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November  2013, 18(9): 2355-2376. doi: 10.3934/dcdsb.2013.18.2355

## The infected frontier in an SEIR epidemic model with infinite delay

 1 School of Mathematical Science, Yangzhou University, Yangzhou 225002, China 2 Department of Mathematics, MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240

Received  February 2013 Revised  June 2013 Published  September 2013

An SEIR epidemic model with infinite delay and the Neumann boundary condition is investigated, as well as the corresponding free boundary problem, in which the free boundary exactly describes the spreading frontier of the disease. For the problem in a fixed domain with null Neumann boundary condition, the transmission dynamics of the disease is determined by the basic reproduction number $R_0$. More specifically, whether the disease will die out or not depends on $R_0<1$ or $R_0>1$; while for the free boundary problem, we show that under certain conditions the disease will die out even $R_0>1$. Our results indicate that besides the basic reproduction number, the initial size of the infected domain and the diffusivity of the disease in a new region also produce a non-negligible influence to the disease transmission, and it seems more reasonable and acceptable.
Citation: Zhigui Lin, Yinan Zhao, Peng Zhou. The infected frontier in an SEIR epidemic model with infinite delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2355-2376. doi: 10.3934/dcdsb.2013.18.2355
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