Heterogeneously Structured Compartmental Models of Epidemiological Systems: From Individual‐Level Processes to Population‐Scale Dynamics

We develop a general modeling 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 integrodifferential 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, , in terms of key parameters and functions of the underlying microscopic model, so as to illustrate how such a modeling framework makes it possible to draw connections between fundamental individual‐level processes and population‐scale dynamics. Finally, we apply the modeling 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.