title: A sharp threshold for disease persistence in host metapopulations.
abstract: A sharp threshold is established that separates disease persistence from the extinction of small disease outbreaks in an S→E→I→R→S type metapopulation model. The travel rates between patches depend on disease prevalence. The threshold is formulated in terms of a basic replacement ratio (disease reproduction number), ℛ(0), and, equivalently, in terms of the spectral bound of a transmission and travel matrix. Since frequency-dependent (standard) incidence is assumed, the threshold results do not require knowledge of a disease-free equilibrium. As a trade-off, for ℛ(0)>1, only uniform weak disease persistence is shown in general, while uniform strong persistence is proved for the special case of constant recruitment of susceptibles into the patch populations. For ℛ(0)<1, Lyapunov's direct stability method shows that small disease outbreaks do not spread much and eventually die out.
authors.affiliation: Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804, USA.
authors.forename: Thanate
authors.initials: T
authors.lastname: Dhirasakdanon
authors.forename: Horst R
authors.initials: HR
authors.lastname: Thieme
authors.forename: P
authors.initials: P
authors.lastname: Van Den Driessche
mesh_terms: D000818:Animals; D003141:Communicable Diseases / Q000453:epidemiology*; D004196:Disease Outbreaks / Q000706:statistics & numerical data; D004199:Disease Vectors; D019353:Endemic Diseases / Q000706:statistics & numerical data; D006801:Humans; D008954:Models, Biological; D011157:Population Dynamics