title: Oscillations in a patchy environment disease model.
abstract: For a single patch SIRS model with a period of immunity of fixed length, recruitment-death demographics, disease related deaths and mass action incidence, the basic reproduction number R(0) is identified. It is shown that the disease-free equilibrium is globally asymptotically stable if R(0)<1. For R(0)>1, local stability of the endemic equilibrium and Hopf bifurcation analysis about this equilibrium are carried out. Moreover, a practical numerical approach to locate the bifurcation values for a characteristic equation with delay-dependent coefficients is provided. For a two patch SIRS model with travel, it is shown that there are several threshold quantities determining its dynamic behavior and that travel can reduce oscillations in both patches; travel may enhance oscillations in both patches; or travel can switch oscillations from one patch to another.
authors.affiliation: Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V8N 3R4.
authors.forename: Fred
authors.initials: F
authors.lastname: Brauer
authors.forename: P
authors.initials: P
authors.lastname: van den Driessche
authors.forename: Lin
authors.initials: L
authors.lastname: Wang
mesh_terms: D004194:Disease; D004196:Disease Outbreaks / Q000706:statistics & numerical data; D018562:Disease Transmission, Infectious / Q000706:statistics & numerical data; D006801:Humans; D008433:Mathematics; D008954:Models, Biological; D014195:Travel