Modeling and Analysis of Monkeypox Outbreak Using a New Time Series Ensemble Technique
<p>Monkeypox infections forecasting: A complete proposed time series ensemble approach layout.</p> "> Figure 2
<p>Comparison of daily confirmed monkeypox cases in four most affected countries from 1 June 2022 to 31 July 2023.</p> "> Figure 3
<p>Comparison of daily cumulative confirmed monkeypox cases with superimposed the nonlinear trend component in four most affected countries from 1 June 2022 to 31 July 2023.</p> "> Figure 4
<p>Residual series after extracting the nonlinear trend component in the four most affected countries and the entire world case within the proposed forecasting technique.</p> ">
Abstract
:1. Introduction
2. The Proposed Ensemble Approach for Time Series
2.1. Preparation of Data
2.1.1. Autoregressive Model
2.1.2. Exponential Smoothing Model
2.1.3. Autoregressive Moving Average Model
2.1.4. Nonparametric Autoregressive Model
2.1.5. Theta Model
2.1.6. The Proposed Ensemble Models
2.2. Evaluation Criteria
- In the first step, the cumulative Mpox-confirmed cases time series () is preprocessed (to address the issue of stabilizing the variance and standard deviation and remove the nonlinear long-run trend component), discussed in detail in Section 2.1.
- In the second step, we divide the stochastic (short-run dynamic) residual component () into three parts: training, validation, and testing datasets. Let represent the residual series of the cumulative Mpox-confirmed cases time series. Then, training dataset: ; validation dataset: ; testing dataset: ; where is the total number of data points.
- In the third step, model the train data using single models, i.e., AR, ARMA, ESM, NAR, and Theta models.
- In the fourth step, calculate the one-day-ahead forecast using the expanding window technique. The forecast values, for j = 1, 2, 3, 4, 5, are obtained by the models listed in step 3.
- In the fifth step, the output of a basic ensemble method is mathematically described by Equation (10).
- In the sixth step, obtain the one-day-ahead forecast values using equations for the EnsE, the EnsT, and the EnsV models.
- In the last step, evaluate the model based on accuracy and average errors (see Table 1).
3. Case Study Results
4. Discussion
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Asadi Noghabi, F.; GRizk, J.; Makkar, D.; Roozbeh, N.; Ghelichpour, S.; Zarei, A. Managing Monkeypox Virus Infections: A Contemporary Review. Iran. J. Med. Sci. 2024, 49, 1–9. [Google Scholar] [CrossRef] [PubMed]
- Leonard, C.M.; Poortinga, K.; Nguyen, E. Brote de Mpox: Condado de Los Ángeles, California, del 4 de mayo al 17 de agosto de 2023. MMWR Morb. Mortal. Wkly. Rep. 2024, 73, 44–48. [Google Scholar] [CrossRef] [PubMed]
- Alqahtani, R.T.; Musa, S.S.; Inc, M. Modeling the role of public health intervention measures in halting the transmission of monkeypox virus. Aims Math. 2023, 8, 14142–14166. [Google Scholar] [CrossRef]
- Bustanji, Y.; Shihab, K.H.A.; El-Huneidi, W.; Semreen, M.H.; Abu-Gharbieh, E.; Alzoubi, K.H.; Alqudah, M.A.Y.; Abuhelwa, A.Y.; Abu-Rish, E.Y.; Bajes, H.; et al. Analysis and mapping of global scientific research on human monkeypox over the past 20 years. Vet. World 2023, 16, 693–703. [Google Scholar] [CrossRef] [PubMed]
