Heterogeneously structured compartmental models of epidemiological systems: from individual-level processes to population-scale dynamics

We develop a general modelling framework for compartmental epidemiological systems structured by continuous variables which are linked to the levels of expression of compartment-specific traits. We start by formulating an individual-based model that describes the dynamics of single individuals in terms of stochastic processes. Then we formally derive: (i) the mesoscopic counterpart of this model, which is formulated as a system of integro-differential equations for the distributions of individuals over the structuring-variable domains of the different compartments; (ii) the corresponding macroscopic model, which takes the form of a system of ordinary differential equations for the fractions of individuals in the different compartments and the mean levels of expression of the traits represented by the structuring variables. We employ a reduced version of the macroscopic model to obtain a general formula for the basic reproduction number, $\mathcal{R}_0$, in terms of key parameters and functions of the underlying microscopic model, so as to illustrate how such a modelling framework makes it possible to draw connections between fundamental individual-level processes and population-scale dynamics. Finally we apply the modelling framework to case studies based on classical compartmental epidemiological systems, for each of which we report on Monte Carlo simulations of the individual-based model as well as on analytical results and numerical solutions of the macroscopic model.