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Optimal Control of Vaccination in an Age-Structured Cholera Model

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Mathematical and Statistical Modeling for Emerging and Re-emerging Infectious Diseases

Abstract

A cholera model with continuous age structure is given as a system of hyperbolic (first-order) partial differential equations (PDEs) in combination with ordinary differential equations. Asymptomatic infected and susceptibles with partial immunity are included in this epidemiology model with vaccination rate as a control; minimizing the symptomatic infecteds while minimizing the cost of the vaccinations represents the goal. With the method of characteristics and a fixed point argument, the existence of a solution to our nonlinear state system is achieved. The representation and existence of a unique optimal control are derived. The steps to justify the optimal control results for such a system with first order PDEs are given. Numerical results illustrate the effect of age structure on optimal vaccination rates.

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Acknowledgments

This work was funded by the National Science Foundation DMS-0813563. Lenhart’s support also included funding from the National Institute for Mathematical and Biological Synthesis NSF EF-0832858. Wang’s work was partially supported by National Science Foundation DMS-1412826. We thanks Boloye Gomero for her initial work with the graphic for the description of the model.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Murray State University, Murray, KY, 42071, USA

    K. Renee Fister

  2. Department of Biological Sciences, Old Dominion University, Norfolk, VA, 23929, USA

    Holly Gaff

  3. Department of Mathematics, University of Tennessee, Knoxville, TN, 37996, USA

    Suzanne Lenhart

  4. Department of Mathematics, Augusta University, Augusta, GA, 30912, USA

    Eric Numfor

  5. Department of Mathematics, Marymount University, Arlington, VA, 22207, USA

    Elsa Schaefer

  6. Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN, 37403, USA

    Jin Wang

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  1. K. Renee Fister
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  2. Holly Gaff
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  3. Suzanne Lenhart
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  4. Eric Numfor
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  5. Elsa Schaefer
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  6. Jin Wang
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Corresponding author

Correspondence to Suzanne Lenhart .

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Editors and Affiliations

  1. School of Public Health, Georgia State University, Atlanta, Georgia, USA

    Gerardo Chowell

  2. Department of Mathematics, Tulane University, New Orleans, Louisiana, USA

    James M. Hyman

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Fister, K.R., Gaff, H., Lenhart, S., Numfor, E., Schaefer, E., Wang, J. (2016). Optimal Control of Vaccination in an Age-Structured Cholera Model. In: Chowell, G., Hyman, J. (eds) Mathematical and Statistical Modeling for Emerging and Re-emerging Infectious Diseases. Springer, Cham. https://doi.org/10.1007/978-3-319-40413-4_14

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