- Usman, S.; Adamu, I.I. Modeling the transmission dynamics of the monkeypox virus infection with treatment and vaccination interventions. J. Appl. Math. Phys. 2017, 5, 2335. [Google Scholar] [CrossRef]
- Ipinnimo, T.M.; Adeniyi, I.O.; Ehizibue, P.E.; Dan-Ugbomoiko, O.S. Monkeypox Outbreak—Is This Another Pandemic? Niger. J. Parasitol. 2023, 44, 78–86. [Google Scholar] [CrossRef]
- Çolakoğlu, Ö.; Kamran, M.; Bonyah, E. M-Polynomial and NM-Polynomial of Used Drugs against Monkeypox. J. Math. 2022, 2022, 9971255. [Google Scholar] [CrossRef]
- Zhu, C. An Adaptive Agent Decision Model Based on Deep Reinforcement Learning and Autonomous Learning. J. Logist. Inform. Serv. Sci. 2023, 10, 107–118. [Google Scholar]
- Vazquez, C.; Fonseca, V.; de la Fuente, A.G.; Gonzalez, S.; Fleitas, F.; Lima, M.; Guimarães, N.R.; Iani, F.C.M.; Rojas, A.; Alfonso, T.; et al. Exploring the Genomic Dynamics of the Monkeypox Epidemic in Paraguay. Viruses 2024, 16, 83. [Google Scholar] [CrossRef]
- Addai, E.; Ngungu, M.; Omoloye, M.A.; Marinda, E. Modelling the impact of vaccination and environmental transmission on the dynamics of monkeypox virus under Caputo operator. Math. Biosci. Eng. 2023, 20, 10174–10199. [Google Scholar] [CrossRef]
- León-Figueroa, D.A.; Barboza, J.J.; Valladares-Garrido, M.J. Sources of information on monkeypox virus infection. A systematic review with meta-analysis. BMC Public Health 2024, 24, 276. [Google Scholar] [CrossRef] [PubMed]
- Qurashi, M.A.; Rashid, S.; Alshehri, A.M.; Jarad, F.; Safdar, F. New numerical dynamics of the fractional monkeypox virus model transmission pertaining to nonsingular kernels. Math. Biosci. Eng. 2023, 20, 402–436. [Google Scholar] [CrossRef]
- Nimbi, F.M.; Baiocco, R.; Giovanardi, G.; Tanzilli, A.; Lingiardi, V. Who Is Afraid of Monkeypox? Analysis of Psychosocial Factors Associated with the First Reactions of Fear of Monkeypox in the Italian Population. Behav. Sci. 2023, 13, 235. [Google Scholar] [CrossRef] [PubMed]
- Sudsutad, W.; Thaiprayoon, C.; Kongson, J.; Sae-dan, W. A mathematical model for fractal-fractional monkeypox disease and its application to real data. Aims Math. 2024, 9, 8516–8563. [Google Scholar] [CrossRef]
- Wang, Q.; Jiang, Q.; Yang, Y.; Pan, J. The burden of travel for care and its influencing factors in China: An inpatient-based study of travel time. J. Transp. Health 2022, 25, 101353. [Google Scholar] [CrossRef]
- Crosato, V.; Formenti, B.; Gulletta, M.; Odolini, S.; Compostella, S.; Tomasoni, L.R.; Matteelli, A.; Castelli, F. Perception and Awareness about Monkeypox and Vaccination Acceptance in an At-Risk Population in Brescia, Italy: An Investigative Survey. Aids Behav. 2024, 28, 1594–1600. [Google Scholar] [CrossRef] [PubMed]
- Khan, A.; Sabbar, Y.; Din, A. Stochastic modeling of the Monkeypox 2022 epidemic with cross-infection hypothesis in a highly disturbed environment. Math. Biosci. Eng. 2022, 19, 13560–13581. [Google Scholar] [CrossRef] [PubMed]
- Kumar, P.; Chaudhary, B.; Yadav, N.; Devi, S.; Pareek, A.; Alla, S.; Kajal, F.; Nowrouzi-Kia, B.; Chattu, V.K.; Gupta, M.M. Recent Advances in Research and Management of Human Monkeypox Virus: An Emerging Global Health Threat. Viruses 2023, 15, 937. [Google Scholar] [CrossRef]
