Optimal control on hybrid ode systems with application to a tick disease model

Area of Science:

• Mathematical modeling
• Control theory
• Epidemiology

Background:

• Hybrid systems combine ordinary differential equations (ODEs) with discrete-time elements.
• Seasonal variations in vector dynamics pose challenges for disease control.
• Optimal control is crucial for managing complex biological systems.

Purpose of the Study:

• To establish existence, necessary conditions, and uniqueness for optimal control in a specific hybrid system.
• To apply this optimal control framework to a tick-transmitted disease model with seasonal tick dynamics.
• To determine an optimal control strategy for maximizing disease-free ticks and minimizing infected ticks.

Main Methods:

• Formulation of an optimal control problem for a hybrid system with seasonal and continuous dynamics.
• Mathematical analysis to prove existence, necessary conditions, and uniqueness of the optimal control.
• Application to a tick-borne disease model incorporating age structure and seasonal tick variations.
• Numerical simulations to illustrate the efficacy of the proposed control strategy.

Main Results:

• Existence, necessary conditions, and uniqueness of the optimal control are mathematically established.
• The optimal control strategy aims to maximize disease-free tick populations.
• The strategy seeks to minimize infected tick populations through targeted acaricide application.
• Numerical examples validate the theoretical findings and demonstrate practical applicability.

Conclusions:

• The developed optimal control approach is effective for hybrid systems with seasonal dynamics.
• The strategy provides a framework for managing tick-transmitted diseases by optimizing acaricide interventions.
• This research contributes to understanding and controlling vector-borne diseases through mathematical modeling and control theory.