Visible and Real Sizes of New COVID-19 Pandemic Waves in Ukraine
DOI:
https://doi.org/10.20535/ibb.2021.5.2.230487Keywords:
COVID-19 pandemic, epidemic dynamics in Ukraine, mathematical modeling of infection diseases, SIR model, Parameter identification, statistical methodsAbstract
Background. To simulate the COVID-19 pandemic dynamics, various data sets and different mathematical models can be used. In particular, previous simulations for Ukraine were based on smoothing of the dependence of the number of cases on time, classical and the generalized SIR (susceptible-infected-removed) models. Different simulation and comparison methods were based on official accumulated number of laboratory confirmed cases and the data reported by Johns Hopkins University. Since both datasets are incomplete (a very large percentage of infected persons are asymptomatic), the accuracy of calculations and predictions is limited. In this paper we will try to assess the degree of data incompleteness and correct the relevant forecasts.
Objective. We aimed to estimate the real sizes of two new epidemic waves in Ukraine and compare them with visible dynamics based on the official number of laboratory confirmed cases. We also aimed to estimate the epidemic durations and final numbers of cases.
Methods. In this study we use the generalized SIR model for the epidemic dynamics and its known exact solution. The known statistical approach is adopted in order to identify both the degree of data incompleteness and parameters of SIR model.
Results. We have improved the method of estimating the unknown parameters of the generalized SIR model and calculated the optimal values of the parameters. In particular, the visibility coefficients and the optimal values of the model parameters were estimated for two pandemic waves in Ukraine occurred in December 2020–March 2021. The real number of cases and the real number of patients spreading the infection versus time were calculated. Predictions of the real final sizes and durations of the pandemic in Ukraine are presented. If current trends continue, the end of the pandemic should be expected no earlier than in August 2022.
Conclusions. New method of the unknown parameters identification for the generalized SIR model was proposed, which allows estimating the coefficients of data incompleteness as well. Its application for two pandemic waves in Ukraine has demonstrated that the real number of COVID-19 cases is approximately four times higher than those shown in official statistics. Probably, this situation is typical for other countries. The reassessments of the COVID-19 pandemic dynamics in other countries and clarification of world forecasts are necessary.
References
Coronavirus Disease (COVID-19): Situation Reports. Who.int. Available from: https://www.who.int/emergencies/diseases/novel-coronavirus-2019/situation-reports/
Li Q, Guan X, Wu P, Wang X, Zhou L, Tong Y, et al. Early Transmission Dynamics in Wuhan, China, of Novel Coronavirus-Infected Pneumonia. New England Journal of Medicine. 2020;382(13):1199-207. DOI: 10.1056/NEJMoa2001316
Zhen L. Coronavirus: "strange pneumonia" seen in Lombardy in November, leading Italian doctor says. Scmp.com, 2020. Available from: https://www.scmp.com/news/china/society/article/3076334/coronavirus-strange-pneumonia-seen-lombardy-november-leading
Lescure F-X, Bouadma L, Nguyen D, Parisey M, Wicky P-H, Behillil S, et al. Clinical and virological data of the first cases of COVID-19 in Europe: a case series. The Lancet Infectious Diseases. 2020;20(6):697-706. DOI: 10.1016/S1473-3099(20)30200-0
Militärweltspiele in Wuhan: „Wir sind alle erkrankt“. Faz.net, 2020. Available from: https://faz.net/aktuell/sport/mehr-sport/militaerweltspiele-2019-in-wuhan-damals-schon-corona-faelle-16758894.html