- Liu, K.; Nie, G.; Jiao, S.; Gao, B.; Ma, H.; Fu, J.; Wu, G. Research on fault diagnosis method of vehicle cable terminal based on time series segmentation for graph neural network model. Measurement 2024, 237, 114999. [Google Scholar] [CrossRef]
- Alzubaidi, A.M.; Othman, H.A.; Ullah, S.; Ahmad, N.; Alam, M.M. Analysis of Monkeypox viral infection with human to animal transmission via a fractional and Fractal-fractional operators with power law kernel. Math. Biosci. Eng. 2023, 20, 6666–6690. [Google Scholar] [CrossRef]
- Al-Shomrani, M.M.; Musa, S.S.; Yusuf, A. Unfolding the transmission dynamics of monkeypox virus: An epidemiological modelling analysis. Mathematics 2023, 11, 1121. [Google Scholar] [CrossRef]
- Alshehri, A.; Ullah, S. Optimal control analysis of Monkeypox disease with the impact of environmental transmission. Aims Math. 2023, 8, 16926–16960. [Google Scholar] [CrossRef]
- Minhaj, F.S.; Singh, V.; Cohen, S.E.; Townsend, M.B.; Scott, H.; Szumowski, J.; Hare, C.B.; Upadhyay, P.; Reddy, J.; Alexander, B.; et al. Prevalence of Undiagnosed Monkeypox Virus Infections during Global Mpox Outbreak, United States, June-September 2022. Emerg. Infect. Dis. 2023, 29, 2307–2314. [Google Scholar] [CrossRef] [PubMed]
- Spirito, F.; Guida, A.; Caponio, V.C.A.; Lo Muzio, L. Monkeypox: A New Challenge for Global Health System? Life 2023, 13, 1250. [Google Scholar] [CrossRef] [PubMed]
- Cooper, L.N.; Radunsky, A.P.; Hanna, J.J.; Most, Z.M.; Perl, T.M.; Lehmann, C.U.; Medford, R.J. Analyzing an Emerging Pandemic on Twitter: Monkeypox. Open Forum Infect. Dis. 2023, 10, ofad142. [Google Scholar] [CrossRef] [PubMed]
- Gaertner, F.; Preissner, S.; Petri, W.A., Jr.; Atolani, O.; Heil, M.; Nahles, S.; Preissner, R.; Hertel, M. Comparison of the risk of hospital admission, need for ventilation, sepsis, pneumonitis, and death among the recent monkeypox outbreak and historical outbreaks. BMC Infect. Dis. 2023, 23, 610. [Google Scholar] [CrossRef] [PubMed]
- Yasmin, F.; Hassan, M.M.; Zaman, S.; Aung, S.T.; Karim, A.; Azam, S. A Forecasting Prognosis of the Monkeypox Outbreak Based on a Comprehensive Statistical and Regression Analysis. Computation 2022, 10, 177. [Google Scholar] [CrossRef]
- Hussain, S.; Ghouse, S. Detection and prediction of monkey pox disease by enhanced convolutional neural network approach. Int. J. Public Health Sci. 2023, 12, 673. [Google Scholar] [CrossRef]
- Iftikhar, H.; Daniyal, M.; Qureshi, M.; Tawaiah, K.; Ansah, R.K.; Afriyie, J.K. A hybrid forecasting technique for infection and death from the mpox virus. Digit. Health 2023, 9, 20552076231204748. [Google Scholar] [CrossRef]
- Manohar, B.; Das, R. Artificial Neural Networks for the Prediction of Monkeypox Outbreak. Trop. Med. Infect. Dis. 2022, 7, 424. [Google Scholar] [CrossRef]
- Muñoz-Saavedra, L.; Escobar-Linero, E.; Civit-Masot, J.; Luna-Perejón, F.; Civit, A.; Domínguez-Morales, M. A Robust Ensemble of Convolutional Neural Networks for the Detection of Monkeypox Disease from Skin Images. Sensors 2023, 23, 7134. [Google Scholar] [CrossRef]
