Main Article Content
A severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) ap- peared in the Chinese region of Wuhan at the end of 2019. Since then, the virus spread to other countries, including most of Europe and USA. This work provides an overview on deterministic and stochastic models that have previously been proposed by us to study the transmission dy- namics of the Coronavirus Disease 2019 (COVID-19) in Europe and USA. Briefly, we describe realistic deterministic and stochastic models for the evolution of the COVID-19 pandemic, subject to the lockdown and quar- antine measures, which take into account the time-delay for recovery or death processes. Realistic dynamic equations for the entire process are derived by adopting the so-called kinetic-type reactions approach. The lockdown and the quarantine measures are modelled by some kind of in- hibitor reactions where susceptible and infected individuals can be trapped into inactive states. The dynamics for the recovered people is obtained by accounting people who are only traced back to hospitalised infected people. To model the role of the Hospitals we take inspiration from the Michaelis-Menten’s enzyme-substrate reaction model (the so-called MM reaction) where the enzyme is associated to the available hospital beds, the substrate to the infected people, and the product to the recovered peo- ple, respectively. In other words, everything happens as if the hospitals beds act as a catalyser in the hospital recovery process. The statistical properties of the models, in particular the relevant correlation functions and the probability density functions, have duly been evaluated. We val- idate our theoretical predictions with a large series of experimental data for Italy, Germany, France, Belgium and United States, and we also compare data for Italy and Belgium with the theoretical predictions of the logistic model. We have found that our predictions are in good agreement with the real world since the onset of COVID 19, contrary to the logistics model that only applies in the first days of the pandemic. In the final part of the work, we can find the (theoretical) relationships that should be satisfied to obtain the disappearance of the virus (corresponding to a value of the effective reproduction number of the infection lower than 1).
The Medical Research Archives grants authors the right to publish and reproduce the unrevised contribution in whole or in part at any time and in any form for any scholarly non-commercial purpose with the condition that all publications of the contribution include a full citation to the journal as published by the Medical Research Archives.
2. Bailey, Norman T. J., The mathematical theory of infectious diseases and its applications (2nd ed.). London: Griffin. ISBN 0-85264-231-8 (1975).
3. Sonia A. and Nunn, C. Infectious diseases in primates: behaviour, ecology and evolution. Oxford Series in Ecology and Evolution. Oxford [Oxford- shire]: Oxford University Press. ISBN 0-19-856585-2 (2006).
4. Vynnycky E. and White R. G., An Introduction to Infectious Disease Mod- elling. Oxford University Press. ISBN 978-0-19-856576-5 (eds. 2010).
5. Gleick J., Chaos: The Making of a New Science, Viking Press, New York (1987).
6. Coullet P., The Covid-19 epidemic as a simple dynamical system, Comptes Rendus M´ecanique, 0000, 1, no 0, p. 000-000 DOI unassigned yet, Under Review (2022).
7. Sonnino G,, The COVID-19 - The Infectious Disease Caused by the Latest Discovered Coronavirus (SARS-CoV-2), European Commission, ARES (2020)1530456. Available online: https://europa.eu/european- union/contact/write-to-us en (accessed on 16 March 2020).
8. Sonnino G. and Nardone P., Dynamics of the COVID-1 - Comparison be- tween the Theoretical Predictions and the Real Data, and Predictions about Returning to Normal Life, Ann Clin and Med Case Rep, 2020; 4:1-21 (9). ISSN 2639-8109. DOI: http://dx.doi.org/10.47829/ACMCR.2020.4902.ß
9. Ladyzhets B., What to Know About the Newest, Most Contagious Omicron Sub-variants, Time (15 June 2022). https://time.com/6187762/ba-4-ba-5- omicron-subvariants-symptoms-risk/
10. Sonnino G., Mora F., and Nardone P, A Stochastic Kinetic Type Reactions Model for COVID-19, MDPI-Mathematics: Math Mod and Ana in Bio and Med, 2021; 9, (Issue 11), 1221. https://doi.org/10.3390/math9111221
11. Sonnino G., Peeters P., and Nardone P, Modelling the Coronavirus Second Wave in Presence of the Lockdown and Quarantine Measures, International Conference on Complex Systems (CCS2020) -December 2020 (page 278). DOI reference: https://doi.org/10.5281/zenodo.4419178
12. Sonnino G., Peeters P., and Nardone P, Modelling the Spreading of the SARS-CoV-2 in Presence of the Lockdown and Quarantine Measures by a Kinetic-Type Reactions Approach, Oxford University Press, Math Med and Bio - A Journal of the IMA, 2021; 39:105–125, Issue 2, June 2022. https://doi.org/10.1093/imammb/dqab017
13. Prigogine I., Etude Thermodynamique des Ph´enom´enes Irr´eversibles, Th´ese d’Aggr´egation de l’Einseignement Sup´erieur de l’Universit´e Libre de Brux- elles (U.L.B.) (1947).
