# Fractional-Order Modelling and Optimal Control of Cholera Transmission

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## Abstract

**:**

## 1. Introduction

## 2. Fractional-Order Cholera Model

## 3. Main Results

#### 3.1. Sensitivity Analysis

**Definition**

**1**

#### 3.2. Fractional Optimal Control of the Model

#### 3.3. Numerical Results and Cost-Effectiveness Analysis

#### 3.3.1. Optimal Control Strategies and Cost-Effectiveness Analysis

- strategy A—implementing the vaccination control, v;
- strategy B—implementing proper hygiene control, m;
- strategy C—implementing both controls, v and m.

#### 3.3.2. The Variable-Order FractInt System

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Evolution of the sensitivity index of parameter ${\beta}_{1}$, evaluated for $\alpha $, with respect to ${{R}_{0}}_{1}$ (first community); (

**b**) Evolution of the sensitivity index of parameter ${\varrho}_{2}$, evaluated for $\alpha $, with respect to ${{R}_{0}}_{2}$ (second community).

**Figure 3.**Evolution of the sensitivity index for the basic reproduction numbers of both communities, $R{{}_{0}}_{1}$ and ${{R}_{0}}_{2}$, with the variation of the derivative order, $\alpha $.

**Figure 4.**Sensitivity index of the basic reproduction numbers with respect to the control variables u (

**top left**), v (

**top right**), and m (

**bottom**).

**Figure 11.**Evolution of the efficacy function for the FractInt system and for the FOCP with $\alpha =1$ and $\alpha =0.68$, considering the parameter values from Table 1.

Parameter | Value | Source |
---|---|---|

${\beta}_{1}$ | 0.00125 | [10] |

${\beta}_{2}$ | 0.0125 | [10] |

K | ${10}^{6}$ | [8] |

${\mu}_{1},{\mu}_{2}$ | $8.4\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-5}$ | [20,21] |

${\delta}_{1}$ | 0.0125 | [12,22] |

${\delta}_{2}$ | 0.045 | [12,22] |

${\gamma}_{1}$ | 0.045 | [12,23] |

${\gamma}_{2}$ | 0.035 | [12,23] |

${\mu}_{p}$ | 1.06 | [8,24,25,26] |

${g}_{1},{g}_{2}$ | 0.73 | [10] |

${\varrho}_{1}$ | 0.102 | – |

${\sigma}_{1},{\sigma}_{2}$ | 50 | [10] |

**Table 2.**Sensitivity of ${{R}_{0}}_{1}$ (top) and ${{R}_{0}}_{2}$ (bottom), evaluated for the parameter values given in Table 1 with $\alpha =1$.

Parameter | ${\mathrm{Y}}_{\xb7}^{{{R}_{0}}_{1}}$ | Parameter | ${\mathrm{Y}}_{\xb7}^{{{R}_{0}}_{1}}$ | Parameter | ${\mathrm{Y}}_{\xb7}^{{{R}_{0}}_{1}}$ |

${\pi}_{1},{\pi}_{2}$ | 0.500 | ${\mu}_{2},{a}_{1}$ | $-0.454$ | ${\beta}_{1},{\sigma}_{1}$ | $2\times {10}^{-6}$ |

${\varrho}_{1}$ | 0.999 | K | $-2\times {10}^{-6}$ | ${\mu}_{1}$ | $-0.547$ |

${a}_{2}$ | 0.454 | ${b}_{1}$ | $-0.343$ | ${\delta}_{1}$ | $-0.143$ |

${\gamma}_{1}$ | $-0.514$ | ${\mu}_{p}$ | $-6\times {10}^{-6}$ | ${g}_{1}$ | $4\times {10}^{-6}$ |

Parameter | ${\mathrm{Y}}_{\xb7}^{{{R}_{0}}_{2}}$ | Parameter | ${\mathrm{Y}}_{\xb7}^{{{R}_{0}}_{2}}$ | Parameter | ${\mathrm{Y}}_{\xb7}^{{{R}_{0}}_{2}}$ |