- Eliwa, E.H.I.; El Koshiry, A.M.; Abd El-Hafeez, T.; Farghaly, H.M. Utilizing convolutional neural networks to classify monkeypox skin lesions. Sci. Rep. 2023, 13, 14495. [Google Scholar] [CrossRef] [PubMed]
- Munir, T.; Khan, M.; Cheema, S.A.; Khan, F.; Usmani, A.; Nazir, M. Time series analysis and short-term forecasting of monkeypox outbreak trends in the 10 major affected countries. BMC Infect. Dis. 2024, 24, 16. [Google Scholar] [CrossRef] [PubMed]
- Pathan, R.K.; Uddin, M.A.; Paul, A.M.; Uddin, M.I.; Hamd, Z.Y.; Aljuaid, H.; Khandaker, M.U. Monkeypox genome mutation analysis using a time series model based on long short-term memory. PLoS ONE 2023, 18, e0290045. [Google Scholar] [CrossRef] [PubMed]
- Iftikhar, H.; Khan, M.; Khan, M.S.; Khan, M. Short-Term Forecasting of Monkeypox Cases Using a Novel Filtering and Combining Technique. Diagnostics 2023, 13, 1923. [Google Scholar] [CrossRef]
- Priyadarshini, I.; Mohanty, P.; Kumar, R.; Taniar, D. Monkeypox Outbreak Analysis: An Extensive Study Using Machine Learning Models and Time Series Analysis. Computers 2023, 12, 36. [Google Scholar] [CrossRef]
- Iftikhar, H.; Khan, M.; Khan, Z.; Khan, F.; Alshanbari, H.M.; Ahmad, Z. A comparative analysis of machine learning models: A case study in predicting chronic kidney disease. Sustainability 2023, 15, 2754. [Google Scholar] [CrossRef]
- Dada, E.G.; Oyewola, D.O.; Joseph, S.B.; Emebo, O.; Oluwagbemi, O.O. Ensemble Machine Learning for Monkeypox Transmission Time Series Forecasting. Appl. Sci. 2022, 12, 12128. [Google Scholar] [CrossRef]
- Qureshi, M.; Khan, S.; Bantan, R.A.R.; Daniyal, M.; Elgarhy, M.; Marzo, R.R.; Lin, Y. Modeling and Forecasting Monkeypox Cases Using Stochastic Models. J. Clin. Med. 2022, 11, 6555. [Google Scholar] [CrossRef]
- Shah, I.; Iftikhar, H.; Ali, S. Modeling and forecasting medium-term electricity consumption using component estimation technique. Forecasting 2020, 2, 163–179. [Google Scholar] [CrossRef]
- Iftikhar, H.; Bibi, N.; Canas Rodrigues, P.; López-Gonzales, J.L. Multiple novel decomposition techniques for time series forecasting: Application to monthly forecasting of electricity consumption in Pakistan. Energies 2023, 16, 2579. [Google Scholar] [CrossRef]
- Shah, I.; Iftikhar, H.; Ali, S. Modeling and forecasting electricity demand and prices: A comparison of alternative approaches. J. Math. 2022, 2022, 3581037. [Google Scholar] [CrossRef]
- Diebold, F.X.; Mariano, R.S. Comparing predictive accuracy. J. Bus. Econ. Stat. 2002, 20, 134–144. [Google Scholar] [CrossRef]
- Iftikhar, H.; Khan, M.; Turpo-Chaparro, J.E.; Rodrigues, P.C.; López-Gonzales, J.L. Forecasting stock prices using a novel filtering-combination technique: Application to the Pakistan stock exchange. Aims Math. 2024, 9, 3264–3288. [Google Scholar] [CrossRef]
- Alshanbari, H.M.; Iftikhar, H.; Khan, F.; Rind, M.; Ahmad, Z.; El-Bagoury, A.A.A.H. On the implementation of the artificial neural network approach for forecasting different healthcare events. Diagnostics 2023, 13, 1310. [Google Scholar] [CrossRef] [PubMed]
- Iftikhar, H.; Zafar, A.; Turpo-Chaparro, J.E.; Canas Rodrigues, P.; López-Gonzales, J.L. Forecasting day-ahead brent crude oil prices using hybrid combinations of time series models. Mathematics 2023, 11, 3548. [Google Scholar] [CrossRef]
