Impulsive Reaction-Diffusion Delayed Models in Biology: Integral Manifolds Approach.

Area of Science:

• Mathematical Biology
• Epidemiology
• Dynamical Systems

Background:

• Reaction-diffusion models are crucial for understanding spatial spread of phenomena like epidemics.
• Existing models often lack considerations for impulsive effects and time delays, limiting their real-world applicability.
• Integral manifolds offer a powerful framework for analyzing multi-stable systems, extending traditional state-based analyses.

Purpose of the Study:

• To develop and analyze a novel impulsive delayed reaction-diffusion model for biological applications, specifically epidemic modeling.
• To introduce and utilize the concept of integral manifolds within this generalized model.
• To establish criteria for the boundedness, permanence, and stability of the system's integral manifolds.

Main Methods:

• Generalization of existing delayed reaction-diffusion epidemic models to incorporate impulsive effects.
• Application of an extended Lyapunov method to prove the existence of integral manifolds.
• Utilizing Lyapunov functions and a Poincarè-type inequality to derive qualitative criteria for system behavior.

Main Results:

• Established the existence of integral manifolds for the impulsive delayed reaction-diffusion model.
• Presented qualitative criteria for boundedness, permanence, and stability of these integral manifolds.
• Demonstrated the model's relevance to optimal control problems in epidemic modeling.

Conclusions:

• The proposed impulsive control model provides a robust framework for studying and managing epidemic dynamics.
• The integral manifolds approach offers significant potential for developing effective therapeutic control strategies in epidemiology.
• Results are broadly applicable to various qualitative investigations within mathematical biology and epidemiology.