title: Large number of endemic equilibria for disease transmission models in patchy environment.
abstract: We show that disease transmission models in a spatially heterogeneous environment can have a large number of coexisting endemic equilibria. A general compartmental model is considered to describe the spread of an infectious disease in a population distributed over several patches. For disconnected regions, many boundary equilibria may exist with mixed disease free and endemic components, but these steady states usually disappear in the presence of spatial dispersal. However, if backward bifurcations can occur in the regions, some partially endemic equilibria of the disconnected system move into the interior of the nonnegative cone and persist with the introduction of mobility between the patches. We provide a mathematical procedure that precisely describes in terms of the local reproduction numbers and the connectivity network of the patches, whether a steady state of the disconnected system is preserved or ceases to exist for low volumes of travel. Our results are illustrated on a patchy HIV transmission model with subthreshold endemic equilibria and backward bifurcation. We demonstrate the rich dynamical behavior (i.e., creation and destruction of steady states) and the presence of multiple stable endemic equilibria for various connection networks.
authors.affiliation: MTA-SZTE Analysis and Stochastic Research Group, University of Szeged, Aradi vértanúk tere 1, Szeged 6720, Hungary. Electronic address: knipl@math.u-szeged.hu.
authors.forename: D H
authors.initials: DH
authors.lastname: Knipl
authors.affiliation: Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged 6720, Hungary. Electronic address: rost@math.u-szeged.hu.
authors.forename: G
authors.initials: G
authors.lastname: Röst
mesh_terms: D019353:Endemic Diseases; D015658:HIV Infections / Q000453:epidemiology* / Q000635:transmission*; D006801:Humans; D008962:Models, Theoretical; D016571:Neural Networks, Computer
keywords: Compartmental patch model; Differential equations; Epidemic spread; Large number of steady states