S.No | Error | Formula |
---|---|---|
1 | MAE | |
2 | MAPE | |
3 | RMSE | |
4 | RMSLE |
Estimator | Minimum | 25% | 50% | Mean | 75% | Maximum | Variance | Standard Deviation | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|---|---|---|
Brazil | 1.00 | 3705.50 | 9606.00 | 7418.42 | 10,758.00 | 10,915.00 | 17,131,328.25 | 4139.00 | −0.86 | 2.06 |
France | 17.00 | 2889.00 | 4102.00 | 3329.43 | 4128.00 | 4144.00 | 1,848,070.37 | 1359.44 | −1.49 | 3.59 |
Spain | 142.00 | 6284.00 | 7392.00 | 6155.19 | 7531.75 | 7549.00 | 5,487,833.98 | 2342.61 | −1.64 | 4.13 |
USA | 25.00 | 16,965.00 | 28,809.00 | 22,030.52 | 29,933.00 | 30,154.00 | 125,647,213.44 | 11,209.25 | −1.11 | 2.52 |
World | 756.00 | 47,108.75 | 80,054.50 | 63,284.54 | 85,658.75 | 87,294.00 | 894,676,742.03 | 29,911.15 | −1.06 | 2.53 |
log (Brazil) | 0.00 | 8.21 | 9.17 | 8.05 | 9.28 | 9.30 | 5.67 | 2.38 | −2.25 | 7.10 |
log (France) | 2.83 | 7.97 | 8.32 | 7.84 | 8.33 | 8.33 | 1.13 | 1.06 | −2.52 | 8.73 |
log (Spain) | 4.96 | 8.75 | 8.91 | 8.50 | 8.93 | 8.93 | 0.85 | 0.92 | −2.44 | 7.89 |
log (USA) | 3.22 | 9.74 | 10.27 | 9.37 | 10.31 | 10.31 | 3.34 | 1.83 | −2.11 | 6.24 |
log (World) | 6.63 | 10.76 | 11.29 | 10.72 | 11.36 | 11.38 | 1.29 | 1.14 | −2.03 | 6.12 |
Test | Nonstatinay Outcomes | |
---|---|---|
Series | Statistic | p-value |
Brazil | 1.3999 | −0.9900 |
France | −13.3770 | −0.0086 |
Spain | −1.7358 | −0.6881 |
USA | −15.9970 | −0.0061 |
World | −2.2683 | −0.4635 |
log (Brazil) | −17.6770 | −0.0041 |
log (France) | 0.3030 | −0.9910 |
log (Spain) | −14.6470 | −0.0049 |
log (USA) | 0.1666 | −0.9919 |
log (World) | −19.83 | −0.0056 |
Brazil | ||||
Model | MAPE | MAE | RMLSE | RMSE |
AR | 0.000037 | 0.4055247 | 0.0001216 | 1.332314 |
ARIMA | 0.00041 | 4.4930502 | 0.0005071 | 5.557751 |
ESM | 0.0000798 | 0.874099 | 0.0001807 | 1.98015 |
NPAR | 0.0002975 | 3.2604158 | 0.0003833 | 4.199699 |
Theta | 0.0000156 | 0.1711713 | 0.0001092 | 1.196843 |
EnsE | 0.00016798 | 1.8408522 | 0.00026038 | 2.8533514 |
EnsT | 0.0000121 | 0.1708917 | 0.0001002 | 1.18923 |
EnsV | 0.0000111 | 0.1681917 | 0.0000992 | 1.1861 |
France | ||||
Model | MAPE | MAE | RMLSE | RMSE |
AR | 0.00002361 | 0.00031809 | 0.00000107 | 0.00086441 |
ARIMA | 0.00000025 | 0.00000309 | 0.00000216 | 0.00000649 |
ESM | 0.00001608 | 0.06575863 | 0.00002243 | 0.09174509 |
NPAR | 0.00040145 | 1.64194309 | 0.00040924 | 1.67457669 |
Theta | 0.00000034 | 0.00137947 | 0.00000262 | 0.01071651 |
EnsE | 0.000088326 | 0.341923274 | 0.000087104 | 0.355713038 |
EnsT | 0.00000023 | 0.00000219 | 0.00000201 | 0.00000581 |
EnsV | 0.00000019 | 0.00000199 | 0.00000191 | 0.00000553 |
Spain | ||||
Model | MAPE | MAE | RMLSE | RMSE |
AR | 0.00006945 | 0.5258639 | 0.00026101 | 1.977389 |
ARIMA | 0.00007903 | 0.5984467 | 0.00025369 | 1.921771 |
ESM | 0.00004015 | 0.3040281 | 0.00020946 | 1.586085 |
NPAR | 0.00011278 | 0.8540059 | 0.00031197 | 2.362472 |
Theta | 0.00005783 | 0.4378565 | 0.00021667 | 1.640763 |
EnsE | 0.000071848 | 0.54404022 | 0.00025056 | 1.897696 |
EnsT | 0.00003758 | 0.2845781 | 0.00020426 | 1.546656 |
EnsV | 0.00003314 | 0.2417941 | 0.00019113 | 1.498126 |
USA | ||||
Model | MAPE | MAE | RMLSE | RMSE |
