Diffusion-Reaction Epidemic Model with a Free Boundary

TL;DR

This work develops a diffusion-reaction SEIS framework with a free boundary to model spatial epidemic spread under rotational symmetry. It proves local (and, under equal diffusion rates, global) well-posedness, and defines a basic reproductive number $\mathcal{R}_0$ that thresholds vanishing versus persisting dynamics. The analysis shows global stability of the disease-free state when $\mathcal{R}_0^{\max}<1$ and characterizes persistence and front dynamics under $\mathcal{R}_0^{\min}>1$, including how initial domain size influences spread. Convergence rates of the infected and exposed compartments are quantified via a nonlinear elliptic eigenvalue problem, yielding exponential decay governed by the gap between leading eigenvalues. Together, the results advance PDE-based epidemic modeling with moving fronts and provide threshold-informed insights for control of spatial disease spread.

Abstract

This study investigates an SEIS PDE model with a free boundary, which captures the dynamics of epidemic transmission, including diseases like COVID-19. This parabolic PDE system is analyzed in a rotationally symmetric domain, and the existence and uniqueness of the local solution are established through the straightening lemma. Furthermore, the existence and uniqueness of the global solution are established under specific conditions on the diffusion coefficients. Then the model introduces the basic reproductive number, $R_0$, which provides sufficient conditions for determining whether the disease will vanish or spread. Notably, when $R_0<1$, the disease-free equilibrium(DFE) is shown to be globally stable, and when $R_0>1$, the DFE is unstable. Lastly, we investigate the convergence speed of solutions by applying nonlinear elliptic eigenvalue techniques to the associated parabolic PDE system.

Theorems & Definitions (43)

• Theorem 3.1
• Lemma 3.2
• proof
• proof
• Proposition 3.3
• proof
• Theorem 3.4
• Lemma 3.5
• proof
• Lemma 3.6