The infected frontier in an SEIR epidemic model with infinite delay

Abstract

An SEIR epidemic model with infinite delay and the Neumannboundary condition is investigated, as well as the correspondingfree boundary problem, in which the free boundary exactly describesthe spreading frontier of the disease. For the problem in a fixed domain with null Neumann boundary condition, thetransmission dynamics of the disease is determined by the basicreproduction number $R_0$. More specifically, whether the diseasewill die out or not depends on $R_01$; while for the free boundary problem, we show thatunder certain conditions the disease will die out even $R_0>1$. Our results indicate that besides the basic reproductionnumber, the initial size of the infected domain and the diffusivityof the disease in a new region also produce a non-negligibleinfluence to the disease transmission, and it seems more reasonableand acceptable.