Weinberger DM, Cohen T, Crawford FW, Mostashari F, Olson D, Pitzer VE, et al. Estimating the early death toll of COVID-19 in the United States. medRxiv, 2020. Preprint. DOI: 10.1101/2020.04.15.20066431
Nesteruk I. Simulations and Predictions of COVID-19 Pandemic With the Use of SIR Model. Innovative Biosystems and Bioengineering. 2020;4(2):110-21. DOI: 10.20535/ibb.2020.4.2.204274
Nesteruk I. COVID-19 Pandemic Dynamics: Mathematical Simulations. Singapore: Springer Singapore; 2021. DOI: 10.1007/978-981-33-6416-5
Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 1927;115(772):700-21. DOI: 10.1098/rspa.1927.0118
Murray JD (eds). Mathematical Biology: I. An Introduction, vol. 17. New York, NY: Springer New York; 2002. DOI: 10.1007/b98868
Langemann D, Nesteruk I, Prestin J. Comparison of mathematical models for the dynamics of the Chernivtsi children disease. Mathematics and Computers in Simulation. 2016;123:68-79. DOI: 10.1016/j.matcom.2016.01.003
Nesteruk IG. Statistics Based Models for the Dynamics of Chernivtsi Children Disease. Research Bulletin of the National Technical University of Ukraine "Kyiv Politechnic Institute". 2017;(5):26-34. DOI: 10.20535/1810-0546.2017.5.108577
Nesteruk I. Statistics-Based Predictions of Coronavirus Epidemic Spreading in Mainland China. Innovative Biosystems and Bioengineering. 2020;4(1):13-8. DOI: 10.20535/ibb.2020.4.1.195074
Coronavirus in Ukraine: Statistics, map of infections, graphs. Index.minfin.com.ua. Available from: https://index.minfin.com.ua/ua/reference/coronavirus/ukraine/
Cabinet of Ministers of Ukraine – Home. Kmu.gov.ua. Available from: https://www.kmu.gov.ua/
COVID-19 Data Repository by the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University. Github.com. Available from: https://github.com/CSSEGISandData/COVID-19
Nesteruk IG, Kydybyn IB, Demelmair G. Global stabilization trends of COVID-19 pandemic. KPI Science News. 2020;(2):55-63. DOI: 10.20535/kpi-sn.2020.2.205124
Nesteruk I. Dynamics of the coronavirus pandemic in Italy and some global predictions. Journal of Allergy & Infectious Diseases. 2020;1(1):5-8. DOI: 10.46439/allergy.1.002
Nesteruk I, Benlagha N. Predictions of COVID-19 Pandemic Dynamics in Ukraine and Qatar Based on Generalized SIR Model. Innovative Biosystems and Bioengineering. 2021;5(1):37-46. DOI: 10.20535/ibb.2021.5.1.228605
An experiment with mass testing for COVID-19 was conducted in Khmelnytskyi. Podillyanews.com. Available from: https://podillyanews.com/2020/12/17/u-shkolah-hmelnytskogo-provely-eksperyment-z-testuvannyam-na-covid-19
Nesteruk I. General SIR Model and Its Exact Solution. In: COVID-19 Pandemic Dynamics: Mathematical Simulations. Singapore: Springer Singapore; 2021, pp. 127-32. DOI: 10.1007/978-981-33-6416-5_9
Nesteruk I. Comparison of the First Waves of the COVID-19 Pandemic in Different Countries and Regions. In: COVID-19 Pandemic Dynamics: Mathematical Simulations. Singapore: Springer Singapore; 2021, pp. 89-107. DOI: 10.1007/978-981-33-6416-5_7
Draper NR, Smith H. Applied Regression Analysis. Wiley; 1998. DOI: 10.1002/9781118625590
Gazzola M, Argentina M, Mahadevan L. Scaling macroscopic aquatic locomotion. Nature Physics. 2014;10(10):758-61. DOI: 10.1038/nphys3078
Nesteruk I. Maximal Speed of Underwater Locomotion. Innovative Biosystems and Bioengineering. 2019;3(3):152-67. DOI: 10.20535/ibb.2019.3.3.177976
Nesteruk I. Procedures of Parameter Identification for the Waves of Epidemics. In: COVID-19 Pandemic Dynamics: Mathematical Simulations. Singapore: Springer Singapore; 2021, pp. 133-39. DOI: 10.1007/978-981-33-6416-5_10
Kottasová I, Etzler T. Slovakia tested most of the country in two days. Here's how they did it and what they found. CNN. 2020. Available from: https://edition.cnn.com/2020/11/02/europe/slovakia-mass-coronavirus-test-intl/index.html
