Modelling coevolutionary dynamics in heterogeneous SI epidemiological systems across scales

TL;DR

This paper introduces a multi-scale structured compartmental model capturing the coevolutionary dynamics between susceptible and infectious individuals in heterogeneous SI epidemiological systems, integrating stochastic, mesoscopic, and macroscopic approaches.

Contribution

It develops a novel multi-scale mathematical framework for SI models with continuous resistance and viral load variables, linking individual stochastic dynamics to population-level equations.

Findings

The macroscopic model accurately predicts long-term behavior.

Monte Carlo simulations validate the analytical results.

The models reveal the impact of heterogeneity on disease dynamics.

Abstract

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.