Bifurcation and stability analyses of a 13-D seic model using normal form reduction and numerical simulation

Abba Gumel, S. M. Moghadas, Y. Yuan, P. Yu

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

Center manifold and normal form theories as well as numerical simulations are employed to analyse the bifurcation and stability of a 13-dimensional deterministic model associated with the transmission dynamics of two diseases within a host population. The stability of the multiple equilibrium solutions together with the associated possible critical points are discussed. It is shown that model can only undergo static bifurcations. Owing to the fact that the system exhibits a double-zero singularity, center manifold and normal form theories are used to reduce the model to a two-dimensional system. A detailed bifurcation and stability analysis is given for the behaviour of the system in the vicinity of the critical point. A robust finite-difference method is developed and used for the solution of the 13-D system. Unlike some conventional numerical schemes, this novel scheme does not suffer from scheme-dependent instabilities. The numerical method gives results that are consistent with the theoretical predictions.

Original languageEnglish (US)
Pages (from-to)317-330
Number of pages14
JournalDynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms
Volume10
Issue number1-3
StatePublished - Feb 2003
Externally publishedYes

Fingerprint

Normal Form
Normal Form Theory
Bifurcation
Center Manifold
Numerical Simulation
Critical point
Computer simulation
Multiple Equilibria
Equilibrium Solution
Two-dimensional Systems
Deterministic Model
Bifurcation Analysis
Numerical Scheme
Difference Method
Stability Analysis
Finite Difference
Numerical Methods
Finite difference method
Model
Singularity

Keywords

  • Bifurcation and stability
  • Center manifold
  • Equilibrium solution
  • Finite-difference method
  • Normal form

ASJC Scopus subject areas

  • Applied Mathematics
  • Discrete Mathematics and Combinatorics

Cite this

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abstract = "Center manifold and normal form theories as well as numerical simulations are employed to analyse the bifurcation and stability of a 13-dimensional deterministic model associated with the transmission dynamics of two diseases within a host population. The stability of the multiple equilibrium solutions together with the associated possible critical points are discussed. It is shown that model can only undergo static bifurcations. Owing to the fact that the system exhibits a double-zero singularity, center manifold and normal form theories are used to reduce the model to a two-dimensional system. A detailed bifurcation and stability analysis is given for the behaviour of the system in the vicinity of the critical point. A robust finite-difference method is developed and used for the solution of the 13-D system. Unlike some conventional numerical schemes, this novel scheme does not suffer from scheme-dependent instabilities. The numerical method gives results that are consistent with the theoretical predictions.",
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AU - Moghadas, S. M.

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N2 - Center manifold and normal form theories as well as numerical simulations are employed to analyse the bifurcation and stability of a 13-dimensional deterministic model associated with the transmission dynamics of two diseases within a host population. The stability of the multiple equilibrium solutions together with the associated possible critical points are discussed. It is shown that model can only undergo static bifurcations. Owing to the fact that the system exhibits a double-zero singularity, center manifold and normal form theories are used to reduce the model to a two-dimensional system. A detailed bifurcation and stability analysis is given for the behaviour of the system in the vicinity of the critical point. A robust finite-difference method is developed and used for the solution of the 13-D system. Unlike some conventional numerical schemes, this novel scheme does not suffer from scheme-dependent instabilities. The numerical method gives results that are consistent with the theoretical predictions.

AB - Center manifold and normal form theories as well as numerical simulations are employed to analyse the bifurcation and stability of a 13-dimensional deterministic model associated with the transmission dynamics of two diseases within a host population. The stability of the multiple equilibrium solutions together with the associated possible critical points are discussed. It is shown that model can only undergo static bifurcations. Owing to the fact that the system exhibits a double-zero singularity, center manifold and normal form theories are used to reduce the model to a two-dimensional system. A detailed bifurcation and stability analysis is given for the behaviour of the system in the vicinity of the critical point. A robust finite-difference method is developed and used for the solution of the 13-D system. Unlike some conventional numerical schemes, this novel scheme does not suffer from scheme-dependent instabilities. The numerical method gives results that are consistent with the theoretical predictions.

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