Slovakia's Second Round of Coronavirus Tests Draws Large Crowds. Voice of America. 2020. Available from: https://www.voanews.com/covid-19-pandemic/slovakias-second-round-coronavirus-tests-draws-large-crowds
Nesteruk I. Classical SIR Model and the Exact Solution of Differential Equations. In: COVID-19 Pandemic Dynamics: Mathematical Simulations. Singapore: Springer Singapore; 2021, pp. 23-32. DOI: 10.1007/978-981-33-6416-5_4
Wu JT, Leung K, Leung GM. Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study. The Lancet. 2020;395(10225):689-97. DOI: 10.1016/S0140-6736(20)30260-9
Zhao S, Lin Q, Ran J, Musa SS, Yang G, Wang W, et al. Preliminary estimation of the basic reproduction number of novel coronavirus (2019-nCoV) in China, from 2019 to 2020: A data-driven analysis in the early phase of the outbreak. International Journal of Infectious Diseases. 2020;92:214-7. DOI: 10.1016/j.ijid.2020.01.050
Byass P. Eco-epidemiological assessment of the COVID-19 epidemic in China, January–February 2020. Global Health Action. 2020;13(1):1760490. DOI: 10.1080/16549716.2020.1760490
Tang B, Bragazzi NL, Li Q, Tang S, Xiao Y, Wu J. An updated estimation of the risk of transmission of the novel coronavirus (2019-nCov). Infectious Disease Modelling. 2020;5:248-55. DOI: 10.1016/j.idm.2020.02.001
Liu Y, Gayle AA, Wilder-Smith A, Rocklöv J. The reproductive number of COVID-19 is higher compared to SARS coronavirus. Journal of Travel Medicine. 2020;27(2):taaa021. DOI: https://doi.org/10.1093/jtm/taaa021
Kucharski AJ, Russell TW, Diamond C, Liu Y, Edmunds J, Funk S, et al. Early dynamics of transmission and control of COVID-19: a mathematical modelling study. The Lancet Infectious Diseases. 2020;20(5):553-8. DOI: 10.1016/S1473-3099(20)30144-4
Batista M. Estimation of the final size of the COVID-19 epidemic. medRxiv, 2020. Preprint. DOI: 10.1101/2020.02.16.20023606
Dehning J, Zierenberg J, Spitzner FP, Wibral M, Neto JP, Wilczek M, Priesemann V. Inferring change points in the COVID-19 spreading reveals the effectiveness of interventions. medRxiv, 2020. Preprint. DOI: 10.1101/2020.04.02.20050922
Chen Y, Cheng J, Jiang Y, Liu K. A Time Delay Dynamical Model for Outbreak of 2019-nCoV and the Parameter Identification. arXiv, 2020. Preprint. DOI: 10.48550/arXiv.2002.00418
Peng L, Yang W, Zhang D, Zhuge C, Hong L. Epidemic analysis of COVID-19 in China by dynamical modeling. medRxiv, 2020. Preprint. DOI: 10.1101/2020.02.16.20023465
Chang SL, Harding N, Zachreson C, Cliff OM, Prokopenko M. Modelling transmission and control of the COVID-19 pandemic in Australia. Nature Communications. 2020;11(1):5710. DOI: 10.1038/s41467-020-19393-6
Maier BF, Brockmann D. Effective containment explains subexponential growth in recent confirmed COVID-19 cases in China. Science. 2020;368(6492):742-6. DOI: 10.1126/science.abb4557
Wang L, Zhou Y, He J, Zhu B, Wang F, Tang L, et al. An epidemiological forecast model and software assessing interventions on the COVID-19 epidemic in China. Journal of Data Science. 2020;18(3):409-32. DOI: 10.6339/JDS.202007_18(3).0003
Chinazzi M, Davis JT, Ajelli M, Gioannini C, Litvinova M, Merler S, et al. The effect of travel restrictions on the spread of the 2019 novel coronavirus (COVID-19) outbreak. Science. 2020;368(6489):395-400. DOI: 10.1126/science.aba9757
Zhang Y, Jiang B, Yuan J, Tao Y. The impact of social distancing and epicenter lockdown on the COVID-19 epidemic in mainland China: A data-driven SEIQR model study. medRxiv, 2020. Preprint. DOI: 10.1101/2020.03.04.20031187
Ghanam R, Boone EL, Abdel-Salam A-SG. SEIRD Model for Qatar Covid-19 Outbreak: A Case Study. Letters in Biomathematics. 2021;8(1):19-28.
