In all countries the political decisions aim to achieve an almost stable configuration with a small number of new infected individuals per day due to Covid-19. When such a condition is reached, the containment effort is usually reduced in favor of a gradual reopening of the social life and of the various economical sectors. However, in this new phase, the infection spread restarts and, moreover, possible mutations of the virus give rise to a large specific growth rate of the infected people. Therefore, a quantitative analysis of the regrowth pattern is very useful. We discuss a macroscopic approach which, on the basis of the collected data in the first lockdown, after few days from the beginning of the new phase, outlines different scenarios of the Covid-19 diffusion for longer time. The purpose of this paper is a demonstration-of-concept: one takes simple growth models, considers the available data and shows how the future trend of the spread can be obtained. The method applies a time dependent carrying capacity, analogously to many macroscopic growth laws in biology, economics and population dynamics. The illustrative cases of France, Italy and United Kingdom are analyzed.

Dipartimento di Fisica e Astronomia Università di Catania Italy

Dipartimento di Medicina clinica e sperimentale Università di Catania Italy

Faculty of Mathematics and Physics Charles University 5 Holešovičkách 2 18000 Prague 8 Czech Republic

INFN Sezione di Catania 1 95123 Catania Italy

Istituto Oncologico del Mediterraneo Viagrande Italy

U O C Malattie Infettive P O Garibaldi Catania Italy

Zobrazit více v PubMed

Backer J.A., Klinkenberg D., Wallinga J. Incubation period of 2019 novel coronavirus (2019-nCoV) infections among travellers from Wuhan, China, 20-28 January 2020. Euro Surveillance. 2020;25(5) PubMed PMC

P. Blanchard, G. F. Bolz and T. Kruger, Mathematical modelling on random graphs of sexually trasmitted disease, in Dynamics and stochastic process - theory and applications, lecture notes in physics, vol. Vol. 355, Springer-Verlag, Berlin.

Castorina P., Blanchard P. Unified approach to growth and aging in biological, technical and biotechnical systems. SpringerPlus. 2012;1:7. PubMed PMC

Castorina P., Delsanto P.P., Guiot C. Classification scheme for phenomenological universalities in growth problems in physics and other sciences. Physical Review Letters. 2006;96:188701. PubMed

Castorina P., Lanteri D., Iorio A. Data analysis on Coronavirus spreading by macroscopic growth laws. International Journal of Modern Physics C. 2003 in press in. arXiv.

Comitato Tecnico Scientifico, Italian Government ( in Italian).

L.Fenga, CoViD19: An Automatic,Semiparametric estimation method for the population infected in Italy, medRxiv preprint. PubMed PMC

Flaxman Seth, Mishra Swapnil, Gandy Axel. Imperial College London; 2020. Estimating the number of infections and the impact of non- pharmaceutical interventions on COVID- 19 in 11 European countries.

Gompertz B. On the nature of the function expressive of the law of human mortality and a new mode of determining life contingencies. Phil. Trans. R. Soc. 1825;115:513. PubMed

Grassly N.C., Fraser C. Mathematical models of infectious disease transmission. Nature Reviews Microbiology. 2008;6:477. PubMed PMC

Guan W.-J. Clinical characteristics of coronavirus disease 2019 in China. New England Journal of Medicine. February 2020 doi: 10.1056/NEJMoa2002032. PubMed DOI PMC

Herzog S.A., Blaizot S., Niel Hens Mathematical models used to inform study design or surveillance systems in infectious diseases: A systematic review. BMC Infectious Diseases. 2017;17:775. PubMed PMC

Lanteri D., Carco’ G., Castorina P. How macroscopic laws describe complex dynamics: Asymptomatic population and CoviD-19 spreading. International Journal of Modern Physics C. 2003:12457. in press in. arXiv.

Lei S. Clinical characteristics and outcomes of patients undergoing surgeries during the incubation period of COVID-19 infection. EClinicalMedicine. April 2020:100331. PubMed PMC

Li Q. Early transmission dynamics in Wuhan, China, of novel coronavirus-infected pneumonia. New England Journal of Medicine. 2020;382(13):1199–1207. PubMed PMC

Meyer P.S., Ausubel J.H. Carrying capacity: A model with logistically varying limits. Technological Forecasting and Social Change. 1999;61(3):209–214.

Novel coronavirus (COVID-19) cases, provided by JHU CSSE.

Pastor-Satorras R., Castellano C., Van Mieghem P., Vespignani A. Epidemic processes in complex network. Reviews of Modern Physics. 2015;87

Pluchino A. 2004. A Novel methodology for epidemic Risk Assessment: The case of COVID-19 outbreak in Italy. arXiv. 02739. PubMed

Royama T. Springer ed. 1992. Analytical population dynamics.

Tuite A.R., Ng V., Rees E., Fisman D. Estimation of COVID-19 outbreak size in Italy. The Lancet Infectious Diseases. 2020 doi: 10.1016/S1473-3099(20)30227-9. Published Online March 19, 2020. PubMed DOI PMC

Verhulst P.F. Notice sur la loi que la population poursuit dans son accroissement. Correspondance Mathématique et Physique. 1838;10:113.

Walters C.E., Mesle’ M.I., Hall I.M. Modelling the global spread of diseases: A review of current practice and capability. Epidemics. 2018;25:1. PubMed PMC

Wheldon T.E. Adam Hilger; 1988. Mathematical model in cancer Research.

World Health Organization, Coronavirus disease (COVID-19) outbreak.

World Health Organization, reportCoronavirus disease (COVID-19), China joint mission on covid 19 final report.

Quantitative Method for Monitoring Tumor Evolution During and After Therapy