The spreading frontiers of avian-human influenza described by the free boundary

Abstract

In this paper, a reaction-diffusion system is proposed to investigate avian-human influenza. Two free boundaries are introduced to describe the spreading frontiers of the avian influenza. The basic reproduction numbers r 0F (t) and R 0F (t) are defined for the bird with the avian influenza and for the human with the mutant avian influenza of the free boundary problem, respectively. Properties of these two time-dependent basic reproduction numbers are obtained. Sufficient conditions both for spreading and for vanishing of the avian influenza are given. It is shown that if r 0F (0) < 1 and the initial number of the infected birds is small, the avian influenza vanishes in the bird world. Furthermore, if r 0F (0) < 1 and R 0F (0) < 1, the avian influenza vanishes in the bird and human worlds. In the case that r 0F (0) < 1 and R 0F (0) > 1, spreading of the mutant avian influenza in the human world is possible. It is also shown that if r 0F (t 0) ≥ 1 for any t 0 ≥ 0, the avian influenza spreads in the bird world. © 2013 Science China Press and Springer-Verlag Berlin Heidelberg.