Modelling the spreading of the SARS-CoV-2 in presence of the lockdown and quarantine measures by a kinetic-type reactions approach.

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 timedelay for recovery or death processes. 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 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 hospitalized infected people. To get the evolution equation 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 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 ordinary differential equations (ODEs) for people tested positive to COVID-19 is simply modelled by the following kinetic scheme $S+I\Rightarrow 2I$ with $I\Rightarrow R$ or $I\Rightarrow D$, with $S$, $I$, $R$ and $D$ denoting the compartments susceptible, infected, recovered and deceased people, respectively. The resulting kinetic-type equations provide the ODEs, 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. 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.