Udomsamuthirun P, Chanilkul G, Tongkhonburi P, Meesubthong C. The reproductive index from SEIR model of Covid-19 epidemic in Asean. medRxiv, 2020. Preprint. DOI: 10.1101/2020.04.24.20078287
Pereira IG, Guerin JM, Silva Júnior AG, Garcia GS, Piscitelli P, Miani A, et al. Forecasting Covid-19 Dynamics in Brazil: A Data Driven Approach. International Journal of Environmental Research and Public Health. 2020;17(14):5115. DOI: 10.3390/ijerph17145115
Linka K, Peirlinck M, Kuhl E. The reproduction number of COVID-19 and its correlation with public health interventions. Computational Mechanics. 2020;66(4):1035-50. DOI: 10.1007/s00466-020-01880-8
Distante C, Pereira IG, Goncalves LMG, Piscitelli P, Miani A. Forecasting Covid-19 outbreak progression in Italian regions: A model based on neural network training from Chinese data. medRxiv, 2020. Preprint. DOI: 10.1101/2020.04.09.20059055
Hamzah FAB, Lau CH, Nazri H, Ligot DV, Lee G, Tan CL, et al. CoronaTracker: World-wide COVID-19 Outbreak Data Analysis and Prediction. Bulletin of the World Health Organization. 2020. DOI: 10.2471/BLT.20.255695
Fanelli D, Piazza F. Analysis and forecast of COVID-19 spreading in China, Italy and France. Chaos, Solitons & Fractals. 2020;134:109761. DOI: 10.1016/j.chaos.2020.109761
Liu Z, Magal P, Seydi O, Webb G. A model to predict COVID-19 epidemics with applications to South Korea, Italy, and Spain. medRxiv, 2020. Preprint. DOI: 10.1101/2020.04.07.20056945
Bastos SB, Cajueiro DO. Modeling and forecasting the early evolution of the Covid-19 pandemic in Brazil. Scientific Reports. 2020;10(1):19457. DOI: 10.1038/s41598-020-76257-1
Grant A. Dynamics of COVID-19 epidemics: SEIR models underestimate peak infection rates and overestimate epidemic duration. medRxiv, 2020. Preprint. DOI: 10.1101/2020.04.02.20050674
Piccolomini EL, Zama F. Monitoring Italian COVID-19 spread by an adaptive SEIRD model. medRxiv, 2020. Preprint. DOI: 10.1101/2020.04.03.20049734
Bärwolff G. A Contribution to the Mathematical Modeling of the Corona/COVID-19 Pandemic. medRxiv, 2020. Preprint. DOI: 10.1101/2020.04.01.20050229
Distante C, Piscitelli P, Miani A. Covid-19 Outbreak Progression in Italian Regions: Approaching the Peak by the End of March in Northern Italy and First Week of April in Southern Italy. International Journal of Environmental Research and Public Health. 2020;17(9):3025. DOI: 10.3390/ijerph17093025
te Vrugt M, Bickmann J, Wittkowski R. Effects of social distancing and isolation on epidemic spreading modeled via dynamical density functional theory. Nature Communications. 2020;11(1):5576. DOI: 10.1038/s41467-020-19024-0
Yang Z, Zeng Z, Wang K, Wong S-S, Liang W, Zanin M, et al. Modified SEIR and AI prediction of the epidemics trend of COVID-19 in China under public health interventions. Journal of Thoracic Disease. 2020;12(3):165-74. DOI: 10.21037/jtd.2020.02.64
Roda WC, Varughese MB, Han D, Li MY. Why is it difficult to accurately predict the COVID-19 epidemic? Infectious Disease Modelling. 2020;5:271-81. DOI: 10.1016/j.idm.2020.03.001
Otunuga OM, Ogunsolu MO. Qualitative analysis of a stochastic SEITR epidemic model with multiple stages of infection and treatment. Infectious Disease Modelling. 2020;5:61-90. DOI: 10.1016/j.idm.2019.12.003
Chatterjee K, Chatterjee K, Kumar A, Shankar S. Healthcare impact of COVID-19 epidemic in India: A stochastic mathematical model. Medical Journal Armed Forces India. 2020;76(2):147-55. DOI: 10.1016/j.mjafi.2020.03.022