AR | 0.00022263 | 6.778124 | 0.00045471 | 13.8483 |
ARIMA | 0.00033699 | 10.259437 | 0.00047854 | 14.57806 |
ESM | 0.00064046 | 19.507443 | 0.00085277 | 26.00396 |
NPAR | 0.0026346 | 20.142995 | 0.00093493 | 33.12454 |
Theta | 0.00010858 | 3.306309 | 0.00042027 | 12.79745 |
EnsE | 0.000788652 | 11.9988616 | 0.000628244 | 20.070462 |
EnsT | 0.00010219 | 3.100101 | 0.00040191 | 12.12132 |
EnsV | 0.00010156 | 2.996107 | 0.00039027 | 11.99131 |
World | ||||
Model | MAPE | MAE | RMLSE | RMSE |
AR | 0.00039134 | 34.98612 | 0.00085887 | 76.997 |
ARIMA | 0.0009302 | 83.32306 | 0.00161852 | 145.44294 |
ESM | 0.00026482 | 23.63597 | 0.00080848 | 72.37218 |
NPAR | 0.00188915 | 168.46651 | 0.0024785 | 221.42033 |
Theta | 0.00045734 | 40.91939 | 0.0009391 | 84.17509 |
EnsE | 0.00078657 | 70.26621 | 0.001340694 | 120.081508 |
EnsT | 0.00025961 | 23.17451 | 0.00080099 | 71.71362 |
EnsV | 0.00021311 | 21.92421 | 0.00079817 | 70.9131 |
Brazil | ||||||||
Model | AR | ARIMA | ESM | NPAR | Theta | EnsE | EnsT | EnsV |
AR | 0 | 0.9038 | 0.86576 | 0.90261 | 0.11822 | 0.89295 | 0.1206 | 0.02124 |
ARIMA | 0.0962 | 0 | 0.095 | 0.09497 | 0.09634 | 0.09441 | 0.09635 | 0.04636 |
ESM | 0.13424 | 0.905 | 0 | 0.90497 | 0.13173 | 0.90073 | 0.13206 | 0.03214 |
NPAR | 0.09739 | 0.90503 | 0.09503 | 0 | 0.09765 | 0.09371 | 0.09769 | 0.0377 |
Theta | 0.88178 | 0.90366 | 0.86827 | 0.90235 | 0 | 0.89248 | 0.19405 | 0.08491 |
EnsE | 0.10705 | 0.90559 | 0.09927 | 0.90629 | 0.10752 | 0 | 0.10764 | 0.0767 |
EnsT | 0.8794 | 0.90365 | 0.86794 | 0.90231 | 0.80595 | 0.89236 | 0 | 0.16451 |
EnsV | 0.87876 | 0.90364 | 0.86786 | 0.9023 | 0.81509 | 0.89233 | 0.83549 | 0 |
France | ||||||||
Model | AR | ARIMA | ESM | NPAR | Theta | EnsE | EnsT | EnsV |
AR | 0 | 0.1528 | 0.89355 | 0.90909 | 0.8088 | 0.90891 | 0.16157 | 0.05157 |
ARIMA | 0.8472 | 0 | 0.89355 | 0.90909 | 0.80892 | 0.90891 | 0.16814 | 0.05214 |
ESM | 0.10645 | 0.10645 | 0 | 0.9091 | 0.1059 | 0.90906 | 0.10645 | 0.04645 |
NPAR | 0.09091 | 0.09091 | 0.0909 | 0 | 0.09091 | 0.0909 | 0.09091 | 0.02091 |
Theta | 0.1912 | 0.19108 | 0.8941 | 0.90909 | 0 | 0.90891 | 0.19098 | 0.01098 |
EnsE | 0.09109 | 0.09109 | 0.09094 | 0.9091 | 0.09109 | 0 | 0.09109 | 0.0109 |
EnsT | 0.83843 | 0.83186 | 0.89355 | 0.90909 | 0.80902 | 0.90891 | 0 | 0.10876 |
EnsV | 0.83843 | 0.83186 | 0.89355 | 0.90909 | 0.80902 | 0.90891 | 0.89124 | 0 |
Spain | ||||||||
Model | AR | ARIMA | ESM | NPAR | Theta | EnsE | EnsT | EnsV |
AR | 0 | 0.31686 | 0.16252 | 0.86438 | 0.17723 | 0.21356 | 0.16323 | 0.05276 |
ARIMA | 0.68314 | 0 | 0.1435 | 0.85099 | 0.15514 | 0.09668 | 0.14585 | 0.04703 |
ESM | 0.83748 | 0.8565 | 0 | 0.85315 | 0.89717 | 0.8492 | 0.17161 | 0.04404 |
NPAR | 0.13562 | 0.14901 | 0.14685 | 0 | 0.15109 | 0.14476 | 0.14771 | 0.04814 |
Theta | 0.82277 | 0.84486 | 0.10283 | 0.84891 | 0 | 0.83356 | 0.12132 | 0.03136 |
EnsE | 0.78644 | 0.90332 | 0.1508 | 0.85524 | 0.16644 | 0 | 0.15276 | 0.0332 |
EnsT | 0.83677 | 0.85415 | 0.82839 | 0.85229 | 0.87868 | 0.84724 | 0 | 0.15806 |
EnsV | 0.83724 | 0.85297 | 0.83596 | 0.85186 | 0.86864 | 0.84668 | 0.84194 | 0 |
USA | ||||||||
Model | AR | ARIMA | ESM | NPAR | Theta | EnsE | EnsT | EnsV |
AR | 0 | 0.87614 | 0.90392 | 0.88212 | 0.0928 | 0.88923 | 0.09259 | 0.0135 |
