  We propose a realistic model for the evolution of the COVID-19 pandemic
subject to the lockdown and quarantine measures, which takes into account the
time-delay for recovery or death processes. The dynamic equations for the
entire process are derived by adopting a kinetic-type reaction approach. More
specifically, the lockdown and the quarantine measures are modelled 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 hospitalised infected people.
To get the evolution equation we take inspiration from the Michaelis- Menten
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 people, respectively. In other words,
everything happens as if the hospitals beds act as a catalyser in the hospital
recovery process. Of course, in our case the reverse MM reactions 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 modelled by the
following kinetic scheme S+I=>2I with I=>R or I=>D, with S, I, R, and D
denoting the compartments Susceptible, Infected, Recovered, and Deceased
people, respectively. The resulting kinetic-type equation} 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 hospitalised, subject to
the lockdown and the quarantine measure, and the total number of deaths. The
model foresees also the second wave of Infection by Coronavirus. The tests
carried out on real data for Belgium, France and Germany confirmed the
correctness of our model.
