# Modeling Sustained Transmission of Wolbachia among Anopheles Mosquitoes: Implications for Malaria Control in Haiti

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model Description

#### 2.1.1. Male Adult Mosquitoes (${M}_{u}$ and ${M}_{w}$)

#### 2.1.2. Female Adult Mosquitoes (${F}_{u}$ and ${F}_{w}$)

#### 2.1.3. Egg Stages (${E}_{u}$ and ${E}_{w}$)

#### 2.1.4. Larvae/Pupae Stages (${L}_{u}$ and ${L}_{w}$)

Description | Value | Reference | |
---|---|---|---|

Specific toAnopheles spp. | |||

$\delta $ | Hatching rate for eggs (=$1/{\tau}_{\delta}$) | 1/3 | [26] |

$\psi $ | Emergence rate for larvae (=$1/{\tau}_{\psi}$) | 1/18 | [6] |

${\mu}_{fu}$ | Death rate for uninfected females (=$1/{\tau}_{fu}$) | 1/13 | [6,26] |

${\mu}_{fw}$ | Death rate for infected females (=$1/{\tau}_{fw}$) | 1/15 | [6,26] |

${\mu}_{mu}$ | Death rate for uninfected males (=$1/{\tau}_{mu}$) | 1/7 | [6,26] |

${\mu}_{mw}$ | Death rate for infected males (=$1/{\tau}_{mw}$) | 1/7 | [6,26] |

${\mu}_{eu}$ | Death rate for uninfected eggs | 0.12 | [6] |

${\mu}_{ew}$ | Death rate for infected eggs | 0.33 | [6] |

${\mu}_{l}$ | Death rate for larvae | 0.01 | [6] |

${\varphi}_{u}$ | Per capita egg laying rate for wild females | 3.8 | [6] |

${\varphi}_{w}$ | Per capita egg laying rate for infected females | 3.3 | [6] |

${v}_{w}$ | wAlbB maternal transmission fraction | 1 | [7] |

${c}_{i}$ | wAlbB CI fraction | 1 | |

Not specific to Anopheles spp. | |||

${b}_{f}$ | Fraction of larvae emerging as females | 0.5 | [27] |

${b}_{m}$ | Fraction of larvae emerging as males | 0.5 | [27] |

${K}_{l}$ | Carrying capacity of larvae/pupae stages | $2\times {10}^{5}$ | Assume |

#### 2.2. Model Analysis

#### 2.2.1. Next-Generation Numbers

#### 2.2.2. Equilibria and Basic Reproductive Number

#### Disease-Free Equilibrium (DFE)

#### Complete-Infection Equilibrium (CIE)

#### Basic Reproductive Number

#### Endemic Equilibrium (EE)

#### 2.2.3. Stability and Bifurcation Analysis

**Theorem 1**

**Theorem 2**

**Theorem 3**

#### 2.3. Parameter Estimations

#### 2.3.1. Maternal Transmission

#### 2.3.2. Mosquito Lifespan

#### 2.3.3. Egg-laying Rates

#### 2.3.4. Egg-hatching Rate and Death Rates

#### 2.3.5. Larvae/Pupae Emergence Rate and Death Rate

## 3. Results

#### 3.1. Sensitivity Analysis

#### 3.2. Compare Prerelease Mitigation Strategies

**Figure 3.**Simulations for different release scenarios. The left figure displays the mosquito populations for a single release of infected mosquitoes when there is no prerelease mitigation. The middle and right figures compare releasing all the mosquitoes in a single release and in multiple batches when there is prerelease mitigation. An equal number of infected male and female mosquitoes are released (release size = 2, relative to baseline female population size at DFE) without or with prerelease mitigation (reduced to 40% in both larvae and adults using hybrid fogging and larviciding, see Table 5). The black line is the percent of infected mosquitoes that are infected as they are released in one batch or multiple batches. The Wolbachia endemic state is established, and the infection reaches $90\%$ infection around 143, 85, and 109 days after the initial release.

#### 3.3. Multiple Releases

#### 3.4. Seasonality

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Theorem 1 (Stability of DFE)

## Appendix B. Proof of Theorem 2 (Stability of CIE)

## Appendix C. Seasonality Fitting

**Table A1.**Seasonality data for the department of Grand Anse in Haiti. Row 1: monthly median rainfall (in mm) from 2016–2020. Row 2: mean aridity index for years 1970–2000. Row 3: mean monthly land surface temperature for 2018–2020 (in Celsius).

Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Nov | Dec | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Rainfall | 59 | 75 | 82 | 131 | 302 | 110 | 94 | 107 | 205 | 178 | 216 | 62 |

Aridity | 0.6 | 0.5 | 0.5 | 0.9 | 1.8 | 1.2 | 0.8 | 1.2 | 1.1 | 2.3 | 1.4 | 0.8 |

Temp. | 25.7 | 27.6 | 28.9 | 29.8 | 29.8 | 29.6 | 29.2 | 29.7 | 29.3 | 27.8 | 26.4 | 25.7 |

**Figure A1.**Fitted seasonality curves. Left axis: rescaled rainfall data and the fitted carrying capacity curve (solid blue line). Right axis: aridity index data and the fitted curve (red dashed line). The raw data values listed in Table A1 and points have been shifted to the center of the month. Both estimated curves are fitted using a periodic smoothing spline, showing a consistent trend.

## References

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**Figure 1.**Maternal transmission of Wolbachia in mosquitoes. The adult population of males and females is divided into compartments based on the infection status. Uninfected females (F

_{u}) produce uninfected eggs (E

_{u}) at an egg-laying rate of ϕ

_{u}. Infected females (F

_{w}) produce a fraction of v

_{w}infected eggs (E

_{w}) with a rate of ϕ

_{w}. Then, eggs develop into the larval stage at a hatching rate of δ. Larvalstage mosquitoes emerge at a rate of ψ and develop into adult mosquitoes. Death rates at different stages are denoted by μ

_{*}.

**Figure 2.**Bifurcation diagram characterizing the threshold condition for establishing a stable infection in mosquitoes, given a perfect maternal transmission rate (${v}_{w}=1$). The solid blue curves represent the stable equilibrium. The red dashed curve corresponds to the unstable equilibrium, which serves as the threshold condition. At the baseline case (vertical dotted line, ${\mathbb{R}}_{0}=0.68$), the threshold infection rate among females was $0.34$.

**Figure 4.**Compare multiple-release strategy for Wolbachia establishment speed without and with prerelease mitigation strategies. The heatmaps indicate the days to achieve $90\%$ infection (color-coded according to the respective color bars). The release size (y-axis) is measured relative to the baseline female population size at DFE, and an equal number of infected males and females are released. When no prerelease mitigation is implemented (

**a**), an optimal number of release batches is observed for large releases sizes. When prerelease mitigation is implemented (

**b**), releasing all infected mosquitoes at once is more efficient than splitting them into multiple batches.

**Figure 5.**Impact of seasonality on the field release. Simulations of releasing the same number of infected mosquitoes at the driest (day = 21, (

**a**)) vs. the wettest (day = 134, (

**b**)) time of the year, indicated by the red vertical bars on the corresponding lower panels. The black lines for the percent of infected mosquitoes show that the infection is successfully established when released in the dry season, while it dies out when released in the wet season.

**Table 1.**Role of Wolbachia infection in mosquito population replacement versus population suppression. CI = cytoplasmic incompatibility.

Population Replacement | Population Suppression | |
---|---|---|

Goal | Replace wild mosquito population with Wolbachia-infected mosquitoes that have significantly lower competence and cannot transmit parasite as efficiently | Introduce male mosquitoes that cannot produce viable offspring, which limits the ability of the mosquito to reproduce and reduces mosquito population [8] |

Role of CI | Infected females can mate successfully with infected males providing them with an evolutionary advantage over uninfected females | The sperm of the infected male is unable to form viable offspring during the egg fertilization process, and as a result, eggs do not hatch |

Release | Release infected males and females | Release infected males only |

**Table 2.**Wolbachia strains, Anopheles species, and corresponding impact on vector and P. falciparum parasite replication. CI = cytoplasmic incompatibility.

Wolbachia Strain | Anopheles Species | Impact on Vector | Impact on P. falciparum | Reference |
---|---|---|---|---|

$wAnga$ | coluzzii | No CI, increases egg laying rate | Reduces sporozoite prevalence | [16,17,18,19] |

funestus | No CI | Unknown | [18] | |

gambiae | No CI | Unknown | [18] | |

arabiensis | No CI | Unknown | [18] | |

$wAlbB$ | stephensi | Almost complete CI, reduces egg hatching rate, perfect maternal transmission, no impact on female lifespan | Reduces sporozoite and oocyst levels | [6,7] |

$wPip$ | gambiae | CI, reduces egg development rate | Unknown | [20] |

$wMelPop$ | gambiae | No effect on lifespan | Significantly reduces oocyst level | [21] |

**Table 4.**Sensitivity Analysis. Sensitivity indices for threshold-related quantities (in the first column) with respect to the model parameters (first row). Threshold (row 3) refers to the threshold level of infection and the time (row 4) measures the time to achieve $90\%$ infection.

