 
Summary: Journal of Biological Dynamics
Vol. 4, No. 5, September 2010, 490505
Effect of a sharp change of the incidence function
on the dynamics of a simple disease
Julien Arinoa
* and C. Connell McCluskeyb
aDepartment of Mathematics, University of Manitoba, Winnipeg, MB, Canada; bDepartment of
Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada
(Received 16 June 2009; final version received 16 March 2010)
We investigate two cases of a sharp change of incidencec functions on the dynamics of a susceptible
infectivesusceptible 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.
Keywords: epidemiology; incidence function
1. Introduction
One of the most critical aspects in modelling disease propagation is the choice of the incidence
