Mathematical Models for the Large Spread of a Contact-Based Infection: A Statistical Mechanics Approach

In this work, we derive a system of Boltzmann-type equations to describe the spread of contact-based infections, such as SARS-CoV-2 virus, at the microscopic scale, that is, by modeling the human-to-human mechanisms of transmission. To this end, we consider two populations, characterized by specific distribution functions, made up of individuals without symptoms (population 1) and infected people with symptoms (population 2). The Boltzmann operators model the interactions between individuals within the same population and among different populations with a probability of transition from one to the other due to contagion or, vice versa, to recovery. In addition, the influence of innate and adaptive immune systems is taken into account. Then, starting from the Boltzmann microscopic description we derive a set of evolution equations for the size and mean state of each population considered. Mathematical properties of such macroscopic equations, as equilibria and their stability, are investigated, and some numerical simulations are performed in order to analyze the ability of our model to reproduce the characteristic features of Covid-19 type pandemics.