On bifurcations of codimension greater than two arising from a childhood disease model

Lih-Ing Wu, Purdue University

Abstract

The Hopf-bifurcation and the homoclinic orbit can occur in an epidemiology model. This thesis analyzes an epidemiological model with a system of four ordinary differential equations. The system describes the transmission dynamics of childhood diseases. Feng and Thieme showed that the Hopf-bifurcation occurs for some parameter values, and the homoclinic orbit was observed numerically. By using mathematical techniques in dynamical systems, including the center manifold, the normal form, and the unfolding, this thesis shows analytically that the homoclinic orbit exists under small perturbations. The center manifold reduction at a bifurcation point has the normal form x˙ = y, y˙ = ζxy + ½ζ(ζ − 1)x2 y, indicating a bifurcation of codimension greater than two. By embedding the normal form into a parameterized system with three sufficiently small parameters, the unfolding x˙ = σ1 x + y, y˙ = (σ2 − σ1)y + αx 2 − ζxy has the Hopf-bifurcation and the homoclinic orbit. By using the Maple program, the perturbing parameters which result in the unfolded, simplified normal form system are found in the original SIQR model. The influence of those small parameters in the original system is discussed. Some difficulty, however, was encountered due to the fact that the higher order terms cannot be ignored. A new method of perturbing the original system is considered, and possible future work and direction are discussed.

Degree

Ph.D.

Advisors

Feng, Purdue University.

Subject Area

Mathematics|Biostatistics

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