  We develop a new structured compartmental model for the coevolutionary
dynamics between susceptible and infectious individuals in heterogeneous SI
epidemiological systems. In this model, the susceptible compartment is
structured by a continuous variable that represents the level of resistance to
infection of susceptible individuals, while the infectious compartment is
structured by a continuous variable that represents the viral load of
infectious individuals. We first formulate an individual-based model wherein
the dynamics of single individuals is described through stochastic processes,
which permits a fine-grain representation of individual dynamics and captures
stochastic variability in evolutionary trajectories amongst individuals. Next
we formally derive the mesoscopic counterpart of this model, which consists of
a system of coupled integro-differential equations for the population density
functions of susceptible and infectious individuals. Then we consider an
appropriately rescaled version of this system and we carry out formal
asymptotic analysis to derive the corresponding macroscopic model, which
comprises a system of coupled ordinary differential equations for the
proportions of susceptible and infectious individuals, the mean level of
resistance to infection of susceptible individuals, and the mean viral load of
infectious individuals. Overall, this leads to a coherent mathematical
representation of the coevolutionary dynamics between susceptible and
infectious individuals across scales. We provide well-posedness results for the
mesoscopic and macroscopic models, and we show that there is excellent
agreement between analytical results on the long-time behaviour of the
components of the solution to the macroscopic model, the results of Monte Carlo
simulations of the individual-based model, and numerical solutions of the
macroscopic model.
