Diffusion-Reaction Epidemic Model with a Free Boundary

TL;DR

This paper develops and analyzes a free boundary PDE model for epidemic spread, establishing conditions for disease extinction or persistence based on the basic reproductive number, with results on solution existence, uniqueness, and convergence speed.

Contribution

It introduces a novel SEIS PDE model with a free boundary, analyzing its solutions and stability criteria, including the impact of the basic reproductive number on disease dynamics.

Findings

Global stability of disease-free equilibrium when R0<1

Unstability of disease-free equilibrium when R0>1

Convergence speed analyzed via nonlinear elliptic eigenvalue techniques

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.