Social contacts and the spread of infectious diseases

G. Dimarco, B. Perthame, G. Toscani, M. Zanella

Preprint arXiv, 2020.

Motivated by the COVID-19 pandemic, we introduce a mathematical description of the impact of sociality in the spread of infectious diseases by integrating an epidemiological dynamic with a kinetic modeling of population-based contacts. The kinetic description leads to study the evolution over time of Boltzmann type equations describing the number densities of social contacts of susceptible, infected and recovered individuals, whose proportions are driven by a classical compartmental model in epidemiology. Explicit calculations show that the spread of the disease is closely related to the mean number of contacts, thus justifying the lockdown strategies assumed by governments to prevent them. Furthermore, the kinetic model allows to clarify how a selective control can be assumed to achieve a minimal lockdown strategy by only reducing individuals undergoing a very large number of daily contacts. This, in turns, could permit to maintain at best the economic activities which would seriously suffer from a total closure policy. Numerical simulations confirm the ability of the model to describe different phenomena characteristic of the rapid spread of an epidemic. A last part is dedicated to fit numerical solutions of the proposed model with experimental data coming from different European countries