THE SEIR MODEL WITH PULSE AND DIFFUSION OF VIRUS IN THE ENVIRONMENT

Abstract

This paper addresses a reaction-diffusion problem featuring impulsive effects under Neumann boundary conditions. The model simulates the periodic eradication of viruses in an environment. Initially, we establish the well-posedness of the reaction-diffusion model. We define the basic reproduction number $R_0$ for the problem in the absence of pulsing and compute the principal eigenvalue of the corresponding elliptic eigenvalue problem. Utilizing Lyapunov functionals and Green's first identity, we derive the global threshold dynamics of the system. Specifically, when $R_0 < 1$, the disease-free equilibrium is globally asymptotically stable; conversely, if $R_0 > 1$, the system exhibits uniform persistence, and the endemic equilibrium is globally asymptotically stable. Additionally, we consider the generalized principal eigenvalues for the problem with pulsing and provide sufficient conditions for the stability of both the disease-free equilibrium and the positive periodic solution. Finally, we corroborate our theoretical findings through numerical simulations, particularly discussing the impacts of periodic environmental cleaning.