ARIMA | 0.12386 | 0 | 0.89914 | 0.87477 | 0.10621 | 0.85232 | 0.09628 | 0.02522 |
ESM | 0.09608 | 0.10086 | 0 | 0.82032 | 0.09496 | 0.09167 | 0.09569 | 0.03581 |
NPAR | 0.11788 | 0.12523 | 0.17968 | 0 | 0.11499 | 0.12034 | 0.11566 | 0.04574 |
Theta | 0.9072 | 0.89379 | 0.90504 | 0.88501 | 0 | 0.89675 | 0.17506 | 0.04038 |
EnsE | 0.11077 | 0.14768 | 0.90833 | 0.87966 | 0.10325 | 0 | 0.10556 | 0.04586 |
EnsT | 0.90741 | 0.90372 | 0.90431 | 0.88434 | 0.82494 | 0.89444 | 0 | 0.14791 |
EnsV | 0.9065 | 0.90478 | 0.90419 | 0.88426 | 0.82962 | 0.89414 | 0.85209 | 0 |
World | ||||||||
Model | AR | ARIMA | ESM | NPAR | Theta | EnsE | EnsT | EnsV |
AR | 0 | 0.87949 | 0.0909 | 0.90117 | 0.88117 | 0.88653 | 0.09156 | 0.0228 |
ARIMA | 0.12051 | 0 | 0.11744 | 0.90647 | 0.12065 | 0.13134 | 0.11756 | 0.01748 |
ESM | 0.9091 | 0.88256 | 0 | 0.90144 | 0.89973 | 0.89038 | 0.14314 | 0.02083 |
NPAR | 0.09883 | 0.09353 | 0.09856 | 0 | 0.09849 | 0.09662 | 0.09861 | 0.04863 |
Theta | 0.11883 | 0.87935 | 0.10027 | 0.90151 | 0 | 0.88731 | 0.10136 | 0.00168 |
EnsE | 0.11347 | 0.86866 | 0.10962 | 0.90338 | 0.11269 | 0 | 0.10985 | 0.00983 |
EnsT | 0.90844 | 0.88244 | 0.85686 | 0.90139 | 0.89864 | 0.89015 | 0 | 0.08811 |
EnsV | 0.90772 | 0.88252 | 0.87917 | 0.90137 | 0.89832 | 0.89017 | 0.89189 | 0 |
Area | Brazil | France | Spain | USA | World | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Date | Actual | Forecasted | PFE | Actual | Forecasted | PFE | Actual | Forecasted | PFE | Actual | Forecasted | PFE | Actual | Forecasted | PFE |
1 May 2023 | 10,915 | 10,918 | 0.02290 | 4144 | 4145 | 0.02413 | 7549 | 7550 | 0.01325 | 30,154 | 30,156 | 0.00663 | 87,305 | 87,320 | 0.01718 |
2 May 2023 | 10,915 | 10,918 | 0.02749 | 4146 | 4146 | 0.00000 | 7551 | 7552 | 0.01324 | 30,154 | 30,157 | 0.00995 | 87,367 | 87,275 | 0.10530 |
3 May 2023 | 10,920 | 10,922 | 0.01832 | 4146 | 4147 | 0.02412 | 7551 | 7554 | 0.03973 | 30,154 | 30,158 | 0.01327 | 87,373 | 87,296 | 0.08813 |
4 May 2023 | 10,920 | 10,924 | 0.03663 | 4146 | 4148 | 0.04824 | 7551 | 7554 | 0.03973 | 30,154 | 30,158 | 0.01327 | 87,425 | 87,419 | 0.00686 |
5 May 2023 | 10,920 | 10,924 | 0.03663 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,154 | 30,158 | 0.01327 | 87,425 | 87,428 | 0.00343 |
6 May 2023 | 10,920 | 10,924 | 0.03663 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,154 | 30,158 | 0.01327 | 87,431 | 87,439 | 0.00915 |
7 May 2023 | 10,920 | 10,924 | 0.03663 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,154 | 30,158 | 0.01327 | 87,431 | 87,448 | 0.01944 |
8 May 2023 | 10,920 | 10,925 | 0.04579 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,154 | 30,158 | 0.01327 | 87,438 | 87,452 | 0.01601 |
9 May 2023 | 10,920 | 10,925 | 0.04579 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,188 | 30,160 | 0.09275 | 87,509 | 87,472 | 0.04228 |
10 May 2023 | 10,929 | 10,930 | 0.00915 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,188 | 30,169 | 0.06294 | 87,510 | 87,489 | 0.02400 |
11 May 2023 | 10,929 | 10,930 | 0.00915 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,188 | 30,176 | 0.03975 | 87,556 | 87,524 | 0.03655 |
