  Mathematical models are formal and simplified representations of the
knowledge related to a phenomenon. In classical epidemic models, a neglected
aspect is the heterogeneity of disease transmission and progression linked to
the viral load of each infectious individual. Here, we attempt to investigate
the interplay between the evolution of individuals' viral load and the epidemic
dynamics from a theoretical point of view. In the framework of multi-agent
systems, we propose a particle stochastic model describing the infection
transmission through interactions among agents and the individual physiological
course of the disease. Agents have a double microscopic state: a discrete
label, that denotes the epidemiological compartment to which they belong and
switches in consequence of a Markovian process, and a microscopic trait,
representing a normalized measure of their viral load, that changes in
consequence of binary interactions or interactions with a background.
Specifically, we consider Susceptible--Infected--Removed--like dynamics where
infectious individuals may be isolated from the general population and the
isolation rate may depend on the viral load sensitivity and frequency of tests.
We derive kinetic evolution equations for the distribution functions of the
viral load of the individuals in each compartment, whence, via suitable
upscaling procedures, we obtain a macroscopic model for the densities and viral
load momentum. We perform then a qualitative analysis of the ensuing
macroscopic model, and we present numerical tests in the case of both constant
and viral load-dependent isolation control. Also, the matching between the
aggregate trends obtained from the macroscopic descriptions and the original
particle dynamics simulated by a Monte Carlo approach is investigated.
