Respiratory particles: from analytical estimates to disease transmission

Respiratory particles containing infectious pathogens are responsible for a large number of diseases. To define health politics and save lives, it is important to study their transmission mechanisms, namely the path of particles once expelled. This path depends on several driving factors as intrinsic properties of particles, environmental aspects and morphology of the scenario. Following physical arguments and taking into account the results of experimental works, we consider a mathematical drift model for the mixture composed by two phases: air and particles. The relative motion between the two phases is described by a kinematic constitutive relation. We prove the stability of the model for fixed times and establish an a priori estimate for the total number of infectious particles. The upper bound of this estimate exhibits sound physical dependencies on the driving factors, in agreement with the experimental literature and mounting epidemiological evidences. Namely, we establish that the amount of particles expelled and their emission rate can explain why some people are superspreaders. Several numerical simulations illustrate the theoretical results.