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Mathematical Biosciences and Engineering

2007, Volume 4, Issue 4: 633-659. doi: 10.3934/mbe.2007.4.633
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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
  • Received: 01 September 2006 Accepted: 29 June 2018 Published: 01 August 2007

  • MSC : 49J15, 34K35, 92D25.

  • 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[J]. Mathematical Biosciences and Engineering, 2007, 4(4): 633-659. doi: 10.3934/mbe.2007.4.633

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  • 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.


  • This article has been cited by:

    1. Milliward Maliyoni, Faraimunashe Chirove, Holly D. Gaff, Keshlan S. Govinder, A Stochastic Tick-Borne Disease Model: Exploring the Probability of Pathogen Persistence, 2017, 79, 0092-8240, 1999, 10.1007/s11538-017-0317-y
    2. Holly D. Gaff, Elsa Schaefer, Suzanne Lenhart, Use of optimal control models to predict treatment time for managing tick-borne disease, 2011, 5, 1751-3758, 517, 10.1080/17513758.2010.535910
    3. Ludovic Mailleret, Valérie Lemesle, A note on semi-discrete modelling in the life sciences, 2009, 367, 1364-503X, 4779, 10.1098/rsta.2009.0153
    4. Preliminary analysis of an agent-based model for a tick-borne disease, 2011, 8, 1551-0018, 463, 10.3934/mbe.2011.8.463
    5. David Gammack, Elsa Schaefer, Holly Gaff, 2013, 9780124157804, 105, 10.1016/B978-0-12-415780-4.00004-1
    6. Prasanta Kumar Mondal, T. K. Kar, Optimal treatment control and bifurcation analysis of a tuberculosis model with effect of multiple re-infections, 2017, 5, 2195-268X, 367, 10.1007/s40435-015-0176-z
    7. Christopher M. Baker, Target the Source: Optimal Spatiotemporal Resource Allocation for Invasive Species Control, 2017, 10, 1755263X, 41, 10.1111/conl.12236
    8. Milliward Maliyoni, Faraimunashe Chirove, Holly D. Gaff, Keshlan S. Govinder, A stochastic epidemic model for the dynamics of two pathogens in a single tick population, 2019, 127, 00405809, 75, 10.1016/j.tpb.2019.04.004
    9. Giphil Cho, Daewook Kim, Sukgeun Jung, Il Hyo Jung, Sangil Kim, Estimating Age-Specific Natural Mortality for Sandfish in the Eastern Coastal Waters of Korea, 2020, 8, 2227-7390, 1612, 10.3390/math8091612
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  • © 2007 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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