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Optimal control applied to vaccination and treatment strategies for various epidemiological models

1. Community and Environmental Health, College of Health Sciences, Old Dominion University, 3133A Technology Building, Norfolk, VA 23529
2. Department of Mathematics, Marymount University, 2807 North Glebe Road, Arlington, VA 22207

Mathematical models provide a powerful tool for investigating the dynamics and control of infectious diseases, but quantifying the underlying epidemic structure can be challenging especially for new and under-studied diseases. Variations of standard SIR, SIRS, and SEIR epidemiological models are considered to determine the sensitivity of these models to various parameter values that may not be fully known when the models are used to investigate emerging diseases. Optimal control theory is applied to suggest the most effective mitigation strategy to minimize the number of individuals who become infected in the course of an infection while efficiently balancing vaccination and treatment applied to the models with various cost scenarios. The optimal control simulations suggest that regardless of the particular epidemiological structure and of the comparative cost of mitigation strategies, vaccination, if available, would be a crucial piece of any intervention plan.
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Keywords vaccination.; SIR; epidemic; SEIR; SIRS; optimal control

Citation: Holly Gaff, Elsa Schaefer. Optimal control applied to vaccination and treatment strategies for various epidemiological models. Mathematical Biosciences and Engineering, 2009, 6(3): 469-492. doi: 10.3934/mbe.2009.6.469


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  • 36. N.H. Sweilam, S.M. Al-Mekhlafi, , Fractional Order Systems, 2018, 63, 10.1016/B978-0-12-816152-4.00003-0
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