After successfully tamping down the first surge of infection and death, Europe is now in the middle of a second coronavirus wave as it moves into winter [1]. We propose a realistic model for the evolution of the COVID-19 pandemic subject to the lockdown and quarantine measures. The dynamic equations for the entire process are derived by adopting a "kinetic-type reactions" approach. More specifically, the lockdown and the quarantine measures are modeled by some kind of inhibitor 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 hospitalized infected people. To get the evolution equation we take inspiration from the Michaelis-Menten’s enzyme-substrate reaction model (the MM reactions) [2] where the enzyme is associated to the available hospital beds, the substrate to the infected people, and the product to the recovered people, respectively. In other words, everything happens as if the hospitals beds act as a “catalyzer” in the hospital recovery process. Of course, in our case the reverse MM reaction has no sense in our case and, consequently, the kinetic constant is equal to zero. Finally, the O.D.E.s for people tested positive to COVID-19 is simply modeled by the following kinetic scheme S+I => 2I with I => R or I=>D, where S, I, R, and D indicate the compartments Susceptible, Infected, Recovered, and Deceased people, respectively [3]. The resulting "kinetic-type equations" provide the O.D.E.s, for elementary “reaction steps”, describing the number of the infected people, the total number of the recovered people previously hospitalized, subject to the lockdown and the quarantine measure, and the total number of deaths.
We show that our model is able to produce predictions not only on the first but also on the second or even the third waves of COVID-19 infections. The theoretical predictions are in agreement with the official number of cases with minimal parameter fitting. We discuss the strengths and limitations of the proposed model regarding the long-term predictions and, above all, the duration of how long the lockdown and the quarantine measures should be taken in force in order to limit as much as possible the intensities of subsequent SARS-CoV-2 infection waves. This task has been carried out by taking into account the theoretical results recently appeared in literature [4].
We are currently incorporating real data into a stochastic model in order to obtain a comparative analysis against the deterministic one, in order to use the new theoretical results to predict the number of new cases of infected people and to propose possible changes to the measures of isolation.
REFERENCES
[1] CACCIAPAGLIA Giacomo, COT Corentin, and SANNINO Francesco, Second wave COVID-19 pandemics in Europe: a temporal playbook, Scientific Reports, 10, Article number: 15514, 2020.
[2] MICHAELIS Leonor and MENTEN Maud, Die Kinetik der Invertinwirkung. Biochem Z. 49, 333–369, 1913.
[3] VYNNYCKY Emilia and WHITE Richard, eds. (2010), An Introduction to Infectious Disease Modelling. Oxford: Oxford University Press. ISBN 978-0-19-856576-5.
[4] SONNINO Giorgio and NARDONE Pasquale, Dynamics of the COVID-19 - Comparison between the Theoretical Predictions and the Real Data, and Predictions about Returning to Normal Life, Annals of Clinical and Medical Case Reports, 4 (9), 1-20, 2020. ISSN 2639-8109.