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May  2003, 3(2): 299-309. doi: 10.3934/dcdsb.2003.3.299

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, United States
2. 

Department of Mathematics and Statistics, University of Victoria, Victoria B.C., Canada V8W 3P4

Received  October 2002 Revised  February 2003 Published  February 2003

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.
Keywords: saddle-node bifurcation, nonlinear incidence function, homoclinic loop bifurcation., Hopf bifurcation, Epidemiological model.
Mathematics Subject Classification: Primary: 92D30; Secondary: 58F2.
Citation: W.R. Derrick, P. van den Driessche. Homoclinic orbits in a disease transmission model with nonlinear incidence and nonconstant population. Discrete and Continuous Dynamical Systems - B, 2003, 3 (2) : 299-309. doi: 10.3934/dcdsb.2003.3.299
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