Bifurcation analysis in an epidemic model on adaptive networks

Area of Science:

• Mathematical epidemiology
• Network dynamics
• Dynamical systems theory

Background:

• Epidemic models often simplify interactions, neglecting spatial and temporal complexities.
• Understanding the impact of time delays in network interactions is crucial for accurate epidemic spread prediction.
• Adaptive network models offer a more realistic representation of evolving contact structures.

Purpose of the Study:

• To analyze a delayed adaptive network epidemic model incorporating time-delay effects on demographic change.
• To investigate the occurrence and characteristics of Hopf bifurcation in the model.
• To derive conditions for bifurcation direction and stability.

Main Methods:

• Utilizing Hopf bifurcation theory to identify critical parameter values.
• Applying the normal form method for analyzing local dynamics near equilibrium.
• Employing central manifold theory to reduce the system's dimensionality.
• Conducting numerical simulations to validate theoretical findings.

Main Results:

• Proving the existence of Hopf bifurcation at a critical delay value (τ₀).
• Deriving explicit criteria for the direction and stability of the bifurcating solutions.
• Demonstrating the feasibility and accuracy of the analytical results through numerical simulations.

Conclusions:

• Time delays in local spatial connections significantly influence epidemic dynamics in adaptive networks.
• Hopf bifurcation is a key phenomenon in this model, leading to oscillatory behaviors.
• The derived criteria provide valuable insights for controlling epidemic spread in complex networks.