title: Effect of a sharp change of the incidence function on the dynamics of a simple disease.
abstract: We investigate two cases of a sharp change of incidencec functions on the dynamics of a susceptible-infective-susceptible epidemic model. In the first case, low population levels have mass action incidence, while high population levels have proportional incidence, the switch occurring when the total population reaches a certain threshold. Using a modified Dulac theorem, we prove that this system has a single equilibrium which attracts all solutions for which the disease is present and the population remains bounded. In the second case, an increase of the number of infectives leads to a mass action term being added to a standard incidence term. We show that this allows a Hopf bifurcation to occur, with periodic orbits being generated when a locally asymptotically stable equilibrium loses stability.
authors.affiliation: Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada. arinoj@cc.umanitoba.ca
authors.forename: Julien
authors.initials: J
authors.lastname: Arino
authors.forename: C Connell
authors.initials: CC
authors.lastname: McCluskey
mesh_terms: D003141:Communicable Diseases / Q000453:epidemiology* / Q000628:therapy; D006801:Humans; D015994:Incidence; D008954:Models, Biological; D011156:Population Density