Ciufolini I, Paolozzi A. Mathematical prediction of the time evolution of the COVID-19 pandemic in Italy by a Gauss error function and Monte Carlo simulations. The European Physical Journal Plus. 2020;135(4):355. DOI: 10.1140/epjp/s13360-020-00383-y
Annas S, Isbar Pratama M, Rifandi M, Sanusi W, Side S. Stability analysis and numerical simulation of SEIR model for pandemic COVID-19 spread in Indonesia. Chaos, Solitons & Fractals. 2020;139:110072. DOI: 10.1016/j.chaos.2020.110072
Yadav RP, Verma R. A numerical simulation of fractional order mathematical modeling of COVID-19 disease in case of Wuhan China. Chaos, Solitons & Fractals. 2020;140:110124. DOI: 10.1016/j.chaos.2020.110124
Ng KY, Gui MM. COVID-19: Development of a robust mathematical model and simulation package with consideration for ageing population and time delay for control action and resusceptibility. Physica D: Nonlinear Phenomena. 2020;411:132599. DOI: 10.1016/j.physd.2020.132599
Ivorra B, Ferrández MR, Vela-Pérez M, Ramos AM. Mathematical modeling of the spread of the coronavirus disease 2019 (COVID-19) taking into account the undetected infections. The case of China. Communications in Nonlinear Science and Numerical Simulation. 2020;88:105303. DOI: 10.1016/j.cnsns.2020.105303
Tuan NH, Mohammadi H, Rezapour S. A mathematical model for COVID-19 transmission by using the Caputo fractional derivative. Chaos, Solitons & Fractals. 2020;140:110107. DOI: 10.1016/j.chaos.2020.110107
Sinkala M, Nkhoma P, Zulu M, Kafita D, Tembo R, Daka V. The COVID-19 Pandemic in Africa: Predictions using the SIR Model. medRxiv, 2020. Preprint. DOI: 10.1101/2020.06.01.20118893
Agbokou K, Gneyou K, Tcharie K. Investigation on the temporal evolution of the covid'19 pandemic: prediction for Togo. Open Journal of Mathematical Sciences. 2020;4:273-9. DOI: 10.30538/oms2020.0118
Pintér G, Felde I, Mosavi A, Ghamisi P, Gloaguen R. COVID-19 Pandemic Prediction for Hungary; A Hybrid Machine Learning Approach. Mathematics. 2020;8(6):890. DOI: 10.3390/math8060890
Rossman H, Shilo S, Meir T, Gorfine M, Shalit U, Segal E. Patterns of COVID-19 pandemic dynamics following deployment of a broad national immunization program. medRxiv, 2021. Preprint. DOI: 10.1101/2021.02.08.21251325
Furati KM, Sarumi IO, Khaliq AQM. Memory-Dependent Model for the Dynamics of COVID-19 Pandemic. medRxiv, 2020. Preprint. DOI: 10.1101/2020.06.26.20141242
Bosch J, Wilson A, O'Neil K, Zimmerman PA. COVID-19Predict – Predicting Pandemic Trends. medRxiv, 2020. Preprint. DOI: 10.1101/2020.09.09.20191593
Asad A, Srivastava S, Verma MK. Evolution of COVID-19 Pandemic in India. Transactions of the Indian National Academy of Engineering. 2020;5(4):711-18. DOI: 10.1007/s41403-020-00166-y
Aries N, Ounis H. Mathematical Modeling of COVID-19 Pandemic in the African Continent. medRxiv, 2020. Preprint. DOI: 10.1101/2020.10.10.20210427
Günther F, Bender A, Katz K, Kuchenhoff H, Hohle M. Nowcasting the COVID-19 pandemic in Bavaria. Biometrical Journal. 2021;63(3):490-502. DOI: 10.1002/bimj.202000112
Yang W, Shaff J, Shaman J. COVID-19 Transmission Dynamics and Effectiveness of Public Health Interventions in New York City during the 2020 Spring Pandemic Wave. medRxiv, 2020. Preprint. DOI: 10.1101/2020.09.08.20190710
Dickman R. A SEIR-like model with a time-dependent contagion factor describes the dynamics of the Covid-19 pandemic. medRxiv, 2020. Preprint. DOI: 10.1101/2020.08.06.20169557
