A distribution-free lattice Boltzmann method for compartmental reaction-diffusion systems with application to epidemic modelling

TL;DR

This paper presents a novel distribution-free lattice Boltzmann method tailored for compartmental reaction-diffusion systems in epidemiology, offering improved accuracy and efficiency over traditional approaches, especially in complex nonlinear regimes.

Contribution

The paper introduces a single-step simplified lattice Boltzmann method that directly evolves macroscopic densities, eliminating the need for particle distributions and streaming, enhancing computational efficiency.

Findings

The SSLBM achieves 2-5 times lower error compared to standard methods.

The approach maintains accuracy across different epidemic regimes.

It effectively handles steep gradients and nonlinear coupling.

Abstract

We introduce a distribution-free lattice Boltzmann formulation for general compartmental reaction--diffusion systems arising in mathematical epidemiology. The proposed scheme, termed a single-step simplified lattice Boltzmann method (SSLBM), evolves directly macroscopic compartment densities, eliminating the need for particle distribution functions and explicit streaming operations. This yields a compact and computationally efficient framework while retaining the kinetic consistency of lattice Boltzmann methodologies. The approach is applied to a SEIRD (Susceptible-Exposed-Infected-Recovered-Deceased) reaction-diffusion model as a representative case. The resulting discrete evolution equations are derived and shown to recover the target macroscopic dynamics. The method is systematically validated against a fourth-order finite difference reference solution and compared with a standard BGK lattice Boltzmann formulation. Numerical results demonstrate that the SSLBM consistently improves accuracy across all compartments and norms. The error reduction is robust with respect to both the basic reproduction number and diffusion strength, typically ranging between factors of approximately two and five depending on the regime. In particular, the method shows enhanced control of localised errors in regimes characterised by strong nonlinear coupling and steep spatial gradients. Our findings indicate that the proposed formulation provides an accurate and efficient alternative to classical lattice Boltzmann approaches for reaction-diffusion systems, with particular advantages in stiff and nonlinear epidemic dynamics.