# Sub-Optimal Control in the Zika Virus Epidemic Model Using Differential Evolution

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## Abstract

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## 1. Introduction

## 2. Dynamical Control Model for ZIKV Infection

## 3. Differential Evolution

## 4. Numerical Simulations for Sub-Optimal Control Problem

#### 4.1. Sub-Optimal Control Problem

#### 4.2. Numerical Simulations

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

WHO | World Health Organization |

GA | Genetic Algorithm |

DE | Differential Evolution |

## References

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**Figure 4.**Simulation results of exposed newborn babies from: (i) the model without controls; (ii) the control model with $N=4$; and (iii) the control model with $N=8$.

**Figure 5.**Simulation results of symptomatically infected newborn babies from: (i) the model without controls; (ii) the control model with $N=4$; and (iii) the control model with $N=8$.

**Figure 6.**Simulation results of exposed adults from: (i) the model without controls; (ii) the control model with $N=4$; and (iii) the control model with $N=8$.

**Figure 7.**Simulation results of symptomatically infected adults from: (i) the model without controls; (ii) the control model with $N=4$; and (iii) the control model with $N=8$.

**Figure 8.**Simulation results of exposed vectors: (i) the model without controls; (ii) the control model with $N=4$; and (iii) the control model with $N=8$.

**Figure 9.**Simulation results of infected vectors from: (i) the model without controls; (ii) the control model with $N=4$; and (iii) the control model with $N=8$.

Variable | Definition |

${S}_{b}(t),{S}_{w}(t)$ | Susceptible newborn babies and adults |

${E}_{b}(t),{E}_{w}(t)$ | Exposed newborn babies and adults |

${A}_{b}(t),{A}_{w}(t)$ | Asymptomatically infected newborn babies and adults |

${I}_{b}(t),{I}_{w}(t)$ | Symptomatically infected newborn babies without microcephaly and symptomatically |

infected adults | |

${I}_{bm}(t),{I}_{wm}(t)$ | Newborn babies and adults with microcephaly |

${R}_{b}(t),{R}_{w}(t)$ | Recovered newborn babies and adults |

${S}_{v}(t)$ | Susceptible vectors |

${E}_{v}(t)$ | Exposed vectors |

${I}_{v}(t)$ | Infected vectors |

Parameter | Definition |

${\pi}_{b}$ | Birth rate of newborn babies |

$\alpha $ | Maturity rate of babies |

p | Fraction of symptomatic infection |

$1-p$ | Remaining fraction of symptomatic infection |

${q}_{E},{q}_{I},{q}_{R}$ | Fraction of newborn babies who are infected from pregnant adult of each class |

$1-r$ | Fraction of newborn babies who are infected with microcephaly |

${\beta}_{b},{\beta}_{w}$ | Transmission probability per contact from infected vectors to susceptible newborn babies |

and adults | |

${\beta}_{v}$ | Transmission probability per contact from infected humans to susceptible vectors |

${\beta}_{s}$ | Transmission probability per sexual contact from infected humans to susceptible humans |

$\eta $ | Exposure modification parameter in babies |

${\rho}_{b},{\rho}_{w}$ | Infectivity modification parameters in exposed babies and adults |

${\rho}_{s}$ | Sexual infectivity modification parameters in exposed adults |

${\sigma}_{b},{\sigma}_{w}$ | Progression rate of exposed newborn babies and adults |

${\gamma}_{b},{\gamma}_{w}$ | Recovery rate of newborn babies and adults |

${\mu}_{b},{\mu}_{w}$ | Natural death rate of newborn babies and adults |

${\pi}_{v}$ | Recruitment rate of mosquitoes |

b | Biting rate of mosquitoes |

${\sigma}_{v}$ | Progression rate of exposed mosquitoes |

${\mu}_{v}$ | Mortality rate of mosquitoes |

Parameter | Value | References |
---|---|---|

${\pi}_{b}$ | $\frac{1}{15\times 365}$ | [34] |

$\alpha $ | $\frac{1}{16\times 365}$ | [13] |

p | 0.5 | [13] |

${q}_{E},{q}_{I},{q}_{R}$ | 0.5 | [13] |

r | 0.5 | [13] |

${\beta}_{b},{\beta}_{w}$ | 0.33 | [13,35] |

${\beta}_{v}$ | 0.5 | [12] |

${\beta}_{s}$ | 0.05 | [12] |

$\eta $ | 0.5 | Assumed |

${\rho}_{b},{\rho}_{w}$ | 0.5 | [13] |

${\rho}_{s}$ | 0.6 | [12] |

${\sigma}_{b},{\sigma}_{w}$ | $\frac{1}{7.5}$ | [36] |

${\gamma}_{b},{\gamma}_{w}$ | $\frac{1}{8.5}$ | [36] |

${\mu}_{b}$ | $\frac{1}{18.60\times 365}$ | [3] |

${\mu}_{w}$ | $\frac{1}{70\times 365}$ | [35] |

${\pi}_{v}$ | 500 | [36] |

b | 0.5 | [35,37,38] |

${\sigma}_{v}$ | $\frac{1}{10}$ | [37] |

${\mu}_{v}$ | $\frac{1}{21}$ | [38] |

${r}_{0}$ | 0.1 | [18] |

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Chaikham, N.; Sawangtong, W.
Sub-Optimal Control in the Zika Virus Epidemic Model Using Differential Evolution. *Axioms* **2018**, *7*, 61.
https://doi.org/10.3390/axioms7030061

**AMA Style**

Chaikham N, Sawangtong W.
Sub-Optimal Control in the Zika Virus Epidemic Model Using Differential Evolution. *Axioms*. 2018; 7(3):61.
https://doi.org/10.3390/axioms7030061

**Chicago/Turabian Style**

Chaikham, Nonthamon, and Wannika Sawangtong.
2018. "Sub-Optimal Control in the Zika Virus Epidemic Model Using Differential Evolution" *Axioms* 7, no. 3: 61.
https://doi.org/10.3390/axioms7030061