Model for heterogeneous reaction-diffusion systems with application to one epidemic

TL;DR

This paper develops and compares models for heterogeneous reaction-diffusion systems, specifically applied to an epidemic scenario, demonstrating that simplified models can approximate detailed ones under certain conditions.

Contribution

It introduces a new reaction-diffusion model for heterogeneous epidemic systems and compares detailed, mean-field, and simplified geometric models using finite volume methods.

Findings

All models behave similarly at high diffusion.

Reduced geometric models match detailed models at moderate diffusion.

Mean-field models have limited accuracy at low and moderate diffusion.

Abstract

The dynamics of ecological as well as chemical systems may depend on heterogeneous configurations. Heterogeneity in reaction-diffusion systems often increase modelling and simulating difficulties when non-linear effects are present. One synthetic epidemic system with short range heterogeneous composition is modelled and its space-time evolution studied using maximum heterogeneity details. Two other modelling alternatives are applied, one of them using elementary mean-field variables, one other using non-localized geometrical parameters, so avoiding the limitations of the used mean-field model, while keeping significant features of more detailed models. Both the detailed and the mean-field models are solved by means of the standard finite volume method. The model with less defined geometry is solved by means of one modified version of the finite volume method. Simulation results of the three models are compared. At the high diffusion range all models behave similarly. At moderate diffusion fluxes, the numerical results of the model with reduced geometric details are in excellent agreement with the results of the detailed model. The simple mean-field model presents limited accuracy at low and moderate values of the diffusion coefficient.