### 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 themethod 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.

Original language | English (US) |
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Title of host publication | Mathematical and Statistical Modeling for Emerging and Re-emerging Infectious Diseases |

Publisher | Springer International Publishing |

Pages | 221-248 |

Number of pages | 28 |

ISBN (Electronic) | 9783319404134 |

ISBN (Print) | 9783319404110 |

DOIs | |

State | Published - Jan 1 2016 |

### Keywords

- Cholera
- Mathematical model
- Optimal control
- Partial differential equation
- Waning immunity

### ASJC Scopus subject areas

- Mathematics(all)
- Medicine(all)

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## Cite this

*Mathematical and Statistical Modeling for Emerging and Re-emerging Infectious Diseases*(pp. 221-248). Springer International Publishing. https://doi.org/10.1007/978-3-319-40413-4_14