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

Alessandro De Rosis

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.

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

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.
Paper Structure (22 sections, 54 equations, 4 figures, 5 tables)

This paper contains 22 sections, 54 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Macroscopic governing equations
  4. BGK lattice Boltzmann method
  5. Simplified lattice Boltzmann method
  6. Relation to finite-difference methods
  7. Kinetic versus phenomenological derivation.
  8. Implicit encoding of transport coefficients.
  9. Structural equivalence and spectral properties.
  10. Extension to complex transport models.
  11. Computational structure.
  12. Scope of analysis
  13. Consistency.
  14. Stability: purely diffusive case ($\Psi = 0$).
  15. Stability: reaction-diffusion case ($\Psi \ne 0$).
  16. ...and 7 more sections

Figures (4)

  • Figure 1: Temporal evolution of SEIRD compartments computed with SSLBM (lines) and FDM (symbols) for increasing initial exposure fraction $\chi$. Excellent agreement is observed across all compartments.
  • Figure 2: Temporal evolution of SEIRD compartments for increasing reproduction number $R_0$. SSLBM (symbols) and FDM (lines) remain in excellent agreement even in highly super-critical regimes.
  • Figure 3: Log--log plot of the $L_2$ error as a function of grid resolution for the diffusion test case. The dashed line indicates second-order convergence. The SSLBM results closely follow the reference slope, confirming the expected $\mathcal{O}(\Delta x^2)$ accuracy.
  • Figure 4: Normalised runtime as a function of total grid points $M$: SSLBM (solid line with squares); BGK LBM (dashed line with circles); second-order FDM (dotted line with triangles); fourth-order FDM (dash-dotted line with inverted triangles). The SSLBM requires the fewest computational resources across all grid sizes tested.