Spread and containment of infectious disease epidemics: a kinetic approach.

In this thesis, we propose a Boltzmann-type kinetic approach in order to study the spread of an infectious disease. In particular, a mass-varying interacting multi-agent system is used to model a population and the confinement strategies that are adopted to stem the on-going epidemic in it, namely quarantine and vaccination. Every agent is characterized by a microscopic state, a “viral load”, and a label, that indicates the group the agent belongs to. The viral load changes as a consequence of interactions, while the label follows a Markov-type state dependent jump process. The quarantine strategy is firstly analyzed and the requirements for it to be effective outlined: a prompt isolation of the infected successfully stem the epidemic. A step further is taken by including a spatial network, e.g. of cities, in which the agents can move. This is accomplished by the means of the same label-based framework. We then focus on the vaccine strategy, numerically evaluating a threshold for the herd immunity condition. We then propose a version of the vaccination model enriched with an age-based distinction of the agents. Different interaction rates based on age are taken into account: the herd immunity threshold is shown to depend on these interaction rates. The four models are discussed analytically and numerically. General kinetic equations are derived in order to describe the statistical distribution of the agents. From those, we also derive macroscopic ODE systems for each model, both for constant and variable parameters. To do so, in the latter case we rely on the hydrodynamic limit. Finally, a Monte Carlo simulation via the Nanbu-Babovsky's scheme is carried out to confirm theoretical findings and to investigate complicated configuration of the parameters.