# Modeling the Influence of Mites on Honey Bee Populations

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

#### 2.1. Adding Mites to the Mathematical Model

#### 2.2. Adding Drones and Infected Bee Populations to the Mathematical Model

#### 2.3. Proportion of Drone Eggs

#### 2.4. Mite Reproduction

#### 2.5. Code and Computational Times

## 3. Results

**differences**in the bee castes are plotted. For example, in regard to hive bees, the difference

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Two-year simulation of a honey bee colony with mites using the simulation parameters in Table 4 and Table 5. The number of eggs, larvae, total pupae, infested pupae, total drones, and infested drones is shown. Mites do not affect the survival rate of any bee caste in this simulation. The number of drones is multiplied by 10 for plotting purposes.

**Figure 2.**Two-year simulation of a honey bee colony with mites using the simulation parameters in Table 4 and Table 5. The number of total hive bees, infested hive bees, total foraging bees, and infested foraging bees is shown. Mites do not affect the survival rate of any bee caste in this simulation. The number of mites closely matches the number of infested hive bees after the summer season ends when one mite remains attached to each infested hive bee.

**Figure 3.**Percentage of eggs that become drone eggs using Equation (15) at latitude 35${}^{\circ}$. The average percent value over the two years is 2.5% during the summer.

**Figure 4.**Daily percent rate of increase in mite population as defined by (16). After a period of oscillation, the rate relaxes. The average rate in the first year is 2.6% and the average rate in the second year is 1.3%.

**Figure 6.**Two-year simulation of a honey bee colony with mites using the simulation parameters in Table 4 and Table 5. Mites increase the mortality rates in Table 3 of infested pupae, hive, drone, and foragers by four times. Only the first year is shown since the colony does not survive past the first year. In this figure, the

**difference**in the bee subpopulations (larvae, total pupae, infested pupae, total drones, infested drones) with unaffected mortality (Figure 1 and Figure 2) and the bee subpopulations with increased mortality is shown.

**Figure 7.**Two-year simulation of a honey bee colony with mites using the simulation parameters in Table 4 and Table 5. Mites increase the mortality rates in Table 3 of infested pupae, hive, drone, and foragers by four times. Only the first year is shown since the colony does not survive past the first year. In this figure, the difference between the number of bee subpopulations (total hive bees, infested hive bees, total foraging bees, and infested foraging bees) with unchanged mortality (Figure 1 and Figure 2) and the bee subpopulations with increased mortality is plotted.

**Figure 8.**Effect of grooming rate $\gamma $ and transmission rate $\beta $ (10)–(13) on the second year maximum hive bee population. The transmission rate models the rate at which uninfested bees become infested. The grooming rate models the rate at which infested bees becomes uninfested. A sharp line divides the conditions under which the colony can be sustained (yellow) versus when the colony perishes (blue).

**Figure 9.**Effect of grooming rate $\gamma $ (10)–(13) and drone reproductive rate on the second year maximum hive bee population. The grooming rate models the rate at which infested bees becomes uninfested. The transmission rate $\beta =0.2$. A sharp line divides the conditions under which the colony can be sustained (yellow) versus when the colony perishes (blue). The figure shows that drones play an important role in sustaining the mite population due to the higher reproductive rate of mites in drone capped cells.

Sum over ${\mathit{B}}_{\mathit{i}}$ | Bee Caste | Number of Days in Caste |
---|---|---|

${E}_{t}\left(i\right)\equiv {\sum}_{i=1}^{i=3}{B}_{i}$ | Egg (E) | 3 |

${L}_{t}\left(i\right)\equiv {\sum}_{i=4}^{i=8}{B}_{i}$ | Brood or larvae (L) | 5 |

${P}_{t}\left(i\right)\equiv {\sum}_{i=9}^{i=20}{B}_{i}$ | Pupae (P) | 12 |

${H}_{t}\left(i\right)\equiv {\sum}_{i=21}^{i=41}{B}_{i}$ | Hive (H) | 21 |

${F}_{t}\left(i\right)\equiv {\sum}_{i=42}^{i=55}{B}_{i}$ | Forager (F) | 14 |

Sum over ${\mathit{B}}_{\mathit{i}}^{\mathit{D}}$ | Bee Caste | Number of Days in Caste |
---|---|---|

${E}_{t}^{D}\left(i\right)\equiv {\sum}_{i=1}^{i=3}{B}_{i}^{D}$ | Drone Egg $\left({E}^{D}\right)$ | 3 |

${L}_{t}^{D}\left(i\right)\equiv {\sum}_{i=4}^{i=10}{B}_{i}^{D}$ | Drone Larvae $\left({L}^{D}\right)$ | 7 |

${P}_{t}^{D}\left(i\right)\equiv {\sum}_{i=11}^{i=24}{B}_{i}^{D}$ | Drone Pupae $\left({P}^{D}\right)$ | 14 |

${D}_{t}\left(i\right)\equiv {\sum}_{i=25}^{i=45}{B}_{i}^{D}$ | Adult Drone $\left(D\right)$ | 21 |

${\mathit{m}}_{\mathit{egg}}$ | ${\mathit{m}}_{\mathit{larvae}}$ | ${\mathit{m}}_{\mathit{pupae}}$ | ${\mathit{m}}_{\mathit{hive}}$ | ${\mathit{m}}_{\mathit{forager}}$ |
---|---|---|---|---|

0.03 | 0.01 | 0.001 | 0.015 | 0.045 |

${\mathit{S}}_{\mathit{egg}}$ | ${\mathit{S}}_{\mathit{larvae}}$ | ${\mathit{S}}_{\mathit{pupae}}$ | ${\mathit{S}}_{\mathit{hive}}$ | ${\mathit{S}}_{\mathit{forager}}$ |

0.97 | 0.99 | 0.999 | 0.985 | 0.955 |

Initial number of hive bees | 8000 |

Simulation start date | 1 February |

Maximum egg laying rate | 1600 |

Ideal hive bee to brood ratio | 2 |

Ideal hive bee to forager ratio | 2.3 |

Hive bee daily survival rate in winter | 0.9947 |

Latitude | 35${}^{\circ}$ |

Time step $\u25b3t$ | 0.1 day |

Length of summer | 6 March to 17 September |

${S}_{b}$, ${S}_{egg}$, ${S}_{peak}$, ${S}_{e}$ | 16, 55, 162, 272 days |

Initial number of mites | 200 |

Lifetime of mite | 27 days |

Mite daily survival rate in summer | 0.994 |

Mite daily survival rate in winter | 0.998 |

Reproductive rate of foundress mite in drone brood | 2.6 |

Reproductive rate of foundress mite in worker brood | 1.5 |

Maximum number of mites in capped brood | 4 |

Maximum number of mites per infested hive or drone bee $\xi $ | 6 |

Grooming rate $\gamma $ | 0.05 |

Transmission rate $\beta $ | 0.25 |

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**MDPI and ACS Style**

Torres, D.J.; Torres, N.A.
Modeling the Influence of Mites on Honey Bee Populations. *Vet. Sci.* **2020**, *7*, 139.
https://doi.org/10.3390/vetsci7030139

**AMA Style**

Torres DJ, Torres NA.
Modeling the Influence of Mites on Honey Bee Populations. *Veterinary Sciences*. 2020; 7(3):139.
https://doi.org/10.3390/vetsci7030139

**Chicago/Turabian Style**

Torres, David J., and Nicholas A. Torres.
2020. "Modeling the Influence of Mites on Honey Bee Populations" *Veterinary Sciences* 7, no. 3: 139.
https://doi.org/10.3390/vetsci7030139