  The viral load is known to be a chief predictor of the risk of transmission
of infectious diseases. In this work, we investigate the role of the
individuals' viral load in the disease transmission by proposing a new
susceptible-infectious-recovered epidemic model for the densities and mean
viral loads of each compartment. To this aim, we formally derive the
compartmental model from an appropriate microscopic one. Firstly, we consider a
multi-agent system in which individuals are identified by the epidemiological
compartment to which they belong and by their viral load. Microscopic rules
describe both the switch of compartment and the evolution of the viral load. In
particular, in the binary interactions between susceptible and infectious
individuals, the probability for the susceptible individual to get infected
depends on the viral load of the infectious individual. Then, we implement the
prescribed microscopic dynamics in appropriate kinetic equations, from which
the macroscopic equations for the densities and viral load momentum of the
compartments are eventually derived. In the macroscopic model, the rate of
disease transmission turns out to be a function of the mean viral load of the
infectious population. We analytically and numerically investigate the case
that the transmission rate linearly depends on the viral load, which is
compared to the classical case of constant transmission rate. A qualitative
analysis is performed based on stability and bifurcation theory. Finally,
numerical investigations concerning the model reproduction number and the
epidemic dynamics are presented.
