMA (1,1,2) models are estimated concerning Germany, Netherlands, and Canada. Further, the US, France, and Brazil showed a lack of auto-regressive part in the modelling of daily reported data with estimated models such as ARIMA (0,1,2), ARIMA (0,1,1) and ARIMA (0,1,1), respectively. The UK data, however, distinctively exhibited the existence of an auto-regressive part only with the estimated model of the order ARIMA (4,0,0). The display of ACF and PACF indicates the linear dependency structure after the launch of the most appropriate models (Figs. 5 and 6).

The agreeable behavioural display can be noticed concerning each country’s estimation of daily reported data. Moreover, the model fitting and appropriateness criteria are compiled in the Table 1. One may notice the legitimacy of the estimated models on various fronts while trading off the multifaceted complexities of the viral flow of Mpox. As long as, the time-series modelling of cumulative cases data is concerned a relatively lesser extent of complexity is observed. This character can be attributed to the availability of more synchronized information. The ARIMA (2,2,2), ARIMA (2,2,1), ARIMA (1,2,1) and ARIMA (1,2,2) are estimated concerning Germany, Netherlands, Brazil, and Canada. Whereas ARIMA (0,2,3), ARIMA (0,2,2), ARIMA (0,1,1) and ARIMA (0,2,1) models gained prominence for Spain, US, UK, and France. The performance capacities of the argued models are substantiated while displaying the correlation formations through ACF and PACF (Figs. 5 and 6) along with relevant numerical summaries regarding the predictive power (Table 2).

Table 1 Estimation of optimal parameters for the best fitted ARIMA model Parameters and AICs of the ARIMA models for 8 countries
Table 2 Accuracy evaluation metrics of ARIMA models for forecasting Monkeypox cases


For each country, the best fit ARIMA model was used to forecast the spread of Mpox in ten countries: United States, Brazil, Spain, France, Colombia, Mexico, Peru, United Kingdom, Germany and Canada. The data used in the models spanned from May 18, 2022, to December 31, 2022, with predictions made for the next 20 days till January 10, 2023, with 95% confidence intervals. The graphical display offered in Figs. 7 and 8 presents the predictive behaviour for daily reported data.

Fig. 7
figure 7

The prediction of the best fitted ARIMA model (95% confidence interval) for the daily new cases (for the next 20 days) from July 26, 2022, to August 13, 2022

Fig. 8
figure 8

The prediction of the best-fitted ARIMA model (95% confidence interval) for the daily cumulative cases (for the next 20 days) from July 26, 2022, to August 13, 2022

The results of the predictions showed a range of expected cases per day, with the United States estimated to have the highest number of cases at an average of 62 per day and the Germany expected to have the lowest at 7 per day. Meanwhile, countries like Peru, Brazil, France, the United Kingdom, and Canada, are forecasted to have averages of 32, 23, 24, 24, and 26 cases per day, respectively. Colombia, and Mexico are expected to have numbers of cases at 12, and 12 cases per day, respectively. The results of the predictive analysis show a need for each country to prepare for their respective levels of risk to control the spread of Mpox.


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