# American Institute of Mathematical Sciences

## Long-time dynamics of a diffusive epidemic model with free boundaries

 School of Science and Technology, University of New England, Armidale, NSW 2351, Australia

* Corresponding author: Yihong Du

Dedicated to Professor Sze-Bi Hsu on the occasion of his 70th birthday

Received  August 2020 Revised  October 2020 Published  December 2020

Fund Project: This work was supported by the Australian Research Council and a PhD scholarship of the University of New England

In this paper, we determine the long-time dynamical behaviour of a reaction-diffusion system with free boundaries, which models the spreading of an epidemic whose moving front is represented by the free boundaries. The system reduces to the epidemic model of Capasso and Maddalena [5] when the boundary is fixed, and it reduces to the model of Ahn et al. [1] if diffusion of the infective host population is ignored. We prove a spreading-vanishing dichotomy and determine exactly when each of the alternatives occurs. If the reproduction number $R_0$ obtained from the corresponding ODE model is no larger than 1, then the epidemic modelled here will vanish, while if $R_0>1$, then the epidemic may vanish or spread depending on its initial size, determined by the dichotomy criteria. Moreover, when spreading happens, we show that the expanding front of the epidemic has an asymptotic spreading speed, which is determined by an associated semi-wave problem.

Citation: Rong Wang, Yihong Du. Long-time dynamics of a diffusive epidemic model with free boundaries. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020360
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