Homoclinic orbits in a disease transmission model with nonlinear incidence and nonconstant population

W.R. Derrick; P. van den Driessche

doi:10.3934/dcdsb.2003.3.299

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2003, Volume 3, Issue 2: 299-309. Doi: 10.3934/dcdsb.2003.3.299

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Homoclinic orbits in a disease transmission model with nonlinear incidence and nonconstant population

W.R. Derrick^1, and
P. van den Driessche^2,

1.
Department of Mathematics, University of Montana, Missoula, MT 59802
2.
Department of Mathematics and Statistics, University of Victoria, Victoria B.C., Canada V8W 3P4

Received: September 30, 2002

Revised: January 31, 2003

Published: April 2003

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Abstract

Periodic oscillations are proved for an SIRS disease transmission model in which the size of the population varies and the incidence rate is a nonlinear function. For this particular incidence function, analytical techniques are used to show that, for some parameter values, periodic solutions can arise through a Hopf bifurcation and disappear through a homoclinic loop bifurcation. The existence of periodicity is important as it may indicate different strategies for controlling disease.

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Mathematics Subject Classification: Primary: 92D30; Secondary: 58F22.

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