# American Institute of Mathematical Sciences

2007, 4(4): 633-659. doi: 10.3934/mbe.2007.4.633

## Optimal control on hybrid ODE Systems with application to a tick disease model

 1 Department of Mathematics, University of Tennessee, 1403 Circle Drive, Knoxville, TN 37996-1300, United States

Received  September 2006 Revised  June 2007 Published  August 2007

We are considering an optimal control problem for a type of hybrid system involving ordinary differential equations and a discrete time feature. One state variable has dynamics in only one season of the year and has a jump condition to obtain the initial condition for that corresponding season in the next year. The other state variable has continuous dynamics. Given a general objective functional, existence, necessary conditions and uniqueness for an optimal control are established. We apply our approach to a tick-transmitted disease model with age structure in which the tick dynamics changes seasonally while hosts have continuous dynamics. The goal is to maximize disease-free ticks and minimize infected ticks through an optimal control strategy of treatment with acaricide. Numerical examples are given to illustrate the results.
Citation: Wandi Ding. Optimal control on hybrid ODE Systems with application to a tick disease model. Mathematical Biosciences & Engineering, 2007, 4 (4) : 633-659. doi: 10.3934/mbe.2007.4.633
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