Kundu LR, Ferdous MZ, Islam US, Sultana M. Forecasting the spread of COVID-19 pandemic in Bangladesh using ARIMA model. medRxiv, 2020. Preprint. DOI: 10.1101/2020.10.22.20217414
Barbastefano R, Carvalho D, Lippi MC, Pastore DH. A novel predictive mathematical model for COVID-19 pandemic with quarantine, contagion dynamics, and environmentally mediated transmission. medRxiv, 2020. Preprint. DOI: 10.1101/2020.07.27.20163063
Biswas MHA, Khatun MS, Paul AK, Khatun MR, Islam MA, Samad SA, Ghosh U. Modeling the Effective Control Strategy for the Transmission Dynamics of Global Pandemic COVID-19. medRxiv, 2020. Preprint. DOI: 10.1101/2020.04.22.20076158
Aviv-Sharon E, Aharoni A. Forecasting COVID-19 pandemic Severity in Asia. medRxiv, 2020. Preprint. DOI: 10.1101/2020.05.15.20102640
Bannur N, Maheshwari H, Jain S, Shetty S, Merugu S, Raval A. Adaptive COVID-19 Forecasting via Bayesian Optimization. In: Proceedings of the 3rd ACM India Joint International Conference on Data Science & Management of Data (8th ACM IKDD CODS & 26th COMAD), Bangalore, India; 2021, p. 432. DOI: 10.1145/3430984.3431047
Honfo SH, Taboe HB, Kakaï RG. Modeling COVID-19 dynamics in the sixteen West African countries. medRxiv, 2020. Preprint. DOI: 10.1101/2020.09.04.20188532
Chruściel PT, Szybka SJ. Universal properties of the dynamics of the Covid-19 pandemic. medRxiv, 2020. Preprint. DOI: 10.1101/2020.08.24.20181214
Reddy BRM, Singh A, Srivastava P. COVID-19 Transmission Dynamics in India with Extended SEIR Model. medRxiv, 2020. Preprint. DOI: 10.1101/2020.08.15.20175703
Huang J, Liu X, Zhang L, Yang K, Chen Y, Huang Z, et al. The amplified second outbreaks of global COVID-19 pandemic. medRxiv, 2020. Preprint. DOI: 10.1101/2020.07.15.20154161
Bhanot G, DeLisi C. Predictions for Europe for the Covid-19 pandemic from a SIR model. medRxiv, 2020. Preprint. DOI: 10.1101/2020.05.26.20114058
Ibrahim MA, Al-Najafi A. Modeling, Control, and Prediction of the Spread of COVID-19 Using Compartmental, Logistic, and Gauss Models: A Case Study in Iraq and Egypt. Processes. 2020;8(11):1400. DOI: 10.3390/pr8111400
Perone G. Comparison of ARIMA, ETS, NNAR and hybrid models to forecast the second wave of COVID-19 hospitalizations in Italy. SSRN Electronic Journal, 2020. Preprint. DOI: 10.2139/ssrn.3716343
Batistela CM, Correa DPF, Bueno AM, Piqueira JRC. SIRSi compartmental model for COVID-19 pandemic with immunity loss. Chaos, Solitons & Fractals. 2021;142:110388. DOI: 10.1016/j.chaos.2020.110388
Fahmy AE, El-desouky MM, Mohamed ASA. Epidemic Analysis of COVID-19 in Egypt, Qatar and Saudi Arabia using the Generalized SEIR Model. medRxiv, 2020. Preprint. DOI: 10.1101/2020.08.19.20178129
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2021 The autror(s)
This work is licensed under a Creative Commons Attribution 4.0 International License.
The ownership of copyright remains with the Authors.
Authors may use their own material in other publications provided that the Journal is acknowledged as the original place of publication and National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute” as the Publisher.
Authors are reminded that it is their responsibility to comply with copyright laws. It is essential to ensure that no part of the text or illustrations have appeared or are due to appear in other publications, without prior permission from the copyright holder.
IBB articles are published under Creative Commons licence:- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under CC BY 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
