# Modelling the Use of Vaccine and Wolbachia on Dengue Transmission Dynamics

## Abstract

**:**

## 1. Introduction

## 2. Methods and Results

#### 2.1. Formulation of the Mathematical Model

#### 2.2. Sensitivity Analysis

#### 2.3. Numerical Simulations

#### 2.3.1. Dengue Reduction

#### 2.3.2. Parameter Exploration

## 3. Discussion and Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Partial Rank Correlation Coefficient (PRCC) values for the model when measured against the increasing number of infected individuals. The positive and negative values indicate the relationship of the parameter and an increase in the number of infected individuals.

**Figure 2.**Numerical simulations of the model with no intervention, vaccine only, Wolbachia only, and both vaccine and Wolbachia. This is for the case ${R}_{A}=2.91$. The vaccine efficacy is 0.536 and the vaccination rate is 0.2. Plot (

**A**): No intervention, Plot (

**B**): Vaccination, Plot (

**C**): Wolbachia, Plot (

**D**): Both Wolbachia and vaccination strategies.

**Figure 3.**Comparing the performance of vaccination and Wolbachia with different vaccination rates. This is for the case ${R}_{A}=2.91$ and the vaccine efficacy is 0.536. The vaccination rates are 0.2 plot (

**A**), 0.5 plot (

**B**), 0.8 Plot (

**C**). Plot (

**D**) is for Wolbachia strategy.

**Figure 4.**Comparing the performance of vaccination and Wolbachia with different vaccine efficacy. This is for the case ${R}_{A}=2.91$ and the vaccine rate is 0.5. The vaccine efficacy is 0.7 plot (

**A**), 0.8 plot (

**B**), 0.95 plot (

**C**). The plot (

**D**) is Wolbachia.

**Figure 5.**Countour plot of vaccine efficacy ($\u03f5)$ and the vaccination rate $\left(p\right)$ against the performance index ($Id{x}_{VW}$).

**Table 1.**Parameter descriptions, values and sources for the model. We use “Non-W” to denote non-Wolbachia mosquitoes and “W” to denote Wolbachia-carrying mosquitoes.

Symbol | Description | Value | Unit | Source |
---|---|---|---|---|

$\alpha $ | Maternal transmission | 0.9 | N/A | [30,31] |

${b}_{N}$ | Biting rate of non-W | 0.63 | day${}^{-1}$ | [32] |

${\gamma}_{H}$ | Progression rate from exposed to infectious human | 1/5.5 | day${}^{-1}$ | [33] |

${\gamma}_{N}$ | Progression from exposed to infectious non-W | 1/10 | day${}^{-1}$ | [34] |

${\gamma}_{W}$ | Progression rate from exposed to infectious | 1/10 | day${}^{-1}$ | [34] |

${\mu}_{N}$ | Adult mosquito death rate (non-W) | 1/14 | day${}^{-1}$ | [35] |

${\mu}_{NA}$ | Death rate of aquatic non-W | 1/14 | day${}^{-1}$ | [35] |

${\mu}_{WA}$ | Aquatic death rate | 1/14 | day${}^{-1}$ | [35] |

${\rho}_{N}$ | Reproductive rate of non-W | 1.25 | day${}^{-1}$ | [36] |

${\rho}_{W}$ | Reproductive rate W | $0.95\times {\rho}_{N}$ | day${}^{-1}$ | [30] |

$\sigma $ | Recovery rate | 1/5 | day${}^{-1}$ | [33] |

${T}_{N}$ | Transmission probability | 0.2614 | N/A | [7] |

${\tau}_{N}$ | Maturation rate of non-W | 1/10 | day${}^{-1}$ | [35] |

${\tau}_{W}$ | Maturation rate of W | 1/10 | day${}^{-1}$ | [35] |

${T}_{HW}$ | Transmission probability from infectious W to human | $0.5\times {T}_{N}$ | N/A | [37] |

${\mu}_{W}$ | Death rate of W | 1.1 $\times {\mu}_{N}$ | day${}^{-1}$ | [30,38] |

${b}_{W}$ | Biting rates of W | $0.95\times {b}_{N}$ | day${}^{-1}$ | [39] |

${\mu}_{H}$ | Natural death rate | $\frac{1}{(66.38\times 365)}$ | day${}^{-1}$ | [40] |

B | Birth rate | $\frac{1}{(66.38\times 365)}$ | day${}^{-1}$ | [40] |

p | Vaccination rate | 0.2 | N/A | [41] |

$\u03f5$ | Vaccine efficacy | 0.538 | N/A | [17,21] |

$\varphi $ | Waning immunity | 0.1 | N/A | [41] |

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**MDPI and ACS Style**

Ndii, M.Z.
Modelling the Use of Vaccine and *Wolbachia* on Dengue Transmission Dynamics. *Trop. Med. Infect. Dis.* **2020**, *5*, 78.
https://doi.org/10.3390/tropicalmed5020078

**AMA Style**

Ndii MZ.
Modelling the Use of Vaccine and *Wolbachia* on Dengue Transmission Dynamics. *Tropical Medicine and Infectious Disease*. 2020; 5(2):78.
https://doi.org/10.3390/tropicalmed5020078

**Chicago/Turabian Style**

Ndii, Meksianis Z.
2020. "Modelling the Use of Vaccine and *Wolbachia* on Dengue Transmission Dynamics" *Tropical Medicine and Infectious Disease* 5, no. 2: 78.
https://doi.org/10.3390/tropicalmed5020078