${\mathit{\nu}}_{\mathit{w}}$ | ${\mathit{\varphi}}_{\mathit{w}}$ | ${\mathit{\varphi}}_{\mathit{u}}$ | ${\mathit{\mu}}_{\mathit{fw}}$ | ${\mathit{\mu}}_{\mathit{fu}}$ | ${\mathit{\mu}}_{\mathit{ew}}$ | ${\mathit{\mu}}_{\mathit{eu}}$ | $\mathit{\delta}$ | $\mathit{\psi}$ | ${\mathit{\mu}}_{\mathit{mu}}$ | ${\mathit{\mu}}_{\mathit{mw}}$ | ${\mathit{\mu}}_{\mathit{l}}$ | ${\mathit{\mu}}_{\mathit{adults}}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathbb{R}}_{0}$ | 1 | 1 | −1 | −1 | 1 | −0.5 | 0.26 | 0.23 | 0 | 0 | 0 | 0 | −2.4 × 10^{−13} |

Threshold | −3.5 | −2.1 | 2.1 | 1.4 | −1.4 | 1 | −0.55 | −0.48 | 0 | 0 | 0 | 0 | 4 × 10^{−13} |

Time | −6.7 | −1.1 | 1.1 | 0.88 | −0.6 | 0.51 | −0.32 | −0.32 | −0.6 | −0.69 | 0.6 | −0.14 | 0.14 |

**Table 5.**Comparison of prerelease mitigation strategies targeting different mosquito life stages (larvae and adults). “Mitigation efficacy” measures the fraction of population reduced given the mitigation approach, and the “release factor” measures the release size of the infected males and females relative to the baseline female population size at DFE (${F}_{u}^{0}$). Threshold release sizes needed to establish Wolbachia within (two months) or without time constraints are identified.

Prerelease Mitigation | Larvae Mitigation Efficacy | Adults Mitigation Efficacy | Threshold Release Factor | Release Factor to Reach $90\%$ by Two Months |
---|---|---|---|---|

No mitigation (DFE) | 0 | 0 | 1.13 | 9.9 |

Thermal fogging | 0 | 0.2 | 1.03 | 9.2 |

0 | 0.4 | 0.93 | 9.2 | |

0 | 0.6 | 0.82 | 7.9 | |

Larviciding | 0.2 | 0 | 1.04 | 8.0 |

0.4 | 0 | 0.96 | 6.5 | |

0.6 | 0 | 0.88 | 5.3 | |

Thermal fogging + larviciding | 0.6 | 0.2 | 0.79 | 4.8 |

0.6 | 0.4 | 0.69 | 4.4 | |

0.6 | 0.6 | 0.60 | 3.9 |

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## Share and Cite

**MDPI and ACS Style**

Florez, D.; Young, A.J.; Bernabé, K.J.; Hyman, J.M.; Qu, Z.
Modeling Sustained Transmission of *Wolbachia* among *Anopheles* Mosquitoes: Implications for Malaria Control in Haiti. *Trop. Med. Infect. Dis.* **2023**, *8*, 162.
https://doi.org/10.3390/tropicalmed8030162

**AMA Style**

Florez D, Young AJ, Bernabé KJ, Hyman JM, Qu Z.
Modeling Sustained Transmission of *Wolbachia* among *Anopheles* Mosquitoes: Implications for Malaria Control in Haiti. *Tropical Medicine and Infectious Disease*. 2023; 8(3):162.
https://doi.org/10.3390/tropicalmed8030162

**Chicago/Turabian Style**

Florez, Daniela, Alyssa J. Young, Kerlly J. Bernabé, James M. Hyman, and Zhuolin Qu.
2023. "Modeling Sustained Transmission of *Wolbachia* among *Anopheles* Mosquitoes: Implications for Malaria Control in Haiti" *Tropical Medicine and Infectious Disease* 8, no. 3: 162.
https://doi.org/10.3390/tropicalmed8030162