12 May 2023 | 10,929 | 10,930 | 0.00915 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,188 | 30,181 | 0.02319 | 87,557 | 87,550 | 0.00799 |
13 May 2023 | 10,929 | 10,931 | 0.01830 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,188 | 30,189 | 0.00331 | 87,571 | 87,559 | 0.01370 |
14 May 2023 | 10,929 | 10,931 | 0.01830 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,188 | 31,092 | 2.99457 | 87,576 | 87,568 | 0.00913 |
15 May 2023 | 10,929 | 10,931 | 0.01830 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,188 | 30,194 | 0.01988 | 87,576 | 87,579 | 0.00343 |
16 May 2023 | 10,929 | 10,932 | 0.02745 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,194 | 30,196 | 0.00662 | 87,615 | 87,599 | 0.01826 |
17 May 2023 | 10,941 | 10,938 | 0.02742 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,194 | 30,200 | 0.01987 | 87,615 | 87,612 | 0.00342 |
18 May 2023 | 10,941 | 10,938 | 0.02742 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,194 | 30,204 | 0.03312 | 87,637 | 87,622 | 0.01712 |
19 May 2023 | 10,941 | 10,942 | 0.00914 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,194 | 30,206 | 0.03974 | 87,642 | 87,639 | 0.00342 |
20 May 2023 | 10,941 | 10,942 | 0.00914 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,194 | 30,208 | 0.04637 | 87,642 | 87,650 | 0.00913 |
21 May 2023 | 10,941 | 10,943 | 0.01828 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,194 | 30,208 | 0.04637 | 87,642 | 87,661 | 0.02168 |
22 May 2023 | 10,941 | 10,944 | 0.02742 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,194 | 30,210 | 0.05299 | 87,667 | 87,672 | 0.00570 |
23 May 2023 | 10,941 | 10,945 | 0.03656 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,225 | 30,215 | 0.03309 | 87,710 | 87,689 | 0.02394 |
24 May 2023 | 10,941 | 10,945 | 0.03656 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,225 | 30,219 | 0.01985 | 87,711 | 87,704 | 0.00798 |
25 May 2023 | 10,941 | 10,945 | 0.03656 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,225 | 30,213 | 0.03970 | 87,733 | 87,725 | 0.00912 |
26 May 2023 | 10,941 | 10,945 | 0.03656 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,225 | 30,218 | 0.02316 | 87,733 | 87,739 | 0.00684 |
27 May 2023 | 10,941 | 10,945 | 0.03656 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,225 | 30,223 | 0.00662 | 87,733 | 87,741 | 0.00912 |
28 May 2023 | 10,941 | 10,945 | 0.03656 | 4146 | 4149 | 0.07236 | 7551 | 7554 | 0.03973 | 30,225 | 30,228 | 0.00993 | 87,733 | 87,748 | 0.01710 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Cuba, W.M.; Huaman Alfaro, J.C.; Iftikhar, H.; López-Gonzales, J.L. Modeling and Analysis of Monkeypox Outbreak Using a New Time Series Ensemble Technique. Axioms 2024, 13, 554. https://doi.org/10.3390/axioms13080554
Cuba WM, Huaman Alfaro JC, Iftikhar H, López-Gonzales JL. Modeling and Analysis of Monkeypox Outbreak Using a New Time Series Ensemble Technique. Axioms. 2024; 13(8):554. https://doi.org/10.3390/axioms13080554
Chicago/Turabian StyleCuba, Wilfredo Meza, Juan Carlos Huaman Alfaro, Hasnain Iftikhar, and Javier Linkolk López-Gonzales. 2024. "Modeling and Analysis of Monkeypox Outbreak Using a New Time Series Ensemble Technique" Axioms 13, no. 8: 554. https://doi.org/10.3390/axioms13080554