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Stability analysis and Hopf bifurcation in a diffusive epidemic model with two delays
Huan Dai1, Yu Ying Liu2, Jun Jie Wei2
1School of Science, Harbin Institute of Technology (Weihai), Weihai 264209, China.
This study analyzes a diffusive epidemic model with delays, confirming the stability of its steady state and investigating Hopf bifurcations. Numerical simulations validate the theoretical findings on disease dynamics.
Area of Science:
- Mathematical Biology
- Epidemiology
- Dynamical Systems
Background:
- Understanding disease spread requires mathematical models.
- Time delays and spatial diffusion are crucial factors in epidemic dynamics.
- Neumann boundary conditions are relevant for closed or semi-closed populations.
Purpose of the Study:
- To analyze a diffusive epidemic model incorporating two time delays.
- To determine the existence and stability of positive constant steady states.
- To investigate the occurrence of Hopf bifurcations and derive normal forms.
Main Methods:
- Analysis of a partial differential equation model with delays.
- Eigenvalue distribution to study stability and bifurcations.
- Center manifold theory to derive normal forms.
- Numerical simulations for validation.
Main Results:
- Existence and stability of the positive constant steady state were established.
- Conditions for Hopf bifurcations were identified through eigenvalue analysis.
- The normal form near the bifurcation singularity was derived.
- Numerical simulations confirmed the theoretical predictions.
Conclusions:
- The diffusive epidemic model with delays exhibits complex dynamics.
- Hopf bifurcations can lead to oscillatory behaviors in disease prevalence.
- The study provides a theoretical framework for understanding delayed spatial epidemics.
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