14. Prigogine I., Thermodynamics of Irreversible Processes, John Wiley & Sons 42 (1954).
15. History and Epidemiology of Global Smallpox Eradication (Archived 2016- 05-10 at the Wayback Machine), a module of the training course Smallpox: Disease, Prevention, and Intervention. The CDC and the World Health Organisation, (2001). Slide 17. This gives sources as Modified from Epid Rev, 15:265-302 (1993), Am J Prev Med, 2000; 20 (4S):88-153 (2001), MMWR 49(SS-9): 27-38.
16. Murray J.D., Mathematical Biology, Springer-Verlag Berlin, Heidelberg GmbH (1993).
17. Anderson R.M. and May R.M., Infectious diseases of humans, Oxford Uni- versity Press (1991).
18. https://populationeducation.org/what-doubling-time-and-how-it- calculated/
19. Steinbrecher G. (University of Craiova - Romania), A short sample in R- program able to compute the estimated value of R0 for 17 infectious diseases. 2020; Private Communication - 20th April 2020).
20. Piovani P., Coronavirus, il pediatra: Questa epidemia durer`a tre mesi, Il messaggero, (24 March 2020).
21. https://sites.me.ucsb.edu/ moehlis/APC514/tutorials/tutorial seasonal/node2.html
22. N. Mathur and G. Shaw, An empirical model on the dynamics of Covid-
19 spread in human population, Department of Theoretical Physics, Tata Institute of Fundamental Research, Mumbai (India) (2020) -
23. https://en.wikipedia.org/wiki/Logistic function
24. Gompertz, B., On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies, Phil Trans of the Royal Soc, 1825; 115:513-585.
25. Hilbe J/M., Practical Guide to Logistic Regression, CRC Press, Taylor & Francis Group, LLC (2015).
26. Il Sole 24 Ore, Coronavirus in Italia, i dati e la mappa (2020) - Website: https://lab24.ilsole24ore.com/coronavirus
27. Sciensano, Epidemologie des maladies infectueuses (2020) - Website: https://epidemio.wiv-isp.be/ID/Pages/2019-nCoV.aspx
28. Wikipedia, the free encyclopedia, 2020 coronavirus pandemic in Belgium,
https://en.wikipedia.org/wiki/2020 coronavirus pandemic in Belgium
29. CDC COVID Data Tracker, Maps, charts, and data provided by the CDC
- Coronavirus Disease 2019 (COVID-19) Center for Disease Control and Prevention (CDC)
30. https://www.nytimes.com/interactive/2020/us/states-reopen-map- coronavirus.html
31. https://www.data.gouv.fr/en/datasets/chiffres-cles-concernant-lepidemie- de-covid19-en-france/ https://www.rainews.it/ran24/speciali/2020/covid19/europe.php
32. https://www.gouvernement.fr/info-coronavirus/les-actions-du- gouvernement
33. https://theconversation.com/the-us-isnt-in-a-second-wave-of-coronavirus- the-first-wave-never-ended-141032
34. Reif F., Fundamentals of Statistical and Thermal Physics, Waveland Press, Inc. ISBN 978-1-57766-612-7 (1965 - Reissued 2009).
35. Michaelis L. and Menten M.L., Die Kinetik der Invertinwirkung. Biochem Z. 1913; 49: 333-369.
36. Srinivasan, Bharath (2020-10-08), Explicit treatment of non Michaelis- Menten and atypical kinetics in early drug discovery. ChemMedChem. (Retrieved 2020-11-09). doi:10.20944/preprints202010.0179.v1. PMID 33231926.
37. Srinivasan, Bharath, Words of advice: teaching enzyme kinetics. The FEBS Journal (2020-09-27), doi:10.1111/febs.15537. ISSN 1742-464X. PMID 32981225.
38. https://www.agi.it/fact-checking/news/2020-03-06/coronavirus-posti- letto-ospedali-7343251/
39. Kermack W. O. and McKendrick, A. G., A Contribution to the Mathemati- cal Theory of Epidemics. Proc of the Royal Soc A. 1927; 115 (772): 700–721. doi:10.1098/rspa.1927.0118
40. Sciensano, https://www.sciensano.be/en/covid-19-data
41. Sant´e Publique France, https://www.santepubliquefrance.fr/
42. Koch Institute, https://dc-covid.site.ined.fr/en/data/germany/
43. Our World in Data, https://ourworldindata.org/covid-models
44. ECDC, Data on country response measures to COVID-19, https://www.ecdc.europa.eu/en/publications-data/download-data- response-measures-covid-19
45. Shampine, L.F. and S. Thompson, Solving DDEs in MATLAB, Appl Num Math, 2001; 37, 441-458.