Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Stability of steady states and existence of travelling waves in a vector-disease model

Shigui Ruana1 and Dongmei Xiaoa2

a1 Department of Mathematics, University of Miami, PO Box 249085, Coral Gables, FL 33124-4250, USA (ruan@math.miami.edu)

a2 Department of Mathematics, Shanghai Jiaotong University, Shanghai 200030, People's Republic of China (xiaodm@sjtu.edu.cn)

Abstract

In this paper, a host-vector model is considered for a disease without immunity in which the current density of infectious vectors is related to the number of infectious hosts at earlier times. Spatial spread in a region is modelled in the partial integro-differential equation by a diffusion term. For the general model, we first study the stability of the steady states using the contracting-convex-sets technique. When the spatial variable is one dimensional and the delay kernel assumes some special form, we establish the existence of travelling wave solutions by using the linear chain trick and the geometric singular perturbation method.

(Received October 15 2003)

(Accepted April 21 2004)