${\pi}_{1},{\pi}_{2}$ | 0.500 | ${\mu}_{2}$ | $-0.456$ | ${\beta}_{2},{\sigma}_{2}$ | $1\times {10}^{-5}$ |

${\varrho}_{2}$ | 0.999 | K | $-1\times {10}^{-5}$ | ${\mu}_{1},{a}_{2}$ | $-0.545$ |

${a}_{1}$ | 0.545 | ${b}_{2}$ | $-0.259$ | ${\delta}_{2}$ | $-0.416$ |

${\gamma}_{2}$ | $-0.324$ | ${\mu}_{p}$ | $-3\times {10}^{-5}$ | ${g}_{2}$ | $2\times {10}^{-5}$ |

**Table 3.**Initial conditions for the fractional optimal control problem of Section 3.2 with parameters given by Table 1, corresponding to the endemic equilibrium of cholera model (3)–(4) with classical derivative order.

${\mathit{S}}_{1}\left(0\right)$ | ${\mathit{I}}_{1}\left(0\right)$ | ${\mathit{R}}_{1}\left(0\right)$ | ${\mathit{B}}_{1}\left(0\right)$ | ${\mathit{S}}_{2}\left(0\right)$ | ${\mathit{I}}_{2}\left(0\right)$ | ${\mathit{R}}_{2}\left(0\right)$ | ${\mathit{B}}_{2}\left(0\right)$ |
---|---|---|---|---|---|---|---|

0.53144 | 0.001997 | 0.01028 | 0.30254 | 0.44222 | 0.002380 | 0.01082 | 0.36065 |

$\mathit{\alpha}$ | $\mathit{AV}$ | $\mathit{TC}$ | $\mathit{ACER}$ | $\overline{\mathit{F}}$ |
---|---|---|---|---|

1.0 | 0.316716 | 0.90049 | 2.84322 | 0.723582 |

0.9 | 0.297175 | 1.17595 | 3.95708 | 0.678938 |

0.8 | 0.280311 | 1.35978 | 4.85099 | 0.640408 |

Strategies | $\mathit{AV}$ | $\mathit{TC}$ | $\mathit{ACER}$ | ICER |
---|---|---|---|---|

B | 0.038888 | 0.0084106 | 0.216276 | 0.216276 |

A | 0.316411 | 0.900865 | 2.84713 | 3.2157877 |

C | 0.316716 | 0.900494 | 2.84322 | $-1.216393$ |

**Table 6.**Incremental cost-effectiveness ratio for strategy B and several derivative orders. Same conditions as Table 5.

$\mathit{\alpha}$ | $\mathit{AV}$ | $\mathit{TC}$ | ICER |
---|---|---|---|

1.0 | 0.038888 | 0.0084106 | 0.216276 |

0.9 | 0.147496 | 0.003549 | $-22.3419$ |

0.8 | 0.203782 | 0.001705 | $-30.5191$ |

0.68 | 0.237211 | 0.000845 | $-38.859$ |

$\mathit{AV}$ | $\mathit{TC}$ | $\mathit{ACER}$ | $\overline{\mathit{F}}$ |
---|---|---|---|

0.332078 | 1.10967 | 3.3416 | 0.758679 |

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**MDPI and ACS Style**

Rosa, S.; Torres, D.F.M.
Fractional-Order Modelling and Optimal Control of Cholera Transmission. *Fractal Fract.* **2021**, *5*, 261.
https://doi.org/10.3390/fractalfract5040261

**AMA Style**

Rosa S, Torres DFM.
Fractional-Order Modelling and Optimal Control of Cholera Transmission. *Fractal and Fractional*. 2021; 5(4):261.
https://doi.org/10.3390/fractalfract5040261

**Chicago/Turabian Style**

Rosa, Silvério, and Delfim F. M. Torres.
2021. "Fractional-Order Modelling and Optimal Control of Cholera Transmission" *Fractal and Fractional* 5, no. 4: 261.
https://doi.org/10.3390/fractalfract5040261